[{"Chapter": "1", "sentence_range": "1-4", "Text": "Chapter One\nELECTRIC CHARGES\nAND FIELDS\n1 1  INTRODUCTION\nAll of us have the experience of seeing a spark or hearing a crackle when\nwe take off our synthetic clothes or sweater, particularly in dry weather Have you ever tried to find any explanation for this phenomenon Another\ncommon example of electric discharge is the lightning that we see in the\nsky during thunderstorms"}, {"Chapter": "1", "sentence_range": "2-5", "Text": "1  INTRODUCTION\nAll of us have the experience of seeing a spark or hearing a crackle when\nwe take off our synthetic clothes or sweater, particularly in dry weather Have you ever tried to find any explanation for this phenomenon Another\ncommon example of electric discharge is the lightning that we see in the\nsky during thunderstorms We also experience a sensation of an electric\nshock either while opening the door of a car or holding the iron bar of a\nbus after sliding from our seat"}, {"Chapter": "1", "sentence_range": "3-6", "Text": "Have you ever tried to find any explanation for this phenomenon Another\ncommon example of electric discharge is the lightning that we see in the\nsky during thunderstorms We also experience a sensation of an electric\nshock either while opening the door of a car or holding the iron bar of a\nbus after sliding from our seat The reason for these experiences is\ndischarge of electric charges through our body, which were accumulated\ndue to rubbing of insulating surfaces"}, {"Chapter": "1", "sentence_range": "4-7", "Text": "Another\ncommon example of electric discharge is the lightning that we see in the\nsky during thunderstorms We also experience a sensation of an electric\nshock either while opening the door of a car or holding the iron bar of a\nbus after sliding from our seat The reason for these experiences is\ndischarge of electric charges through our body, which were accumulated\ndue to rubbing of insulating surfaces You might have also heard that\nthis is due to generation of static electricity"}, {"Chapter": "1", "sentence_range": "5-8", "Text": "We also experience a sensation of an electric\nshock either while opening the door of a car or holding the iron bar of a\nbus after sliding from our seat The reason for these experiences is\ndischarge of electric charges through our body, which were accumulated\ndue to rubbing of insulating surfaces You might have also heard that\nthis is due to generation of static electricity This is precisely the topic we\nare going to discuss in this and the next chapter"}, {"Chapter": "1", "sentence_range": "6-9", "Text": "The reason for these experiences is\ndischarge of electric charges through our body, which were accumulated\ndue to rubbing of insulating surfaces You might have also heard that\nthis is due to generation of static electricity This is precisely the topic we\nare going to discuss in this and the next chapter Static means anything\nthat does not move or change with time"}, {"Chapter": "1", "sentence_range": "7-10", "Text": "You might have also heard that\nthis is due to generation of static electricity This is precisely the topic we\nare going to discuss in this and the next chapter Static means anything\nthat does not move or change with time Electrostatics deals with\nthe study of forces, fields and potentials arising from\nstatic charges"}, {"Chapter": "1", "sentence_range": "8-11", "Text": "This is precisely the topic we\nare going to discuss in this and the next chapter Static means anything\nthat does not move or change with time Electrostatics deals with\nthe study of forces, fields and potentials arising from\nstatic charges 1"}, {"Chapter": "1", "sentence_range": "9-12", "Text": "Static means anything\nthat does not move or change with time Electrostatics deals with\nthe study of forces, fields and potentials arising from\nstatic charges 1 2  ELECTRIC CHARGE\nHistorically the credit of discovery of the fact that amber rubbed with\nwool or silk cloth attracts light objects goes to Thales of Miletus, Greece,\naround 600 BC"}, {"Chapter": "1", "sentence_range": "10-13", "Text": "Electrostatics deals with\nthe study of forces, fields and potentials arising from\nstatic charges 1 2  ELECTRIC CHARGE\nHistorically the credit of discovery of the fact that amber rubbed with\nwool or silk cloth attracts light objects goes to Thales of Miletus, Greece,\naround 600 BC The name electricity is coined from the Greek word\nRationalised 2023-24\n2\nPhysics\nelektron meaning amber"}, {"Chapter": "1", "sentence_range": "11-14", "Text": "1 2  ELECTRIC CHARGE\nHistorically the credit of discovery of the fact that amber rubbed with\nwool or silk cloth attracts light objects goes to Thales of Miletus, Greece,\naround 600 BC The name electricity is coined from the Greek word\nRationalised 2023-24\n2\nPhysics\nelektron meaning amber Many such pairs of materials were known which\non rubbing could attract light objects like straw, pith balls and bits of\npapers"}, {"Chapter": "1", "sentence_range": "12-15", "Text": "2  ELECTRIC CHARGE\nHistorically the credit of discovery of the fact that amber rubbed with\nwool or silk cloth attracts light objects goes to Thales of Miletus, Greece,\naround 600 BC The name electricity is coined from the Greek word\nRationalised 2023-24\n2\nPhysics\nelektron meaning amber Many such pairs of materials were known which\non rubbing could attract light objects like straw, pith balls and bits of\npapers It was observed that if two glass rods rubbed with wool or silk cloth\nare brought close to each other, they repel each other [Fig"}, {"Chapter": "1", "sentence_range": "13-16", "Text": "The name electricity is coined from the Greek word\nRationalised 2023-24\n2\nPhysics\nelektron meaning amber Many such pairs of materials were known which\non rubbing could attract light objects like straw, pith balls and bits of\npapers It was observed that if two glass rods rubbed with wool or silk cloth\nare brought close to each other, they repel each other [Fig 1"}, {"Chapter": "1", "sentence_range": "14-17", "Text": "Many such pairs of materials were known which\non rubbing could attract light objects like straw, pith balls and bits of\npapers It was observed that if two glass rods rubbed with wool or silk cloth\nare brought close to each other, they repel each other [Fig 1 1(a)]"}, {"Chapter": "1", "sentence_range": "15-18", "Text": "It was observed that if two glass rods rubbed with wool or silk cloth\nare brought close to each other, they repel each other [Fig 1 1(a)] The\ntwo strands of wool or two pieces of silk cloth, with which the rods were\nrubbed, also repel each other"}, {"Chapter": "1", "sentence_range": "16-19", "Text": "1 1(a)] The\ntwo strands of wool or two pieces of silk cloth, with which the rods were\nrubbed, also repel each other However, the glass rod and wool attracted\neach other"}, {"Chapter": "1", "sentence_range": "17-20", "Text": "1(a)] The\ntwo strands of wool or two pieces of silk cloth, with which the rods were\nrubbed, also repel each other However, the glass rod and wool attracted\neach other Similarly, two plastic rods rubbed with cat\u2019s fur repelled each\nother [Fig"}, {"Chapter": "1", "sentence_range": "18-21", "Text": "The\ntwo strands of wool or two pieces of silk cloth, with which the rods were\nrubbed, also repel each other However, the glass rod and wool attracted\neach other Similarly, two plastic rods rubbed with cat\u2019s fur repelled each\nother [Fig 1"}, {"Chapter": "1", "sentence_range": "19-22", "Text": "However, the glass rod and wool attracted\neach other Similarly, two plastic rods rubbed with cat\u2019s fur repelled each\nother [Fig 1 1(b)] but attracted the fur"}, {"Chapter": "1", "sentence_range": "20-23", "Text": "Similarly, two plastic rods rubbed with cat\u2019s fur repelled each\nother [Fig 1 1(b)] but attracted the fur On the other hand, the plastic\nrod attracts the glass rod [Fig"}, {"Chapter": "1", "sentence_range": "21-24", "Text": "1 1(b)] but attracted the fur On the other hand, the plastic\nrod attracts the glass rod [Fig 1"}, {"Chapter": "1", "sentence_range": "22-25", "Text": "1(b)] but attracted the fur On the other hand, the plastic\nrod attracts the glass rod [Fig 1 1(c)] and repel the silk or wool with\nwhich the glass rod is rubbed"}, {"Chapter": "1", "sentence_range": "23-26", "Text": "On the other hand, the plastic\nrod attracts the glass rod [Fig 1 1(c)] and repel the silk or wool with\nwhich the glass rod is rubbed The glass rod repels the fur"}, {"Chapter": "1", "sentence_range": "24-27", "Text": "1 1(c)] and repel the silk or wool with\nwhich the glass rod is rubbed The glass rod repels the fur These seemingly simple facts  were established from years of efforts\nand careful experiments and their analyses"}, {"Chapter": "1", "sentence_range": "25-28", "Text": "1(c)] and repel the silk or wool with\nwhich the glass rod is rubbed The glass rod repels the fur These seemingly simple facts  were established from years of efforts\nand careful experiments and their analyses It was concluded, after many\ncareful studies by different scientists, that there were only two kinds of\nan entry which is called the electric charge"}, {"Chapter": "1", "sentence_range": "26-29", "Text": "The glass rod repels the fur These seemingly simple facts  were established from years of efforts\nand careful experiments and their analyses It was concluded, after many\ncareful studies by different scientists, that there were only two kinds of\nan entry which is called the electric charge We say that the bodies like\nglass or plastic rods, silk, fur and pith balls are electrified"}, {"Chapter": "1", "sentence_range": "27-30", "Text": "These seemingly simple facts  were established from years of efforts\nand careful experiments and their analyses It was concluded, after many\ncareful studies by different scientists, that there were only two kinds of\nan entry which is called the electric charge We say that the bodies like\nglass or plastic rods, silk, fur and pith balls are electrified They acquire\nan electric charge on rubbing"}, {"Chapter": "1", "sentence_range": "28-31", "Text": "It was concluded, after many\ncareful studies by different scientists, that there were only two kinds of\nan entry which is called the electric charge We say that the bodies like\nglass or plastic rods, silk, fur and pith balls are electrified They acquire\nan electric charge on rubbing There are two kinds of electrification and\nwe find  that (i) like charges repel and (ii) unlike charges attract each\nother"}, {"Chapter": "1", "sentence_range": "29-32", "Text": "We say that the bodies like\nglass or plastic rods, silk, fur and pith balls are electrified They acquire\nan electric charge on rubbing There are two kinds of electrification and\nwe find  that (i) like charges repel and (ii) unlike charges attract each\nother The property which differentiates the two kinds of charges is called\nthe polarity of charge"}, {"Chapter": "1", "sentence_range": "30-33", "Text": "They acquire\nan electric charge on rubbing There are two kinds of electrification and\nwe find  that (i) like charges repel and (ii) unlike charges attract each\nother The property which differentiates the two kinds of charges is called\nthe polarity of charge When a glass rod is rubbed with silk, the rod acquires one kind of\ncharge and the silk acquires the second kind of charge"}, {"Chapter": "1", "sentence_range": "31-34", "Text": "There are two kinds of electrification and\nwe find  that (i) like charges repel and (ii) unlike charges attract each\nother The property which differentiates the two kinds of charges is called\nthe polarity of charge When a glass rod is rubbed with silk, the rod acquires one kind of\ncharge and the silk acquires the second kind of charge This is true for\nany pair of objects that are rubbed to be electrified"}, {"Chapter": "1", "sentence_range": "32-35", "Text": "The property which differentiates the two kinds of charges is called\nthe polarity of charge When a glass rod is rubbed with silk, the rod acquires one kind of\ncharge and the silk acquires the second kind of charge This is true for\nany pair of objects that are rubbed to be electrified Now if the electrified\nglass rod is brought in contact with silk, with which it was rubbed, they\nno longer attract each other"}, {"Chapter": "1", "sentence_range": "33-36", "Text": "When a glass rod is rubbed with silk, the rod acquires one kind of\ncharge and the silk acquires the second kind of charge This is true for\nany pair of objects that are rubbed to be electrified Now if the electrified\nglass rod is brought in contact with silk, with which it was rubbed, they\nno longer attract each other They also do not attract or repel other light\nobjects as they did on being electrified"}, {"Chapter": "1", "sentence_range": "34-37", "Text": "This is true for\nany pair of objects that are rubbed to be electrified Now if the electrified\nglass rod is brought in contact with silk, with which it was rubbed, they\nno longer attract each other They also do not attract or repel other light\nobjects as they did on being electrified Thus, the charges acquired after rubbing are lost when the charged\nbodies are brought in contact"}, {"Chapter": "1", "sentence_range": "35-38", "Text": "Now if the electrified\nglass rod is brought in contact with silk, with which it was rubbed, they\nno longer attract each other They also do not attract or repel other light\nobjects as they did on being electrified Thus, the charges acquired after rubbing are lost when the charged\nbodies are brought in contact What can you conclude from these\nobservations"}, {"Chapter": "1", "sentence_range": "36-39", "Text": "They also do not attract or repel other light\nobjects as they did on being electrified Thus, the charges acquired after rubbing are lost when the charged\nbodies are brought in contact What can you conclude from these\nobservations It just tells us that unlike charges acquired by the objects\nneutralise or nullify each other\u2019s effect"}, {"Chapter": "1", "sentence_range": "37-40", "Text": "Thus, the charges acquired after rubbing are lost when the charged\nbodies are brought in contact What can you conclude from these\nobservations It just tells us that unlike charges acquired by the objects\nneutralise or nullify each other\u2019s effect Therefore, the charges were named\nas positive and negative by the American scientist Benjamin Franklin"}, {"Chapter": "1", "sentence_range": "38-41", "Text": "What can you conclude from these\nobservations It just tells us that unlike charges acquired by the objects\nneutralise or nullify each other\u2019s effect Therefore, the charges were named\nas positive and negative by the American scientist Benjamin Franklin By convention, the charge on glass rod or cat\u2019s fur is called positive and\nthat on plastic rod or silk is termed negative"}, {"Chapter": "1", "sentence_range": "39-42", "Text": "It just tells us that unlike charges acquired by the objects\nneutralise or nullify each other\u2019s effect Therefore, the charges were named\nas positive and negative by the American scientist Benjamin Franklin By convention, the charge on glass rod or cat\u2019s fur is called positive and\nthat on plastic rod or silk is termed negative If an object possesses an\nelectric charge, it is said to be electrified or charged"}, {"Chapter": "1", "sentence_range": "40-43", "Text": "Therefore, the charges were named\nas positive and negative by the American scientist Benjamin Franklin By convention, the charge on glass rod or cat\u2019s fur is called positive and\nthat on plastic rod or silk is termed negative If an object possesses an\nelectric charge, it is said to be electrified or charged When it has no charge\nit is said to be electrically neutral"}, {"Chapter": "1", "sentence_range": "41-44", "Text": "By convention, the charge on glass rod or cat\u2019s fur is called positive and\nthat on plastic rod or silk is termed negative If an object possesses an\nelectric charge, it is said to be electrified or charged When it has no charge\nit is said to be electrically neutral FIGURE 1"}, {"Chapter": "1", "sentence_range": "42-45", "Text": "If an object possesses an\nelectric charge, it is said to be electrified or charged When it has no charge\nit is said to be electrically neutral FIGURE 1 1 Rods: like charges repel and unlike charges attract each other"}, {"Chapter": "1", "sentence_range": "43-46", "Text": "When it has no charge\nit is said to be electrically neutral FIGURE 1 1 Rods: like charges repel and unlike charges attract each other Rationalised 2023-24\nElectric Charges\nand Fields\n3\nA simple apparatus to detect charge on a body is the gold-leaf\nelectroscope [Fig"}, {"Chapter": "1", "sentence_range": "44-47", "Text": "FIGURE 1 1 Rods: like charges repel and unlike charges attract each other Rationalised 2023-24\nElectric Charges\nand Fields\n3\nA simple apparatus to detect charge on a body is the gold-leaf\nelectroscope [Fig 1"}, {"Chapter": "1", "sentence_range": "45-48", "Text": "1 Rods: like charges repel and unlike charges attract each other Rationalised 2023-24\nElectric Charges\nand Fields\n3\nA simple apparatus to detect charge on a body is the gold-leaf\nelectroscope [Fig 1 2(a)]"}, {"Chapter": "1", "sentence_range": "46-49", "Text": "Rationalised 2023-24\nElectric Charges\nand Fields\n3\nA simple apparatus to detect charge on a body is the gold-leaf\nelectroscope [Fig 1 2(a)] It consists of a vertical metal rod housed in a\nbox, with two thin gold leaves attached to its bottom end"}, {"Chapter": "1", "sentence_range": "47-50", "Text": "1 2(a)] It consists of a vertical metal rod housed in a\nbox, with two thin gold leaves attached to its bottom end When a charged\nobject touches the metal knob at the top of the rod, charge flows on to\nthe leaves and they diverge"}, {"Chapter": "1", "sentence_range": "48-51", "Text": "2(a)] It consists of a vertical metal rod housed in a\nbox, with two thin gold leaves attached to its bottom end When a charged\nobject touches the metal knob at the top of the rod, charge flows on to\nthe leaves and they diverge The degree of divergance is an indicator of\nthe amount of charge"}, {"Chapter": "1", "sentence_range": "49-52", "Text": "It consists of a vertical metal rod housed in a\nbox, with two thin gold leaves attached to its bottom end When a charged\nobject touches the metal knob at the top of the rod, charge flows on to\nthe leaves and they diverge The degree of divergance is an indicator of\nthe amount of charge Try to understand why material bodies acquire charge"}, {"Chapter": "1", "sentence_range": "50-53", "Text": "When a charged\nobject touches the metal knob at the top of the rod, charge flows on to\nthe leaves and they diverge The degree of divergance is an indicator of\nthe amount of charge Try to understand why material bodies acquire charge You know that\nall matter is made up of atoms and/or molecules"}, {"Chapter": "1", "sentence_range": "51-54", "Text": "The degree of divergance is an indicator of\nthe amount of charge Try to understand why material bodies acquire charge You know that\nall matter is made up of atoms and/or molecules Although normally the\nmaterials are electrically neutral,  they do contain charges; but their charges\nare exactly balanced"}, {"Chapter": "1", "sentence_range": "52-55", "Text": "Try to understand why material bodies acquire charge You know that\nall matter is made up of atoms and/or molecules Although normally the\nmaterials are electrically neutral,  they do contain charges; but their charges\nare exactly balanced Forces that hold the molecules together, forces that\nhold atoms together in a solid, the adhesive force of glue, forces associated\nwith surface tension, all are basically electrical in nature, arising from the\nforces between charged particles"}, {"Chapter": "1", "sentence_range": "53-56", "Text": "You know that\nall matter is made up of atoms and/or molecules Although normally the\nmaterials are electrically neutral,  they do contain charges; but their charges\nare exactly balanced Forces that hold the molecules together, forces that\nhold atoms together in a solid, the adhesive force of glue, forces associated\nwith surface tension, all are basically electrical in nature, arising from the\nforces between charged particles Thus the electric force is all pervasive and\nit encompasses almost each and every field associated with our life"}, {"Chapter": "1", "sentence_range": "54-57", "Text": "Although normally the\nmaterials are electrically neutral,  they do contain charges; but their charges\nare exactly balanced Forces that hold the molecules together, forces that\nhold atoms together in a solid, the adhesive force of glue, forces associated\nwith surface tension, all are basically electrical in nature, arising from the\nforces between charged particles Thus the electric force is all pervasive and\nit encompasses almost each and every field associated with our life It is\ntherefore essential that we learn more about such a force"}, {"Chapter": "1", "sentence_range": "55-58", "Text": "Forces that hold the molecules together, forces that\nhold atoms together in a solid, the adhesive force of glue, forces associated\nwith surface tension, all are basically electrical in nature, arising from the\nforces between charged particles Thus the electric force is all pervasive and\nit encompasses almost each and every field associated with our life It is\ntherefore essential that we learn more about such a force To electrify a neutral body, we need to add or remove one kind of\ncharge"}, {"Chapter": "1", "sentence_range": "56-59", "Text": "Thus the electric force is all pervasive and\nit encompasses almost each and every field associated with our life It is\ntherefore essential that we learn more about such a force To electrify a neutral body, we need to add or remove one kind of\ncharge When we say that a body is charged, we always refer to this\nexcess charge or deficit of charge"}, {"Chapter": "1", "sentence_range": "57-60", "Text": "It is\ntherefore essential that we learn more about such a force To electrify a neutral body, we need to add or remove one kind of\ncharge When we say that a body is charged, we always refer to this\nexcess charge or deficit of charge In solids,  some of the electrons, being\nless tightly bound in the atom, are the charges which are transferred\nfrom one body to the other"}, {"Chapter": "1", "sentence_range": "58-61", "Text": "To electrify a neutral body, we need to add or remove one kind of\ncharge When we say that a body is charged, we always refer to this\nexcess charge or deficit of charge In solids,  some of the electrons, being\nless tightly bound in the atom, are the charges which are transferred\nfrom one body to the other A body can thus be charged positively by\nlosing some of its electrons"}, {"Chapter": "1", "sentence_range": "59-62", "Text": "When we say that a body is charged, we always refer to this\nexcess charge or deficit of charge In solids,  some of the electrons, being\nless tightly bound in the atom, are the charges which are transferred\nfrom one body to the other A body can thus be charged positively by\nlosing some of its electrons Similarly, a body can be charged negatively\nby gaining electrons"}, {"Chapter": "1", "sentence_range": "60-63", "Text": "In solids,  some of the electrons, being\nless tightly bound in the atom, are the charges which are transferred\nfrom one body to the other A body can thus be charged positively by\nlosing some of its electrons Similarly, a body can be charged negatively\nby gaining electrons When we rub a glass rod with silk, some of the\nelectrons from the rod are transferred to the silk cloth"}, {"Chapter": "1", "sentence_range": "61-64", "Text": "A body can thus be charged positively by\nlosing some of its electrons Similarly, a body can be charged negatively\nby gaining electrons When we rub a glass rod with silk, some of the\nelectrons from the rod are transferred to the silk cloth Thus the rod gets\npositively charged and the silk gets negatively charged"}, {"Chapter": "1", "sentence_range": "62-65", "Text": "Similarly, a body can be charged negatively\nby gaining electrons When we rub a glass rod with silk, some of the\nelectrons from the rod are transferred to the silk cloth Thus the rod gets\npositively charged and the silk gets negatively charged No new charge is\ncreated in the process of rubbing"}, {"Chapter": "1", "sentence_range": "63-66", "Text": "When we rub a glass rod with silk, some of the\nelectrons from the rod are transferred to the silk cloth Thus the rod gets\npositively charged and the silk gets negatively charged No new charge is\ncreated in the process of rubbing Also the number of electrons, that are\ntransferred, is a very small fraction of the total number of electrons in the\nmaterial body"}, {"Chapter": "1", "sentence_range": "64-67", "Text": "Thus the rod gets\npositively charged and the silk gets negatively charged No new charge is\ncreated in the process of rubbing Also the number of electrons, that are\ntransferred, is a very small fraction of the total number of electrons in the\nmaterial body 1"}, {"Chapter": "1", "sentence_range": "65-68", "Text": "No new charge is\ncreated in the process of rubbing Also the number of electrons, that are\ntransferred, is a very small fraction of the total number of electrons in the\nmaterial body 1 3  CONDUCTORS AND INSULATORS\nSome substances readily allow passage of electricity through them, others\ndo not"}, {"Chapter": "1", "sentence_range": "66-69", "Text": "Also the number of electrons, that are\ntransferred, is a very small fraction of the total number of electrons in the\nmaterial body 1 3  CONDUCTORS AND INSULATORS\nSome substances readily allow passage of electricity through them, others\ndo not Those which allow electricity to pass through them easily are\ncalled conductors"}, {"Chapter": "1", "sentence_range": "67-70", "Text": "1 3  CONDUCTORS AND INSULATORS\nSome substances readily allow passage of electricity through them, others\ndo not Those which allow electricity to pass through them easily are\ncalled conductors They have electric charges (electrons) that are\ncomparatively free to move inside the material"}, {"Chapter": "1", "sentence_range": "68-71", "Text": "3  CONDUCTORS AND INSULATORS\nSome substances readily allow passage of electricity through them, others\ndo not Those which allow electricity to pass through them easily are\ncalled conductors They have electric charges (electrons) that are\ncomparatively free to move inside the material Metals, human and animal\nbodies and earth are conductors"}, {"Chapter": "1", "sentence_range": "69-72", "Text": "Those which allow electricity to pass through them easily are\ncalled conductors They have electric charges (electrons) that are\ncomparatively free to move inside the material Metals, human and animal\nbodies and earth are conductors Most of the non-metals like glass,\nporcelain, plastic, nylon, wood offer high resistance to the passage of\nelectricity through them"}, {"Chapter": "1", "sentence_range": "70-73", "Text": "They have electric charges (electrons) that are\ncomparatively free to move inside the material Metals, human and animal\nbodies and earth are conductors Most of the non-metals like glass,\nporcelain, plastic, nylon, wood offer high resistance to the passage of\nelectricity through them They are called insulators"}, {"Chapter": "1", "sentence_range": "71-74", "Text": "Metals, human and animal\nbodies and earth are conductors Most of the non-metals like glass,\nporcelain, plastic, nylon, wood offer high resistance to the passage of\nelectricity through them They are called insulators Most  substances\nfall into one of the two classes stated above*"}, {"Chapter": "1", "sentence_range": "72-75", "Text": "Most of the non-metals like glass,\nporcelain, plastic, nylon, wood offer high resistance to the passage of\nelectricity through them They are called insulators Most  substances\nfall into one of the two classes stated above* When some charge is transferred to a conductor, it readily gets\ndistributed over the entire surface of the conductor"}, {"Chapter": "1", "sentence_range": "73-76", "Text": "They are called insulators Most  substances\nfall into one of the two classes stated above* When some charge is transferred to a conductor, it readily gets\ndistributed over the entire surface of the conductor In contrast, if some\ncharge is put on an insulator, it stays at the same place"}, {"Chapter": "1", "sentence_range": "74-77", "Text": "Most  substances\nfall into one of the two classes stated above* When some charge is transferred to a conductor, it readily gets\ndistributed over the entire surface of the conductor In contrast, if some\ncharge is put on an insulator, it stays at the same place You will learn\nwhy this happens in the next chapter"}, {"Chapter": "1", "sentence_range": "75-78", "Text": "When some charge is transferred to a conductor, it readily gets\ndistributed over the entire surface of the conductor In contrast, if some\ncharge is put on an insulator, it stays at the same place You will learn\nwhy this happens in the next chapter This property of the materials tells you why a nylon or plastic comb\ngets electrified on combing dry hair or on rubbing, but a metal article\n*\nThere is a third category called semiconductors, which offer resistance to the\nmovement of charges which is intermediate between the conductors and\ninsulators"}, {"Chapter": "1", "sentence_range": "76-79", "Text": "In contrast, if some\ncharge is put on an insulator, it stays at the same place You will learn\nwhy this happens in the next chapter This property of the materials tells you why a nylon or plastic comb\ngets electrified on combing dry hair or on rubbing, but a metal article\n*\nThere is a third category called semiconductors, which offer resistance to the\nmovement of charges which is intermediate between the conductors and\ninsulators Rationalised 2023-24\n4\nPhysics\nlike spoon does not"}, {"Chapter": "1", "sentence_range": "77-80", "Text": "You will learn\nwhy this happens in the next chapter This property of the materials tells you why a nylon or plastic comb\ngets electrified on combing dry hair or on rubbing, but a metal article\n*\nThere is a third category called semiconductors, which offer resistance to the\nmovement of charges which is intermediate between the conductors and\ninsulators Rationalised 2023-24\n4\nPhysics\nlike spoon does not The charges on metal leak through\nour body to the ground as both are conductors of\nelectricity"}, {"Chapter": "1", "sentence_range": "78-81", "Text": "This property of the materials tells you why a nylon or plastic comb\ngets electrified on combing dry hair or on rubbing, but a metal article\n*\nThere is a third category called semiconductors, which offer resistance to the\nmovement of charges which is intermediate between the conductors and\ninsulators Rationalised 2023-24\n4\nPhysics\nlike spoon does not The charges on metal leak through\nour body to the ground as both are conductors of\nelectricity However, if a metal rod with a wooden or plastic\nhandle is rubbed without touching its metal part, it shows\nsigns of charging"}, {"Chapter": "1", "sentence_range": "79-82", "Text": "Rationalised 2023-24\n4\nPhysics\nlike spoon does not The charges on metal leak through\nour body to the ground as both are conductors of\nelectricity However, if a metal rod with a wooden or plastic\nhandle is rubbed without touching its metal part, it shows\nsigns of charging 1"}, {"Chapter": "1", "sentence_range": "80-83", "Text": "The charges on metal leak through\nour body to the ground as both are conductors of\nelectricity However, if a metal rod with a wooden or plastic\nhandle is rubbed without touching its metal part, it shows\nsigns of charging 1 4  BASIC PROPERTIES OF ELECTRIC\nCHARGE\nWe have seen that there are two types of charges, namely\npositive and negative and their effects tend to cancel each\nother"}, {"Chapter": "1", "sentence_range": "81-84", "Text": "However, if a metal rod with a wooden or plastic\nhandle is rubbed without touching its metal part, it shows\nsigns of charging 1 4  BASIC PROPERTIES OF ELECTRIC\nCHARGE\nWe have seen that there are two types of charges, namely\npositive and negative and their effects tend to cancel each\nother Here,  we shall now describe some other properties\nof the electric charge"}, {"Chapter": "1", "sentence_range": "82-85", "Text": "1 4  BASIC PROPERTIES OF ELECTRIC\nCHARGE\nWe have seen that there are two types of charges, namely\npositive and negative and their effects tend to cancel each\nother Here,  we shall now describe some other properties\nof the electric charge If the sizes of charged bodies are very small as\ncompared to the distances between them, we treat them\nas point charges"}, {"Chapter": "1", "sentence_range": "83-86", "Text": "4  BASIC PROPERTIES OF ELECTRIC\nCHARGE\nWe have seen that there are two types of charges, namely\npositive and negative and their effects tend to cancel each\nother Here,  we shall now describe some other properties\nof the electric charge If the sizes of charged bodies are very small as\ncompared to the distances between them, we treat them\nas point charges All the charge content of the body is\nassumed to be concentrated at one point in space"}, {"Chapter": "1", "sentence_range": "84-87", "Text": "Here,  we shall now describe some other properties\nof the electric charge If the sizes of charged bodies are very small as\ncompared to the distances between them, we treat them\nas point charges All the charge content of the body is\nassumed to be concentrated at one point in space 1"}, {"Chapter": "1", "sentence_range": "85-88", "Text": "If the sizes of charged bodies are very small as\ncompared to the distances between them, we treat them\nas point charges All the charge content of the body is\nassumed to be concentrated at one point in space 1 4"}, {"Chapter": "1", "sentence_range": "86-89", "Text": "All the charge content of the body is\nassumed to be concentrated at one point in space 1 4 1  Additivity of charges\nWe have not as yet given a quantitative definition of a\ncharge; we shall follow it up in the next section"}, {"Chapter": "1", "sentence_range": "87-90", "Text": "1 4 1  Additivity of charges\nWe have not as yet given a quantitative definition of a\ncharge; we shall follow it up in the next section We shall\ntentatively assume that this can be done and proceed"}, {"Chapter": "1", "sentence_range": "88-91", "Text": "4 1  Additivity of charges\nWe have not as yet given a quantitative definition of a\ncharge; we shall follow it up in the next section We shall\ntentatively assume that this can be done and proceed If\na system contains two point charges q1 and q2, the total\ncharge of the system is obtained simply by adding\nalgebraically q1 and q2 , i"}, {"Chapter": "1", "sentence_range": "89-92", "Text": "1  Additivity of charges\nWe have not as yet given a quantitative definition of a\ncharge; we shall follow it up in the next section We shall\ntentatively assume that this can be done and proceed If\na system contains two point charges q1 and q2, the total\ncharge of the system is obtained simply by adding\nalgebraically q1 and q2 , i e"}, {"Chapter": "1", "sentence_range": "90-93", "Text": "We shall\ntentatively assume that this can be done and proceed If\na system contains two point charges q1 and q2, the total\ncharge of the system is obtained simply by adding\nalgebraically q1 and q2 , i e , charges add up like real numbers or they\nare scalars like the mass of a body"}, {"Chapter": "1", "sentence_range": "91-94", "Text": "If\na system contains two point charges q1 and q2, the total\ncharge of the system is obtained simply by adding\nalgebraically q1 and q2 , i e , charges add up like real numbers or they\nare scalars like the mass of a body If a system contains n charges q1,\nq2, q3, \u2026, qn, then the total charge of the system is q1 + q2 + q3 + \u2026 + qn"}, {"Chapter": "1", "sentence_range": "92-95", "Text": "e , charges add up like real numbers or they\nare scalars like the mass of a body If a system contains n charges q1,\nq2, q3, \u2026, qn, then the total charge of the system is q1 + q2 + q3 + \u2026 + qn Charge has  magnitude but no direction, similar to mass"}, {"Chapter": "1", "sentence_range": "93-96", "Text": ", charges add up like real numbers or they\nare scalars like the mass of a body If a system contains n charges q1,\nq2, q3, \u2026, qn, then the total charge of the system is q1 + q2 + q3 + \u2026 + qn Charge has  magnitude but no direction, similar to mass However,\nthere is one difference between mass and charge"}, {"Chapter": "1", "sentence_range": "94-97", "Text": "If a system contains n charges q1,\nq2, q3, \u2026, qn, then the total charge of the system is q1 + q2 + q3 + \u2026 + qn Charge has  magnitude but no direction, similar to mass However,\nthere is one difference between mass and charge Mass of a body is\nalways positive whereas a charge can be either positive or negative"}, {"Chapter": "1", "sentence_range": "95-98", "Text": "Charge has  magnitude but no direction, similar to mass However,\nthere is one difference between mass and charge Mass of a body is\nalways positive whereas a charge can be either positive or negative Proper signs have to be used while adding the charges in a system"}, {"Chapter": "1", "sentence_range": "96-99", "Text": "However,\nthere is one difference between mass and charge Mass of a body is\nalways positive whereas a charge can be either positive or negative Proper signs have to be used while adding the charges in a system For\nexample, the total charge of a system containing five charges +1, +2, \u20133,\n+4 and \u20135, in some arbitrary unit, is (+1) + (+2) + (\u20133) + (+4) + (\u20135) = \u20131 in\nthe same unit"}, {"Chapter": "1", "sentence_range": "97-100", "Text": "Mass of a body is\nalways positive whereas a charge can be either positive or negative Proper signs have to be used while adding the charges in a system For\nexample, the total charge of a system containing five charges +1, +2, \u20133,\n+4 and \u20135, in some arbitrary unit, is (+1) + (+2) + (\u20133) + (+4) + (\u20135) = \u20131 in\nthe same unit 1"}, {"Chapter": "1", "sentence_range": "98-101", "Text": "Proper signs have to be used while adding the charges in a system For\nexample, the total charge of a system containing five charges +1, +2, \u20133,\n+4 and \u20135, in some arbitrary unit, is (+1) + (+2) + (\u20133) + (+4) + (\u20135) = \u20131 in\nthe same unit 1 4"}, {"Chapter": "1", "sentence_range": "99-102", "Text": "For\nexample, the total charge of a system containing five charges +1, +2, \u20133,\n+4 and \u20135, in some arbitrary unit, is (+1) + (+2) + (\u20133) + (+4) + (\u20135) = \u20131 in\nthe same unit 1 4 2  Charge is conserved\nWe have already  hinted to the fact that when bodies are charged by\nrubbing, there is transfer of electrons from one body to the other; no new\ncharges are either created or destroyed"}, {"Chapter": "1", "sentence_range": "100-103", "Text": "1 4 2  Charge is conserved\nWe have already  hinted to the fact that when bodies are charged by\nrubbing, there is transfer of electrons from one body to the other; no new\ncharges are either created or destroyed A picture of particles of electric\ncharge enables us to understand the idea of conservation of charge"}, {"Chapter": "1", "sentence_range": "101-104", "Text": "4 2  Charge is conserved\nWe have already  hinted to the fact that when bodies are charged by\nrubbing, there is transfer of electrons from one body to the other; no new\ncharges are either created or destroyed A picture of particles of electric\ncharge enables us to understand the idea of conservation of charge When\nwe rub two bodies, what one body gains in charge the other body loses"}, {"Chapter": "1", "sentence_range": "102-105", "Text": "2  Charge is conserved\nWe have already  hinted to the fact that when bodies are charged by\nrubbing, there is transfer of electrons from one body to the other; no new\ncharges are either created or destroyed A picture of particles of electric\ncharge enables us to understand the idea of conservation of charge When\nwe rub two bodies, what one body gains in charge the other body loses Within an isolated system consisting of many charged bodies, due to\ninteractions among the bodies, charges may get redistributed but it is\nfound that the total charge of the isolated system is always conserved"}, {"Chapter": "1", "sentence_range": "103-106", "Text": "A picture of particles of electric\ncharge enables us to understand the idea of conservation of charge When\nwe rub two bodies, what one body gains in charge the other body loses Within an isolated system consisting of many charged bodies, due to\ninteractions among the bodies, charges may get redistributed but it is\nfound that the total charge of the isolated system is always conserved Conservation of charge has been established experimentally"}, {"Chapter": "1", "sentence_range": "104-107", "Text": "When\nwe rub two bodies, what one body gains in charge the other body loses Within an isolated system consisting of many charged bodies, due to\ninteractions among the bodies, charges may get redistributed but it is\nfound that the total charge of the isolated system is always conserved Conservation of charge has been established experimentally It is not possible to create or destroy net charge carried by any isolated\nsystem although the charge carrying particles may be created or destroyed\nFIGURE 1"}, {"Chapter": "1", "sentence_range": "105-108", "Text": "Within an isolated system consisting of many charged bodies, due to\ninteractions among the bodies, charges may get redistributed but it is\nfound that the total charge of the isolated system is always conserved Conservation of charge has been established experimentally It is not possible to create or destroy net charge carried by any isolated\nsystem although the charge carrying particles may be created or destroyed\nFIGURE 1 2 Electroscopes: (a)\nThe gold leaf electroscope, (b)\nSchematics of a simple\nelectroscope"}, {"Chapter": "1", "sentence_range": "106-109", "Text": "Conservation of charge has been established experimentally It is not possible to create or destroy net charge carried by any isolated\nsystem although the charge carrying particles may be created or destroyed\nFIGURE 1 2 Electroscopes: (a)\nThe gold leaf electroscope, (b)\nSchematics of a simple\nelectroscope Rationalised 2023-24\nElectric Charges\nand Fields\n5\nin a process"}, {"Chapter": "1", "sentence_range": "107-110", "Text": "It is not possible to create or destroy net charge carried by any isolated\nsystem although the charge carrying particles may be created or destroyed\nFIGURE 1 2 Electroscopes: (a)\nThe gold leaf electroscope, (b)\nSchematics of a simple\nelectroscope Rationalised 2023-24\nElectric Charges\nand Fields\n5\nin a process Sometimes nature creates charged particles: a neutron turns\ninto a proton and an electron"}, {"Chapter": "1", "sentence_range": "108-111", "Text": "2 Electroscopes: (a)\nThe gold leaf electroscope, (b)\nSchematics of a simple\nelectroscope Rationalised 2023-24\nElectric Charges\nand Fields\n5\nin a process Sometimes nature creates charged particles: a neutron turns\ninto a proton and an electron The proton and electron thus created have\nequal and opposite charges and the total charge is zero before and after\nthe creation"}, {"Chapter": "1", "sentence_range": "109-112", "Text": "Rationalised 2023-24\nElectric Charges\nand Fields\n5\nin a process Sometimes nature creates charged particles: a neutron turns\ninto a proton and an electron The proton and electron thus created have\nequal and opposite charges and the total charge is zero before and after\nthe creation 1"}, {"Chapter": "1", "sentence_range": "110-113", "Text": "Sometimes nature creates charged particles: a neutron turns\ninto a proton and an electron The proton and electron thus created have\nequal and opposite charges and the total charge is zero before and after\nthe creation 1 4"}, {"Chapter": "1", "sentence_range": "111-114", "Text": "The proton and electron thus created have\nequal and opposite charges and the total charge is zero before and after\nthe creation 1 4 3  Quantisation of charge\nExperimentally it is established that all free charges are integral multiples\nof a basic unit of charge denoted by e"}, {"Chapter": "1", "sentence_range": "112-115", "Text": "1 4 3  Quantisation of charge\nExperimentally it is established that all free charges are integral multiples\nof a basic unit of charge denoted by e Thus charge q on a body is always\ngiven by\nq = ne\nwhere n is any integer, positive or negative"}, {"Chapter": "1", "sentence_range": "113-116", "Text": "4 3  Quantisation of charge\nExperimentally it is established that all free charges are integral multiples\nof a basic unit of charge denoted by e Thus charge q on a body is always\ngiven by\nq = ne\nwhere n is any integer, positive or negative This basic unit of charge is\nthe charge that an electron or proton carries"}, {"Chapter": "1", "sentence_range": "114-117", "Text": "3  Quantisation of charge\nExperimentally it is established that all free charges are integral multiples\nof a basic unit of charge denoted by e Thus charge q on a body is always\ngiven by\nq = ne\nwhere n is any integer, positive or negative This basic unit of charge is\nthe charge that an electron or proton carries By convention, the charge\non an electron is taken to be negative; therefore charge on an electron is\nwritten as \u2013e and that on a proton as +e"}, {"Chapter": "1", "sentence_range": "115-118", "Text": "Thus charge q on a body is always\ngiven by\nq = ne\nwhere n is any integer, positive or negative This basic unit of charge is\nthe charge that an electron or proton carries By convention, the charge\non an electron is taken to be negative; therefore charge on an electron is\nwritten as \u2013e and that on a proton as +e The fact that electric charge is always an integral multiple of e is termed\nas quantisation of charge"}, {"Chapter": "1", "sentence_range": "116-119", "Text": "This basic unit of charge is\nthe charge that an electron or proton carries By convention, the charge\non an electron is taken to be negative; therefore charge on an electron is\nwritten as \u2013e and that on a proton as +e The fact that electric charge is always an integral multiple of e is termed\nas quantisation of charge There are a large number of situations in physics\nwhere certain physical quantities are quantised"}, {"Chapter": "1", "sentence_range": "117-120", "Text": "By convention, the charge\non an electron is taken to be negative; therefore charge on an electron is\nwritten as \u2013e and that on a proton as +e The fact that electric charge is always an integral multiple of e is termed\nas quantisation of charge There are a large number of situations in physics\nwhere certain physical quantities are quantised The quantisation of charge\nwas first suggested by the experimental laws of electrolysis discovered by\nEnglish experimentalist Faraday"}, {"Chapter": "1", "sentence_range": "118-121", "Text": "The fact that electric charge is always an integral multiple of e is termed\nas quantisation of charge There are a large number of situations in physics\nwhere certain physical quantities are quantised The quantisation of charge\nwas first suggested by the experimental laws of electrolysis discovered by\nEnglish experimentalist Faraday It was experimentally demonstrated by\nMillikan in 1912"}, {"Chapter": "1", "sentence_range": "119-122", "Text": "There are a large number of situations in physics\nwhere certain physical quantities are quantised The quantisation of charge\nwas first suggested by the experimental laws of electrolysis discovered by\nEnglish experimentalist Faraday It was experimentally demonstrated by\nMillikan in 1912 In the International System (SI) of Units, a unit of charge is called a\ncoulomb and is denoted by the symbol C"}, {"Chapter": "1", "sentence_range": "120-123", "Text": "The quantisation of charge\nwas first suggested by the experimental laws of electrolysis discovered by\nEnglish experimentalist Faraday It was experimentally demonstrated by\nMillikan in 1912 In the International System (SI) of Units, a unit of charge is called a\ncoulomb and is denoted by the symbol C A coulomb is defined in terms\nthe unit of the electric current which you are going to learn in a\nsubsequent chapter"}, {"Chapter": "1", "sentence_range": "121-124", "Text": "It was experimentally demonstrated by\nMillikan in 1912 In the International System (SI) of Units, a unit of charge is called a\ncoulomb and is denoted by the symbol C A coulomb is defined in terms\nthe unit of the electric current which you are going to learn in a\nsubsequent chapter In terms of this definition, one coulomb is the charge\nflowing through a wire in 1 s if the current is 1 A (ampere), (see Chapter 1\nof Class XI, Physics Textbook , Part I)"}, {"Chapter": "1", "sentence_range": "122-125", "Text": "In the International System (SI) of Units, a unit of charge is called a\ncoulomb and is denoted by the symbol C A coulomb is defined in terms\nthe unit of the electric current which you are going to learn in a\nsubsequent chapter In terms of this definition, one coulomb is the charge\nflowing through a wire in 1 s if the current is 1 A (ampere), (see Chapter 1\nof Class XI, Physics Textbook , Part I) In this system, the value of the\nbasic unit of charge is\ne = 1"}, {"Chapter": "1", "sentence_range": "123-126", "Text": "A coulomb is defined in terms\nthe unit of the electric current which you are going to learn in a\nsubsequent chapter In terms of this definition, one coulomb is the charge\nflowing through a wire in 1 s if the current is 1 A (ampere), (see Chapter 1\nof Class XI, Physics Textbook , Part I) In this system, the value of the\nbasic unit of charge is\ne = 1 602192 \u00d7 10\u201319 C\nThus, there are about 6 \u00d7 1018 electrons in a charge of  \u20131C"}, {"Chapter": "1", "sentence_range": "124-127", "Text": "In terms of this definition, one coulomb is the charge\nflowing through a wire in 1 s if the current is 1 A (ampere), (see Chapter 1\nof Class XI, Physics Textbook , Part I) In this system, the value of the\nbasic unit of charge is\ne = 1 602192 \u00d7 10\u201319 C\nThus, there are about 6 \u00d7 1018 electrons in a charge of  \u20131C In\nelectrostatics, charges of this large magnitude are seldom encountered\nand hence we use smaller units 1 mC (micro coulomb) = 10\u20136 C or 1 mC\n(milli coulomb) = 10\u20133 C"}, {"Chapter": "1", "sentence_range": "125-128", "Text": "In this system, the value of the\nbasic unit of charge is\ne = 1 602192 \u00d7 10\u201319 C\nThus, there are about 6 \u00d7 1018 electrons in a charge of  \u20131C In\nelectrostatics, charges of this large magnitude are seldom encountered\nand hence we use smaller units 1 mC (micro coulomb) = 10\u20136 C or 1 mC\n(milli coulomb) = 10\u20133 C If the protons and electrons are the only basic charges in the\nuniverse, all the observable charges have to be integral multiples of e"}, {"Chapter": "1", "sentence_range": "126-129", "Text": "602192 \u00d7 10\u201319 C\nThus, there are about 6 \u00d7 1018 electrons in a charge of  \u20131C In\nelectrostatics, charges of this large magnitude are seldom encountered\nand hence we use smaller units 1 mC (micro coulomb) = 10\u20136 C or 1 mC\n(milli coulomb) = 10\u20133 C If the protons and electrons are the only basic charges in the\nuniverse, all the observable charges have to be integral multiples of e Thus, if a body contains n1 electrons and n2 protons, the total amount\nof charge on the body is n2 \u00d7 e + n1 \u00d7 (\u2013e) = (n2 \u2013 n1) e"}, {"Chapter": "1", "sentence_range": "127-130", "Text": "In\nelectrostatics, charges of this large magnitude are seldom encountered\nand hence we use smaller units 1 mC (micro coulomb) = 10\u20136 C or 1 mC\n(milli coulomb) = 10\u20133 C If the protons and electrons are the only basic charges in the\nuniverse, all the observable charges have to be integral multiples of e Thus, if a body contains n1 electrons and n2 protons, the total amount\nof charge on the body is n2 \u00d7 e + n1 \u00d7 (\u2013e) = (n2 \u2013 n1) e Since n1 and n2\nare integers, their difference is also an integer"}, {"Chapter": "1", "sentence_range": "128-131", "Text": "If the protons and electrons are the only basic charges in the\nuniverse, all the observable charges have to be integral multiples of e Thus, if a body contains n1 electrons and n2 protons, the total amount\nof charge on the body is n2 \u00d7 e + n1 \u00d7 (\u2013e) = (n2 \u2013 n1) e Since n1 and n2\nare integers, their difference is also an integer Thus the charge on any\nbody is always an integral multiple of e and can be increased or\ndecreased also in steps of e"}, {"Chapter": "1", "sentence_range": "129-132", "Text": "Thus, if a body contains n1 electrons and n2 protons, the total amount\nof charge on the body is n2 \u00d7 e + n1 \u00d7 (\u2013e) = (n2 \u2013 n1) e Since n1 and n2\nare integers, their difference is also an integer Thus the charge on any\nbody is always an integral multiple of e and can be increased or\ndecreased also in steps of e The step size e is, however, very small because at the macroscopic\nlevel, we deal with charges of a few mC"}, {"Chapter": "1", "sentence_range": "130-133", "Text": "Since n1 and n2\nare integers, their difference is also an integer Thus the charge on any\nbody is always an integral multiple of e and can be increased or\ndecreased also in steps of e The step size e is, however, very small because at the macroscopic\nlevel, we deal with charges of a few mC At this scale the fact that charge of\na body can increase or decrease in units of e is not visible"}, {"Chapter": "1", "sentence_range": "131-134", "Text": "Thus the charge on any\nbody is always an integral multiple of e and can be increased or\ndecreased also in steps of e The step size e is, however, very small because at the macroscopic\nlevel, we deal with charges of a few mC At this scale the fact that charge of\na body can increase or decrease in units of e is not visible In this respect,\nthe grainy nature of the charge is lost and it appears to be continuous"}, {"Chapter": "1", "sentence_range": "132-135", "Text": "The step size e is, however, very small because at the macroscopic\nlevel, we deal with charges of a few mC At this scale the fact that charge of\na body can increase or decrease in units of e is not visible In this respect,\nthe grainy nature of the charge is lost and it appears to be continuous This situation can be compared with the geometrical concepts of points\nand lines"}, {"Chapter": "1", "sentence_range": "133-136", "Text": "At this scale the fact that charge of\na body can increase or decrease in units of e is not visible In this respect,\nthe grainy nature of the charge is lost and it appears to be continuous This situation can be compared with the geometrical concepts of points\nand lines A dotted line viewed from a distance appears continuous to\nus but is not continuous in reality"}, {"Chapter": "1", "sentence_range": "134-137", "Text": "In this respect,\nthe grainy nature of the charge is lost and it appears to be continuous This situation can be compared with the geometrical concepts of points\nand lines A dotted line viewed from a distance appears continuous to\nus but is not continuous in reality As many points very close to\nRationalised 2023-24\n6\nPhysics\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "135-138", "Text": "This situation can be compared with the geometrical concepts of points\nand lines A dotted line viewed from a distance appears continuous to\nus but is not continuous in reality As many points very close to\nRationalised 2023-24\n6\nPhysics\n EXAMPLE 1 2\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "136-139", "Text": "A dotted line viewed from a distance appears continuous to\nus but is not continuous in reality As many points very close to\nRationalised 2023-24\n6\nPhysics\n EXAMPLE 1 2\n EXAMPLE 1 1\neach other normally give an impression of a continuous line, many\nsmall charges taken together appear as a continuous charge distribution"}, {"Chapter": "1", "sentence_range": "137-140", "Text": "As many points very close to\nRationalised 2023-24\n6\nPhysics\n EXAMPLE 1 2\n EXAMPLE 1 1\neach other normally give an impression of a continuous line, many\nsmall charges taken together appear as a continuous charge distribution At the macroscopic level, one deals with charges that are enormous\ncompared to the magnitude of charge e"}, {"Chapter": "1", "sentence_range": "138-141", "Text": "2\n EXAMPLE 1 1\neach other normally give an impression of a continuous line, many\nsmall charges taken together appear as a continuous charge distribution At the macroscopic level, one deals with charges that are enormous\ncompared to the magnitude of charge e Since e = 1"}, {"Chapter": "1", "sentence_range": "139-142", "Text": "1\neach other normally give an impression of a continuous line, many\nsmall charges taken together appear as a continuous charge distribution At the macroscopic level, one deals with charges that are enormous\ncompared to the magnitude of charge e Since e = 1 6 \u00d7 10\u201319 C, a charge\nof magnituOde, say 1 mC, contains something like 1013 times the electronic\ncharge"}, {"Chapter": "1", "sentence_range": "140-143", "Text": "At the macroscopic level, one deals with charges that are enormous\ncompared to the magnitude of charge e Since e = 1 6 \u00d7 10\u201319 C, a charge\nof magnituOde, say 1 mC, contains something like 1013 times the electronic\ncharge At this scale, the fact that charge can increase or decrease only in\nunits of e is not very different from saying that charge can take continuous\nvalues"}, {"Chapter": "1", "sentence_range": "141-144", "Text": "Since e = 1 6 \u00d7 10\u201319 C, a charge\nof magnituOde, say 1 mC, contains something like 1013 times the electronic\ncharge At this scale, the fact that charge can increase or decrease only in\nunits of e is not very different from saying that charge can take continuous\nvalues Thus, at the macroscopic level, the quantisation of charge has no\npractical consequence and can be ignored"}, {"Chapter": "1", "sentence_range": "142-145", "Text": "6 \u00d7 10\u201319 C, a charge\nof magnituOde, say 1 mC, contains something like 1013 times the electronic\ncharge At this scale, the fact that charge can increase or decrease only in\nunits of e is not very different from saying that charge can take continuous\nvalues Thus, at the macroscopic level, the quantisation of charge has no\npractical consequence and can be ignored However, at the microscopic\nlevel, where the charges involved are of the order of a few tens or hundreds\nof e, i"}, {"Chapter": "1", "sentence_range": "143-146", "Text": "At this scale, the fact that charge can increase or decrease only in\nunits of e is not very different from saying that charge can take continuous\nvalues Thus, at the macroscopic level, the quantisation of charge has no\npractical consequence and can be ignored However, at the microscopic\nlevel, where the charges involved are of the order of a few tens or hundreds\nof e, i e"}, {"Chapter": "1", "sentence_range": "144-147", "Text": "Thus, at the macroscopic level, the quantisation of charge has no\npractical consequence and can be ignored However, at the microscopic\nlevel, where the charges involved are of the order of a few tens or hundreds\nof e, i e , they can be counted, they appear in discrete lumps and\nquantisation of charge cannot be ignored"}, {"Chapter": "1", "sentence_range": "145-148", "Text": "However, at the microscopic\nlevel, where the charges involved are of the order of a few tens or hundreds\nof e, i e , they can be counted, they appear in discrete lumps and\nquantisation of charge cannot be ignored It is the magnitude of scale\ninvolved that is very important"}, {"Chapter": "1", "sentence_range": "146-149", "Text": "e , they can be counted, they appear in discrete lumps and\nquantisation of charge cannot be ignored It is the magnitude of scale\ninvolved that is very important Example 1"}, {"Chapter": "1", "sentence_range": "147-150", "Text": ", they can be counted, they appear in discrete lumps and\nquantisation of charge cannot be ignored It is the magnitude of scale\ninvolved that is very important Example 1 1 If 109 electrons move out of a body to another body\nevery second, how much time is required to get a total charge of 1 C\non the other body"}, {"Chapter": "1", "sentence_range": "148-151", "Text": "It is the magnitude of scale\ninvolved that is very important Example 1 1 If 109 electrons move out of a body to another body\nevery second, how much time is required to get a total charge of 1 C\non the other body Solution In one second 109 electrons move out of the body"}, {"Chapter": "1", "sentence_range": "149-152", "Text": "Example 1 1 If 109 electrons move out of a body to another body\nevery second, how much time is required to get a total charge of 1 C\non the other body Solution In one second 109 electrons move out of the body Therefore\nthe charge given out in one second is 1"}, {"Chapter": "1", "sentence_range": "150-153", "Text": "1 If 109 electrons move out of a body to another body\nevery second, how much time is required to get a total charge of 1 C\non the other body Solution In one second 109 electrons move out of the body Therefore\nthe charge given out in one second is 1 6 \u00d7 10\u201319 \u00d7 109 C = 1"}, {"Chapter": "1", "sentence_range": "151-154", "Text": "Solution In one second 109 electrons move out of the body Therefore\nthe charge given out in one second is 1 6 \u00d7 10\u201319 \u00d7 109 C = 1 6 \u00d7 10\u201310 C"}, {"Chapter": "1", "sentence_range": "152-155", "Text": "Therefore\nthe charge given out in one second is 1 6 \u00d7 10\u201319 \u00d7 109 C = 1 6 \u00d7 10\u201310 C The time required to accumulate a charge of 1 C can then be estimated\nto be 1 C \u00f7 (1"}, {"Chapter": "1", "sentence_range": "153-156", "Text": "6 \u00d7 10\u201319 \u00d7 109 C = 1 6 \u00d7 10\u201310 C The time required to accumulate a charge of 1 C can then be estimated\nto be 1 C \u00f7 (1 6 \u00d7 10\u201310 C/s) = 6"}, {"Chapter": "1", "sentence_range": "154-157", "Text": "6 \u00d7 10\u201310 C The time required to accumulate a charge of 1 C can then be estimated\nto be 1 C \u00f7 (1 6 \u00d7 10\u201310 C/s) = 6 25 \u00d7 109 s = 6"}, {"Chapter": "1", "sentence_range": "155-158", "Text": "The time required to accumulate a charge of 1 C can then be estimated\nto be 1 C \u00f7 (1 6 \u00d7 10\u201310 C/s) = 6 25 \u00d7 109 s = 6 25 \u00d7 109 \u00f7 (365  \u00d7 24 \u00d7\n3600) years = 198 years"}, {"Chapter": "1", "sentence_range": "156-159", "Text": "6 \u00d7 10\u201310 C/s) = 6 25 \u00d7 109 s = 6 25 \u00d7 109 \u00f7 (365  \u00d7 24 \u00d7\n3600) years = 198 years Thus to collect a charge of one coulomb,\nfrom a body from which 109 electrons move out every second, we will\nneed approximately 200 years"}, {"Chapter": "1", "sentence_range": "157-160", "Text": "25 \u00d7 109 s = 6 25 \u00d7 109 \u00f7 (365  \u00d7 24 \u00d7\n3600) years = 198 years Thus to collect a charge of one coulomb,\nfrom a body from which 109 electrons move out every second, we will\nneed approximately 200 years One coulomb is, therefore, a very large\nunit for many practical purposes"}, {"Chapter": "1", "sentence_range": "158-161", "Text": "25 \u00d7 109 \u00f7 (365  \u00d7 24 \u00d7\n3600) years = 198 years Thus to collect a charge of one coulomb,\nfrom a body from which 109 electrons move out every second, we will\nneed approximately 200 years One coulomb is, therefore, a very large\nunit for many practical purposes It is, however, also important to know what is roughly the number of\nelectrons contained in a piece of one cubic centimetre of a material"}, {"Chapter": "1", "sentence_range": "159-162", "Text": "Thus to collect a charge of one coulomb,\nfrom a body from which 109 electrons move out every second, we will\nneed approximately 200 years One coulomb is, therefore, a very large\nunit for many practical purposes It is, however, also important to know what is roughly the number of\nelectrons contained in a piece of one cubic centimetre of a material A cubic piece of copper of side 1 cm contains about 2"}, {"Chapter": "1", "sentence_range": "160-163", "Text": "One coulomb is, therefore, a very large\nunit for many practical purposes It is, however, also important to know what is roughly the number of\nelectrons contained in a piece of one cubic centimetre of a material A cubic piece of copper of side 1 cm contains about 2 5 \u00d7 1024\nelectrons"}, {"Chapter": "1", "sentence_range": "161-164", "Text": "It is, however, also important to know what is roughly the number of\nelectrons contained in a piece of one cubic centimetre of a material A cubic piece of copper of side 1 cm contains about 2 5 \u00d7 1024\nelectrons Example 1"}, {"Chapter": "1", "sentence_range": "162-165", "Text": "A cubic piece of copper of side 1 cm contains about 2 5 \u00d7 1024\nelectrons Example 1 2 How much positive and negative charge is there in a\ncup of water"}, {"Chapter": "1", "sentence_range": "163-166", "Text": "5 \u00d7 1024\nelectrons Example 1 2 How much positive and negative charge is there in a\ncup of water Solution Let us assume that the mass of one cup of water is\n250 g"}, {"Chapter": "1", "sentence_range": "164-167", "Text": "Example 1 2 How much positive and negative charge is there in a\ncup of water Solution Let us assume that the mass of one cup of water is\n250 g The molecular mass of water is 18g"}, {"Chapter": "1", "sentence_range": "165-168", "Text": "2 How much positive and negative charge is there in a\ncup of water Solution Let us assume that the mass of one cup of water is\n250 g The molecular mass of water is 18g Thus, one mole\n(= 6"}, {"Chapter": "1", "sentence_range": "166-169", "Text": "Solution Let us assume that the mass of one cup of water is\n250 g The molecular mass of water is 18g Thus, one mole\n(= 6 02 \u00d7 1023 molecules) of water is 18 g"}, {"Chapter": "1", "sentence_range": "167-170", "Text": "The molecular mass of water is 18g Thus, one mole\n(= 6 02 \u00d7 1023 molecules) of water is 18 g Therefore the number of\nmolecules in one cup of water is (250/18) \u00d7 6"}, {"Chapter": "1", "sentence_range": "168-171", "Text": "Thus, one mole\n(= 6 02 \u00d7 1023 molecules) of water is 18 g Therefore the number of\nmolecules in one cup of water is (250/18) \u00d7 6 02 \u00d7 1023"}, {"Chapter": "1", "sentence_range": "169-172", "Text": "02 \u00d7 1023 molecules) of water is 18 g Therefore the number of\nmolecules in one cup of water is (250/18) \u00d7 6 02 \u00d7 1023 Each molecule of water contains two hydrogen atoms and one oxygen\natom, i"}, {"Chapter": "1", "sentence_range": "170-173", "Text": "Therefore the number of\nmolecules in one cup of water is (250/18) \u00d7 6 02 \u00d7 1023 Each molecule of water contains two hydrogen atoms and one oxygen\natom, i e"}, {"Chapter": "1", "sentence_range": "171-174", "Text": "02 \u00d7 1023 Each molecule of water contains two hydrogen atoms and one oxygen\natom, i e , 10 electrons and 10 protons"}, {"Chapter": "1", "sentence_range": "172-175", "Text": "Each molecule of water contains two hydrogen atoms and one oxygen\natom, i e , 10 electrons and 10 protons Hence the total positive and\ntotal negative charge has the same magnitude"}, {"Chapter": "1", "sentence_range": "173-176", "Text": "e , 10 electrons and 10 protons Hence the total positive and\ntotal negative charge has the same magnitude It is equal to\n(250/18) \u00d7 6"}, {"Chapter": "1", "sentence_range": "174-177", "Text": ", 10 electrons and 10 protons Hence the total positive and\ntotal negative charge has the same magnitude It is equal to\n(250/18) \u00d7 6 02 \u00d7 1023 \u00d7 10 \u00d7 1"}, {"Chapter": "1", "sentence_range": "175-178", "Text": "Hence the total positive and\ntotal negative charge has the same magnitude It is equal to\n(250/18) \u00d7 6 02 \u00d7 1023 \u00d7 10 \u00d7 1 6 \u00d7 10\u201319 C = 1"}, {"Chapter": "1", "sentence_range": "176-179", "Text": "It is equal to\n(250/18) \u00d7 6 02 \u00d7 1023 \u00d7 10 \u00d7 1 6 \u00d7 10\u201319 C = 1 34 \u00d7 107 C"}, {"Chapter": "1", "sentence_range": "177-180", "Text": "02 \u00d7 1023 \u00d7 10 \u00d7 1 6 \u00d7 10\u201319 C = 1 34 \u00d7 107 C 1"}, {"Chapter": "1", "sentence_range": "178-181", "Text": "6 \u00d7 10\u201319 C = 1 34 \u00d7 107 C 1 5  COULOMB\u2019S LAW\nCoulomb\u2019s law is a quantitative statement about the force between two\npoint charges"}, {"Chapter": "1", "sentence_range": "179-182", "Text": "34 \u00d7 107 C 1 5  COULOMB\u2019S LAW\nCoulomb\u2019s law is a quantitative statement about the force between two\npoint charges When the linear size of charged bodies are much smaller\nthan the distance separating them, the size may be ignored and the\ncharged bodies are treated as point charges"}, {"Chapter": "1", "sentence_range": "180-183", "Text": "1 5  COULOMB\u2019S LAW\nCoulomb\u2019s law is a quantitative statement about the force between two\npoint charges When the linear size of charged bodies are much smaller\nthan the distance separating them, the size may be ignored and the\ncharged bodies are treated as point charges Coulomb measured the\nforce between two point charges and found that it varied inversely as\nthe square of the distance between the charges and was directly\nproportional to the product of the magnitude of the two charges and\nRationalised 2023-24\nElectric Charges\nand Fields\n7\nacted along the line joining the two charges"}, {"Chapter": "1", "sentence_range": "181-184", "Text": "5  COULOMB\u2019S LAW\nCoulomb\u2019s law is a quantitative statement about the force between two\npoint charges When the linear size of charged bodies are much smaller\nthan the distance separating them, the size may be ignored and the\ncharged bodies are treated as point charges Coulomb measured the\nforce between two point charges and found that it varied inversely as\nthe square of the distance between the charges and was directly\nproportional to the product of the magnitude of the two charges and\nRationalised 2023-24\nElectric Charges\nand Fields\n7\nacted along the line joining the two charges Thus, if two\npoint charges q1, q2 are separated by a distance r in vacuum,\nthe magnitude of the force (F) between them is given by\n2\n1\n2\nq\nq\nF\nk\nr\n=\n(1"}, {"Chapter": "1", "sentence_range": "182-185", "Text": "When the linear size of charged bodies are much smaller\nthan the distance separating them, the size may be ignored and the\ncharged bodies are treated as point charges Coulomb measured the\nforce between two point charges and found that it varied inversely as\nthe square of the distance between the charges and was directly\nproportional to the product of the magnitude of the two charges and\nRationalised 2023-24\nElectric Charges\nand Fields\n7\nacted along the line joining the two charges Thus, if two\npoint charges q1, q2 are separated by a distance r in vacuum,\nthe magnitude of the force (F) between them is given by\n2\n1\n2\nq\nq\nF\nk\nr\n=\n(1 1)\nHow did Coulomb arrive at this law from his experiments"}, {"Chapter": "1", "sentence_range": "183-186", "Text": "Coulomb measured the\nforce between two point charges and found that it varied inversely as\nthe square of the distance between the charges and was directly\nproportional to the product of the magnitude of the two charges and\nRationalised 2023-24\nElectric Charges\nand Fields\n7\nacted along the line joining the two charges Thus, if two\npoint charges q1, q2 are separated by a distance r in vacuum,\nthe magnitude of the force (F) between them is given by\n2\n1\n2\nq\nq\nF\nk\nr\n=\n(1 1)\nHow did Coulomb arrive at this law from his experiments Coulomb used a torsion balance*  for measuring the force\nbetween two charged metallic spheres"}, {"Chapter": "1", "sentence_range": "184-187", "Text": "Thus, if two\npoint charges q1, q2 are separated by a distance r in vacuum,\nthe magnitude of the force (F) between them is given by\n2\n1\n2\nq\nq\nF\nk\nr\n=\n(1 1)\nHow did Coulomb arrive at this law from his experiments Coulomb used a torsion balance*  for measuring the force\nbetween two charged metallic spheres When the separation\nbetween two spheres is much larger than the radius of each\nsphere, the charged spheres may be regarded as point charges"}, {"Chapter": "1", "sentence_range": "185-188", "Text": "1)\nHow did Coulomb arrive at this law from his experiments Coulomb used a torsion balance*  for measuring the force\nbetween two charged metallic spheres When the separation\nbetween two spheres is much larger than the radius of each\nsphere, the charged spheres may be regarded as point charges However, the charges on the spheres were unknown, to begin\nwith"}, {"Chapter": "1", "sentence_range": "186-189", "Text": "Coulomb used a torsion balance*  for measuring the force\nbetween two charged metallic spheres When the separation\nbetween two spheres is much larger than the radius of each\nsphere, the charged spheres may be regarded as point charges However, the charges on the spheres were unknown, to begin\nwith How then could he discover a relation like Eq"}, {"Chapter": "1", "sentence_range": "187-190", "Text": "When the separation\nbetween two spheres is much larger than the radius of each\nsphere, the charged spheres may be regarded as point charges However, the charges on the spheres were unknown, to begin\nwith How then could he discover a relation like Eq (1"}, {"Chapter": "1", "sentence_range": "188-191", "Text": "However, the charges on the spheres were unknown, to begin\nwith How then could he discover a relation like Eq (1 1)"}, {"Chapter": "1", "sentence_range": "189-192", "Text": "How then could he discover a relation like Eq (1 1) Coulomb thought of the following simple way: Suppose the\ncharge on a metallic sphere is q"}, {"Chapter": "1", "sentence_range": "190-193", "Text": "(1 1) Coulomb thought of the following simple way: Suppose the\ncharge on a metallic sphere is q If the sphere is put in contact\nwith an identical uncharged sphere, the charge will spread over\nthe two spheres"}, {"Chapter": "1", "sentence_range": "191-194", "Text": "1) Coulomb thought of the following simple way: Suppose the\ncharge on a metallic sphere is q If the sphere is put in contact\nwith an identical uncharged sphere, the charge will spread over\nthe two spheres By symmetry, the charge on each sphere will\nbe q/2*"}, {"Chapter": "1", "sentence_range": "192-195", "Text": "Coulomb thought of the following simple way: Suppose the\ncharge on a metallic sphere is q If the sphere is put in contact\nwith an identical uncharged sphere, the charge will spread over\nthe two spheres By symmetry, the charge on each sphere will\nbe q/2* Repeating this process, we can get charges q/2, q/4,\netc"}, {"Chapter": "1", "sentence_range": "193-196", "Text": "If the sphere is put in contact\nwith an identical uncharged sphere, the charge will spread over\nthe two spheres By symmetry, the charge on each sphere will\nbe q/2* Repeating this process, we can get charges q/2, q/4,\netc Coulomb varied the distance for a fixed pair of charges and\nmeasured the force for different separations"}, {"Chapter": "1", "sentence_range": "194-197", "Text": "By symmetry, the charge on each sphere will\nbe q/2* Repeating this process, we can get charges q/2, q/4,\netc Coulomb varied the distance for a fixed pair of charges and\nmeasured the force for different separations He then varied the\ncharges in pairs, keeping the distance fixed for each pair"}, {"Chapter": "1", "sentence_range": "195-198", "Text": "Repeating this process, we can get charges q/2, q/4,\netc Coulomb varied the distance for a fixed pair of charges and\nmeasured the force for different separations He then varied the\ncharges in pairs, keeping the distance fixed for each pair Comparing forces for different pairs of charges at different\ndistances, Coulomb arrived at the relation, Eq"}, {"Chapter": "1", "sentence_range": "196-199", "Text": "Coulomb varied the distance for a fixed pair of charges and\nmeasured the force for different separations He then varied the\ncharges in pairs, keeping the distance fixed for each pair Comparing forces for different pairs of charges at different\ndistances, Coulomb arrived at the relation, Eq (1"}, {"Chapter": "1", "sentence_range": "197-200", "Text": "He then varied the\ncharges in pairs, keeping the distance fixed for each pair Comparing forces for different pairs of charges at different\ndistances, Coulomb arrived at the relation, Eq (1 1)"}, {"Chapter": "1", "sentence_range": "198-201", "Text": "Comparing forces for different pairs of charges at different\ndistances, Coulomb arrived at the relation, Eq (1 1) Coulomb\u2019s law, a simple mathematical statement,  was\ninitially experimentally arrived at  in the manner described\nabove"}, {"Chapter": "1", "sentence_range": "199-202", "Text": "(1 1) Coulomb\u2019s law, a simple mathematical statement,  was\ninitially experimentally arrived at  in the manner described\nabove While the original experiments established it at a\nmacroscopic scale, it has also been established down to\nsubatomic level  (r ~ 10\u201310 m)"}, {"Chapter": "1", "sentence_range": "200-203", "Text": "1) Coulomb\u2019s law, a simple mathematical statement,  was\ninitially experimentally arrived at  in the manner described\nabove While the original experiments established it at a\nmacroscopic scale, it has also been established down to\nsubatomic level  (r ~ 10\u201310 m) Coulomb discovered his law without knowing the explicit\nmagnitude of the charge"}, {"Chapter": "1", "sentence_range": "201-204", "Text": "Coulomb\u2019s law, a simple mathematical statement,  was\ninitially experimentally arrived at  in the manner described\nabove While the original experiments established it at a\nmacroscopic scale, it has also been established down to\nsubatomic level  (r ~ 10\u201310 m) Coulomb discovered his law without knowing the explicit\nmagnitude of the charge In fact, it is the other way round:\nCoulomb\u2019s law can now be employed to furnish a definition\nfor a unit of charge"}, {"Chapter": "1", "sentence_range": "202-205", "Text": "While the original experiments established it at a\nmacroscopic scale, it has also been established down to\nsubatomic level  (r ~ 10\u201310 m) Coulomb discovered his law without knowing the explicit\nmagnitude of the charge In fact, it is the other way round:\nCoulomb\u2019s law can now be employed to furnish a definition\nfor a unit of charge In the relation, Eq"}, {"Chapter": "1", "sentence_range": "203-206", "Text": "Coulomb discovered his law without knowing the explicit\nmagnitude of the charge In fact, it is the other way round:\nCoulomb\u2019s law can now be employed to furnish a definition\nfor a unit of charge In the relation, Eq (1"}, {"Chapter": "1", "sentence_range": "204-207", "Text": "In fact, it is the other way round:\nCoulomb\u2019s law can now be employed to furnish a definition\nfor a unit of charge In the relation, Eq (1 1), k is so far\narbitrary"}, {"Chapter": "1", "sentence_range": "205-208", "Text": "In the relation, Eq (1 1), k is so far\narbitrary We can choose any positive value of k"}, {"Chapter": "1", "sentence_range": "206-209", "Text": "(1 1), k is so far\narbitrary We can choose any positive value of k The choice\nof k determines the size of the unit of charge"}, {"Chapter": "1", "sentence_range": "207-210", "Text": "1), k is so far\narbitrary We can choose any positive value of k The choice\nof k determines the size of the unit of charge In SI units, the\nvalue of k is about 9 \u00d7 109 \n2\n2\nNm\nC"}, {"Chapter": "1", "sentence_range": "208-211", "Text": "We can choose any positive value of k The choice\nof k determines the size of the unit of charge In SI units, the\nvalue of k is about 9 \u00d7 109 \n2\n2\nNm\nC The unit of charge that\nresults from this choice is called a coulomb which we defined\nearlier in Section 1"}, {"Chapter": "1", "sentence_range": "209-212", "Text": "The choice\nof k determines the size of the unit of charge In SI units, the\nvalue of k is about 9 \u00d7 109 \n2\n2\nNm\nC The unit of charge that\nresults from this choice is called a coulomb which we defined\nearlier in Section 1 4"}, {"Chapter": "1", "sentence_range": "210-213", "Text": "In SI units, the\nvalue of k is about 9 \u00d7 109 \n2\n2\nNm\nC The unit of charge that\nresults from this choice is called a coulomb which we defined\nearlier in Section 1 4 Putting this value of k in Eq"}, {"Chapter": "1", "sentence_range": "211-214", "Text": "The unit of charge that\nresults from this choice is called a coulomb which we defined\nearlier in Section 1 4 Putting this value of k in Eq (1"}, {"Chapter": "1", "sentence_range": "212-215", "Text": "4 Putting this value of k in Eq (1 1), we\nsee that for q1 = q2 = 1 C, r = 1 m\nF = 9 \u00d7 109 N\nThat is, 1 C is the charge that when placed at a distance\nof 1 m from another charge of the same magnitude in vacuum\nexperiences an electrical force of repulsion of magnitude\n*\nA torsion balance is a sensitive device to measure force"}, {"Chapter": "1", "sentence_range": "213-216", "Text": "Putting this value of k in Eq (1 1), we\nsee that for q1 = q2 = 1 C, r = 1 m\nF = 9 \u00d7 109 N\nThat is, 1 C is the charge that when placed at a distance\nof 1 m from another charge of the same magnitude in vacuum\nexperiences an electrical force of repulsion of magnitude\n*\nA torsion balance is a sensitive device to measure force It was also used later\nby Cavendish to measure the very feeble gravitational force between two objects,\nto verify Newton\u2019s Law of Gravitation"}, {"Chapter": "1", "sentence_range": "214-217", "Text": "(1 1), we\nsee that for q1 = q2 = 1 C, r = 1 m\nF = 9 \u00d7 109 N\nThat is, 1 C is the charge that when placed at a distance\nof 1 m from another charge of the same magnitude in vacuum\nexperiences an electrical force of repulsion of magnitude\n*\nA torsion balance is a sensitive device to measure force It was also used later\nby Cavendish to measure the very feeble gravitational force between two objects,\nto verify Newton\u2019s Law of Gravitation *\nImplicit in this is the assumption of additivity of charges and conservation:\ntwo charges (q/2 each) add up to make a total charge q"}, {"Chapter": "1", "sentence_range": "215-218", "Text": "1), we\nsee that for q1 = q2 = 1 C, r = 1 m\nF = 9 \u00d7 109 N\nThat is, 1 C is the charge that when placed at a distance\nof 1 m from another charge of the same magnitude in vacuum\nexperiences an electrical force of repulsion of magnitude\n*\nA torsion balance is a sensitive device to measure force It was also used later\nby Cavendish to measure the very feeble gravitational force between two objects,\nto verify Newton\u2019s Law of Gravitation *\nImplicit in this is the assumption of additivity of charges and conservation:\ntwo charges (q/2 each) add up to make a total charge q Charles Augustin de\nCoulomb (1736 \u2013 1806)\nCoulomb, a French\nphysicist, began his\ncareer as a military\nengineer in the West\nIndies"}, {"Chapter": "1", "sentence_range": "216-219", "Text": "It was also used later\nby Cavendish to measure the very feeble gravitational force between two objects,\nto verify Newton\u2019s Law of Gravitation *\nImplicit in this is the assumption of additivity of charges and conservation:\ntwo charges (q/2 each) add up to make a total charge q Charles Augustin de\nCoulomb (1736 \u2013 1806)\nCoulomb, a French\nphysicist, began his\ncareer as a military\nengineer in the West\nIndies In 1776, he\nreturned to Paris and\nretired to a small estate\nto do his scientific\nresearch"}, {"Chapter": "1", "sentence_range": "217-220", "Text": "*\nImplicit in this is the assumption of additivity of charges and conservation:\ntwo charges (q/2 each) add up to make a total charge q Charles Augustin de\nCoulomb (1736 \u2013 1806)\nCoulomb, a French\nphysicist, began his\ncareer as a military\nengineer in the West\nIndies In 1776, he\nreturned to Paris and\nretired to a small estate\nto do his scientific\nresearch He invented a\ntorsion \nbalance \nto\nmeasure the quantity of\na force and used it for\ndetermination of forces\nof electric attraction or\nrepulsion between small\ncharged spheres"}, {"Chapter": "1", "sentence_range": "218-221", "Text": "Charles Augustin de\nCoulomb (1736 \u2013 1806)\nCoulomb, a French\nphysicist, began his\ncareer as a military\nengineer in the West\nIndies In 1776, he\nreturned to Paris and\nretired to a small estate\nto do his scientific\nresearch He invented a\ntorsion \nbalance \nto\nmeasure the quantity of\na force and used it for\ndetermination of forces\nof electric attraction or\nrepulsion between small\ncharged spheres He\nthus arrived in 1785 at\nthe inverse square law\nrelation, now known as\nCoulomb\u2019s law"}, {"Chapter": "1", "sentence_range": "219-222", "Text": "In 1776, he\nreturned to Paris and\nretired to a small estate\nto do his scientific\nresearch He invented a\ntorsion \nbalance \nto\nmeasure the quantity of\na force and used it for\ndetermination of forces\nof electric attraction or\nrepulsion between small\ncharged spheres He\nthus arrived in 1785 at\nthe inverse square law\nrelation, now known as\nCoulomb\u2019s law The law\nhad been anticipated by\nPriestley and also by\nCavendish \nearlier,\nthough \nCavendish\nnever published his\nresults"}, {"Chapter": "1", "sentence_range": "220-223", "Text": "He invented a\ntorsion \nbalance \nto\nmeasure the quantity of\na force and used it for\ndetermination of forces\nof electric attraction or\nrepulsion between small\ncharged spheres He\nthus arrived in 1785 at\nthe inverse square law\nrelation, now known as\nCoulomb\u2019s law The law\nhad been anticipated by\nPriestley and also by\nCavendish \nearlier,\nthough \nCavendish\nnever published his\nresults Coulomb also\nfound \nthe \ninverse\nsquare law of force\nbetween unlike and like\nmagnetic poles"}, {"Chapter": "1", "sentence_range": "221-224", "Text": "He\nthus arrived in 1785 at\nthe inverse square law\nrelation, now known as\nCoulomb\u2019s law The law\nhad been anticipated by\nPriestley and also by\nCavendish \nearlier,\nthough \nCavendish\nnever published his\nresults Coulomb also\nfound \nthe \ninverse\nsquare law of force\nbetween unlike and like\nmagnetic poles CHARLES AUGUSTIN DE COULOMB  (1736 \u20131806)\nRationalised 2023-24\n8\nPhysics\n9 \u00d7 109 N"}, {"Chapter": "1", "sentence_range": "222-225", "Text": "The law\nhad been anticipated by\nPriestley and also by\nCavendish \nearlier,\nthough \nCavendish\nnever published his\nresults Coulomb also\nfound \nthe \ninverse\nsquare law of force\nbetween unlike and like\nmagnetic poles CHARLES AUGUSTIN DE COULOMB  (1736 \u20131806)\nRationalised 2023-24\n8\nPhysics\n9 \u00d7 109 N One coulomb is evidently too big a unit to\nbe used"}, {"Chapter": "1", "sentence_range": "223-226", "Text": "Coulomb also\nfound \nthe \ninverse\nsquare law of force\nbetween unlike and like\nmagnetic poles CHARLES AUGUSTIN DE COULOMB  (1736 \u20131806)\nRationalised 2023-24\n8\nPhysics\n9 \u00d7 109 N One coulomb is evidently too big a unit to\nbe used In practice, in electrostatics, one uses\nsmaller units like 1 mC or 1 mC"}, {"Chapter": "1", "sentence_range": "224-227", "Text": "CHARLES AUGUSTIN DE COULOMB  (1736 \u20131806)\nRationalised 2023-24\n8\nPhysics\n9 \u00d7 109 N One coulomb is evidently too big a unit to\nbe used In practice, in electrostatics, one uses\nsmaller units like 1 mC or 1 mC The constant k in Eq"}, {"Chapter": "1", "sentence_range": "225-228", "Text": "One coulomb is evidently too big a unit to\nbe used In practice, in electrostatics, one uses\nsmaller units like 1 mC or 1 mC The constant k in Eq (1"}, {"Chapter": "1", "sentence_range": "226-229", "Text": "In practice, in electrostatics, one uses\nsmaller units like 1 mC or 1 mC The constant k in Eq (1 1) is usually put as\nk = 1/4pe0 for later convenience, so that Coulomb\u2019s\nlaw is written as\n0\n1\n2\n2\n1\n4\nq q\nF\nr\n\u03b5\n=\n\u03c0\n(1"}, {"Chapter": "1", "sentence_range": "227-230", "Text": "The constant k in Eq (1 1) is usually put as\nk = 1/4pe0 for later convenience, so that Coulomb\u2019s\nlaw is written as\n0\n1\n2\n2\n1\n4\nq q\nF\nr\n\u03b5\n=\n\u03c0\n(1 2)\ne0 is called the permittivity of free space"}, {"Chapter": "1", "sentence_range": "228-231", "Text": "(1 1) is usually put as\nk = 1/4pe0 for later convenience, so that Coulomb\u2019s\nlaw is written as\n0\n1\n2\n2\n1\n4\nq q\nF\nr\n\u03b5\n=\n\u03c0\n(1 2)\ne0 is called the permittivity of free space The value\nof e0 in SI units is\n0\n\uf065 = 8"}, {"Chapter": "1", "sentence_range": "229-232", "Text": "1) is usually put as\nk = 1/4pe0 for later convenience, so that Coulomb\u2019s\nlaw is written as\n0\n1\n2\n2\n1\n4\nq q\nF\nr\n\u03b5\n=\n\u03c0\n(1 2)\ne0 is called the permittivity of free space The value\nof e0 in SI units is\n0\n\uf065 = 8 854 \u00d7 10\u201312 C2 N\u20131m\u20132\nSince force is a vector, it is better to write\nCoulomb\u2019s law in the vector notation"}, {"Chapter": "1", "sentence_range": "230-233", "Text": "2)\ne0 is called the permittivity of free space The value\nof e0 in SI units is\n0\n\uf065 = 8 854 \u00d7 10\u201312 C2 N\u20131m\u20132\nSince force is a vector, it is better to write\nCoulomb\u2019s law in the vector notation Let the position\nvectors of charges q1 and q2 be r1 and r2 respectively\n[see Fig"}, {"Chapter": "1", "sentence_range": "231-234", "Text": "The value\nof e0 in SI units is\n0\n\uf065 = 8 854 \u00d7 10\u201312 C2 N\u20131m\u20132\nSince force is a vector, it is better to write\nCoulomb\u2019s law in the vector notation Let the position\nvectors of charges q1 and q2 be r1 and r2 respectively\n[see Fig 1"}, {"Chapter": "1", "sentence_range": "232-235", "Text": "854 \u00d7 10\u201312 C2 N\u20131m\u20132\nSince force is a vector, it is better to write\nCoulomb\u2019s law in the vector notation Let the position\nvectors of charges q1 and q2 be r1 and r2 respectively\n[see Fig 1 3(a)]"}, {"Chapter": "1", "sentence_range": "233-236", "Text": "Let the position\nvectors of charges q1 and q2 be r1 and r2 respectively\n[see Fig 1 3(a)] We denote force on q1 due to q2 by\nF12 and force on q2 due to q1 by F21"}, {"Chapter": "1", "sentence_range": "234-237", "Text": "1 3(a)] We denote force on q1 due to q2 by\nF12 and force on q2 due to q1 by F21 The two point\ncharges q1 and q2 have been numbered 1 and 2 for\nconvenience and the vector leading from 1 to 2 is\ndenoted by r21:\nr21 = r2 \u2013 r1\nIn the same way, the vector leading from 2 to 1 is denoted by r12:\nr12 = r1 \u2013 r2 = \u2013 r21\nThe magnitude of the vectors r21 and r12 is denoted by r21 and r12,\nrespectively (r12 = r21)"}, {"Chapter": "1", "sentence_range": "235-238", "Text": "3(a)] We denote force on q1 due to q2 by\nF12 and force on q2 due to q1 by F21 The two point\ncharges q1 and q2 have been numbered 1 and 2 for\nconvenience and the vector leading from 1 to 2 is\ndenoted by r21:\nr21 = r2 \u2013 r1\nIn the same way, the vector leading from 2 to 1 is denoted by r12:\nr12 = r1 \u2013 r2 = \u2013 r21\nThe magnitude of the vectors r21 and r12 is denoted by r21 and r12,\nrespectively (r12 = r21) The direction of a vector is specified by a unit\nvector along the vector"}, {"Chapter": "1", "sentence_range": "236-239", "Text": "We denote force on q1 due to q2 by\nF12 and force on q2 due to q1 by F21 The two point\ncharges q1 and q2 have been numbered 1 and 2 for\nconvenience and the vector leading from 1 to 2 is\ndenoted by r21:\nr21 = r2 \u2013 r1\nIn the same way, the vector leading from 2 to 1 is denoted by r12:\nr12 = r1 \u2013 r2 = \u2013 r21\nThe magnitude of the vectors r21 and r12 is denoted by r21 and r12,\nrespectively (r12 = r21) The direction of a vector is specified by a unit\nvector along the vector To denote the direction from 1 to 2 (or from 2 to\n1), we define the unit vectors:\n\u0275\n21\n21\n21\n= r\nr\nr\n, \u0275\n\u0275\n\u0275\n12\n12\n21\n12\n12\n,\n=\n\u2212\nr\nr\nr\nr\nr\nCoulomb\u2019s force law between two point charges q1 and q2 located at\nr1 and r2, respectively is then expressed as\n\u0275\n1\n2\n21\n21\n2\n21\n41\n\u03b5\n=\n\u03c0\nF\nr\no\nq\nrq\n(1"}, {"Chapter": "1", "sentence_range": "237-240", "Text": "The two point\ncharges q1 and q2 have been numbered 1 and 2 for\nconvenience and the vector leading from 1 to 2 is\ndenoted by r21:\nr21 = r2 \u2013 r1\nIn the same way, the vector leading from 2 to 1 is denoted by r12:\nr12 = r1 \u2013 r2 = \u2013 r21\nThe magnitude of the vectors r21 and r12 is denoted by r21 and r12,\nrespectively (r12 = r21) The direction of a vector is specified by a unit\nvector along the vector To denote the direction from 1 to 2 (or from 2 to\n1), we define the unit vectors:\n\u0275\n21\n21\n21\n= r\nr\nr\n, \u0275\n\u0275\n\u0275\n12\n12\n21\n12\n12\n,\n=\n\u2212\nr\nr\nr\nr\nr\nCoulomb\u2019s force law between two point charges q1 and q2 located at\nr1 and r2, respectively is then expressed as\n\u0275\n1\n2\n21\n21\n2\n21\n41\n\u03b5\n=\n\u03c0\nF\nr\no\nq\nrq\n(1 3)\nSome remarks on Eq"}, {"Chapter": "1", "sentence_range": "238-241", "Text": "The direction of a vector is specified by a unit\nvector along the vector To denote the direction from 1 to 2 (or from 2 to\n1), we define the unit vectors:\n\u0275\n21\n21\n21\n= r\nr\nr\n, \u0275\n\u0275\n\u0275\n12\n12\n21\n12\n12\n,\n=\n\u2212\nr\nr\nr\nr\nr\nCoulomb\u2019s force law between two point charges q1 and q2 located at\nr1 and r2, respectively is then expressed as\n\u0275\n1\n2\n21\n21\n2\n21\n41\n\u03b5\n=\n\u03c0\nF\nr\no\nq\nrq\n(1 3)\nSome remarks on Eq (1"}, {"Chapter": "1", "sentence_range": "239-242", "Text": "To denote the direction from 1 to 2 (or from 2 to\n1), we define the unit vectors:\n\u0275\n21\n21\n21\n= r\nr\nr\n, \u0275\n\u0275\n\u0275\n12\n12\n21\n12\n12\n,\n=\n\u2212\nr\nr\nr\nr\nr\nCoulomb\u2019s force law between two point charges q1 and q2 located at\nr1 and r2, respectively is then expressed as\n\u0275\n1\n2\n21\n21\n2\n21\n41\n\u03b5\n=\n\u03c0\nF\nr\no\nq\nrq\n(1 3)\nSome remarks on Eq (1 3) are relevant:\n\u00b7\nEquation (1"}, {"Chapter": "1", "sentence_range": "240-243", "Text": "3)\nSome remarks on Eq (1 3) are relevant:\n\u00b7\nEquation (1 3) is valid for any sign of q1 and q2 whether positive or\nnegative"}, {"Chapter": "1", "sentence_range": "241-244", "Text": "(1 3) are relevant:\n\u00b7\nEquation (1 3) is valid for any sign of q1 and q2 whether positive or\nnegative If q1 and q2 are of the same sign (either both positive or both\nnegative), F21 is along \u02c6r 21, which denotes repulsion, as it should be for\nlike charges"}, {"Chapter": "1", "sentence_range": "242-245", "Text": "3) are relevant:\n\u00b7\nEquation (1 3) is valid for any sign of q1 and q2 whether positive or\nnegative If q1 and q2 are of the same sign (either both positive or both\nnegative), F21 is along \u02c6r 21, which denotes repulsion, as it should be for\nlike charges If q1 and q2 are of opposite signs, F21 is along \u2013 \u0275r 21(= \u0275r 12),\nwhich denotes attraction, as expected for unlike charges"}, {"Chapter": "1", "sentence_range": "243-246", "Text": "3) is valid for any sign of q1 and q2 whether positive or\nnegative If q1 and q2 are of the same sign (either both positive or both\nnegative), F21 is along \u02c6r 21, which denotes repulsion, as it should be for\nlike charges If q1 and q2 are of opposite signs, F21 is along \u2013 \u0275r 21(= \u0275r 12),\nwhich denotes attraction, as expected for unlike charges Thus, we do\nnot have to write separate equations for the cases of like and unlike\ncharges"}, {"Chapter": "1", "sentence_range": "244-247", "Text": "If q1 and q2 are of the same sign (either both positive or both\nnegative), F21 is along \u02c6r 21, which denotes repulsion, as it should be for\nlike charges If q1 and q2 are of opposite signs, F21 is along \u2013 \u0275r 21(= \u0275r 12),\nwhich denotes attraction, as expected for unlike charges Thus, we do\nnot have to write separate equations for the cases of like and unlike\ncharges Equation (1"}, {"Chapter": "1", "sentence_range": "245-248", "Text": "If q1 and q2 are of opposite signs, F21 is along \u2013 \u0275r 21(= \u0275r 12),\nwhich denotes attraction, as expected for unlike charges Thus, we do\nnot have to write separate equations for the cases of like and unlike\ncharges Equation (1 3) takes care of both cases correctly [Fig"}, {"Chapter": "1", "sentence_range": "246-249", "Text": "Thus, we do\nnot have to write separate equations for the cases of like and unlike\ncharges Equation (1 3) takes care of both cases correctly [Fig 1"}, {"Chapter": "1", "sentence_range": "247-250", "Text": "Equation (1 3) takes care of both cases correctly [Fig 1 3(b)]"}, {"Chapter": "1", "sentence_range": "248-251", "Text": "3) takes care of both cases correctly [Fig 1 3(b)] FIGURE 1"}, {"Chapter": "1", "sentence_range": "249-252", "Text": "1 3(b)] FIGURE 1 3 (a) Geometry and\n(b) Forces between charges"}, {"Chapter": "1", "sentence_range": "250-253", "Text": "3(b)] FIGURE 1 3 (a) Geometry and\n(b) Forces between charges Rationalised 2023-24\nElectric Charges\nand Fields\n9\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "251-254", "Text": "FIGURE 1 3 (a) Geometry and\n(b) Forces between charges Rationalised 2023-24\nElectric Charges\nand Fields\n9\n EXAMPLE 1 3\nInteractive animation on Coulomb\u2019s law:\nhttp://webphysics"}, {"Chapter": "1", "sentence_range": "252-255", "Text": "3 (a) Geometry and\n(b) Forces between charges Rationalised 2023-24\nElectric Charges\nand Fields\n9\n EXAMPLE 1 3\nInteractive animation on Coulomb\u2019s law:\nhttp://webphysics davidson"}, {"Chapter": "1", "sentence_range": "253-256", "Text": "Rationalised 2023-24\nElectric Charges\nand Fields\n9\n EXAMPLE 1 3\nInteractive animation on Coulomb\u2019s law:\nhttp://webphysics davidson edu/physlet_resources/bu_semester2/menu_semester2"}, {"Chapter": "1", "sentence_range": "254-257", "Text": "3\nInteractive animation on Coulomb\u2019s law:\nhttp://webphysics davidson edu/physlet_resources/bu_semester2/menu_semester2 html\n\u00b7\nThe force F12  on charge q1 due to charge q2, is obtained from Eq"}, {"Chapter": "1", "sentence_range": "255-258", "Text": "davidson edu/physlet_resources/bu_semester2/menu_semester2 html\n\u00b7\nThe force F12  on charge q1 due to charge q2, is obtained from Eq (1"}, {"Chapter": "1", "sentence_range": "256-259", "Text": "edu/physlet_resources/bu_semester2/menu_semester2 html\n\u00b7\nThe force F12  on charge q1 due to charge q2, is obtained from Eq (1 3),\nby simply interchanging 1 and 2, i"}, {"Chapter": "1", "sentence_range": "257-260", "Text": "html\n\u00b7\nThe force F12  on charge q1 due to charge q2, is obtained from Eq (1 3),\nby simply interchanging 1 and 2, i e"}, {"Chapter": "1", "sentence_range": "258-261", "Text": "(1 3),\nby simply interchanging 1 and 2, i e ,\n1\n2\n12\n12\n21\n2\n0\n12\n1\n\u02c6\n4\n\u03b5\n=\n= \u2212\n\u03c0\nF\nr\nF\nq\nq\nr\nThus, Coulomb\u2019s law agrees with the Newton\u2019s third law"}, {"Chapter": "1", "sentence_range": "259-262", "Text": "3),\nby simply interchanging 1 and 2, i e ,\n1\n2\n12\n12\n21\n2\n0\n12\n1\n\u02c6\n4\n\u03b5\n=\n= \u2212\n\u03c0\nF\nr\nF\nq\nq\nr\nThus, Coulomb\u2019s law agrees with the Newton\u2019s third law \u00b7\nCoulomb\u2019s law [Eq"}, {"Chapter": "1", "sentence_range": "260-263", "Text": "e ,\n1\n2\n12\n12\n21\n2\n0\n12\n1\n\u02c6\n4\n\u03b5\n=\n= \u2212\n\u03c0\nF\nr\nF\nq\nq\nr\nThus, Coulomb\u2019s law agrees with the Newton\u2019s third law \u00b7\nCoulomb\u2019s law [Eq (1"}, {"Chapter": "1", "sentence_range": "261-264", "Text": ",\n1\n2\n12\n12\n21\n2\n0\n12\n1\n\u02c6\n4\n\u03b5\n=\n= \u2212\n\u03c0\nF\nr\nF\nq\nq\nr\nThus, Coulomb\u2019s law agrees with the Newton\u2019s third law \u00b7\nCoulomb\u2019s law [Eq (1 3)] gives the force between two charges q1 and\nq2 in vacuum"}, {"Chapter": "1", "sentence_range": "262-265", "Text": "\u00b7\nCoulomb\u2019s law [Eq (1 3)] gives the force between two charges q1 and\nq2 in vacuum If the charges are placed in matter or the intervening\nspace has matter, the situation gets complicated due to the presence\nof charged constituents of matter"}, {"Chapter": "1", "sentence_range": "263-266", "Text": "(1 3)] gives the force between two charges q1 and\nq2 in vacuum If the charges are placed in matter or the intervening\nspace has matter, the situation gets complicated due to the presence\nof charged constituents of matter We shall consider electrostatics in\nmatter in the next chapter"}, {"Chapter": "1", "sentence_range": "264-267", "Text": "3)] gives the force between two charges q1 and\nq2 in vacuum If the charges are placed in matter or the intervening\nspace has matter, the situation gets complicated due to the presence\nof charged constituents of matter We shall consider electrostatics in\nmatter in the next chapter Example 1"}, {"Chapter": "1", "sentence_range": "265-268", "Text": "If the charges are placed in matter or the intervening\nspace has matter, the situation gets complicated due to the presence\nof charged constituents of matter We shall consider electrostatics in\nmatter in the next chapter Example 1 3 Coulomb\u2019s law for electrostatic force between two point\ncharges and Newton\u2019s law for gravitational force between two stationary\npoint masses, both have inverse-square dependence on the distance\nbetween the charges and masses respectively"}, {"Chapter": "1", "sentence_range": "266-269", "Text": "We shall consider electrostatics in\nmatter in the next chapter Example 1 3 Coulomb\u2019s law for electrostatic force between two point\ncharges and Newton\u2019s law for gravitational force between two stationary\npoint masses, both have inverse-square dependence on the distance\nbetween the charges and masses respectively (a) Compare the strength\nof these forces by determining the ratio of their magnitudes (i) for an\nelectron and a proton and (ii) for two protons"}, {"Chapter": "1", "sentence_range": "267-270", "Text": "Example 1 3 Coulomb\u2019s law for electrostatic force between two point\ncharges and Newton\u2019s law for gravitational force between two stationary\npoint masses, both have inverse-square dependence on the distance\nbetween the charges and masses respectively (a) Compare the strength\nof these forces by determining the ratio of their magnitudes (i) for an\nelectron and a proton and (ii) for two protons (b) Estimate the\naccelerations of electron and proton due to the electrical force of their\nmutual attraction when they are 1 \u00c5 (= 10-10 m) apart"}, {"Chapter": "1", "sentence_range": "268-271", "Text": "3 Coulomb\u2019s law for electrostatic force between two point\ncharges and Newton\u2019s law for gravitational force between two stationary\npoint masses, both have inverse-square dependence on the distance\nbetween the charges and masses respectively (a) Compare the strength\nof these forces by determining the ratio of their magnitudes (i) for an\nelectron and a proton and (ii) for two protons (b) Estimate the\naccelerations of electron and proton due to the electrical force of their\nmutual attraction when they are 1 \u00c5 (= 10-10 m) apart (mp = 1"}, {"Chapter": "1", "sentence_range": "269-272", "Text": "(a) Compare the strength\nof these forces by determining the ratio of their magnitudes (i) for an\nelectron and a proton and (ii) for two protons (b) Estimate the\naccelerations of electron and proton due to the electrical force of their\nmutual attraction when they are 1 \u00c5 (= 10-10 m) apart (mp = 1 67 \u00d7\n10\u201327 kg, me = 9"}, {"Chapter": "1", "sentence_range": "270-273", "Text": "(b) Estimate the\naccelerations of electron and proton due to the electrical force of their\nmutual attraction when they are 1 \u00c5 (= 10-10 m) apart (mp = 1 67 \u00d7\n10\u201327 kg, me = 9 11 \u00d7 10\u201331 kg)\nSolution\n(a) (i) The electric force between an electron and a proton at a distance\nr apart is:\n2\n2\n0\n41\ne\ne\nF\nr\n\u03b5\n= \u2212\n\u03c0\nwhere the negative sign indicates that the force is attractive"}, {"Chapter": "1", "sentence_range": "271-274", "Text": "(mp = 1 67 \u00d7\n10\u201327 kg, me = 9 11 \u00d7 10\u201331 kg)\nSolution\n(a) (i) The electric force between an electron and a proton at a distance\nr apart is:\n2\n2\n0\n41\ne\ne\nF\nr\n\u03b5\n= \u2212\n\u03c0\nwhere the negative sign indicates that the force is attractive The\ncorresponding gravitational force (always attractive) is:\n2\np\ne\nG\nm\nm\nF\nG\nr\n= \u2212\nwhere mp and me are the masses of a proton and an electron\nrespectively"}, {"Chapter": "1", "sentence_range": "272-275", "Text": "67 \u00d7\n10\u201327 kg, me = 9 11 \u00d7 10\u201331 kg)\nSolution\n(a) (i) The electric force between an electron and a proton at a distance\nr apart is:\n2\n2\n0\n41\ne\ne\nF\nr\n\u03b5\n= \u2212\n\u03c0\nwhere the negative sign indicates that the force is attractive The\ncorresponding gravitational force (always attractive) is:\n2\np\ne\nG\nm\nm\nF\nG\nr\n= \u2212\nwhere mp and me are the masses of a proton and an electron\nrespectively 2\n39\n0\n2"}, {"Chapter": "1", "sentence_range": "273-276", "Text": "11 \u00d7 10\u201331 kg)\nSolution\n(a) (i) The electric force between an electron and a proton at a distance\nr apart is:\n2\n2\n0\n41\ne\ne\nF\nr\n\u03b5\n= \u2212\n\u03c0\nwhere the negative sign indicates that the force is attractive The\ncorresponding gravitational force (always attractive) is:\n2\np\ne\nG\nm\nm\nF\nG\nr\n= \u2212\nwhere mp and me are the masses of a proton and an electron\nrespectively 2\n39\n0\n2 4\n10\n4\ne\nG\np\ne\nF\ne\nF\n\u03b5Gm m\n=\n=\n\u00d7\n\u03c0\n(ii) On similar lines, the ratio of the magnitudes of electric force\nto the gravitational force between two protons at a distance r\napart is:\nFF\ne\nGm m\ne\nG\np\np\n=\n=\n2\n4\u03c0\u03b50\n 1"}, {"Chapter": "1", "sentence_range": "274-277", "Text": "The\ncorresponding gravitational force (always attractive) is:\n2\np\ne\nG\nm\nm\nF\nG\nr\n= \u2212\nwhere mp and me are the masses of a proton and an electron\nrespectively 2\n39\n0\n2 4\n10\n4\ne\nG\np\ne\nF\ne\nF\n\u03b5Gm m\n=\n=\n\u00d7\n\u03c0\n(ii) On similar lines, the ratio of the magnitudes of electric force\nto the gravitational force between two protons at a distance r\napart is:\nFF\ne\nGm m\ne\nG\np\np\n=\n=\n2\n4\u03c0\u03b50\n 1 3 \u00d7 1036\nHowever, it may be mentioned here that the signs of the two forces\nare different"}, {"Chapter": "1", "sentence_range": "275-278", "Text": "2\n39\n0\n2 4\n10\n4\ne\nG\np\ne\nF\ne\nF\n\u03b5Gm m\n=\n=\n\u00d7\n\u03c0\n(ii) On similar lines, the ratio of the magnitudes of electric force\nto the gravitational force between two protons at a distance r\napart is:\nFF\ne\nGm m\ne\nG\np\np\n=\n=\n2\n4\u03c0\u03b50\n 1 3 \u00d7 1036\nHowever, it may be mentioned here that the signs of the two forces\nare different For two protons,  the gravitational force is attractive\nin nature and the Coulomb force is repulsive"}, {"Chapter": "1", "sentence_range": "276-279", "Text": "4\n10\n4\ne\nG\np\ne\nF\ne\nF\n\u03b5Gm m\n=\n=\n\u00d7\n\u03c0\n(ii) On similar lines, the ratio of the magnitudes of electric force\nto the gravitational force between two protons at a distance r\napart is:\nFF\ne\nGm m\ne\nG\np\np\n=\n=\n2\n4\u03c0\u03b50\n 1 3 \u00d7 1036\nHowever, it may be mentioned here that the signs of the two forces\nare different For two protons,  the gravitational force is attractive\nin nature and the Coulomb force is repulsive The actual values\nof these forces between two protons inside a nucleus (distance\nbetween two protons is ~ 10-15 m inside a nucleus) are Fe ~ 230 N,\nwhereas, FG ~ 1"}, {"Chapter": "1", "sentence_range": "277-280", "Text": "3 \u00d7 1036\nHowever, it may be mentioned here that the signs of the two forces\nare different For two protons,  the gravitational force is attractive\nin nature and the Coulomb force is repulsive The actual values\nof these forces between two protons inside a nucleus (distance\nbetween two protons is ~ 10-15 m inside a nucleus) are Fe ~ 230 N,\nwhereas, FG ~ 1 9 \u00d7 10\u201334 N"}, {"Chapter": "1", "sentence_range": "278-281", "Text": "For two protons,  the gravitational force is attractive\nin nature and the Coulomb force is repulsive The actual values\nof these forces between two protons inside a nucleus (distance\nbetween two protons is ~ 10-15 m inside a nucleus) are Fe ~ 230 N,\nwhereas, FG ~ 1 9 \u00d7 10\u201334 N The (dimensionless) ratio of the two forces shows that electrical\nforces are enormously stronger than the gravitational forces"}, {"Chapter": "1", "sentence_range": "279-282", "Text": "The actual values\nof these forces between two protons inside a nucleus (distance\nbetween two protons is ~ 10-15 m inside a nucleus) are Fe ~ 230 N,\nwhereas, FG ~ 1 9 \u00d7 10\u201334 N The (dimensionless) ratio of the two forces shows that electrical\nforces are enormously stronger than the gravitational forces Rationalised 2023-24\n10\nPhysics\n (b) The electric force F exerted by a proton on an electron is same in\nmagnitude to the force exerted by an electron on a proton; however,\nthe masses of an electron and a proton are different"}, {"Chapter": "1", "sentence_range": "280-283", "Text": "9 \u00d7 10\u201334 N The (dimensionless) ratio of the two forces shows that electrical\nforces are enormously stronger than the gravitational forces Rationalised 2023-24\n10\nPhysics\n (b) The electric force F exerted by a proton on an electron is same in\nmagnitude to the force exerted by an electron on a proton; however,\nthe masses of an electron and a proton are different Thus, the\nmagnitude of force is\n|F| = \n41\n0\n2\n2\n\u03c0\u03b5\nre\n = 8"}, {"Chapter": "1", "sentence_range": "281-284", "Text": "The (dimensionless) ratio of the two forces shows that electrical\nforces are enormously stronger than the gravitational forces Rationalised 2023-24\n10\nPhysics\n (b) The electric force F exerted by a proton on an electron is same in\nmagnitude to the force exerted by an electron on a proton; however,\nthe masses of an electron and a proton are different Thus, the\nmagnitude of force is\n|F| = \n41\n0\n2\n2\n\u03c0\u03b5\nre\n = 8 987 \u00d7 109 Nm2/C2 \u00d7 (1"}, {"Chapter": "1", "sentence_range": "282-285", "Text": "Rationalised 2023-24\n10\nPhysics\n (b) The electric force F exerted by a proton on an electron is same in\nmagnitude to the force exerted by an electron on a proton; however,\nthe masses of an electron and a proton are different Thus, the\nmagnitude of force is\n|F| = \n41\n0\n2\n2\n\u03c0\u03b5\nre\n = 8 987 \u00d7 109 Nm2/C2 \u00d7 (1 6 \u00d710\u201319C)2 / (10\u201310m)2\n        = 2"}, {"Chapter": "1", "sentence_range": "283-286", "Text": "Thus, the\nmagnitude of force is\n|F| = \n41\n0\n2\n2\n\u03c0\u03b5\nre\n = 8 987 \u00d7 109 Nm2/C2 \u00d7 (1 6 \u00d710\u201319C)2 / (10\u201310m)2\n        = 2 3 \u00d7 10\u20138 N\nUsing Newton\u2019s second law of motion, F = ma, the acceleration\nthat an electron will undergo is\na = 2"}, {"Chapter": "1", "sentence_range": "284-287", "Text": "987 \u00d7 109 Nm2/C2 \u00d7 (1 6 \u00d710\u201319C)2 / (10\u201310m)2\n        = 2 3 \u00d7 10\u20138 N\nUsing Newton\u2019s second law of motion, F = ma, the acceleration\nthat an electron will undergo is\na = 2 3\u00d710\u20138 N / 9"}, {"Chapter": "1", "sentence_range": "285-288", "Text": "6 \u00d710\u201319C)2 / (10\u201310m)2\n        = 2 3 \u00d7 10\u20138 N\nUsing Newton\u2019s second law of motion, F = ma, the acceleration\nthat an electron will undergo is\na = 2 3\u00d710\u20138 N / 9 11 \u00d710\u201331 kg = 2"}, {"Chapter": "1", "sentence_range": "286-289", "Text": "3 \u00d7 10\u20138 N\nUsing Newton\u2019s second law of motion, F = ma, the acceleration\nthat an electron will undergo is\na = 2 3\u00d710\u20138 N / 9 11 \u00d710\u201331 kg = 2 5 \u00d7 1022 m/s2\nComparing this with the value of acceleration due to gravity, we\ncan conclude that the effect of gravitational field is negligible on\nthe motion of electron and it undergoes very large accelerations\nunder the action of Coulomb force due to a proton"}, {"Chapter": "1", "sentence_range": "287-290", "Text": "3\u00d710\u20138 N / 9 11 \u00d710\u201331 kg = 2 5 \u00d7 1022 m/s2\nComparing this with the value of acceleration due to gravity, we\ncan conclude that the effect of gravitational field is negligible on\nthe motion of electron and it undergoes very large accelerations\nunder the action of Coulomb force due to a proton The value for acceleration of the proton is\n2"}, {"Chapter": "1", "sentence_range": "288-291", "Text": "11 \u00d710\u201331 kg = 2 5 \u00d7 1022 m/s2\nComparing this with the value of acceleration due to gravity, we\ncan conclude that the effect of gravitational field is negligible on\nthe motion of electron and it undergoes very large accelerations\nunder the action of Coulomb force due to a proton The value for acceleration of the proton is\n2 3 \u00d7 10\u20138 N / 1"}, {"Chapter": "1", "sentence_range": "289-292", "Text": "5 \u00d7 1022 m/s2\nComparing this with the value of acceleration due to gravity, we\ncan conclude that the effect of gravitational field is negligible on\nthe motion of electron and it undergoes very large accelerations\nunder the action of Coulomb force due to a proton The value for acceleration of the proton is\n2 3 \u00d7 10\u20138 N / 1 67 \u00d7 10\u201327 kg = 1"}, {"Chapter": "1", "sentence_range": "290-293", "Text": "The value for acceleration of the proton is\n2 3 \u00d7 10\u20138 N / 1 67 \u00d7 10\u201327 kg = 1 4 \u00d7 1019 m/s2\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "291-294", "Text": "3 \u00d7 10\u20138 N / 1 67 \u00d7 10\u201327 kg = 1 4 \u00d7 1019 m/s2\n EXAMPLE 1 3\nFIGURE 1"}, {"Chapter": "1", "sentence_range": "292-295", "Text": "67 \u00d7 10\u201327 kg = 1 4 \u00d7 1019 m/s2\n EXAMPLE 1 3\nFIGURE 1 4\nExample 1"}, {"Chapter": "1", "sentence_range": "293-296", "Text": "4 \u00d7 1019 m/s2\n EXAMPLE 1 3\nFIGURE 1 4\nExample 1 4 A charged metallic sphere A is suspended by a nylon\nthread"}, {"Chapter": "1", "sentence_range": "294-297", "Text": "3\nFIGURE 1 4\nExample 1 4 A charged metallic sphere A is suspended by a nylon\nthread Another charged metallic sphere B held by an insulating\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "295-298", "Text": "4\nExample 1 4 A charged metallic sphere A is suspended by a nylon\nthread Another charged metallic sphere B held by an insulating\n EXAMPLE 1 4\nRationalised 2023-24\nElectric Charges\nand Fields\n11\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "296-299", "Text": "4 A charged metallic sphere A is suspended by a nylon\nthread Another charged metallic sphere B held by an insulating\n EXAMPLE 1 4\nRationalised 2023-24\nElectric Charges\nand Fields\n11\n EXAMPLE 1 4\nhandle is brought close to A such that the distance between their\ncentres is 10 cm, as shown in Fig"}, {"Chapter": "1", "sentence_range": "297-300", "Text": "Another charged metallic sphere B held by an insulating\n EXAMPLE 1 4\nRationalised 2023-24\nElectric Charges\nand Fields\n11\n EXAMPLE 1 4\nhandle is brought close to A such that the distance between their\ncentres is 10 cm, as shown in Fig 1"}, {"Chapter": "1", "sentence_range": "298-301", "Text": "4\nRationalised 2023-24\nElectric Charges\nand Fields\n11\n EXAMPLE 1 4\nhandle is brought close to A such that the distance between their\ncentres is 10 cm, as shown in Fig 1 4(a)"}, {"Chapter": "1", "sentence_range": "299-302", "Text": "4\nhandle is brought close to A such that the distance between their\ncentres is 10 cm, as shown in Fig 1 4(a) The resulting repulsion of A\nis noted (for example, by shining a beam of light and measuring the\ndeflection of its shadow on a screen)"}, {"Chapter": "1", "sentence_range": "300-303", "Text": "1 4(a) The resulting repulsion of A\nis noted (for example, by shining a beam of light and measuring the\ndeflection of its shadow on a screen) Spheres A and B are touched\nby uncharged spheres C and D respectively, as shown in Fig"}, {"Chapter": "1", "sentence_range": "301-304", "Text": "4(a) The resulting repulsion of A\nis noted (for example, by shining a beam of light and measuring the\ndeflection of its shadow on a screen) Spheres A and B are touched\nby uncharged spheres C and D respectively, as shown in Fig 1"}, {"Chapter": "1", "sentence_range": "302-305", "Text": "The resulting repulsion of A\nis noted (for example, by shining a beam of light and measuring the\ndeflection of its shadow on a screen) Spheres A and B are touched\nby uncharged spheres C and D respectively, as shown in Fig 1 4(b)"}, {"Chapter": "1", "sentence_range": "303-306", "Text": "Spheres A and B are touched\nby uncharged spheres C and D respectively, as shown in Fig 1 4(b) C and D are then removed and B is brought closer to A to a\ndistance of 5"}, {"Chapter": "1", "sentence_range": "304-307", "Text": "1 4(b) C and D are then removed and B is brought closer to A to a\ndistance of 5 0 cm between their centres, as shown in Fig"}, {"Chapter": "1", "sentence_range": "305-308", "Text": "4(b) C and D are then removed and B is brought closer to A to a\ndistance of 5 0 cm between their centres, as shown in Fig 1"}, {"Chapter": "1", "sentence_range": "306-309", "Text": "C and D are then removed and B is brought closer to A to a\ndistance of 5 0 cm between their centres, as shown in Fig 1 4(c)"}, {"Chapter": "1", "sentence_range": "307-310", "Text": "0 cm between their centres, as shown in Fig 1 4(c) What is the expected repulsion of A on the basis of Coulomb\u2019s law"}, {"Chapter": "1", "sentence_range": "308-311", "Text": "1 4(c) What is the expected repulsion of A on the basis of Coulomb\u2019s law Spheres A and C and spheres B and D have identical sizes"}, {"Chapter": "1", "sentence_range": "309-312", "Text": "4(c) What is the expected repulsion of A on the basis of Coulomb\u2019s law Spheres A and C and spheres B and D have identical sizes Ignore\nthe sizes of A and B in comparison to the separation between their\ncentres"}, {"Chapter": "1", "sentence_range": "310-313", "Text": "What is the expected repulsion of A on the basis of Coulomb\u2019s law Spheres A and C and spheres B and D have identical sizes Ignore\nthe sizes of A and B in comparison to the separation between their\ncentres Solution Let the original charge on sphere A be q and that on B be\nq\u00a2"}, {"Chapter": "1", "sentence_range": "311-314", "Text": "Spheres A and C and spheres B and D have identical sizes Ignore\nthe sizes of A and B in comparison to the separation between their\ncentres Solution Let the original charge on sphere A be q and that on B be\nq\u00a2 At a distance r between their centres, the magnitude of the\nelectrostatic force on each is given by\nF\nrqq\n=\n\u2032\n41\n0\n2\n\u03c0\u03b5\nneglecting the sizes of spheres A and B in comparison to r"}, {"Chapter": "1", "sentence_range": "312-315", "Text": "Ignore\nthe sizes of A and B in comparison to the separation between their\ncentres Solution Let the original charge on sphere A be q and that on B be\nq\u00a2 At a distance r between their centres, the magnitude of the\nelectrostatic force on each is given by\nF\nrqq\n=\n\u2032\n41\n0\n2\n\u03c0\u03b5\nneglecting the sizes of spheres A and B in comparison to r When an\nidentical but uncharged sphere C touches A, the charges redistribute\non A and C and, by symmetry, each sphere carries a charge q/2"}, {"Chapter": "1", "sentence_range": "313-316", "Text": "Solution Let the original charge on sphere A be q and that on B be\nq\u00a2 At a distance r between their centres, the magnitude of the\nelectrostatic force on each is given by\nF\nrqq\n=\n\u2032\n41\n0\n2\n\u03c0\u03b5\nneglecting the sizes of spheres A and B in comparison to r When an\nidentical but uncharged sphere C touches A, the charges redistribute\non A and C and, by symmetry, each sphere carries a charge q/2 Similarly, after D touches B, the redistributed charge on each is\nq\u00a2/2"}, {"Chapter": "1", "sentence_range": "314-317", "Text": "At a distance r between their centres, the magnitude of the\nelectrostatic force on each is given by\nF\nrqq\n=\n\u2032\n41\n0\n2\n\u03c0\u03b5\nneglecting the sizes of spheres A and B in comparison to r When an\nidentical but uncharged sphere C touches A, the charges redistribute\non A and C and, by symmetry, each sphere carries a charge q/2 Similarly, after D touches B, the redistributed charge on each is\nq\u00a2/2 Now, if the separation between A and B is halved, the magnitude\nof the electrostatic force on each is\n\u2032 =\n\u2032\n=\n\u2032 =\nF\nq\nq\nr\nrqq\nF\n41\n2\n2\n2\n41\n0\n2\n0\n2\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n( / )(\n( / )/ )\n(\n)\nThus the electrostatic force on A, due to B, remains unaltered"}, {"Chapter": "1", "sentence_range": "315-318", "Text": "When an\nidentical but uncharged sphere C touches A, the charges redistribute\non A and C and, by symmetry, each sphere carries a charge q/2 Similarly, after D touches B, the redistributed charge on each is\nq\u00a2/2 Now, if the separation between A and B is halved, the magnitude\nof the electrostatic force on each is\n\u2032 =\n\u2032\n=\n\u2032 =\nF\nq\nq\nr\nrqq\nF\n41\n2\n2\n2\n41\n0\n2\n0\n2\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n( / )(\n( / )/ )\n(\n)\nThus the electrostatic force on A, due to B, remains unaltered 1"}, {"Chapter": "1", "sentence_range": "316-319", "Text": "Similarly, after D touches B, the redistributed charge on each is\nq\u00a2/2 Now, if the separation between A and B is halved, the magnitude\nof the electrostatic force on each is\n\u2032 =\n\u2032\n=\n\u2032 =\nF\nq\nq\nr\nrqq\nF\n41\n2\n2\n2\n41\n0\n2\n0\n2\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n( / )(\n( / )/ )\n(\n)\nThus the electrostatic force on A, due to B, remains unaltered 1 6  FORCES BETWEEN MULTIPLE CHARGES\nThe mutual electric force between two charges is given by\nCoulomb\u2019s law"}, {"Chapter": "1", "sentence_range": "317-320", "Text": "Now, if the separation between A and B is halved, the magnitude\nof the electrostatic force on each is\n\u2032 =\n\u2032\n=\n\u2032 =\nF\nq\nq\nr\nrqq\nF\n41\n2\n2\n2\n41\n0\n2\n0\n2\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n( / )(\n( / )/ )\n(\n)\nThus the electrostatic force on A, due to B, remains unaltered 1 6  FORCES BETWEEN MULTIPLE CHARGES\nThe mutual electric force between two charges is given by\nCoulomb\u2019s law How to calculate the force on a charge where\nthere are not one but several charges around"}, {"Chapter": "1", "sentence_range": "318-321", "Text": "1 6  FORCES BETWEEN MULTIPLE CHARGES\nThe mutual electric force between two charges is given by\nCoulomb\u2019s law How to calculate the force on a charge where\nthere are not one but several charges around Consider a\nsystem of n stationary charges q1, q2, q3,"}, {"Chapter": "1", "sentence_range": "319-322", "Text": "6  FORCES BETWEEN MULTIPLE CHARGES\nThe mutual electric force between two charges is given by\nCoulomb\u2019s law How to calculate the force on a charge where\nthere are not one but several charges around Consider a\nsystem of n stationary charges q1, q2, q3, , qn in vacuum"}, {"Chapter": "1", "sentence_range": "320-323", "Text": "How to calculate the force on a charge where\nthere are not one but several charges around Consider a\nsystem of n stationary charges q1, q2, q3, , qn in vacuum What is the force on q1 due to q2, q3,"}, {"Chapter": "1", "sentence_range": "321-324", "Text": "Consider a\nsystem of n stationary charges q1, q2, q3, , qn in vacuum What is the force on q1 due to q2, q3, , qn"}, {"Chapter": "1", "sentence_range": "322-325", "Text": ", qn in vacuum What is the force on q1 due to q2, q3, , qn Coulomb\u2019s law is\nnot enough to answer this question"}, {"Chapter": "1", "sentence_range": "323-326", "Text": "What is the force on q1 due to q2, q3, , qn Coulomb\u2019s law is\nnot enough to answer this question Recall that forces of\nmechanical origin add according to the parallelogram law of\naddition"}, {"Chapter": "1", "sentence_range": "324-327", "Text": ", qn Coulomb\u2019s law is\nnot enough to answer this question Recall that forces of\nmechanical origin add according to the parallelogram law of\naddition Is the same true for forces of electrostatic origin"}, {"Chapter": "1", "sentence_range": "325-328", "Text": "Coulomb\u2019s law is\nnot enough to answer this question Recall that forces of\nmechanical origin add according to the parallelogram law of\naddition Is the same true for forces of electrostatic origin Experimentally, it is verified that force on any charge due\nto a number of other charges is the vector sum of all the forces\non that charge due to the other charges, taken one at a time"}, {"Chapter": "1", "sentence_range": "326-329", "Text": "Recall that forces of\nmechanical origin add according to the parallelogram law of\naddition Is the same true for forces of electrostatic origin Experimentally, it is verified that force on any charge due\nto a number of other charges is the vector sum of all the forces\non that charge due to the other charges, taken one at a time The individual forces are unaffected due to the presence of\nother charges"}, {"Chapter": "1", "sentence_range": "327-330", "Text": "Is the same true for forces of electrostatic origin Experimentally, it is verified that force on any charge due\nto a number of other charges is the vector sum of all the forces\non that charge due to the other charges, taken one at a time The individual forces are unaffected due to the presence of\nother charges This is termed as the principle of superposition"}, {"Chapter": "1", "sentence_range": "328-331", "Text": "Experimentally, it is verified that force on any charge due\nto a number of other charges is the vector sum of all the forces\non that charge due to the other charges, taken one at a time The individual forces are unaffected due to the presence of\nother charges This is termed as the principle of superposition To better understand the concept, consider a system of\nthree charges q1, q2 and q3, as shown in Fig"}, {"Chapter": "1", "sentence_range": "329-332", "Text": "The individual forces are unaffected due to the presence of\nother charges This is termed as the principle of superposition To better understand the concept, consider a system of\nthree charges q1, q2 and q3, as shown in Fig 1"}, {"Chapter": "1", "sentence_range": "330-333", "Text": "This is termed as the principle of superposition To better understand the concept, consider a system of\nthree charges q1, q2 and q3, as shown in Fig 1 5(a)"}, {"Chapter": "1", "sentence_range": "331-334", "Text": "To better understand the concept, consider a system of\nthree charges q1, q2 and q3, as shown in Fig 1 5(a) The force\non one charge, say q1, due to two other charges q2, q3 can\ntherefore be obtained by performing a vector addition of the\nforces due to each one of these charges"}, {"Chapter": "1", "sentence_range": "332-335", "Text": "1 5(a) The force\non one charge, say q1, due to two other charges q2, q3 can\ntherefore be obtained by performing a vector addition of the\nforces due to each one of these charges Thus, if the force on q1\ndue to q2 is denoted by F12, F12 is given by Eq"}, {"Chapter": "1", "sentence_range": "333-336", "Text": "5(a) The force\non one charge, say q1, due to two other charges q2, q3 can\ntherefore be obtained by performing a vector addition of the\nforces due to each one of these charges Thus, if the force on q1\ndue to q2 is denoted by F12, F12 is given by Eq (1"}, {"Chapter": "1", "sentence_range": "334-337", "Text": "The force\non one charge, say q1, due to two other charges q2, q3 can\ntherefore be obtained by performing a vector addition of the\nforces due to each one of these charges Thus, if the force on q1\ndue to q2 is denoted by F12, F12 is given by Eq (1 3)  even\nthough other charges are present"}, {"Chapter": "1", "sentence_range": "335-338", "Text": "Thus, if the force on q1\ndue to q2 is denoted by F12, F12 is given by Eq (1 3)  even\nthough other charges are present Thus,\nF12 =\n41\n0\n1\n2\n12\n2\n12\n\u03c0\u03b5\nrq q\n\u02c6r\nFIGURE 1"}, {"Chapter": "1", "sentence_range": "336-339", "Text": "(1 3)  even\nthough other charges are present Thus,\nF12 =\n41\n0\n1\n2\n12\n2\n12\n\u03c0\u03b5\nrq q\n\u02c6r\nFIGURE 1 5 A system of\n(a) three charges\n(b) multiple charges"}, {"Chapter": "1", "sentence_range": "337-340", "Text": "3)  even\nthough other charges are present Thus,\nF12 =\n41\n0\n1\n2\n12\n2\n12\n\u03c0\u03b5\nrq q\n\u02c6r\nFIGURE 1 5 A system of\n(a) three charges\n(b) multiple charges Rationalised 2023-24\n12\nPhysics\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "338-341", "Text": "Thus,\nF12 =\n41\n0\n1\n2\n12\n2\n12\n\u03c0\u03b5\nrq q\n\u02c6r\nFIGURE 1 5 A system of\n(a) three charges\n(b) multiple charges Rationalised 2023-24\n12\nPhysics\n EXAMPLE 1 5\nIn the same way, the force on q1 due to q3, denoted by F13, is given by\nF\nr\n13\n0\n1\n3\n13\n2\n13\n=41\n\u03c0\u03b5\nrq q\n\u02c6\nwhich again is the Coulomb force on q1 due to q3, even though other\ncharge q2 is present"}, {"Chapter": "1", "sentence_range": "339-342", "Text": "5 A system of\n(a) three charges\n(b) multiple charges Rationalised 2023-24\n12\nPhysics\n EXAMPLE 1 5\nIn the same way, the force on q1 due to q3, denoted by F13, is given by\nF\nr\n13\n0\n1\n3\n13\n2\n13\n=41\n\u03c0\u03b5\nrq q\n\u02c6\nwhich again is the Coulomb force on q1 due to q3, even though other\ncharge q2 is present Thus the total force F1 on q1 due to the two charges q2 and q3 is\ngiven as\nF\nF\nF\nr\nr\n1\n12\n13\n0\n1\n2\n12\n2\n12\n0\n1\n3\n13\n2\n13\n41\n41\n=\n+\n=\n+\n\u03c0\n\u03c0\n\u03b5\n\u03b5\nrq q\nrq q\n\u02c6\n\u02c6\n(1"}, {"Chapter": "1", "sentence_range": "340-343", "Text": "Rationalised 2023-24\n12\nPhysics\n EXAMPLE 1 5\nIn the same way, the force on q1 due to q3, denoted by F13, is given by\nF\nr\n13\n0\n1\n3\n13\n2\n13\n=41\n\u03c0\u03b5\nrq q\n\u02c6\nwhich again is the Coulomb force on q1 due to q3, even though other\ncharge q2 is present Thus the total force F1 on q1 due to the two charges q2 and q3 is\ngiven as\nF\nF\nF\nr\nr\n1\n12\n13\n0\n1\n2\n12\n2\n12\n0\n1\n3\n13\n2\n13\n41\n41\n=\n+\n=\n+\n\u03c0\n\u03c0\n\u03b5\n\u03b5\nrq q\nrq q\n\u02c6\n\u02c6\n(1 4)\nThe above calculation of force can be generalised to a system of\ncharges more than three, as shown in Fig"}, {"Chapter": "1", "sentence_range": "341-344", "Text": "5\nIn the same way, the force on q1 due to q3, denoted by F13, is given by\nF\nr\n13\n0\n1\n3\n13\n2\n13\n=41\n\u03c0\u03b5\nrq q\n\u02c6\nwhich again is the Coulomb force on q1 due to q3, even though other\ncharge q2 is present Thus the total force F1 on q1 due to the two charges q2 and q3 is\ngiven as\nF\nF\nF\nr\nr\n1\n12\n13\n0\n1\n2\n12\n2\n12\n0\n1\n3\n13\n2\n13\n41\n41\n=\n+\n=\n+\n\u03c0\n\u03c0\n\u03b5\n\u03b5\nrq q\nrq q\n\u02c6\n\u02c6\n(1 4)\nThe above calculation of force can be generalised to a system of\ncharges more than three, as shown in Fig 1"}, {"Chapter": "1", "sentence_range": "342-345", "Text": "Thus the total force F1 on q1 due to the two charges q2 and q3 is\ngiven as\nF\nF\nF\nr\nr\n1\n12\n13\n0\n1\n2\n12\n2\n12\n0\n1\n3\n13\n2\n13\n41\n41\n=\n+\n=\n+\n\u03c0\n\u03c0\n\u03b5\n\u03b5\nrq q\nrq q\n\u02c6\n\u02c6\n(1 4)\nThe above calculation of force can be generalised to a system of\ncharges more than three, as shown in Fig 1 5(b)"}, {"Chapter": "1", "sentence_range": "343-346", "Text": "4)\nThe above calculation of force can be generalised to a system of\ncharges more than three, as shown in Fig 1 5(b) The principle of superposition says that in a system of charges q1,\nq2,"}, {"Chapter": "1", "sentence_range": "344-347", "Text": "1 5(b) The principle of superposition says that in a system of charges q1,\nq2, , qn, the force on q1 due to q2 is the same as given by Coulomb\u2019s law,\ni"}, {"Chapter": "1", "sentence_range": "345-348", "Text": "5(b) The principle of superposition says that in a system of charges q1,\nq2, , qn, the force on q1 due to q2 is the same as given by Coulomb\u2019s law,\ni e"}, {"Chapter": "1", "sentence_range": "346-349", "Text": "The principle of superposition says that in a system of charges q1,\nq2, , qn, the force on q1 due to q2 is the same as given by Coulomb\u2019s law,\ni e , it is unaffected by the presence of the other charges q3, q4,"}, {"Chapter": "1", "sentence_range": "347-350", "Text": ", qn, the force on q1 due to q2 is the same as given by Coulomb\u2019s law,\ni e , it is unaffected by the presence of the other charges q3, q4, , qn"}, {"Chapter": "1", "sentence_range": "348-351", "Text": "e , it is unaffected by the presence of the other charges q3, q4, , qn The\ntotal force F1 on the charge q1, due to all other charges, is then given by\nthe vector sum of the forces F12, F13,"}, {"Chapter": "1", "sentence_range": "349-352", "Text": ", it is unaffected by the presence of the other charges q3, q4, , qn The\ntotal force F1 on the charge q1, due to all other charges, is then given by\nthe vector sum of the forces F12, F13, ,  F1n:\ni"}, {"Chapter": "1", "sentence_range": "350-353", "Text": ", qn The\ntotal force F1 on the charge q1, due to all other charges, is then given by\nthe vector sum of the forces F12, F13, ,  F1n:\ni e"}, {"Chapter": "1", "sentence_range": "351-354", "Text": "The\ntotal force F1 on the charge q1, due to all other charges, is then given by\nthe vector sum of the forces F12, F13, ,  F1n:\ni e ,\nF\nF\nF\nF\nr\nr\n1\n12\n13\n1n\n =  \n + \n +"}, {"Chapter": "1", "sentence_range": "352-355", "Text": ",  F1n:\ni e ,\nF\nF\nF\nF\nr\nr\n1\n12\n13\n1n\n =  \n + \n + +  \n=\n+\n41\n0\n1\n2\n12\n2\n12\n1\n3\n13\n2\n13\n\u03c0\u03b5\nrq q\nrq q\n\u02c6\n\u02c6\n+\n+\n\uf8ee\n\uf8ef\uf8f0\n\uf8f9\n\uf8fa\uf8fb"}, {"Chapter": "1", "sentence_range": "353-356", "Text": "e ,\nF\nF\nF\nF\nr\nr\n1\n12\n13\n1n\n =  \n + \n + +  \n=\n+\n41\n0\n1\n2\n12\n2\n12\n1\n3\n13\n2\n13\n\u03c0\u03b5\nrq q\nrq q\n\u02c6\n\u02c6\n+\n+\n\uf8ee\n\uf8ef\uf8f0\n\uf8f9\n\uf8fa\uf8fb \u02c6\nq q\nr\nn\nn\nn\n1\n1\n2\n1r\n=\n=\u2211\nq\nq\nr\ni\ni\ni\nn\ni\n1\n0\n1\n2\n2\n1\n4\u03c0\u03b5\n\u02c6r\n(1"}, {"Chapter": "1", "sentence_range": "354-357", "Text": ",\nF\nF\nF\nF\nr\nr\n1\n12\n13\n1n\n =  \n + \n + +  \n=\n+\n41\n0\n1\n2\n12\n2\n12\n1\n3\n13\n2\n13\n\u03c0\u03b5\nrq q\nrq q\n\u02c6\n\u02c6\n+\n+\n\uf8ee\n\uf8ef\uf8f0\n\uf8f9\n\uf8fa\uf8fb \u02c6\nq q\nr\nn\nn\nn\n1\n1\n2\n1r\n=\n=\u2211\nq\nq\nr\ni\ni\ni\nn\ni\n1\n0\n1\n2\n2\n1\n4\u03c0\u03b5\n\u02c6r\n(1 5)\nThe vector sum is obtained as usual by the parallelogram law of\naddition of vectors"}, {"Chapter": "1", "sentence_range": "355-358", "Text": "+  \n=\n+\n41\n0\n1\n2\n12\n2\n12\n1\n3\n13\n2\n13\n\u03c0\u03b5\nrq q\nrq q\n\u02c6\n\u02c6\n+\n+\n\uf8ee\n\uf8ef\uf8f0\n\uf8f9\n\uf8fa\uf8fb \u02c6\nq q\nr\nn\nn\nn\n1\n1\n2\n1r\n=\n=\u2211\nq\nq\nr\ni\ni\ni\nn\ni\n1\n0\n1\n2\n2\n1\n4\u03c0\u03b5\n\u02c6r\n(1 5)\nThe vector sum is obtained as usual by the parallelogram law of\naddition of vectors All of electrostatics is basically a consequence of\nCoulomb\u2019s law and the superposition principle"}, {"Chapter": "1", "sentence_range": "356-359", "Text": "\u02c6\nq q\nr\nn\nn\nn\n1\n1\n2\n1r\n=\n=\u2211\nq\nq\nr\ni\ni\ni\nn\ni\n1\n0\n1\n2\n2\n1\n4\u03c0\u03b5\n\u02c6r\n(1 5)\nThe vector sum is obtained as usual by the parallelogram law of\naddition of vectors All of electrostatics is basically a consequence of\nCoulomb\u2019s law and the superposition principle Example 1"}, {"Chapter": "1", "sentence_range": "357-360", "Text": "5)\nThe vector sum is obtained as usual by the parallelogram law of\naddition of vectors All of electrostatics is basically a consequence of\nCoulomb\u2019s law and the superposition principle Example 1 5 Consider three charges q1, q2, q3 each equal to q at the\nvertices of an equilateral triangle of side l"}, {"Chapter": "1", "sentence_range": "358-361", "Text": "All of electrostatics is basically a consequence of\nCoulomb\u2019s law and the superposition principle Example 1 5 Consider three charges q1, q2, q3 each equal to q at the\nvertices of an equilateral triangle of side l What is the force on a\ncharge Q (with the same sign as q) placed at the centroid of the\ntriangle, as shown in Fig"}, {"Chapter": "1", "sentence_range": "359-362", "Text": "Example 1 5 Consider three charges q1, q2, q3 each equal to q at the\nvertices of an equilateral triangle of side l What is the force on a\ncharge Q (with the same sign as q) placed at the centroid of the\ntriangle, as shown in Fig 1"}, {"Chapter": "1", "sentence_range": "360-363", "Text": "5 Consider three charges q1, q2, q3 each equal to q at the\nvertices of an equilateral triangle of side l What is the force on a\ncharge Q (with the same sign as q) placed at the centroid of the\ntriangle, as shown in Fig 1 6"}, {"Chapter": "1", "sentence_range": "361-364", "Text": "What is the force on a\ncharge Q (with the same sign as q) placed at the centroid of the\ntriangle, as shown in Fig 1 6 FIGURE 1"}, {"Chapter": "1", "sentence_range": "362-365", "Text": "1 6 FIGURE 1 6\nSolution In the given equilateral triangle ABC of sides of length l, if\nwe draw a perpendicular AD to the side BC,\nAD = AC cos 30\u00ba = ( 3 /2 ) l  and the distance AO of the centroid O\nfrom A is (2/3) AD = (1/ 3 ) l"}, {"Chapter": "1", "sentence_range": "363-366", "Text": "6 FIGURE 1 6\nSolution In the given equilateral triangle ABC of sides of length l, if\nwe draw a perpendicular AD to the side BC,\nAD = AC cos 30\u00ba = ( 3 /2 ) l  and the distance AO of the centroid O\nfrom A is (2/3) AD = (1/ 3 ) l By symmatry AO = BO = CO"}, {"Chapter": "1", "sentence_range": "364-367", "Text": "FIGURE 1 6\nSolution In the given equilateral triangle ABC of sides of length l, if\nwe draw a perpendicular AD to the side BC,\nAD = AC cos 30\u00ba = ( 3 /2 ) l  and the distance AO of the centroid O\nfrom A is (2/3) AD = (1/ 3 ) l By symmatry AO = BO = CO Rationalised 2023-24\nElectric Charges\nand Fields\n13\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "365-368", "Text": "6\nSolution In the given equilateral triangle ABC of sides of length l, if\nwe draw a perpendicular AD to the side BC,\nAD = AC cos 30\u00ba = ( 3 /2 ) l  and the distance AO of the centroid O\nfrom A is (2/3) AD = (1/ 3 ) l By symmatry AO = BO = CO Rationalised 2023-24\nElectric Charges\nand Fields\n13\n EXAMPLE 1 5\nThus,\nForce F1 on Q due to charge q at A  = \n43\n0\n2\n\u03c0\u03b5\nlQq\n along AO\nForce F2 on Q due to charge q at B  = \n43\n0\n2\n\u03c0\u03b5\nlQq\n along BO\nForce F3 on Q due to charge q at C  = \n43\n0\n2\n\u03c0\u03b5\nlQq\n along CO\nThe resultant of forces F2 and F3 is \n43\n0\n2\n\u03c0\u03b5\nlQq\n along OA, by the\nparallelogram law"}, {"Chapter": "1", "sentence_range": "366-369", "Text": "By symmatry AO = BO = CO Rationalised 2023-24\nElectric Charges\nand Fields\n13\n EXAMPLE 1 5\nThus,\nForce F1 on Q due to charge q at A  = \n43\n0\n2\n\u03c0\u03b5\nlQq\n along AO\nForce F2 on Q due to charge q at B  = \n43\n0\n2\n\u03c0\u03b5\nlQq\n along BO\nForce F3 on Q due to charge q at C  = \n43\n0\n2\n\u03c0\u03b5\nlQq\n along CO\nThe resultant of forces F2 and F3 is \n43\n0\n2\n\u03c0\u03b5\nlQq\n along OA, by the\nparallelogram law Therefore, the total force on Q = \n43\n0\n2\n\u03c0\u03b5\nlQq\n\u02c6\n\u02c6\nr\n(\u2212r\n)\n = 0, where \u02c6r\nis the unit vector along OA"}, {"Chapter": "1", "sentence_range": "367-370", "Text": "Rationalised 2023-24\nElectric Charges\nand Fields\n13\n EXAMPLE 1 5\nThus,\nForce F1 on Q due to charge q at A  = \n43\n0\n2\n\u03c0\u03b5\nlQq\n along AO\nForce F2 on Q due to charge q at B  = \n43\n0\n2\n\u03c0\u03b5\nlQq\n along BO\nForce F3 on Q due to charge q at C  = \n43\n0\n2\n\u03c0\u03b5\nlQq\n along CO\nThe resultant of forces F2 and F3 is \n43\n0\n2\n\u03c0\u03b5\nlQq\n along OA, by the\nparallelogram law Therefore, the total force on Q = \n43\n0\n2\n\u03c0\u03b5\nlQq\n\u02c6\n\u02c6\nr\n(\u2212r\n)\n = 0, where \u02c6r\nis the unit vector along OA It is clear also by symmetry that the three forces will sum to zero"}, {"Chapter": "1", "sentence_range": "368-371", "Text": "5\nThus,\nForce F1 on Q due to charge q at A  = \n43\n0\n2\n\u03c0\u03b5\nlQq\n along AO\nForce F2 on Q due to charge q at B  = \n43\n0\n2\n\u03c0\u03b5\nlQq\n along BO\nForce F3 on Q due to charge q at C  = \n43\n0\n2\n\u03c0\u03b5\nlQq\n along CO\nThe resultant of forces F2 and F3 is \n43\n0\n2\n\u03c0\u03b5\nlQq\n along OA, by the\nparallelogram law Therefore, the total force on Q = \n43\n0\n2\n\u03c0\u03b5\nlQq\n\u02c6\n\u02c6\nr\n(\u2212r\n)\n = 0, where \u02c6r\nis the unit vector along OA It is clear also by symmetry that the three forces will sum to zero Suppose that the resultant force was non-zero but in some direction"}, {"Chapter": "1", "sentence_range": "369-372", "Text": "Therefore, the total force on Q = \n43\n0\n2\n\u03c0\u03b5\nlQq\n\u02c6\n\u02c6\nr\n(\u2212r\n)\n = 0, where \u02c6r\nis the unit vector along OA It is clear also by symmetry that the three forces will sum to zero Suppose that the resultant force was non-zero but in some direction Consider what would happen if the system was rotated through 60\u00b0\nabout O"}, {"Chapter": "1", "sentence_range": "370-373", "Text": "It is clear also by symmetry that the three forces will sum to zero Suppose that the resultant force was non-zero but in some direction Consider what would happen if the system was rotated through 60\u00b0\nabout O Example 1"}, {"Chapter": "1", "sentence_range": "371-374", "Text": "Suppose that the resultant force was non-zero but in some direction Consider what would happen if the system was rotated through 60\u00b0\nabout O Example 1 6 Consider the charges q, q, and \u2013q placed at the vertices\nof an equilateral triangle, as shown in Fig"}, {"Chapter": "1", "sentence_range": "372-375", "Text": "Consider what would happen if the system was rotated through 60\u00b0\nabout O Example 1 6 Consider the charges q, q, and \u2013q placed at the vertices\nof an equilateral triangle, as shown in Fig 1"}, {"Chapter": "1", "sentence_range": "373-376", "Text": "Example 1 6 Consider the charges q, q, and \u2013q placed at the vertices\nof an equilateral triangle, as shown in Fig 1 7"}, {"Chapter": "1", "sentence_range": "374-377", "Text": "6 Consider the charges q, q, and \u2013q placed at the vertices\nof an equilateral triangle, as shown in Fig 1 7 What is the force on\neach charge"}, {"Chapter": "1", "sentence_range": "375-378", "Text": "1 7 What is the force on\neach charge FIGURE 1"}, {"Chapter": "1", "sentence_range": "376-379", "Text": "7 What is the force on\neach charge FIGURE 1 7\nSolution  The forces acting on charge q at A due to charges q at B\nand \u2013q at C are F12 along BA and F13 along AC respectively, as shown\nin Fig"}, {"Chapter": "1", "sentence_range": "377-380", "Text": "What is the force on\neach charge FIGURE 1 7\nSolution  The forces acting on charge q at A due to charges q at B\nand \u2013q at C are F12 along BA and F13 along AC respectively, as shown\nin Fig 1"}, {"Chapter": "1", "sentence_range": "378-381", "Text": "FIGURE 1 7\nSolution  The forces acting on charge q at A due to charges q at B\nand \u2013q at C are F12 along BA and F13 along AC respectively, as shown\nin Fig 1 7"}, {"Chapter": "1", "sentence_range": "379-382", "Text": "7\nSolution  The forces acting on charge q at A due to charges q at B\nand \u2013q at C are F12 along BA and F13 along AC respectively, as shown\nin Fig 1 7 By the parallelogram law, the total force F1 on the charge\nq at A is given by\nF1 = F \n1\u02c6r  where \n1\u02c6r  is a unit vector along BC"}, {"Chapter": "1", "sentence_range": "380-383", "Text": "1 7 By the parallelogram law, the total force F1 on the charge\nq at A is given by\nF1 = F \n1\u02c6r  where \n1\u02c6r  is a unit vector along BC The force of attraction or repulsion for each pair of charges has the\nsame magnitude F\nq\n=\n2\n0\n2\n4 \u03c0 \u03b5 l\nThe total force F2 on charge q at B is thus F2 = F \u02c6r 2, where \u02c6r 2 is a\nunit vector along AC"}, {"Chapter": "1", "sentence_range": "381-384", "Text": "7 By the parallelogram law, the total force F1 on the charge\nq at A is given by\nF1 = F \n1\u02c6r  where \n1\u02c6r  is a unit vector along BC The force of attraction or repulsion for each pair of charges has the\nsame magnitude F\nq\n=\n2\n0\n2\n4 \u03c0 \u03b5 l\nThe total force F2 on charge q at B is thus F2 = F \u02c6r 2, where \u02c6r 2 is a\nunit vector along AC EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "382-385", "Text": "By the parallelogram law, the total force F1 on the charge\nq at A is given by\nF1 = F \n1\u02c6r  where \n1\u02c6r  is a unit vector along BC The force of attraction or repulsion for each pair of charges has the\nsame magnitude F\nq\n=\n2\n0\n2\n4 \u03c0 \u03b5 l\nThe total force F2 on charge q at B is thus F2 = F \u02c6r 2, where \u02c6r 2 is a\nunit vector along AC EXAMPLE 1 6\nRationalised 2023-24\n14\nPhysics\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "383-386", "Text": "The force of attraction or repulsion for each pair of charges has the\nsame magnitude F\nq\n=\n2\n0\n2\n4 \u03c0 \u03b5 l\nThe total force F2 on charge q at B is thus F2 = F \u02c6r 2, where \u02c6r 2 is a\nunit vector along AC EXAMPLE 1 6\nRationalised 2023-24\n14\nPhysics\n EXAMPLE 1 6\nSimilarly the total force on charge \u2013q at C is F3  = \n3  F \u02c6n , where \u02c6n is\nthe unit vector along the direction bisecting the \u00d0BCA"}, {"Chapter": "1", "sentence_range": "384-387", "Text": "EXAMPLE 1 6\nRationalised 2023-24\n14\nPhysics\n EXAMPLE 1 6\nSimilarly the total force on charge \u2013q at C is F3  = \n3  F \u02c6n , where \u02c6n is\nthe unit vector along the direction bisecting the \u00d0BCA It is interesting to see that the sum of the forces on the three charges\nis zero, i"}, {"Chapter": "1", "sentence_range": "385-388", "Text": "6\nRationalised 2023-24\n14\nPhysics\n EXAMPLE 1 6\nSimilarly the total force on charge \u2013q at C is F3  = \n3  F \u02c6n , where \u02c6n is\nthe unit vector along the direction bisecting the \u00d0BCA It is interesting to see that the sum of the forces on the three charges\nis zero, i e"}, {"Chapter": "1", "sentence_range": "386-389", "Text": "6\nSimilarly the total force on charge \u2013q at C is F3  = \n3  F \u02c6n , where \u02c6n is\nthe unit vector along the direction bisecting the \u00d0BCA It is interesting to see that the sum of the forces on the three charges\nis zero, i e ,\nF1 + F2 + F3 = 0\nThe result is not at all surprising"}, {"Chapter": "1", "sentence_range": "387-390", "Text": "It is interesting to see that the sum of the forces on the three charges\nis zero, i e ,\nF1 + F2 + F3 = 0\nThe result is not at all surprising It follows straight from the fact\nthat Coulomb\u2019s law is consistent with Newton\u2019s third law"}, {"Chapter": "1", "sentence_range": "388-391", "Text": "e ,\nF1 + F2 + F3 = 0\nThe result is not at all surprising It follows straight from the fact\nthat Coulomb\u2019s law is consistent with Newton\u2019s third law The proof\nis left to you as an exercise"}, {"Chapter": "1", "sentence_range": "389-392", "Text": ",\nF1 + F2 + F3 = 0\nThe result is not at all surprising It follows straight from the fact\nthat Coulomb\u2019s law is consistent with Newton\u2019s third law The proof\nis left to you as an exercise 1"}, {"Chapter": "1", "sentence_range": "390-393", "Text": "It follows straight from the fact\nthat Coulomb\u2019s law is consistent with Newton\u2019s third law The proof\nis left to you as an exercise 1 7  ELECTRIC FIELD\nLet us consider a point charge Q placed in vacuum, at the origin O"}, {"Chapter": "1", "sentence_range": "391-394", "Text": "The proof\nis left to you as an exercise 1 7  ELECTRIC FIELD\nLet us consider a point charge Q placed in vacuum, at the origin O If we\nplace another point charge q at a point P, where OP = r, then the charge Q\nwill exert a force on q as per Coulomb\u2019s law"}, {"Chapter": "1", "sentence_range": "392-395", "Text": "1 7  ELECTRIC FIELD\nLet us consider a point charge Q placed in vacuum, at the origin O If we\nplace another point charge q at a point P, where OP = r, then the charge Q\nwill exert a force on q as per Coulomb\u2019s law We may ask the question: If\ncharge q is removed, then what is left in the surrounding"}, {"Chapter": "1", "sentence_range": "393-396", "Text": "7  ELECTRIC FIELD\nLet us consider a point charge Q placed in vacuum, at the origin O If we\nplace another point charge q at a point P, where OP = r, then the charge Q\nwill exert a force on q as per Coulomb\u2019s law We may ask the question: If\ncharge q is removed, then what is left in the surrounding Is there\nnothing"}, {"Chapter": "1", "sentence_range": "394-397", "Text": "If we\nplace another point charge q at a point P, where OP = r, then the charge Q\nwill exert a force on q as per Coulomb\u2019s law We may ask the question: If\ncharge q is removed, then what is left in the surrounding Is there\nnothing If there is nothing at the point P, then how does a force act\nwhen we place the charge q at P"}, {"Chapter": "1", "sentence_range": "395-398", "Text": "We may ask the question: If\ncharge q is removed, then what is left in the surrounding Is there\nnothing If there is nothing at the point P, then how does a force act\nwhen we place the charge q at P In order to answer such questions, the\nearly scientists introduced the concept of field"}, {"Chapter": "1", "sentence_range": "396-399", "Text": "Is there\nnothing If there is nothing at the point P, then how does a force act\nwhen we place the charge q at P In order to answer such questions, the\nearly scientists introduced the concept of field According to this, we say\nthat the charge Q produces an electric field everywhere in the surrounding"}, {"Chapter": "1", "sentence_range": "397-400", "Text": "If there is nothing at the point P, then how does a force act\nwhen we place the charge q at P In order to answer such questions, the\nearly scientists introduced the concept of field According to this, we say\nthat the charge Q produces an electric field everywhere in the surrounding When another charge q is brought at some point P, the field there acts on\nit and produces a force"}, {"Chapter": "1", "sentence_range": "398-401", "Text": "In order to answer such questions, the\nearly scientists introduced the concept of field According to this, we say\nthat the charge Q produces an electric field everywhere in the surrounding When another charge q is brought at some point P, the field there acts on\nit and produces a force The electric field produced by the charge Q  at a\npoint r is given as\nE r\nr\nr\n( ) =\n=\n41\n41\n0\n2\n0\n2\n\u03c0\n\u03c0\n\u03b5\n\u03b5\nrQ\nrQ\n\u02c6\n\u02c6\n(1"}, {"Chapter": "1", "sentence_range": "399-402", "Text": "According to this, we say\nthat the charge Q produces an electric field everywhere in the surrounding When another charge q is brought at some point P, the field there acts on\nit and produces a force The electric field produced by the charge Q  at a\npoint r is given as\nE r\nr\nr\n( ) =\n=\n41\n41\n0\n2\n0\n2\n\u03c0\n\u03c0\n\u03b5\n\u03b5\nrQ\nrQ\n\u02c6\n\u02c6\n(1 6)\nwhere \u02c6 =\nr\n r/r, is a unit vector from the origin to the point r"}, {"Chapter": "1", "sentence_range": "400-403", "Text": "When another charge q is brought at some point P, the field there acts on\nit and produces a force The electric field produced by the charge Q  at a\npoint r is given as\nE r\nr\nr\n( ) =\n=\n41\n41\n0\n2\n0\n2\n\u03c0\n\u03c0\n\u03b5\n\u03b5\nrQ\nrQ\n\u02c6\n\u02c6\n(1 6)\nwhere \u02c6 =\nr\n r/r, is a unit vector from the origin to the point r Thus, Eq"}, {"Chapter": "1", "sentence_range": "401-404", "Text": "The electric field produced by the charge Q  at a\npoint r is given as\nE r\nr\nr\n( ) =\n=\n41\n41\n0\n2\n0\n2\n\u03c0\n\u03c0\n\u03b5\n\u03b5\nrQ\nrQ\n\u02c6\n\u02c6\n(1 6)\nwhere \u02c6 =\nr\n r/r, is a unit vector from the origin to the point r Thus, Eq (1"}, {"Chapter": "1", "sentence_range": "402-405", "Text": "6)\nwhere \u02c6 =\nr\n r/r, is a unit vector from the origin to the point r Thus, Eq (1 6)\nspecifies the value of the electric field for each value of the position\nvector r"}, {"Chapter": "1", "sentence_range": "403-406", "Text": "Thus, Eq (1 6)\nspecifies the value of the electric field for each value of the position\nvector r The word \u201cfield\u201d signifies how some distributed quantity (which\ncould be a scalar or a vector) varies with position"}, {"Chapter": "1", "sentence_range": "404-407", "Text": "(1 6)\nspecifies the value of the electric field for each value of the position\nvector r The word \u201cfield\u201d signifies how some distributed quantity (which\ncould be a scalar or a vector) varies with position The effect of the charge\nhas been incorporated in the existence of the electric field"}, {"Chapter": "1", "sentence_range": "405-408", "Text": "6)\nspecifies the value of the electric field for each value of the position\nvector r The word \u201cfield\u201d signifies how some distributed quantity (which\ncould be a scalar or a vector) varies with position The effect of the charge\nhas been incorporated in the existence of the electric field We obtain the\nforce F exerted by a charge Q on a charge q, as\nF\nr\n=\n41\n0\n2\n\u03c0\u03b5\nrQq\n\u02c6\n(1"}, {"Chapter": "1", "sentence_range": "406-409", "Text": "The word \u201cfield\u201d signifies how some distributed quantity (which\ncould be a scalar or a vector) varies with position The effect of the charge\nhas been incorporated in the existence of the electric field We obtain the\nforce F exerted by a charge Q on a charge q, as\nF\nr\n=\n41\n0\n2\n\u03c0\u03b5\nrQq\n\u02c6\n(1 7)\nNote that the charge q also exerts an equal and opposite force on the\ncharge Q"}, {"Chapter": "1", "sentence_range": "407-410", "Text": "The effect of the charge\nhas been incorporated in the existence of the electric field We obtain the\nforce F exerted by a charge Q on a charge q, as\nF\nr\n=\n41\n0\n2\n\u03c0\u03b5\nrQq\n\u02c6\n(1 7)\nNote that the charge q also exerts an equal and opposite force on the\ncharge Q The electrostatic force between the charges Q and q can be\nlooked upon as an interaction between charge q and the electric field of\nQ and vice versa"}, {"Chapter": "1", "sentence_range": "408-411", "Text": "We obtain the\nforce F exerted by a charge Q on a charge q, as\nF\nr\n=\n41\n0\n2\n\u03c0\u03b5\nrQq\n\u02c6\n(1 7)\nNote that the charge q also exerts an equal and opposite force on the\ncharge Q The electrostatic force between the charges Q and q can be\nlooked upon as an interaction between charge q and the electric field of\nQ and vice versa If we denote the position of charge q by the vector r, it\nexperiences a force F equal to the charge q multiplied by the electric\nfield E at the location of q"}, {"Chapter": "1", "sentence_range": "409-412", "Text": "7)\nNote that the charge q also exerts an equal and opposite force on the\ncharge Q The electrostatic force between the charges Q and q can be\nlooked upon as an interaction between charge q and the electric field of\nQ and vice versa If we denote the position of charge q by the vector r, it\nexperiences a force F equal to the charge q multiplied by the electric\nfield E at the location of q Thus,\nF(r) = q E(r)\n(1"}, {"Chapter": "1", "sentence_range": "410-413", "Text": "The electrostatic force between the charges Q and q can be\nlooked upon as an interaction between charge q and the electric field of\nQ and vice versa If we denote the position of charge q by the vector r, it\nexperiences a force F equal to the charge q multiplied by the electric\nfield E at the location of q Thus,\nF(r) = q E(r)\n(1 8)\nEquation (1"}, {"Chapter": "1", "sentence_range": "411-414", "Text": "If we denote the position of charge q by the vector r, it\nexperiences a force F equal to the charge q multiplied by the electric\nfield E at the location of q Thus,\nF(r) = q E(r)\n(1 8)\nEquation (1 8) defines the SI unit of electric field as N/C*"}, {"Chapter": "1", "sentence_range": "412-415", "Text": "Thus,\nF(r) = q E(r)\n(1 8)\nEquation (1 8) defines the SI unit of electric field as N/C* Some important remarks may be made here:\n(i)\nFrom Eq"}, {"Chapter": "1", "sentence_range": "413-416", "Text": "8)\nEquation (1 8) defines the SI unit of electric field as N/C* Some important remarks may be made here:\n(i)\nFrom Eq (1"}, {"Chapter": "1", "sentence_range": "414-417", "Text": "8) defines the SI unit of electric field as N/C* Some important remarks may be made here:\n(i)\nFrom Eq (1 8), we can infer that if q is unity, the electric field due to\na charge Q is numerically equal to the force exerted by it"}, {"Chapter": "1", "sentence_range": "415-418", "Text": "Some important remarks may be made here:\n(i)\nFrom Eq (1 8), we can infer that if q is unity, the electric field due to\na charge Q is numerically equal to the force exerted by it Thus, the\nelectric field due to a charge Q at a point in space may be defined\nas the force that a unit positive charge would experience if placed\n*\nAn alternate unit V/m will be introduced in the next chapter"}, {"Chapter": "1", "sentence_range": "416-419", "Text": "(1 8), we can infer that if q is unity, the electric field due to\na charge Q is numerically equal to the force exerted by it Thus, the\nelectric field due to a charge Q at a point in space may be defined\nas the force that a unit positive charge would experience if placed\n*\nAn alternate unit V/m will be introduced in the next chapter FIGURE 1"}, {"Chapter": "1", "sentence_range": "417-420", "Text": "8), we can infer that if q is unity, the electric field due to\na charge Q is numerically equal to the force exerted by it Thus, the\nelectric field due to a charge Q at a point in space may be defined\nas the force that a unit positive charge would experience if placed\n*\nAn alternate unit V/m will be introduced in the next chapter FIGURE 1 8 Electric\nfield (a) due to a\ncharge Q, (b) due to a\ncharge \u2013Q"}, {"Chapter": "1", "sentence_range": "418-421", "Text": "Thus, the\nelectric field due to a charge Q at a point in space may be defined\nas the force that a unit positive charge would experience if placed\n*\nAn alternate unit V/m will be introduced in the next chapter FIGURE 1 8 Electric\nfield (a) due to a\ncharge Q, (b) due to a\ncharge \u2013Q Rationalised 2023-24\nElectric Charges\nand Fields\n15\nat that point"}, {"Chapter": "1", "sentence_range": "419-422", "Text": "FIGURE 1 8 Electric\nfield (a) due to a\ncharge Q, (b) due to a\ncharge \u2013Q Rationalised 2023-24\nElectric Charges\nand Fields\n15\nat that point The charge Q, which is producing the electric field, is\ncalled a source charge and the charge q, which tests the effect of a\nsource charge, is called a test charge"}, {"Chapter": "1", "sentence_range": "420-423", "Text": "8 Electric\nfield (a) due to a\ncharge Q, (b) due to a\ncharge \u2013Q Rationalised 2023-24\nElectric Charges\nand Fields\n15\nat that point The charge Q, which is producing the electric field, is\ncalled a source charge and the charge q, which tests the effect of a\nsource charge, is called a test charge Note that the source charge Q\nmust remain at its original location"}, {"Chapter": "1", "sentence_range": "421-424", "Text": "Rationalised 2023-24\nElectric Charges\nand Fields\n15\nat that point The charge Q, which is producing the electric field, is\ncalled a source charge and the charge q, which tests the effect of a\nsource charge, is called a test charge Note that the source charge Q\nmust remain at its original location However, if a charge q is brought\nat any point around Q, Q itself is bound to experience an electrical\nforce due to q and will tend to move"}, {"Chapter": "1", "sentence_range": "422-425", "Text": "The charge Q, which is producing the electric field, is\ncalled a source charge and the charge q, which tests the effect of a\nsource charge, is called a test charge Note that the source charge Q\nmust remain at its original location However, if a charge q is brought\nat any point around Q, Q itself is bound to experience an electrical\nforce due to q and will tend to move A way out of this difficulty is to\nmake q negligibly small"}, {"Chapter": "1", "sentence_range": "423-426", "Text": "Note that the source charge Q\nmust remain at its original location However, if a charge q is brought\nat any point around Q, Q itself is bound to experience an electrical\nforce due to q and will tend to move A way out of this difficulty is to\nmake q negligibly small The force F is then negligibly small but the\nratio F/q is finite and defines the electric field:\nE\nF\n=\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\u2192\nqlim\nq\n0\n(1"}, {"Chapter": "1", "sentence_range": "424-427", "Text": "However, if a charge q is brought\nat any point around Q, Q itself is bound to experience an electrical\nforce due to q and will tend to move A way out of this difficulty is to\nmake q negligibly small The force F is then negligibly small but the\nratio F/q is finite and defines the electric field:\nE\nF\n=\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\u2192\nqlim\nq\n0\n(1 9)\nA practical way to get around the problem (of keeping Q undisturbed\nin the presence of q) is to hold Q to its location by unspecified forces"}, {"Chapter": "1", "sentence_range": "425-428", "Text": "A way out of this difficulty is to\nmake q negligibly small The force F is then negligibly small but the\nratio F/q is finite and defines the electric field:\nE\nF\n=\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\u2192\nqlim\nq\n0\n(1 9)\nA practical way to get around the problem (of keeping Q undisturbed\nin the presence of q) is to hold Q to its location by unspecified forces This may look strange but actually this is what happens in practice"}, {"Chapter": "1", "sentence_range": "426-429", "Text": "The force F is then negligibly small but the\nratio F/q is finite and defines the electric field:\nE\nF\n=\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\u2192\nqlim\nq\n0\n(1 9)\nA practical way to get around the problem (of keeping Q undisturbed\nin the presence of q) is to hold Q to its location by unspecified forces This may look strange but actually this is what happens in practice When we are considering the electric force on a test charge q due to a\ncharged planar sheet (Section 1"}, {"Chapter": "1", "sentence_range": "427-430", "Text": "9)\nA practical way to get around the problem (of keeping Q undisturbed\nin the presence of q) is to hold Q to its location by unspecified forces This may look strange but actually this is what happens in practice When we are considering the electric force on a test charge q due to a\ncharged planar sheet (Section 1 14), the charges on the sheet are held to\ntheir locations by the forces due to the unspecified charged constituents\ninside the sheet"}, {"Chapter": "1", "sentence_range": "428-431", "Text": "This may look strange but actually this is what happens in practice When we are considering the electric force on a test charge q due to a\ncharged planar sheet (Section 1 14), the charges on the sheet are held to\ntheir locations by the forces due to the unspecified charged constituents\ninside the sheet (ii) Note that the electric field E due to Q, though defined operationally in\nterms of some test charge q, is independent of q"}, {"Chapter": "1", "sentence_range": "429-432", "Text": "When we are considering the electric force on a test charge q due to a\ncharged planar sheet (Section 1 14), the charges on the sheet are held to\ntheir locations by the forces due to the unspecified charged constituents\ninside the sheet (ii) Note that the electric field E due to Q, though defined operationally in\nterms of some test charge q, is independent of q This is because\nF is proportional to q, so the ratio F/q does not depend on q"}, {"Chapter": "1", "sentence_range": "430-433", "Text": "14), the charges on the sheet are held to\ntheir locations by the forces due to the unspecified charged constituents\ninside the sheet (ii) Note that the electric field E due to Q, though defined operationally in\nterms of some test charge q, is independent of q This is because\nF is proportional to q, so the ratio F/q does not depend on q The\nforce F on the charge q due to the charge Q depends on the particular\nlocation of charge q which may take any value in the space around\nthe charge Q"}, {"Chapter": "1", "sentence_range": "431-434", "Text": "(ii) Note that the electric field E due to Q, though defined operationally in\nterms of some test charge q, is independent of q This is because\nF is proportional to q, so the ratio F/q does not depend on q The\nforce F on the charge q due to the charge Q depends on the particular\nlocation of charge q which may take any value in the space around\nthe charge Q Thus, the electric field E due to Q is also dependent on\nthe space coordinate r"}, {"Chapter": "1", "sentence_range": "432-435", "Text": "This is because\nF is proportional to q, so the ratio F/q does not depend on q The\nforce F on the charge q due to the charge Q depends on the particular\nlocation of charge q which may take any value in the space around\nthe charge Q Thus, the electric field E due to Q is also dependent on\nthe space coordinate r For different positions of the charge q all over\nthe space, we get different values of electric field E"}, {"Chapter": "1", "sentence_range": "433-436", "Text": "The\nforce F on the charge q due to the charge Q depends on the particular\nlocation of charge q which may take any value in the space around\nthe charge Q Thus, the electric field E due to Q is also dependent on\nthe space coordinate r For different positions of the charge q all over\nthe space, we get different values of electric field E The field exists at\nevery point in three-dimensional space"}, {"Chapter": "1", "sentence_range": "434-437", "Text": "Thus, the electric field E due to Q is also dependent on\nthe space coordinate r For different positions of the charge q all over\nthe space, we get different values of electric field E The field exists at\nevery point in three-dimensional space (iii) For a positive charge, the electric field will be directed radially\noutwards from the charge"}, {"Chapter": "1", "sentence_range": "435-438", "Text": "For different positions of the charge q all over\nthe space, we get different values of electric field E The field exists at\nevery point in three-dimensional space (iii) For a positive charge, the electric field will be directed radially\noutwards from the charge On the other hand, if the source charge is\nnegative, the electric field vector, at each point, points radially inwards"}, {"Chapter": "1", "sentence_range": "436-439", "Text": "The field exists at\nevery point in three-dimensional space (iii) For a positive charge, the electric field will be directed radially\noutwards from the charge On the other hand, if the source charge is\nnegative, the electric field vector, at each point, points radially inwards (iv) Since the magnitude of the force F on charge q due to charge Q\ndepends only on the distance r of the charge q from charge Q,\nthe magnitude of the electric field E will also depend only on the\ndistance r"}, {"Chapter": "1", "sentence_range": "437-440", "Text": "(iii) For a positive charge, the electric field will be directed radially\noutwards from the charge On the other hand, if the source charge is\nnegative, the electric field vector, at each point, points radially inwards (iv) Since the magnitude of the force F on charge q due to charge Q\ndepends only on the distance r of the charge q from charge Q,\nthe magnitude of the electric field E will also depend only on the\ndistance r Thus at equal distances from the charge Q, the magnitude\nof its electric field E is same"}, {"Chapter": "1", "sentence_range": "438-441", "Text": "On the other hand, if the source charge is\nnegative, the electric field vector, at each point, points radially inwards (iv) Since the magnitude of the force F on charge q due to charge Q\ndepends only on the distance r of the charge q from charge Q,\nthe magnitude of the electric field E will also depend only on the\ndistance r Thus at equal distances from the charge Q, the magnitude\nof its electric field E is same The magnitude of electric field E due to\na point charge is thus same on a sphere with the point charge at its\ncentre; in other words, it has a spherical symmetry"}, {"Chapter": "1", "sentence_range": "439-442", "Text": "(iv) Since the magnitude of the force F on charge q due to charge Q\ndepends only on the distance r of the charge q from charge Q,\nthe magnitude of the electric field E will also depend only on the\ndistance r Thus at equal distances from the charge Q, the magnitude\nof its electric field E is same The magnitude of electric field E due to\na point charge is thus same on a sphere with the point charge at its\ncentre; in other words, it has a spherical symmetry 1"}, {"Chapter": "1", "sentence_range": "440-443", "Text": "Thus at equal distances from the charge Q, the magnitude\nof its electric field E is same The magnitude of electric field E due to\na point charge is thus same on a sphere with the point charge at its\ncentre; in other words, it has a spherical symmetry 1 7"}, {"Chapter": "1", "sentence_range": "441-444", "Text": "The magnitude of electric field E due to\na point charge is thus same on a sphere with the point charge at its\ncentre; in other words, it has a spherical symmetry 1 7 1  Electric field due to a system of charges\nConsider a system of charges q1, q2,"}, {"Chapter": "1", "sentence_range": "442-445", "Text": "1 7 1  Electric field due to a system of charges\nConsider a system of charges q1, q2, , qn with  position vectors r1,\nr2,"}, {"Chapter": "1", "sentence_range": "443-446", "Text": "7 1  Electric field due to a system of charges\nConsider a system of charges q1, q2, , qn with  position vectors r1,\nr2, , rn relative  to some origin O"}, {"Chapter": "1", "sentence_range": "444-447", "Text": "1  Electric field due to a system of charges\nConsider a system of charges q1, q2, , qn with  position vectors r1,\nr2, , rn relative  to some origin O Like the electric field at a point in\nspace due to a single charge, electric field at a point in space due to the\nsystem of charges is defined to be the force experienced by a unit\ntest charge placed at that point, without disturbing the original\npositions of charges q1, q2,"}, {"Chapter": "1", "sentence_range": "445-448", "Text": ", qn with  position vectors r1,\nr2, , rn relative  to some origin O Like the electric field at a point in\nspace due to a single charge, electric field at a point in space due to the\nsystem of charges is defined to be the force experienced by a unit\ntest charge placed at that point, without disturbing the original\npositions of charges q1, q2, , qn"}, {"Chapter": "1", "sentence_range": "446-449", "Text": ", rn relative  to some origin O Like the electric field at a point in\nspace due to a single charge, electric field at a point in space due to the\nsystem of charges is defined to be the force experienced by a unit\ntest charge placed at that point, without disturbing the original\npositions of charges q1, q2, , qn We can use Coulomb\u2019s law and the\nsuperposition principle to determine this field at a point P denoted by\nposition vector r"}, {"Chapter": "1", "sentence_range": "447-450", "Text": "Like the electric field at a point in\nspace due to a single charge, electric field at a point in space due to the\nsystem of charges is defined to be the force experienced by a unit\ntest charge placed at that point, without disturbing the original\npositions of charges q1, q2, , qn We can use Coulomb\u2019s law and the\nsuperposition principle to determine this field at a point P denoted by\nposition vector r Rationalised 2023-24\n16\nPhysics\nElectric field E1 at r due to q1 at r1 is given by\nE1 = \n41\n0\n1\n1\n2\n\u03c0\u03b5\nq\nr P\n1P\n\u02c6r\nwhere \n1P\n\u02c6r  is a unit vector in the direction from q1 to P,\nand r1P is the distance between q1 and P"}, {"Chapter": "1", "sentence_range": "448-451", "Text": ", qn We can use Coulomb\u2019s law and the\nsuperposition principle to determine this field at a point P denoted by\nposition vector r Rationalised 2023-24\n16\nPhysics\nElectric field E1 at r due to q1 at r1 is given by\nE1 = \n41\n0\n1\n1\n2\n\u03c0\u03b5\nq\nr P\n1P\n\u02c6r\nwhere \n1P\n\u02c6r  is a unit vector in the direction from q1 to P,\nand r1P is the distance between q1 and P In the same manner, electric field E2 at r due to q2 at\nr2 is\nE2 = \n41\n0\n2\n2\n2\n\u03c0\u03b5\nq\nr P\n2P\n\u02c6r\nwhere \n\u02c6r2P\n is a unit vector in the direction from q2 to P\nand r2P is the distance between q2 and P"}, {"Chapter": "1", "sentence_range": "449-452", "Text": "We can use Coulomb\u2019s law and the\nsuperposition principle to determine this field at a point P denoted by\nposition vector r Rationalised 2023-24\n16\nPhysics\nElectric field E1 at r due to q1 at r1 is given by\nE1 = \n41\n0\n1\n1\n2\n\u03c0\u03b5\nq\nr P\n1P\n\u02c6r\nwhere \n1P\n\u02c6r  is a unit vector in the direction from q1 to P,\nand r1P is the distance between q1 and P In the same manner, electric field E2 at r due to q2 at\nr2 is\nE2 = \n41\n0\n2\n2\n2\n\u03c0\u03b5\nq\nr P\n2P\n\u02c6r\nwhere \n\u02c6r2P\n is a unit vector in the direction from q2 to P\nand r2P is the distance between q2 and P Similar\nexpressions hold good for fields E3, E4,"}, {"Chapter": "1", "sentence_range": "450-453", "Text": "Rationalised 2023-24\n16\nPhysics\nElectric field E1 at r due to q1 at r1 is given by\nE1 = \n41\n0\n1\n1\n2\n\u03c0\u03b5\nq\nr P\n1P\n\u02c6r\nwhere \n1P\n\u02c6r  is a unit vector in the direction from q1 to P,\nand r1P is the distance between q1 and P In the same manner, electric field E2 at r due to q2 at\nr2 is\nE2 = \n41\n0\n2\n2\n2\n\u03c0\u03b5\nq\nr P\n2P\n\u02c6r\nwhere \n\u02c6r2P\n is a unit vector in the direction from q2 to P\nand r2P is the distance between q2 and P Similar\nexpressions hold good for fields E3, E4, , En due to\ncharges q3, q4,"}, {"Chapter": "1", "sentence_range": "451-454", "Text": "In the same manner, electric field E2 at r due to q2 at\nr2 is\nE2 = \n41\n0\n2\n2\n2\n\u03c0\u03b5\nq\nr P\n2P\n\u02c6r\nwhere \n\u02c6r2P\n is a unit vector in the direction from q2 to P\nand r2P is the distance between q2 and P Similar\nexpressions hold good for fields E3, E4, , En due to\ncharges q3, q4, , qn"}, {"Chapter": "1", "sentence_range": "452-455", "Text": "Similar\nexpressions hold good for fields E3, E4, , En due to\ncharges q3, q4, , qn By the superposition principle, the electric field E at r\ndue to the system of charges is (as shown in Fig"}, {"Chapter": "1", "sentence_range": "453-456", "Text": ", En due to\ncharges q3, q4, , qn By the superposition principle, the electric field E at r\ndue to the system of charges is (as shown in Fig 1"}, {"Chapter": "1", "sentence_range": "454-457", "Text": ", qn By the superposition principle, the electric field E at r\ndue to the system of charges is (as shown in Fig 1 9)\nE(r)  =  E1 (r) +  E2 (r)  + \u2026  +  En(r)\n        =\n41\n41\n41\n0\n1\n1\n2\n1\n0\n2\n2\n2\n2\n0\n2\n\u03c0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n\u03b5\nrq\nrq\nq\nr\nn\nn\nn\nP\nP\nP\nP\nP\nP\n\u02c6\n\u02c6"}, {"Chapter": "1", "sentence_range": "455-458", "Text": "By the superposition principle, the electric field E at r\ndue to the system of charges is (as shown in Fig 1 9)\nE(r)  =  E1 (r) +  E2 (r)  + \u2026  +  En(r)\n        =\n41\n41\n41\n0\n1\n1\n2\n1\n0\n2\n2\n2\n2\n0\n2\n\u03c0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n\u03b5\nrq\nrq\nq\nr\nn\nn\nn\nP\nP\nP\nP\nP\nP\n\u02c6\n\u02c6 \u02c6\nr\nr\nr\n+\n+\n+\nE(r) =\n=\u2211\n \n41\n0\nP\ni P\n\u03c0\u03b5\nq\nr\ni\ni\ni\nn\n2\n1\n\u02c6r\n(1"}, {"Chapter": "1", "sentence_range": "456-459", "Text": "1 9)\nE(r)  =  E1 (r) +  E2 (r)  + \u2026  +  En(r)\n        =\n41\n41\n41\n0\n1\n1\n2\n1\n0\n2\n2\n2\n2\n0\n2\n\u03c0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n\u03b5\nrq\nrq\nq\nr\nn\nn\nn\nP\nP\nP\nP\nP\nP\n\u02c6\n\u02c6 \u02c6\nr\nr\nr\n+\n+\n+\nE(r) =\n=\u2211\n \n41\n0\nP\ni P\n\u03c0\u03b5\nq\nr\ni\ni\ni\nn\n2\n1\n\u02c6r\n(1 10)\nE is a vector quantity that varies from one point to another point in space\nand is determined from the positions of the source charges"}, {"Chapter": "1", "sentence_range": "457-460", "Text": "9)\nE(r)  =  E1 (r) +  E2 (r)  + \u2026  +  En(r)\n        =\n41\n41\n41\n0\n1\n1\n2\n1\n0\n2\n2\n2\n2\n0\n2\n\u03c0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n\u03b5\nrq\nrq\nq\nr\nn\nn\nn\nP\nP\nP\nP\nP\nP\n\u02c6\n\u02c6 \u02c6\nr\nr\nr\n+\n+\n+\nE(r) =\n=\u2211\n \n41\n0\nP\ni P\n\u03c0\u03b5\nq\nr\ni\ni\ni\nn\n2\n1\n\u02c6r\n(1 10)\nE is a vector quantity that varies from one point to another point in space\nand is determined from the positions of the source charges 1"}, {"Chapter": "1", "sentence_range": "458-461", "Text": "\u02c6\nr\nr\nr\n+\n+\n+\nE(r) =\n=\u2211\n \n41\n0\nP\ni P\n\u03c0\u03b5\nq\nr\ni\ni\ni\nn\n2\n1\n\u02c6r\n(1 10)\nE is a vector quantity that varies from one point to another point in space\nand is determined from the positions of the source charges 1 7"}, {"Chapter": "1", "sentence_range": "459-462", "Text": "10)\nE is a vector quantity that varies from one point to another point in space\nand is determined from the positions of the source charges 1 7 2  Physical significance of electric field\nYou may wonder why the notion of electric field has been introduced\nhere at all"}, {"Chapter": "1", "sentence_range": "460-463", "Text": "1 7 2  Physical significance of electric field\nYou may wonder why the notion of electric field has been introduced\nhere at all After all, for any system of charges, the measurable quantity\nis the force on a charge which can be directly determined using Coulomb\u2019s\nlaw and the superposition principle [Eq"}, {"Chapter": "1", "sentence_range": "461-464", "Text": "7 2  Physical significance of electric field\nYou may wonder why the notion of electric field has been introduced\nhere at all After all, for any system of charges, the measurable quantity\nis the force on a charge which can be directly determined using Coulomb\u2019s\nlaw and the superposition principle [Eq (1"}, {"Chapter": "1", "sentence_range": "462-465", "Text": "2  Physical significance of electric field\nYou may wonder why the notion of electric field has been introduced\nhere at all After all, for any system of charges, the measurable quantity\nis the force on a charge which can be directly determined using Coulomb\u2019s\nlaw and the superposition principle [Eq (1 5)]"}, {"Chapter": "1", "sentence_range": "463-466", "Text": "After all, for any system of charges, the measurable quantity\nis the force on a charge which can be directly determined using Coulomb\u2019s\nlaw and the superposition principle [Eq (1 5)] Why then introduce this\nintermediate quantity called the electric field"}, {"Chapter": "1", "sentence_range": "464-467", "Text": "(1 5)] Why then introduce this\nintermediate quantity called the electric field For electrostatics, the concept of electric field is convenient, but not\nreally necessary"}, {"Chapter": "1", "sentence_range": "465-468", "Text": "5)] Why then introduce this\nintermediate quantity called the electric field For electrostatics, the concept of electric field is convenient, but not\nreally necessary Electric field is an elegant way of characterising the\nelectrical environment of a system of charges"}, {"Chapter": "1", "sentence_range": "466-469", "Text": "Why then introduce this\nintermediate quantity called the electric field For electrostatics, the concept of electric field is convenient, but not\nreally necessary Electric field is an elegant way of characterising the\nelectrical environment of a system of charges Electric field at a point in\nthe space around a system of charges tells you the force a unit positive\ntest charge would experience if placed at that point (without disturbing\nthe system)"}, {"Chapter": "1", "sentence_range": "467-470", "Text": "For electrostatics, the concept of electric field is convenient, but not\nreally necessary Electric field is an elegant way of characterising the\nelectrical environment of a system of charges Electric field at a point in\nthe space around a system of charges tells you the force a unit positive\ntest charge would experience if placed at that point (without disturbing\nthe system) Electric field is a characteristic of the system of charges and\nis independent of the test charge that you place at a point to determine\nthe field"}, {"Chapter": "1", "sentence_range": "468-471", "Text": "Electric field is an elegant way of characterising the\nelectrical environment of a system of charges Electric field at a point in\nthe space around a system of charges tells you the force a unit positive\ntest charge would experience if placed at that point (without disturbing\nthe system) Electric field is a characteristic of the system of charges and\nis independent of the test charge that you place at a point to determine\nthe field The term field in physics generally refers to a quantity that is\ndefined at every point in space and may vary from point to point"}, {"Chapter": "1", "sentence_range": "469-472", "Text": "Electric field at a point in\nthe space around a system of charges tells you the force a unit positive\ntest charge would experience if placed at that point (without disturbing\nthe system) Electric field is a characteristic of the system of charges and\nis independent of the test charge that you place at a point to determine\nthe field The term field in physics generally refers to a quantity that is\ndefined at every point in space and may vary from point to point Electric\nfield is a vector field, since force is a vector quantity"}, {"Chapter": "1", "sentence_range": "470-473", "Text": "Electric field is a characteristic of the system of charges and\nis independent of the test charge that you place at a point to determine\nthe field The term field in physics generally refers to a quantity that is\ndefined at every point in space and may vary from point to point Electric\nfield is a vector field, since force is a vector quantity The true physical significance of the concept of electric field, however,\nemerges only when we go beyond electrostatics and deal with time-\ndependent electromagnetic phenomena"}, {"Chapter": "1", "sentence_range": "471-474", "Text": "The term field in physics generally refers to a quantity that is\ndefined at every point in space and may vary from point to point Electric\nfield is a vector field, since force is a vector quantity The true physical significance of the concept of electric field, however,\nemerges only when we go beyond electrostatics and deal with time-\ndependent electromagnetic phenomena Suppose we consider the force\nbetween two distant charges q1, q2 in accelerated motion"}, {"Chapter": "1", "sentence_range": "472-475", "Text": "Electric\nfield is a vector field, since force is a vector quantity The true physical significance of the concept of electric field, however,\nemerges only when we go beyond electrostatics and deal with time-\ndependent electromagnetic phenomena Suppose we consider the force\nbetween two distant charges q1, q2 in accelerated motion Now the greatest\nspeed with which a signal or information can go from one point to another\nis c, the speed of light"}, {"Chapter": "1", "sentence_range": "473-476", "Text": "The true physical significance of the concept of electric field, however,\nemerges only when we go beyond electrostatics and deal with time-\ndependent electromagnetic phenomena Suppose we consider the force\nbetween two distant charges q1, q2 in accelerated motion Now the greatest\nspeed with which a signal or information can go from one point to another\nis c, the speed of light Thus, the effect of any motion of q1 on q2 cannot\nFIGURE 1"}, {"Chapter": "1", "sentence_range": "474-477", "Text": "Suppose we consider the force\nbetween two distant charges q1, q2 in accelerated motion Now the greatest\nspeed with which a signal or information can go from one point to another\nis c, the speed of light Thus, the effect of any motion of q1 on q2 cannot\nFIGURE 1 9 Electric field at a point\ndue to a system of charges is the\nvector sum of the electric fields at\nthe point due to individual charges"}, {"Chapter": "1", "sentence_range": "475-478", "Text": "Now the greatest\nspeed with which a signal or information can go from one point to another\nis c, the speed of light Thus, the effect of any motion of q1 on q2 cannot\nFIGURE 1 9 Electric field at a point\ndue to a system of charges is the\nvector sum of the electric fields at\nthe point due to individual charges Rationalised 2023-24\nElectric Charges\nand Fields\n17\narise instantaneously"}, {"Chapter": "1", "sentence_range": "476-479", "Text": "Thus, the effect of any motion of q1 on q2 cannot\nFIGURE 1 9 Electric field at a point\ndue to a system of charges is the\nvector sum of the electric fields at\nthe point due to individual charges Rationalised 2023-24\nElectric Charges\nand Fields\n17\narise instantaneously There will be some time delay between the effect\n(force on q2) and the cause (motion of q1)"}, {"Chapter": "1", "sentence_range": "477-480", "Text": "9 Electric field at a point\ndue to a system of charges is the\nvector sum of the electric fields at\nthe point due to individual charges Rationalised 2023-24\nElectric Charges\nand Fields\n17\narise instantaneously There will be some time delay between the effect\n(force on q2) and the cause (motion of q1) It is precisely here that the\nnotion of electric field (strictly, electromagnetic field) is natural and very\nuseful"}, {"Chapter": "1", "sentence_range": "478-481", "Text": "Rationalised 2023-24\nElectric Charges\nand Fields\n17\narise instantaneously There will be some time delay between the effect\n(force on q2) and the cause (motion of q1) It is precisely here that the\nnotion of electric field (strictly, electromagnetic field) is natural and very\nuseful The field picture is this: the accelerated motion of charge q1\nproduces electromagnetic waves, which then propagate with the speed\nc, reach q2 and cause a force on q2"}, {"Chapter": "1", "sentence_range": "479-482", "Text": "There will be some time delay between the effect\n(force on q2) and the cause (motion of q1) It is precisely here that the\nnotion of electric field (strictly, electromagnetic field) is natural and very\nuseful The field picture is this: the accelerated motion of charge q1\nproduces electromagnetic waves, which then propagate with the speed\nc, reach q2 and cause a force on q2 The notion of field elegantly accounts\nfor the time delay"}, {"Chapter": "1", "sentence_range": "480-483", "Text": "It is precisely here that the\nnotion of electric field (strictly, electromagnetic field) is natural and very\nuseful The field picture is this: the accelerated motion of charge q1\nproduces electromagnetic waves, which then propagate with the speed\nc, reach q2 and cause a force on q2 The notion of field elegantly accounts\nfor the time delay Thus, even though electric and magnetic fields can be\ndetected only by their effects (forces) on charges, they are regarded as\nphysical entities, not merely mathematical constructs"}, {"Chapter": "1", "sentence_range": "481-484", "Text": "The field picture is this: the accelerated motion of charge q1\nproduces electromagnetic waves, which then propagate with the speed\nc, reach q2 and cause a force on q2 The notion of field elegantly accounts\nfor the time delay Thus, even though electric and magnetic fields can be\ndetected only by their effects (forces) on charges, they are regarded as\nphysical entities, not merely mathematical constructs They have an\nindependent dynamics of their own,  i"}, {"Chapter": "1", "sentence_range": "482-485", "Text": "The notion of field elegantly accounts\nfor the time delay Thus, even though electric and magnetic fields can be\ndetected only by their effects (forces) on charges, they are regarded as\nphysical entities, not merely mathematical constructs They have an\nindependent dynamics of their own,  i e"}, {"Chapter": "1", "sentence_range": "483-486", "Text": "Thus, even though electric and magnetic fields can be\ndetected only by their effects (forces) on charges, they are regarded as\nphysical entities, not merely mathematical constructs They have an\nindependent dynamics of their own,  i e , they evolve according to laws\nof their own"}, {"Chapter": "1", "sentence_range": "484-487", "Text": "They have an\nindependent dynamics of their own,  i e , they evolve according to laws\nof their own They can also transport energy"}, {"Chapter": "1", "sentence_range": "485-488", "Text": "e , they evolve according to laws\nof their own They can also transport energy Thus, a source of time-\ndependent electromagnetic fields, turned on for a short interval of time\nand then switched off, leaves behind propagating electromagnetic fields\ntransporting energy"}, {"Chapter": "1", "sentence_range": "486-489", "Text": ", they evolve according to laws\nof their own They can also transport energy Thus, a source of time-\ndependent electromagnetic fields, turned on for a short interval of time\nand then switched off, leaves behind propagating electromagnetic fields\ntransporting energy The concept of field was first introduced by Faraday\nand is now among the central concepts in physics"}, {"Chapter": "1", "sentence_range": "487-490", "Text": "They can also transport energy Thus, a source of time-\ndependent electromagnetic fields, turned on for a short interval of time\nand then switched off, leaves behind propagating electromagnetic fields\ntransporting energy The concept of field was first introduced by Faraday\nand is now among the central concepts in physics Example 1"}, {"Chapter": "1", "sentence_range": "488-491", "Text": "Thus, a source of time-\ndependent electromagnetic fields, turned on for a short interval of time\nand then switched off, leaves behind propagating electromagnetic fields\ntransporting energy The concept of field was first introduced by Faraday\nand is now among the central concepts in physics Example 1 7 An electron falls through a distance of 1"}, {"Chapter": "1", "sentence_range": "489-492", "Text": "The concept of field was first introduced by Faraday\nand is now among the central concepts in physics Example 1 7 An electron falls through a distance of 1 5 cm in a\nuniform electric field of magnitude 2"}, {"Chapter": "1", "sentence_range": "490-493", "Text": "Example 1 7 An electron falls through a distance of 1 5 cm in a\nuniform electric field of magnitude 2 0 \u00d7 104 N C\u20131 [Fig"}, {"Chapter": "1", "sentence_range": "491-494", "Text": "7 An electron falls through a distance of 1 5 cm in a\nuniform electric field of magnitude 2 0 \u00d7 104 N C\u20131 [Fig 1"}, {"Chapter": "1", "sentence_range": "492-495", "Text": "5 cm in a\nuniform electric field of magnitude 2 0 \u00d7 104 N C\u20131 [Fig 1 10(a)]"}, {"Chapter": "1", "sentence_range": "493-496", "Text": "0 \u00d7 104 N C\u20131 [Fig 1 10(a)] The\ndirection of the field is reversed keeping its magnitude unchanged\nand a proton falls through the same distance [Fig"}, {"Chapter": "1", "sentence_range": "494-497", "Text": "1 10(a)] The\ndirection of the field is reversed keeping its magnitude unchanged\nand a proton falls through the same distance [Fig 1"}, {"Chapter": "1", "sentence_range": "495-498", "Text": "10(a)] The\ndirection of the field is reversed keeping its magnitude unchanged\nand a proton falls through the same distance [Fig 1 10(b)]"}, {"Chapter": "1", "sentence_range": "496-499", "Text": "The\ndirection of the field is reversed keeping its magnitude unchanged\nand a proton falls through the same distance [Fig 1 10(b)] Compute\nthe time of fall in each case"}, {"Chapter": "1", "sentence_range": "497-500", "Text": "1 10(b)] Compute\nthe time of fall in each case Contrast the situation with that of \u2018free\nfall under gravity\u2019"}, {"Chapter": "1", "sentence_range": "498-501", "Text": "10(b)] Compute\nthe time of fall in each case Contrast the situation with that of \u2018free\nfall under gravity\u2019 FIGURE 1"}, {"Chapter": "1", "sentence_range": "499-502", "Text": "Compute\nthe time of fall in each case Contrast the situation with that of \u2018free\nfall under gravity\u2019 FIGURE 1 10\nSolution In Fig"}, {"Chapter": "1", "sentence_range": "500-503", "Text": "Contrast the situation with that of \u2018free\nfall under gravity\u2019 FIGURE 1 10\nSolution In Fig 1"}, {"Chapter": "1", "sentence_range": "501-504", "Text": "FIGURE 1 10\nSolution In Fig 1 10(a) the field is upward, so the negatively charged\nelectron experiences a downward force of magnitude eE where E is\nthe magnitude of the electric field"}, {"Chapter": "1", "sentence_range": "502-505", "Text": "10\nSolution In Fig 1 10(a) the field is upward, so the negatively charged\nelectron experiences a downward force of magnitude eE where E is\nthe magnitude of the electric field The acceleration of the electron is\nae  =  eE/me\nwhere me is the mass of the electron"}, {"Chapter": "1", "sentence_range": "503-506", "Text": "1 10(a) the field is upward, so the negatively charged\nelectron experiences a downward force of magnitude eE where E is\nthe magnitude of the electric field The acceleration of the electron is\nae  =  eE/me\nwhere me is the mass of the electron Starting from rest, the time required by the electron to fall through a\ndistance h is given by  \n2\n2\ne\ne\ne\nh m\nh\nt\na\ne E\n=\n=\nFor e = 1"}, {"Chapter": "1", "sentence_range": "504-507", "Text": "10(a) the field is upward, so the negatively charged\nelectron experiences a downward force of magnitude eE where E is\nthe magnitude of the electric field The acceleration of the electron is\nae  =  eE/me\nwhere me is the mass of the electron Starting from rest, the time required by the electron to fall through a\ndistance h is given by  \n2\n2\ne\ne\ne\nh m\nh\nt\na\ne E\n=\n=\nFor e = 1 6 \u00d7 10\u201319C, me = 9"}, {"Chapter": "1", "sentence_range": "505-508", "Text": "The acceleration of the electron is\nae  =  eE/me\nwhere me is the mass of the electron Starting from rest, the time required by the electron to fall through a\ndistance h is given by  \n2\n2\ne\ne\ne\nh m\nh\nt\na\ne E\n=\n=\nFor e = 1 6 \u00d7 10\u201319C, me = 9 11 \u00d7 10\u201331 kg,\n     E = 2"}, {"Chapter": "1", "sentence_range": "506-509", "Text": "Starting from rest, the time required by the electron to fall through a\ndistance h is given by  \n2\n2\ne\ne\ne\nh m\nh\nt\na\ne E\n=\n=\nFor e = 1 6 \u00d7 10\u201319C, me = 9 11 \u00d7 10\u201331 kg,\n     E = 2 0 \u00d7 104 N C\u20131, h = 1"}, {"Chapter": "1", "sentence_range": "507-510", "Text": "6 \u00d7 10\u201319C, me = 9 11 \u00d7 10\u201331 kg,\n     E = 2 0 \u00d7 104 N C\u20131, h = 1 5 \u00d7 10\u20132 m,\n  te = 2"}, {"Chapter": "1", "sentence_range": "508-511", "Text": "11 \u00d7 10\u201331 kg,\n     E = 2 0 \u00d7 104 N C\u20131, h = 1 5 \u00d7 10\u20132 m,\n  te = 2 9 \u00d7 10\u20139s\nIn Fig"}, {"Chapter": "1", "sentence_range": "509-512", "Text": "0 \u00d7 104 N C\u20131, h = 1 5 \u00d7 10\u20132 m,\n  te = 2 9 \u00d7 10\u20139s\nIn Fig 1"}, {"Chapter": "1", "sentence_range": "510-513", "Text": "5 \u00d7 10\u20132 m,\n  te = 2 9 \u00d7 10\u20139s\nIn Fig 1 10 (b), the field is downward, and the positively charged\nproton experiences a downward force of magnitude eE"}, {"Chapter": "1", "sentence_range": "511-514", "Text": "9 \u00d7 10\u20139s\nIn Fig 1 10 (b), the field is downward, and the positively charged\nproton experiences a downward force of magnitude eE The\nacceleration of the proton is\nap  =  eE/mp\nwhere mp is the mass of the proton; mp = 1"}, {"Chapter": "1", "sentence_range": "512-515", "Text": "1 10 (b), the field is downward, and the positively charged\nproton experiences a downward force of magnitude eE The\nacceleration of the proton is\nap  =  eE/mp\nwhere mp is the mass of the proton; mp = 1 67 \u00d7 10\u201327 kg"}, {"Chapter": "1", "sentence_range": "513-516", "Text": "10 (b), the field is downward, and the positively charged\nproton experiences a downward force of magnitude eE The\nacceleration of the proton is\nap  =  eE/mp\nwhere mp is the mass of the proton; mp = 1 67 \u00d7 10\u201327 kg The time of\nfall for the proton is\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "514-517", "Text": "The\nacceleration of the proton is\nap  =  eE/mp\nwhere mp is the mass of the proton; mp = 1 67 \u00d7 10\u201327 kg The time of\nfall for the proton is\n EXAMPLE 1 7\nRationalised 2023-24\n18\nPhysics\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "515-518", "Text": "67 \u00d7 10\u201327 kg The time of\nfall for the proton is\n EXAMPLE 1 7\nRationalised 2023-24\n18\nPhysics\n EXAMPLE 1 8\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "516-519", "Text": "The time of\nfall for the proton is\n EXAMPLE 1 7\nRationalised 2023-24\n18\nPhysics\n EXAMPLE 1 8\n EXAMPLE 1 7\n\u20137\n2\n2\n1 3 10\ns\np\np\np\nh m\nh\nt"}, {"Chapter": "1", "sentence_range": "517-520", "Text": "7\nRationalised 2023-24\n18\nPhysics\n EXAMPLE 1 8\n EXAMPLE 1 7\n\u20137\n2\n2\n1 3 10\ns\np\np\np\nh m\nh\nt a\ne E\n=\n=\n=\n\u00d7\nThus, the heavier particle (proton) takes a greater time to fall through\nthe same distance"}, {"Chapter": "1", "sentence_range": "518-521", "Text": "8\n EXAMPLE 1 7\n\u20137\n2\n2\n1 3 10\ns\np\np\np\nh m\nh\nt a\ne E\n=\n=\n=\n\u00d7\nThus, the heavier particle (proton) takes a greater time to fall through\nthe same distance This is in basic contrast to the situation of \u2018free\nfall under gravity\u2019 where the time of fall is independent of the mass of\nthe body"}, {"Chapter": "1", "sentence_range": "519-522", "Text": "7\n\u20137\n2\n2\n1 3 10\ns\np\np\np\nh m\nh\nt a\ne E\n=\n=\n=\n\u00d7\nThus, the heavier particle (proton) takes a greater time to fall through\nthe same distance This is in basic contrast to the situation of \u2018free\nfall under gravity\u2019 where the time of fall is independent of the mass of\nthe body Note that in this example we have ignored the acceleration\ndue to gravity in calculating the time of fall"}, {"Chapter": "1", "sentence_range": "520-523", "Text": "a\ne E\n=\n=\n=\n\u00d7\nThus, the heavier particle (proton) takes a greater time to fall through\nthe same distance This is in basic contrast to the situation of \u2018free\nfall under gravity\u2019 where the time of fall is independent of the mass of\nthe body Note that in this example we have ignored the acceleration\ndue to gravity in calculating the time of fall To see if this is justified,\nlet us calculate the acceleration of the proton in the given electric\nfield:\np\np\ne E\na\nm\n=\n     \n19\n4\n1\n27\n(1 6\n10\nC)\n(2 0\n10\nN C\n)\n1 67\n10\nkg"}, {"Chapter": "1", "sentence_range": "521-524", "Text": "This is in basic contrast to the situation of \u2018free\nfall under gravity\u2019 where the time of fall is independent of the mass of\nthe body Note that in this example we have ignored the acceleration\ndue to gravity in calculating the time of fall To see if this is justified,\nlet us calculate the acceleration of the proton in the given electric\nfield:\np\np\ne E\na\nm\n=\n     \n19\n4\n1\n27\n(1 6\n10\nC)\n(2 0\n10\nN C\n)\n1 67\n10\nkg \u2212\n\u2212\n\u2212\n\u00d7\n\u00d7\n\u00d7\n=\n\u00d7\n     \n12\n\u20132\n1 9\n10\nm s"}, {"Chapter": "1", "sentence_range": "522-525", "Text": "Note that in this example we have ignored the acceleration\ndue to gravity in calculating the time of fall To see if this is justified,\nlet us calculate the acceleration of the proton in the given electric\nfield:\np\np\ne E\na\nm\n=\n     \n19\n4\n1\n27\n(1 6\n10\nC)\n(2 0\n10\nN C\n)\n1 67\n10\nkg \u2212\n\u2212\n\u2212\n\u00d7\n\u00d7\n\u00d7\n=\n\u00d7\n     \n12\n\u20132\n1 9\n10\nm s =\n\u00d7\nwhich is enormous compared to the value of g (9"}, {"Chapter": "1", "sentence_range": "523-526", "Text": "To see if this is justified,\nlet us calculate the acceleration of the proton in the given electric\nfield:\np\np\ne E\na\nm\n=\n     \n19\n4\n1\n27\n(1 6\n10\nC)\n(2 0\n10\nN C\n)\n1 67\n10\nkg \u2212\n\u2212\n\u2212\n\u00d7\n\u00d7\n\u00d7\n=\n\u00d7\n     \n12\n\u20132\n1 9\n10\nm s =\n\u00d7\nwhich is enormous compared to the value of g (9 8 m s\u20132), the\nacceleration due to gravity"}, {"Chapter": "1", "sentence_range": "524-527", "Text": "\u2212\n\u2212\n\u2212\n\u00d7\n\u00d7\n\u00d7\n=\n\u00d7\n     \n12\n\u20132\n1 9\n10\nm s =\n\u00d7\nwhich is enormous compared to the value of g (9 8 m s\u20132), the\nacceleration due to gravity The acceleration of the electron is even\ngreater"}, {"Chapter": "1", "sentence_range": "525-528", "Text": "=\n\u00d7\nwhich is enormous compared to the value of g (9 8 m s\u20132), the\nacceleration due to gravity The acceleration of the electron is even\ngreater Thus, the effect of acceleration due to gravity can be ignored\nin this example"}, {"Chapter": "1", "sentence_range": "526-529", "Text": "8 m s\u20132), the\nacceleration due to gravity The acceleration of the electron is even\ngreater Thus, the effect of acceleration due to gravity can be ignored\nin this example Example 1"}, {"Chapter": "1", "sentence_range": "527-530", "Text": "The acceleration of the electron is even\ngreater Thus, the effect of acceleration due to gravity can be ignored\nin this example Example 1 8 Two point charges q1 and q2, of magnitude +10\u20138 C and\n\u201310\u20138 C, respectively, are placed 0"}, {"Chapter": "1", "sentence_range": "528-531", "Text": "Thus, the effect of acceleration due to gravity can be ignored\nin this example Example 1 8 Two point charges q1 and q2, of magnitude +10\u20138 C and\n\u201310\u20138 C, respectively, are placed 0 1 m apart"}, {"Chapter": "1", "sentence_range": "529-532", "Text": "Example 1 8 Two point charges q1 and q2, of magnitude +10\u20138 C and\n\u201310\u20138 C, respectively, are placed 0 1 m apart Calculate the electric\nfields at points A, B and C shown in Fig"}, {"Chapter": "1", "sentence_range": "530-533", "Text": "8 Two point charges q1 and q2, of magnitude +10\u20138 C and\n\u201310\u20138 C, respectively, are placed 0 1 m apart Calculate the electric\nfields at points A, B and C shown in Fig 1"}, {"Chapter": "1", "sentence_range": "531-534", "Text": "1 m apart Calculate the electric\nfields at points A, B and C shown in Fig 1 11"}, {"Chapter": "1", "sentence_range": "532-535", "Text": "Calculate the electric\nfields at points A, B and C shown in Fig 1 11 FIGURE 1"}, {"Chapter": "1", "sentence_range": "533-536", "Text": "1 11 FIGURE 1 11\nSolution The electric field vector E1A at A due to the positive charge\nq1 points towards the right and has a magnitude\n9\n2\n-2\n8\n1A\n2\n(9\n10 Nm C )\n(10\nC)\n(0"}, {"Chapter": "1", "sentence_range": "534-537", "Text": "11 FIGURE 1 11\nSolution The electric field vector E1A at A due to the positive charge\nq1 points towards the right and has a magnitude\n9\n2\n-2\n8\n1A\n2\n(9\n10 Nm C )\n(10\nC)\n(0 05m)\nE\n\u2212\n\u00d7\n\u00d7\n=\n =  3"}, {"Chapter": "1", "sentence_range": "535-538", "Text": "FIGURE 1 11\nSolution The electric field vector E1A at A due to the positive charge\nq1 points towards the right and has a magnitude\n9\n2\n-2\n8\n1A\n2\n(9\n10 Nm C )\n(10\nC)\n(0 05m)\nE\n\u2212\n\u00d7\n\u00d7\n=\n =  3 6 \u00d7 104  N C\u20131\nThe electric field vector E2A at A due to the negative charge q2 points\ntowards the right and has the same magnitude"}, {"Chapter": "1", "sentence_range": "536-539", "Text": "11\nSolution The electric field vector E1A at A due to the positive charge\nq1 points towards the right and has a magnitude\n9\n2\n-2\n8\n1A\n2\n(9\n10 Nm C )\n(10\nC)\n(0 05m)\nE\n\u2212\n\u00d7\n\u00d7\n=\n =  3 6 \u00d7 104  N C\u20131\nThe electric field vector E2A at A due to the negative charge q2 points\ntowards the right and has the same magnitude Hence the magnitude\nof the total electric field EA at A is\nEA = E1A + E2A = 7"}, {"Chapter": "1", "sentence_range": "537-540", "Text": "05m)\nE\n\u2212\n\u00d7\n\u00d7\n=\n =  3 6 \u00d7 104  N C\u20131\nThe electric field vector E2A at A due to the negative charge q2 points\ntowards the right and has the same magnitude Hence the magnitude\nof the total electric field EA at A is\nEA = E1A + E2A = 7 2 \u00d7 104 N C\u20131\nEA is directed toward the right"}, {"Chapter": "1", "sentence_range": "538-541", "Text": "6 \u00d7 104  N C\u20131\nThe electric field vector E2A at A due to the negative charge q2 points\ntowards the right and has the same magnitude Hence the magnitude\nof the total electric field EA at A is\nEA = E1A + E2A = 7 2 \u00d7 104 N C\u20131\nEA is directed toward the right Rationalised 2023-24\nElectric Charges\nand Fields\n19\nThe electric field vector E1B at B due to the positive charge q1 points\ntowards the left and has a magnitude\n9\n2\n\u20132\n8\n1B\n2\n(9\n10 Nm C\n)\n(10\nC)\n(0"}, {"Chapter": "1", "sentence_range": "539-542", "Text": "Hence the magnitude\nof the total electric field EA at A is\nEA = E1A + E2A = 7 2 \u00d7 104 N C\u20131\nEA is directed toward the right Rationalised 2023-24\nElectric Charges\nand Fields\n19\nThe electric field vector E1B at B due to the positive charge q1 points\ntowards the left and has a magnitude\n9\n2\n\u20132\n8\n1B\n2\n(9\n10 Nm C\n)\n(10\nC)\n(0 05 m)\nE\n\u2212\n\u00d7\n\u00d7\n=\n= 3"}, {"Chapter": "1", "sentence_range": "540-543", "Text": "2 \u00d7 104 N C\u20131\nEA is directed toward the right Rationalised 2023-24\nElectric Charges\nand Fields\n19\nThe electric field vector E1B at B due to the positive charge q1 points\ntowards the left and has a magnitude\n9\n2\n\u20132\n8\n1B\n2\n(9\n10 Nm C\n)\n(10\nC)\n(0 05 m)\nE\n\u2212\n\u00d7\n\u00d7\n=\n= 3 6 \u00d7 104 N C\u20131\nThe electric field vector E2B at B due to the negative charge q2 points\ntowards the right and has a magnitude\n9\n2\n\u20132\n8\n2B\n2\n(9\n10 Nm C\n)\n(10\nC)\n(0"}, {"Chapter": "1", "sentence_range": "541-544", "Text": "Rationalised 2023-24\nElectric Charges\nand Fields\n19\nThe electric field vector E1B at B due to the positive charge q1 points\ntowards the left and has a magnitude\n9\n2\n\u20132\n8\n1B\n2\n(9\n10 Nm C\n)\n(10\nC)\n(0 05 m)\nE\n\u2212\n\u00d7\n\u00d7\n=\n= 3 6 \u00d7 104 N C\u20131\nThe electric field vector E2B at B due to the negative charge q2 points\ntowards the right and has a magnitude\n9\n2\n\u20132\n8\n2B\n2\n(9\n10 Nm C\n)\n(10\nC)\n(0 15 m)\nE\n\u2212\n\u00d7\n\u00d7\n=\n=  4 \u00d7 103  N C\u20131\nThe magnitude of the total electric field at B is\nEB = E1B \u2013  E2B = 3"}, {"Chapter": "1", "sentence_range": "542-545", "Text": "05 m)\nE\n\u2212\n\u00d7\n\u00d7\n=\n= 3 6 \u00d7 104 N C\u20131\nThe electric field vector E2B at B due to the negative charge q2 points\ntowards the right and has a magnitude\n9\n2\n\u20132\n8\n2B\n2\n(9\n10 Nm C\n)\n(10\nC)\n(0 15 m)\nE\n\u2212\n\u00d7\n\u00d7\n=\n=  4 \u00d7 103  N C\u20131\nThe magnitude of the total electric field at B is\nEB = E1B \u2013  E2B = 3 2 \u00d7 104 N C\u20131\nEB is directed towards the left"}, {"Chapter": "1", "sentence_range": "543-546", "Text": "6 \u00d7 104 N C\u20131\nThe electric field vector E2B at B due to the negative charge q2 points\ntowards the right and has a magnitude\n9\n2\n\u20132\n8\n2B\n2\n(9\n10 Nm C\n)\n(10\nC)\n(0 15 m)\nE\n\u2212\n\u00d7\n\u00d7\n=\n=  4 \u00d7 103  N C\u20131\nThe magnitude of the total electric field at B is\nEB = E1B \u2013  E2B = 3 2 \u00d7 104 N C\u20131\nEB is directed towards the left The magnitude of each electric field vector at point C, due to charge\nq1 and q2 is\n     \n9\n2\n\u20132\n8\n1C\n2C\n2\n(9\n10 Nm C\n)\n(10\nC)\n(0"}, {"Chapter": "1", "sentence_range": "544-547", "Text": "15 m)\nE\n\u2212\n\u00d7\n\u00d7\n=\n=  4 \u00d7 103  N C\u20131\nThe magnitude of the total electric field at B is\nEB = E1B \u2013  E2B = 3 2 \u00d7 104 N C\u20131\nEB is directed towards the left The magnitude of each electric field vector at point C, due to charge\nq1 and q2 is\n     \n9\n2\n\u20132\n8\n1C\n2C\n2\n(9\n10 Nm C\n)\n(10\nC)\n(0 10 m)\nE\nE\n\u2212\n\u00d7\n\u00d7\n=\n=\n =  9 \u00d7 103  N C\u20131\nThe directions in which these two vectors point are indicated in\nFig"}, {"Chapter": "1", "sentence_range": "545-548", "Text": "2 \u00d7 104 N C\u20131\nEB is directed towards the left The magnitude of each electric field vector at point C, due to charge\nq1 and q2 is\n     \n9\n2\n\u20132\n8\n1C\n2C\n2\n(9\n10 Nm C\n)\n(10\nC)\n(0 10 m)\nE\nE\n\u2212\n\u00d7\n\u00d7\n=\n=\n =  9 \u00d7 103  N C\u20131\nThe directions in which these two vectors point are indicated in\nFig 1"}, {"Chapter": "1", "sentence_range": "546-549", "Text": "The magnitude of each electric field vector at point C, due to charge\nq1 and q2 is\n     \n9\n2\n\u20132\n8\n1C\n2C\n2\n(9\n10 Nm C\n)\n(10\nC)\n(0 10 m)\nE\nE\n\u2212\n\u00d7\n\u00d7\n=\n=\n =  9 \u00d7 103  N C\u20131\nThe directions in which these two vectors point are indicated in\nFig 1 11"}, {"Chapter": "1", "sentence_range": "547-550", "Text": "10 m)\nE\nE\n\u2212\n\u00d7\n\u00d7\n=\n=\n =  9 \u00d7 103  N C\u20131\nThe directions in which these two vectors point are indicated in\nFig 1 11 The resultant of these two vectors is\n1\n2\ncos\ncos\n3\n3\n\u03c0\n\u03c0\n=\n+\nC\nc\nc\nE\nE\nE\n=  9 \u00d7 103  N C\u20131\nEC points towards the right"}, {"Chapter": "1", "sentence_range": "548-551", "Text": "1 11 The resultant of these two vectors is\n1\n2\ncos\ncos\n3\n3\n\u03c0\n\u03c0\n=\n+\nC\nc\nc\nE\nE\nE\n=  9 \u00d7 103  N C\u20131\nEC points towards the right 1"}, {"Chapter": "1", "sentence_range": "549-552", "Text": "11 The resultant of these two vectors is\n1\n2\ncos\ncos\n3\n3\n\u03c0\n\u03c0\n=\n+\nC\nc\nc\nE\nE\nE\n=  9 \u00d7 103  N C\u20131\nEC points towards the right 1 8  ELECTRIC FIELD LINES\nWe have studied electric field in the last section"}, {"Chapter": "1", "sentence_range": "550-553", "Text": "The resultant of these two vectors is\n1\n2\ncos\ncos\n3\n3\n\u03c0\n\u03c0\n=\n+\nC\nc\nc\nE\nE\nE\n=  9 \u00d7 103  N C\u20131\nEC points towards the right 1 8  ELECTRIC FIELD LINES\nWe have studied electric field in the last section It is a vector quantity\nand can be represented as we represent vectors"}, {"Chapter": "1", "sentence_range": "551-554", "Text": "1 8  ELECTRIC FIELD LINES\nWe have studied electric field in the last section It is a vector quantity\nand can be represented as we represent vectors Let us try to represent E\ndue to a point charge pictorially"}, {"Chapter": "1", "sentence_range": "552-555", "Text": "8  ELECTRIC FIELD LINES\nWe have studied electric field in the last section It is a vector quantity\nand can be represented as we represent vectors Let us try to represent E\ndue to a point charge pictorially Let the point charge be placed at the\norigin"}, {"Chapter": "1", "sentence_range": "553-556", "Text": "It is a vector quantity\nand can be represented as we represent vectors Let us try to represent E\ndue to a point charge pictorially Let the point charge be placed at the\norigin Draw vectors pointing along the direction of the\nelectric field with their lengths proportional to the strength\nof the field at each point"}, {"Chapter": "1", "sentence_range": "554-557", "Text": "Let us try to represent E\ndue to a point charge pictorially Let the point charge be placed at the\norigin Draw vectors pointing along the direction of the\nelectric field with their lengths proportional to the strength\nof the field at each point Since the magnitude of electric\nfield at a point decreases inversely as the square of the\ndistance of that point from the charge, the vector gets\nshorter as one goes away from the origin, always pointing\nradially outward"}, {"Chapter": "1", "sentence_range": "555-558", "Text": "Let the point charge be placed at the\norigin Draw vectors pointing along the direction of the\nelectric field with their lengths proportional to the strength\nof the field at each point Since the magnitude of electric\nfield at a point decreases inversely as the square of the\ndistance of that point from the charge, the vector gets\nshorter as one goes away from the origin, always pointing\nradially outward Figure 1"}, {"Chapter": "1", "sentence_range": "556-559", "Text": "Draw vectors pointing along the direction of the\nelectric field with their lengths proportional to the strength\nof the field at each point Since the magnitude of electric\nfield at a point decreases inversely as the square of the\ndistance of that point from the charge, the vector gets\nshorter as one goes away from the origin, always pointing\nradially outward Figure 1 12 shows such a picture"}, {"Chapter": "1", "sentence_range": "557-560", "Text": "Since the magnitude of electric\nfield at a point decreases inversely as the square of the\ndistance of that point from the charge, the vector gets\nshorter as one goes away from the origin, always pointing\nradially outward Figure 1 12 shows such a picture In\nthis figure, each arrow indicates the electric field, i"}, {"Chapter": "1", "sentence_range": "558-561", "Text": "Figure 1 12 shows such a picture In\nthis figure, each arrow indicates the electric field, i e"}, {"Chapter": "1", "sentence_range": "559-562", "Text": "12 shows such a picture In\nthis figure, each arrow indicates the electric field, i e , the\nforce acting on a unit positive charge, placed at the tail of\nthat arrow"}, {"Chapter": "1", "sentence_range": "560-563", "Text": "In\nthis figure, each arrow indicates the electric field, i e , the\nforce acting on a unit positive charge, placed at the tail of\nthat arrow Connect the arrows pointing in one direction\nand the resulting figure represents a field line"}, {"Chapter": "1", "sentence_range": "561-564", "Text": "e , the\nforce acting on a unit positive charge, placed at the tail of\nthat arrow Connect the arrows pointing in one direction\nand the resulting figure represents a field line We thus\nget many field lines, all pointing outwards from the point\ncharge"}, {"Chapter": "1", "sentence_range": "562-565", "Text": ", the\nforce acting on a unit positive charge, placed at the tail of\nthat arrow Connect the arrows pointing in one direction\nand the resulting figure represents a field line We thus\nget many field lines, all pointing outwards from the point\ncharge Have we lost the information about the strength\nor magnitude of the field now, because it was contained\nin the length of the arrow"}, {"Chapter": "1", "sentence_range": "563-566", "Text": "Connect the arrows pointing in one direction\nand the resulting figure represents a field line We thus\nget many field lines, all pointing outwards from the point\ncharge Have we lost the information about the strength\nor magnitude of the field now, because it was contained\nin the length of the arrow No"}, {"Chapter": "1", "sentence_range": "564-567", "Text": "We thus\nget many field lines, all pointing outwards from the point\ncharge Have we lost the information about the strength\nor magnitude of the field now, because it was contained\nin the length of the arrow No Now the magnitude of the\nfield is represented by the density of field lines"}, {"Chapter": "1", "sentence_range": "565-568", "Text": "Have we lost the information about the strength\nor magnitude of the field now, because it was contained\nin the length of the arrow No Now the magnitude of the\nfield is represented by the density of field lines E is strong\nnear the charge, so the density of field lines is more near\nthe charge and the lines are closer"}, {"Chapter": "1", "sentence_range": "566-569", "Text": "No Now the magnitude of the\nfield is represented by the density of field lines E is strong\nnear the charge, so the density of field lines is more near\nthe charge and the lines are closer Away from the charge,\nthe field gets weaker and the density of field lines is less,\nresulting in well-separated lines"}, {"Chapter": "1", "sentence_range": "567-570", "Text": "Now the magnitude of the\nfield is represented by the density of field lines E is strong\nnear the charge, so the density of field lines is more near\nthe charge and the lines are closer Away from the charge,\nthe field gets weaker and the density of field lines is less,\nresulting in well-separated lines Another person may draw more lines"}, {"Chapter": "1", "sentence_range": "568-571", "Text": "E is strong\nnear the charge, so the density of field lines is more near\nthe charge and the lines are closer Away from the charge,\nthe field gets weaker and the density of field lines is less,\nresulting in well-separated lines Another person may draw more lines But the number of lines is not\nimportant"}, {"Chapter": "1", "sentence_range": "569-572", "Text": "Away from the charge,\nthe field gets weaker and the density of field lines is less,\nresulting in well-separated lines Another person may draw more lines But the number of lines is not\nimportant In fact, an infinite number of lines can be drawn in any region"}, {"Chapter": "1", "sentence_range": "570-573", "Text": "Another person may draw more lines But the number of lines is not\nimportant In fact, an infinite number of lines can be drawn in any region FIGURE 1"}, {"Chapter": "1", "sentence_range": "571-574", "Text": "But the number of lines is not\nimportant In fact, an infinite number of lines can be drawn in any region FIGURE 1 12 Field of a point charge"}, {"Chapter": "1", "sentence_range": "572-575", "Text": "In fact, an infinite number of lines can be drawn in any region FIGURE 1 12 Field of a point charge EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "573-576", "Text": "FIGURE 1 12 Field of a point charge EXAMPLE 1 8\nRationalised 2023-24\n20\nPhysics\nIt is the relative density of lines in different regions which is\nimportant"}, {"Chapter": "1", "sentence_range": "574-577", "Text": "12 Field of a point charge EXAMPLE 1 8\nRationalised 2023-24\n20\nPhysics\nIt is the relative density of lines in different regions which is\nimportant We draw the figure on the plane of paper, i"}, {"Chapter": "1", "sentence_range": "575-578", "Text": "EXAMPLE 1 8\nRationalised 2023-24\n20\nPhysics\nIt is the relative density of lines in different regions which is\nimportant We draw the figure on the plane of paper, i e"}, {"Chapter": "1", "sentence_range": "576-579", "Text": "8\nRationalised 2023-24\n20\nPhysics\nIt is the relative density of lines in different regions which is\nimportant We draw the figure on the plane of paper, i e , in two-\ndimensions but we live in three-dimensions"}, {"Chapter": "1", "sentence_range": "577-580", "Text": "We draw the figure on the plane of paper, i e , in two-\ndimensions but we live in three-dimensions So if one wishes\nto estimate the density of field lines, one has to consider the\nnumber of lines per unit cross-sectional area, perpendicular\nto the lines"}, {"Chapter": "1", "sentence_range": "578-581", "Text": "e , in two-\ndimensions but we live in three-dimensions So if one wishes\nto estimate the density of field lines, one has to consider the\nnumber of lines per unit cross-sectional area, perpendicular\nto the lines Since the electric field decreases as the square of\nthe distance from a point charge and the area enclosing the\ncharge increases as the square of the distance, the number\nof field lines crossing the enclosing area remains constant,\nwhatever may be the distance of the area from the charge"}, {"Chapter": "1", "sentence_range": "579-582", "Text": ", in two-\ndimensions but we live in three-dimensions So if one wishes\nto estimate the density of field lines, one has to consider the\nnumber of lines per unit cross-sectional area, perpendicular\nto the lines Since the electric field decreases as the square of\nthe distance from a point charge and the area enclosing the\ncharge increases as the square of the distance, the number\nof field lines crossing the enclosing area remains constant,\nwhatever may be the distance of the area from the charge We started by saying that the field lines carry information\nabout the direction of electric field at different points in space"}, {"Chapter": "1", "sentence_range": "580-583", "Text": "So if one wishes\nto estimate the density of field lines, one has to consider the\nnumber of lines per unit cross-sectional area, perpendicular\nto the lines Since the electric field decreases as the square of\nthe distance from a point charge and the area enclosing the\ncharge increases as the square of the distance, the number\nof field lines crossing the enclosing area remains constant,\nwhatever may be the distance of the area from the charge We started by saying that the field lines carry information\nabout the direction of electric field at different points in space Having drawn a certain set of field lines, the relative density\n(i"}, {"Chapter": "1", "sentence_range": "581-584", "Text": "Since the electric field decreases as the square of\nthe distance from a point charge and the area enclosing the\ncharge increases as the square of the distance, the number\nof field lines crossing the enclosing area remains constant,\nwhatever may be the distance of the area from the charge We started by saying that the field lines carry information\nabout the direction of electric field at different points in space Having drawn a certain set of field lines, the relative density\n(i e"}, {"Chapter": "1", "sentence_range": "582-585", "Text": "We started by saying that the field lines carry information\nabout the direction of electric field at different points in space Having drawn a certain set of field lines, the relative density\n(i e , closeness) of the field lines at different points indicates\nthe relative strength of electric field at those points"}, {"Chapter": "1", "sentence_range": "583-586", "Text": "Having drawn a certain set of field lines, the relative density\n(i e , closeness) of the field lines at different points indicates\nthe relative strength of electric field at those points The field\nlines crowd where the field is strong and are spaced apart\nwhere it is weak"}, {"Chapter": "1", "sentence_range": "584-587", "Text": "e , closeness) of the field lines at different points indicates\nthe relative strength of electric field at those points The field\nlines crowd where the field is strong and are spaced apart\nwhere it is weak Figure 1"}, {"Chapter": "1", "sentence_range": "585-588", "Text": ", closeness) of the field lines at different points indicates\nthe relative strength of electric field at those points The field\nlines crowd where the field is strong and are spaced apart\nwhere it is weak Figure 1 13 shows a set of field lines"}, {"Chapter": "1", "sentence_range": "586-589", "Text": "The field\nlines crowd where the field is strong and are spaced apart\nwhere it is weak Figure 1 13 shows a set of field lines We\ncan imagine two equal and small elements of area placed at points R and\nS normal to the field lines there"}, {"Chapter": "1", "sentence_range": "587-590", "Text": "Figure 1 13 shows a set of field lines We\ncan imagine two equal and small elements of area placed at points R and\nS normal to the field lines there The number of field lines in our picture\ncutting the area elements is proportional to the magnitude of field at\nthese points"}, {"Chapter": "1", "sentence_range": "588-591", "Text": "13 shows a set of field lines We\ncan imagine two equal and small elements of area placed at points R and\nS normal to the field lines there The number of field lines in our picture\ncutting the area elements is proportional to the magnitude of field at\nthese points The picture shows that the field at R is stronger than at S"}, {"Chapter": "1", "sentence_range": "589-592", "Text": "We\ncan imagine two equal and small elements of area placed at points R and\nS normal to the field lines there The number of field lines in our picture\ncutting the area elements is proportional to the magnitude of field at\nthese points The picture shows that the field at R is stronger than at S To understand the dependence of the field lines on the area, or rather\nthe solid angle subtended by an area element, let us try to relate the\narea with the solid angle, a generalisation of angle to three dimensions"}, {"Chapter": "1", "sentence_range": "590-593", "Text": "The number of field lines in our picture\ncutting the area elements is proportional to the magnitude of field at\nthese points The picture shows that the field at R is stronger than at S To understand the dependence of the field lines on the area, or rather\nthe solid angle subtended by an area element, let us try to relate the\narea with the solid angle, a generalisation of angle to three dimensions Recall how a (plane) angle is defined in two-dimensions"}, {"Chapter": "1", "sentence_range": "591-594", "Text": "The picture shows that the field at R is stronger than at S To understand the dependence of the field lines on the area, or rather\nthe solid angle subtended by an area element, let us try to relate the\narea with the solid angle, a generalisation of angle to three dimensions Recall how a (plane) angle is defined in two-dimensions Let a small\ntransverse line element Dl be placed at a distance r from a point O"}, {"Chapter": "1", "sentence_range": "592-595", "Text": "To understand the dependence of the field lines on the area, or rather\nthe solid angle subtended by an area element, let us try to relate the\narea with the solid angle, a generalisation of angle to three dimensions Recall how a (plane) angle is defined in two-dimensions Let a small\ntransverse line element Dl be placed at a distance r from a point O Then\nthe angle subtended by Dl  at O can be approximated as Dq = Dl/r"}, {"Chapter": "1", "sentence_range": "593-596", "Text": "Recall how a (plane) angle is defined in two-dimensions Let a small\ntransverse line element Dl be placed at a distance r from a point O Then\nthe angle subtended by Dl  at O can be approximated as Dq = Dl/r Likewise, in three-dimensions the solid angle* subtended by a small\nperpendicular plane area DS, at a distance r, can be written as\nDW = DS/r2"}, {"Chapter": "1", "sentence_range": "594-597", "Text": "Let a small\ntransverse line element Dl be placed at a distance r from a point O Then\nthe angle subtended by Dl  at O can be approximated as Dq = Dl/r Likewise, in three-dimensions the solid angle* subtended by a small\nperpendicular plane area DS, at a distance r, can be written as\nDW = DS/r2 We know that in a given solid angle the number of radial\nfield lines is the same"}, {"Chapter": "1", "sentence_range": "595-598", "Text": "Then\nthe angle subtended by Dl  at O can be approximated as Dq = Dl/r Likewise, in three-dimensions the solid angle* subtended by a small\nperpendicular plane area DS, at a distance r, can be written as\nDW = DS/r2 We know that in a given solid angle the number of radial\nfield lines is the same In Fig"}, {"Chapter": "1", "sentence_range": "596-599", "Text": "Likewise, in three-dimensions the solid angle* subtended by a small\nperpendicular plane area DS, at a distance r, can be written as\nDW = DS/r2 We know that in a given solid angle the number of radial\nfield lines is the same In Fig 1"}, {"Chapter": "1", "sentence_range": "597-600", "Text": "We know that in a given solid angle the number of radial\nfield lines is the same In Fig 1 13, for two points P1 and P2 at distances\nr1 and r2 from the charge, the element of area subtending the solid angle\nDW is \n2\n1r DW at P1 and an element of area \n2\n2r DW at P2, respectively"}, {"Chapter": "1", "sentence_range": "598-601", "Text": "In Fig 1 13, for two points P1 and P2 at distances\nr1 and r2 from the charge, the element of area subtending the solid angle\nDW is \n2\n1r DW at P1 and an element of area \n2\n2r DW at P2, respectively The\nnumber of lines (say n) cutting these area elements are the same"}, {"Chapter": "1", "sentence_range": "599-602", "Text": "1 13, for two points P1 and P2 at distances\nr1 and r2 from the charge, the element of area subtending the solid angle\nDW is \n2\n1r DW at P1 and an element of area \n2\n2r DW at P2, respectively The\nnumber of lines (say n) cutting these area elements are the same The\nnumber of field lines, cutting unit area element is therefore n/(\n2\n1r DW) at\nP1 and n/(\n2\n2r DW) at P2, respectively"}, {"Chapter": "1", "sentence_range": "600-603", "Text": "13, for two points P1 and P2 at distances\nr1 and r2 from the charge, the element of area subtending the solid angle\nDW is \n2\n1r DW at P1 and an element of area \n2\n2r DW at P2, respectively The\nnumber of lines (say n) cutting these area elements are the same The\nnumber of field lines, cutting unit area element is therefore n/(\n2\n1r DW) at\nP1 and n/(\n2\n2r DW) at P2, respectively Since n and DW are common, the\nstrength of the field clearly has a 1/r 2 dependence"}, {"Chapter": "1", "sentence_range": "601-604", "Text": "The\nnumber of lines (say n) cutting these area elements are the same The\nnumber of field lines, cutting unit area element is therefore n/(\n2\n1r DW) at\nP1 and n/(\n2\n2r DW) at P2, respectively Since n and DW are common, the\nstrength of the field clearly has a 1/r 2 dependence The picture of field lines was invented by Faraday to develop an\nintuitive non-mathematical way of visualising electric fields around\ncharged configurations"}, {"Chapter": "1", "sentence_range": "602-605", "Text": "The\nnumber of field lines, cutting unit area element is therefore n/(\n2\n1r DW) at\nP1 and n/(\n2\n2r DW) at P2, respectively Since n and DW are common, the\nstrength of the field clearly has a 1/r 2 dependence The picture of field lines was invented by Faraday to develop an\nintuitive non-mathematical way of visualising electric fields around\ncharged configurations Faraday called them lines of force"}, {"Chapter": "1", "sentence_range": "603-606", "Text": "Since n and DW are common, the\nstrength of the field clearly has a 1/r 2 dependence The picture of field lines was invented by Faraday to develop an\nintuitive non-mathematical way of visualising electric fields around\ncharged configurations Faraday called them lines of force This term is\nsomewhat misleading, especially in case of magnetic fields"}, {"Chapter": "1", "sentence_range": "604-607", "Text": "The picture of field lines was invented by Faraday to develop an\nintuitive non-mathematical way of visualising electric fields around\ncharged configurations Faraday called them lines of force This term is\nsomewhat misleading, especially in case of magnetic fields The more\nappropriate term is field lines (electric or magnetic) that we have\nadopted in this book"}, {"Chapter": "1", "sentence_range": "605-608", "Text": "Faraday called them lines of force This term is\nsomewhat misleading, especially in case of magnetic fields The more\nappropriate term is field lines (electric or magnetic) that we have\nadopted in this book Electric field lines are thus a way of pictorially mapping the electric\nfield around a configuration of charges"}, {"Chapter": "1", "sentence_range": "606-609", "Text": "This term is\nsomewhat misleading, especially in case of magnetic fields The more\nappropriate term is field lines (electric or magnetic) that we have\nadopted in this book Electric field lines are thus a way of pictorially mapping the electric\nfield around a configuration of charges An electric field line is, in general,\nFIGURE 1"}, {"Chapter": "1", "sentence_range": "607-610", "Text": "The more\nappropriate term is field lines (electric or magnetic) that we have\nadopted in this book Electric field lines are thus a way of pictorially mapping the electric\nfield around a configuration of charges An electric field line is, in general,\nFIGURE 1 13  Dependence of\nelectric field strength on the\ndistance and its relation to the\nnumber of field lines"}, {"Chapter": "1", "sentence_range": "608-611", "Text": "Electric field lines are thus a way of pictorially mapping the electric\nfield around a configuration of charges An electric field line is, in general,\nFIGURE 1 13  Dependence of\nelectric field strength on the\ndistance and its relation to the\nnumber of field lines *\nSolid angle is a measure of a cone"}, {"Chapter": "1", "sentence_range": "609-612", "Text": "An electric field line is, in general,\nFIGURE 1 13  Dependence of\nelectric field strength on the\ndistance and its relation to the\nnumber of field lines *\nSolid angle is a measure of a cone Consider the intersection of the given cone\nwith a sphere of radius R"}, {"Chapter": "1", "sentence_range": "610-613", "Text": "13  Dependence of\nelectric field strength on the\ndistance and its relation to the\nnumber of field lines *\nSolid angle is a measure of a cone Consider the intersection of the given cone\nwith a sphere of radius R The solid angle DW of the cone is defined to be equal\nto DS/R\n2, where DS is the area on the sphere cut out by the cone"}, {"Chapter": "1", "sentence_range": "611-614", "Text": "*\nSolid angle is a measure of a cone Consider the intersection of the given cone\nwith a sphere of radius R The solid angle DW of the cone is defined to be equal\nto DS/R\n2, where DS is the area on the sphere cut out by the cone Rationalised 2023-24\nElectric Charges\nand Fields\n21\na curve drawn in such a way that the tangent to it at each\npoint is in the direction of the net field at that point"}, {"Chapter": "1", "sentence_range": "612-615", "Text": "Consider the intersection of the given cone\nwith a sphere of radius R The solid angle DW of the cone is defined to be equal\nto DS/R\n2, where DS is the area on the sphere cut out by the cone Rationalised 2023-24\nElectric Charges\nand Fields\n21\na curve drawn in such a way that the tangent to it at each\npoint is in the direction of the net field at that point An\narrow on the curve is obviously necessary to specify the\ndirection of electric field from the two possible directions\nindicated by a tangent to the curve"}, {"Chapter": "1", "sentence_range": "613-616", "Text": "The solid angle DW of the cone is defined to be equal\nto DS/R\n2, where DS is the area on the sphere cut out by the cone Rationalised 2023-24\nElectric Charges\nand Fields\n21\na curve drawn in such a way that the tangent to it at each\npoint is in the direction of the net field at that point An\narrow on the curve is obviously necessary to specify the\ndirection of electric field from the two possible directions\nindicated by a tangent to the curve A field line is a space\ncurve, i"}, {"Chapter": "1", "sentence_range": "614-617", "Text": "Rationalised 2023-24\nElectric Charges\nand Fields\n21\na curve drawn in such a way that the tangent to it at each\npoint is in the direction of the net field at that point An\narrow on the curve is obviously necessary to specify the\ndirection of electric field from the two possible directions\nindicated by a tangent to the curve A field line is a space\ncurve, i e"}, {"Chapter": "1", "sentence_range": "615-618", "Text": "An\narrow on the curve is obviously necessary to specify the\ndirection of electric field from the two possible directions\nindicated by a tangent to the curve A field line is a space\ncurve, i e , a curve in three dimensions"}, {"Chapter": "1", "sentence_range": "616-619", "Text": "A field line is a space\ncurve, i e , a curve in three dimensions Figure 1"}, {"Chapter": "1", "sentence_range": "617-620", "Text": "e , a curve in three dimensions Figure 1 14 shows the field lines around some simple\ncharge configurations"}, {"Chapter": "1", "sentence_range": "618-621", "Text": ", a curve in three dimensions Figure 1 14 shows the field lines around some simple\ncharge configurations As mentioned earlier, the field lines\nare in 3-dimensional space, though the figure shows them\nonly in a plane"}, {"Chapter": "1", "sentence_range": "619-622", "Text": "Figure 1 14 shows the field lines around some simple\ncharge configurations As mentioned earlier, the field lines\nare in 3-dimensional space, though the figure shows them\nonly in a plane The field lines of a single positive charge\nare radially outward while those of a single negative\ncharge are radially inward"}, {"Chapter": "1", "sentence_range": "620-623", "Text": "14 shows the field lines around some simple\ncharge configurations As mentioned earlier, the field lines\nare in 3-dimensional space, though the figure shows them\nonly in a plane The field lines of a single positive charge\nare radially outward while those of a single negative\ncharge are radially inward The field lines around a system\nof two positive charges (q, q) give a vivid pictorial\ndescription of their mutual repulsion, while those around\nthe configuration of two equal and opposite charges\n(q, \u2013q), a dipole, show clearly the mutual attraction\nbetween the charges"}, {"Chapter": "1", "sentence_range": "621-624", "Text": "As mentioned earlier, the field lines\nare in 3-dimensional space, though the figure shows them\nonly in a plane The field lines of a single positive charge\nare radially outward while those of a single negative\ncharge are radially inward The field lines around a system\nof two positive charges (q, q) give a vivid pictorial\ndescription of their mutual repulsion, while those around\nthe configuration of two equal and opposite charges\n(q, \u2013q), a dipole, show clearly the mutual attraction\nbetween the charges The field lines follow some important\ngeneral properties:\n(i)\nField lines start from positive charges and end at\nnegative charges"}, {"Chapter": "1", "sentence_range": "622-625", "Text": "The field lines of a single positive charge\nare radially outward while those of a single negative\ncharge are radially inward The field lines around a system\nof two positive charges (q, q) give a vivid pictorial\ndescription of their mutual repulsion, while those around\nthe configuration of two equal and opposite charges\n(q, \u2013q), a dipole, show clearly the mutual attraction\nbetween the charges The field lines follow some important\ngeneral properties:\n(i)\nField lines start from positive charges and end at\nnegative charges If there is a single charge, they may\nstart or end at infinity"}, {"Chapter": "1", "sentence_range": "623-626", "Text": "The field lines around a system\nof two positive charges (q, q) give a vivid pictorial\ndescription of their mutual repulsion, while those around\nthe configuration of two equal and opposite charges\n(q, \u2013q), a dipole, show clearly the mutual attraction\nbetween the charges The field lines follow some important\ngeneral properties:\n(i)\nField lines start from positive charges and end at\nnegative charges If there is a single charge, they may\nstart or end at infinity (ii) In a charge-free region, electric field lines can be taken\nto be continuous curves without any breaks"}, {"Chapter": "1", "sentence_range": "624-627", "Text": "The field lines follow some important\ngeneral properties:\n(i)\nField lines start from positive charges and end at\nnegative charges If there is a single charge, they may\nstart or end at infinity (ii) In a charge-free region, electric field lines can be taken\nto be continuous curves without any breaks (iii) Two field lines can never cross each other"}, {"Chapter": "1", "sentence_range": "625-628", "Text": "If there is a single charge, they may\nstart or end at infinity (ii) In a charge-free region, electric field lines can be taken\nto be continuous curves without any breaks (iii) Two field lines can never cross each other (If they did,\nthe field at the point of intersection will not have a\nunique direction, which is absurd"}, {"Chapter": "1", "sentence_range": "626-629", "Text": "(ii) In a charge-free region, electric field lines can be taken\nto be continuous curves without any breaks (iii) Two field lines can never cross each other (If they did,\nthe field at the point of intersection will not have a\nunique direction, which is absurd )\n(iv) Electrostatic field lines do not form any closed loops"}, {"Chapter": "1", "sentence_range": "627-630", "Text": "(iii) Two field lines can never cross each other (If they did,\nthe field at the point of intersection will not have a\nunique direction, which is absurd )\n(iv) Electrostatic field lines do not form any closed loops This follows from the conservative nature of electric\nfield (Chapter 2)"}, {"Chapter": "1", "sentence_range": "628-631", "Text": "(If they did,\nthe field at the point of intersection will not have a\nunique direction, which is absurd )\n(iv) Electrostatic field lines do not form any closed loops This follows from the conservative nature of electric\nfield (Chapter 2) 1"}, {"Chapter": "1", "sentence_range": "629-632", "Text": ")\n(iv) Electrostatic field lines do not form any closed loops This follows from the conservative nature of electric\nfield (Chapter 2) 1 9  ELECTRIC FLUX\nConsider flow of a liquid with velocity v, through a small\nflat surface dS, in a direction normal to the surface"}, {"Chapter": "1", "sentence_range": "630-633", "Text": "This follows from the conservative nature of electric\nfield (Chapter 2) 1 9  ELECTRIC FLUX\nConsider flow of a liquid with velocity v, through a small\nflat surface dS, in a direction normal to the surface The\nrate of flow of liquid is given by the volume crossing the\narea per unit time  v dS and represents the flux of liquid\nflowing across the plane"}, {"Chapter": "1", "sentence_range": "631-634", "Text": "1 9  ELECTRIC FLUX\nConsider flow of a liquid with velocity v, through a small\nflat surface dS, in a direction normal to the surface The\nrate of flow of liquid is given by the volume crossing the\narea per unit time  v dS and represents the flux of liquid\nflowing across the plane If the normal to the surface is\nnot parallel to the direction of flow of liquid, i"}, {"Chapter": "1", "sentence_range": "632-635", "Text": "9  ELECTRIC FLUX\nConsider flow of a liquid with velocity v, through a small\nflat surface dS, in a direction normal to the surface The\nrate of flow of liquid is given by the volume crossing the\narea per unit time  v dS and represents the flux of liquid\nflowing across the plane If the normal to the surface is\nnot parallel to the direction of flow of liquid, i e"}, {"Chapter": "1", "sentence_range": "633-636", "Text": "The\nrate of flow of liquid is given by the volume crossing the\narea per unit time  v dS and represents the flux of liquid\nflowing across the plane If the normal to the surface is\nnot parallel to the direction of flow of liquid, i e , to v, but\nmakes an angle q with it, the projected area in a plane\nperpendicular to v is \u03b4 dS cos q"}, {"Chapter": "1", "sentence_range": "634-637", "Text": "If the normal to the surface is\nnot parallel to the direction of flow of liquid, i e , to v, but\nmakes an angle q with it, the projected area in a plane\nperpendicular to v is \u03b4 dS cos q Therefore, the flux going\nout of the surface dS is v"}, {"Chapter": "1", "sentence_range": "635-638", "Text": "e , to v, but\nmakes an angle q with it, the projected area in a plane\nperpendicular to v is \u03b4 dS cos q Therefore, the flux going\nout of the surface dS is v \u02c6n dS"}, {"Chapter": "1", "sentence_range": "636-639", "Text": ", to v, but\nmakes an angle q with it, the projected area in a plane\nperpendicular to v is \u03b4 dS cos q Therefore, the flux going\nout of the surface dS is v \u02c6n dS For the case of  the electric\nfield, we define an analogous quantity and call it electric\nflux"}, {"Chapter": "1", "sentence_range": "637-640", "Text": "Therefore, the flux going\nout of the surface dS is v \u02c6n dS For the case of  the electric\nfield, we define an analogous quantity and call it electric\nflux We should, however, note that there is no flow of a\nphysically observable quantity unlike the case of\nliquid flow"}, {"Chapter": "1", "sentence_range": "638-641", "Text": "\u02c6n dS For the case of  the electric\nfield, we define an analogous quantity and call it electric\nflux We should, however, note that there is no flow of a\nphysically observable quantity unlike the case of\nliquid flow In the picture of electric field lines described above,\nwe saw that the number of field lines crossing a unit area,\nplaced normal to the field at a point is a measure of the\nstrength of electric field at that point"}, {"Chapter": "1", "sentence_range": "639-642", "Text": "For the case of  the electric\nfield, we define an analogous quantity and call it electric\nflux We should, however, note that there is no flow of a\nphysically observable quantity unlike the case of\nliquid flow In the picture of electric field lines described above,\nwe saw that the number of field lines crossing a unit area,\nplaced normal to the field at a point is a measure of the\nstrength of electric field at that point This means that if\nFIGURE 1"}, {"Chapter": "1", "sentence_range": "640-643", "Text": "We should, however, note that there is no flow of a\nphysically observable quantity unlike the case of\nliquid flow In the picture of electric field lines described above,\nwe saw that the number of field lines crossing a unit area,\nplaced normal to the field at a point is a measure of the\nstrength of electric field at that point This means that if\nFIGURE 1 14 Field lines due to\nsome simple charge configurations"}, {"Chapter": "1", "sentence_range": "641-644", "Text": "In the picture of electric field lines described above,\nwe saw that the number of field lines crossing a unit area,\nplaced normal to the field at a point is a measure of the\nstrength of electric field at that point This means that if\nFIGURE 1 14 Field lines due to\nsome simple charge configurations Rationalised 2023-24\n22\nPhysics\nwe place a small planar element of area DS\nnormal to E at a point, the number of field lines\ncrossing it is proportional* to E DS"}, {"Chapter": "1", "sentence_range": "642-645", "Text": "This means that if\nFIGURE 1 14 Field lines due to\nsome simple charge configurations Rationalised 2023-24\n22\nPhysics\nwe place a small planar element of area DS\nnormal to E at a point, the number of field lines\ncrossing it is proportional* to E DS Now\nsuppose we tilt the area element by angle q"}, {"Chapter": "1", "sentence_range": "643-646", "Text": "14 Field lines due to\nsome simple charge configurations Rationalised 2023-24\n22\nPhysics\nwe place a small planar element of area DS\nnormal to E at a point, the number of field lines\ncrossing it is proportional* to E DS Now\nsuppose we tilt the area element by angle q Clearly, the number of field lines crossing the\narea element will be smaller"}, {"Chapter": "1", "sentence_range": "644-647", "Text": "Rationalised 2023-24\n22\nPhysics\nwe place a small planar element of area DS\nnormal to E at a point, the number of field lines\ncrossing it is proportional* to E DS Now\nsuppose we tilt the area element by angle q Clearly, the number of field lines crossing the\narea element will be smaller The projection of\nthe area element normal to E is DS cosq"}, {"Chapter": "1", "sentence_range": "645-648", "Text": "Now\nsuppose we tilt the area element by angle q Clearly, the number of field lines crossing the\narea element will be smaller The projection of\nthe area element normal to E is DS cosq Thus,\nthe number of field lines crossing DS is\nproportional to E DS cosq"}, {"Chapter": "1", "sentence_range": "646-649", "Text": "Clearly, the number of field lines crossing the\narea element will be smaller The projection of\nthe area element normal to E is DS cosq Thus,\nthe number of field lines crossing DS is\nproportional to E DS cosq When q = 90\u00b0, field\nlines will be parallel to DS and will not cross it\nat all (Fig"}, {"Chapter": "1", "sentence_range": "647-650", "Text": "The projection of\nthe area element normal to E is DS cosq Thus,\nthe number of field lines crossing DS is\nproportional to E DS cosq When q = 90\u00b0, field\nlines will be parallel to DS and will not cross it\nat all (Fig 1"}, {"Chapter": "1", "sentence_range": "648-651", "Text": "Thus,\nthe number of field lines crossing DS is\nproportional to E DS cosq When q = 90\u00b0, field\nlines will be parallel to DS and will not cross it\nat all (Fig 1 15)"}, {"Chapter": "1", "sentence_range": "649-652", "Text": "When q = 90\u00b0, field\nlines will be parallel to DS and will not cross it\nat all (Fig 1 15) The orientation of area element and not\nmerely its magnitude is important in many\ncontexts"}, {"Chapter": "1", "sentence_range": "650-653", "Text": "1 15) The orientation of area element and not\nmerely its magnitude is important in many\ncontexts For example, in a stream, the amount\nof water flowing through a ring will naturally\ndepend on how you hold the ring"}, {"Chapter": "1", "sentence_range": "651-654", "Text": "15) The orientation of area element and not\nmerely its magnitude is important in many\ncontexts For example, in a stream, the amount\nof water flowing through a ring will naturally\ndepend on how you hold the ring If you hold\nit normal to the flow, maximum water will flow\nthrough it than if you hold it with some other\norientation"}, {"Chapter": "1", "sentence_range": "652-655", "Text": "The orientation of area element and not\nmerely its magnitude is important in many\ncontexts For example, in a stream, the amount\nof water flowing through a ring will naturally\ndepend on how you hold the ring If you hold\nit normal to the flow, maximum water will flow\nthrough it than if you hold it with some other\norientation This shows that an area element\nshould be treated as a vector"}, {"Chapter": "1", "sentence_range": "653-656", "Text": "For example, in a stream, the amount\nof water flowing through a ring will naturally\ndepend on how you hold the ring If you hold\nit normal to the flow, maximum water will flow\nthrough it than if you hold it with some other\norientation This shows that an area element\nshould be treated as a vector It has a\nmagnitude and also a direction"}, {"Chapter": "1", "sentence_range": "654-657", "Text": "If you hold\nit normal to the flow, maximum water will flow\nthrough it than if you hold it with some other\norientation This shows that an area element\nshould be treated as a vector It has a\nmagnitude and also a direction How to specify the direction of a planar\narea"}, {"Chapter": "1", "sentence_range": "655-658", "Text": "This shows that an area element\nshould be treated as a vector It has a\nmagnitude and also a direction How to specify the direction of a planar\narea Clearly, the normal to the plane specifies the orientation of the\nplane"}, {"Chapter": "1", "sentence_range": "656-659", "Text": "It has a\nmagnitude and also a direction How to specify the direction of a planar\narea Clearly, the normal to the plane specifies the orientation of the\nplane Thus the direction of a planar area vector is along its normal"}, {"Chapter": "1", "sentence_range": "657-660", "Text": "How to specify the direction of a planar\narea Clearly, the normal to the plane specifies the orientation of the\nplane Thus the direction of a planar area vector is along its normal How to associate a vector to the area of a curved surface"}, {"Chapter": "1", "sentence_range": "658-661", "Text": "Clearly, the normal to the plane specifies the orientation of the\nplane Thus the direction of a planar area vector is along its normal How to associate a vector to the area of a curved surface We imagine\ndividing the surface into a large number of very small area elements"}, {"Chapter": "1", "sentence_range": "659-662", "Text": "Thus the direction of a planar area vector is along its normal How to associate a vector to the area of a curved surface We imagine\ndividing the surface into a large number of very small area elements Each small area element may be treated as planar and a vector associated\nwith it, as explained before"}, {"Chapter": "1", "sentence_range": "660-663", "Text": "How to associate a vector to the area of a curved surface We imagine\ndividing the surface into a large number of very small area elements Each small area element may be treated as planar and a vector associated\nwith it, as explained before Notice one ambiguity here"}, {"Chapter": "1", "sentence_range": "661-664", "Text": "We imagine\ndividing the surface into a large number of very small area elements Each small area element may be treated as planar and a vector associated\nwith it, as explained before Notice one ambiguity here The direction of an area element is along\nits normal"}, {"Chapter": "1", "sentence_range": "662-665", "Text": "Each small area element may be treated as planar and a vector associated\nwith it, as explained before Notice one ambiguity here The direction of an area element is along\nits normal But a normal can point in two directions"}, {"Chapter": "1", "sentence_range": "663-666", "Text": "Notice one ambiguity here The direction of an area element is along\nits normal But a normal can point in two directions Which direction do\nwe choose as the direction of the vector associated with the area element"}, {"Chapter": "1", "sentence_range": "664-667", "Text": "The direction of an area element is along\nits normal But a normal can point in two directions Which direction do\nwe choose as the direction of the vector associated with the area element This problem is resolved by some convention appropriate to the given\ncontext"}, {"Chapter": "1", "sentence_range": "665-668", "Text": "But a normal can point in two directions Which direction do\nwe choose as the direction of the vector associated with the area element This problem is resolved by some convention appropriate to the given\ncontext For the case of a closed surface, this convention is very simple"}, {"Chapter": "1", "sentence_range": "666-669", "Text": "Which direction do\nwe choose as the direction of the vector associated with the area element This problem is resolved by some convention appropriate to the given\ncontext For the case of a closed surface, this convention is very simple The vector associated with every area element of a closed surface is taken\nto be in the direction of the outward normal"}, {"Chapter": "1", "sentence_range": "667-670", "Text": "This problem is resolved by some convention appropriate to the given\ncontext For the case of a closed surface, this convention is very simple The vector associated with every area element of a closed surface is taken\nto be in the direction of the outward normal This is the convention used\nin Fig"}, {"Chapter": "1", "sentence_range": "668-671", "Text": "For the case of a closed surface, this convention is very simple The vector associated with every area element of a closed surface is taken\nto be in the direction of the outward normal This is the convention used\nin Fig 1"}, {"Chapter": "1", "sentence_range": "669-672", "Text": "The vector associated with every area element of a closed surface is taken\nto be in the direction of the outward normal This is the convention used\nin Fig 1 16"}, {"Chapter": "1", "sentence_range": "670-673", "Text": "This is the convention used\nin Fig 1 16 Thus, the area element vector DS at a point on a closed\nsurface equals DS \u02c6n  where DS is the magnitude of the area element and\n\u02c6n  is a unit vector in the direction of outward normal at that point"}, {"Chapter": "1", "sentence_range": "671-674", "Text": "1 16 Thus, the area element vector DS at a point on a closed\nsurface equals DS \u02c6n  where DS is the magnitude of the area element and\n\u02c6n  is a unit vector in the direction of outward normal at that point We now come to the definition of electric flux"}, {"Chapter": "1", "sentence_range": "672-675", "Text": "16 Thus, the area element vector DS at a point on a closed\nsurface equals DS \u02c6n  where DS is the magnitude of the area element and\n\u02c6n  is a unit vector in the direction of outward normal at that point We now come to the definition of electric flux Electric flux Df through\nan area element DS is defined by\nDf = E"}, {"Chapter": "1", "sentence_range": "673-676", "Text": "Thus, the area element vector DS at a point on a closed\nsurface equals DS \u02c6n  where DS is the magnitude of the area element and\n\u02c6n  is a unit vector in the direction of outward normal at that point We now come to the definition of electric flux Electric flux Df through\nan area element DS is defined by\nDf = E DS = E DS  cosq\n(1"}, {"Chapter": "1", "sentence_range": "674-677", "Text": "We now come to the definition of electric flux Electric flux Df through\nan area element DS is defined by\nDf = E DS = E DS  cosq\n(1 11)\nwhich, as seen before, is proportional to the number of field lines cutting\nthe area element"}, {"Chapter": "1", "sentence_range": "675-678", "Text": "Electric flux Df through\nan area element DS is defined by\nDf = E DS = E DS  cosq\n(1 11)\nwhich, as seen before, is proportional to the number of field lines cutting\nthe area element The angle q here is the angle between E and DS"}, {"Chapter": "1", "sentence_range": "676-679", "Text": "DS = E DS  cosq\n(1 11)\nwhich, as seen before, is proportional to the number of field lines cutting\nthe area element The angle q here is the angle between E and DS For a\nclosed surface, with the convention stated already, q is the angle between\nE and the outward normal   to the area element"}, {"Chapter": "1", "sentence_range": "677-680", "Text": "11)\nwhich, as seen before, is proportional to the number of field lines cutting\nthe area element The angle q here is the angle between E and DS For a\nclosed surface, with the convention stated already, q is the angle between\nE and the outward normal   to the area element Notice we could look at\nthe expression E DS cosq  in two ways:  E (DS cosq ) i"}, {"Chapter": "1", "sentence_range": "678-681", "Text": "The angle q here is the angle between E and DS For a\nclosed surface, with the convention stated already, q is the angle between\nE and the outward normal   to the area element Notice we could look at\nthe expression E DS cosq  in two ways:  E (DS cosq ) i e"}, {"Chapter": "1", "sentence_range": "679-682", "Text": "For a\nclosed surface, with the convention stated already, q is the angle between\nE and the outward normal   to the area element Notice we could look at\nthe expression E DS cosq  in two ways:  E (DS cosq ) i e , E times the\nFIGURE 1"}, {"Chapter": "1", "sentence_range": "680-683", "Text": "Notice we could look at\nthe expression E DS cosq  in two ways:  E (DS cosq ) i e , E times the\nFIGURE 1 15 Dependence of flux on the\ninclination q between E and \u02c6n"}, {"Chapter": "1", "sentence_range": "681-684", "Text": "e , E times the\nFIGURE 1 15 Dependence of flux on the\ninclination q between E and \u02c6n FIGURE 1"}, {"Chapter": "1", "sentence_range": "682-685", "Text": ", E times the\nFIGURE 1 15 Dependence of flux on the\ninclination q between E and \u02c6n FIGURE 1 16\nConvention for\ndefining normal\n\u02c6n  and DS"}, {"Chapter": "1", "sentence_range": "683-686", "Text": "15 Dependence of flux on the\ninclination q between E and \u02c6n FIGURE 1 16\nConvention for\ndefining normal\n\u02c6n  and DS *\nIt will not be proper to say that the number of field lines is equal to EDS"}, {"Chapter": "1", "sentence_range": "684-687", "Text": "FIGURE 1 16\nConvention for\ndefining normal\n\u02c6n  and DS *\nIt will not be proper to say that the number of field lines is equal to EDS The\nnumber of field lines is after all, a matter of how many field lines we choose to\ndraw"}, {"Chapter": "1", "sentence_range": "685-688", "Text": "16\nConvention for\ndefining normal\n\u02c6n  and DS *\nIt will not be proper to say that the number of field lines is equal to EDS The\nnumber of field lines is after all, a matter of how many field lines we choose to\ndraw What is physically significant is the relative number of field lines crossing\na given area at different points"}, {"Chapter": "1", "sentence_range": "686-689", "Text": "*\nIt will not be proper to say that the number of field lines is equal to EDS The\nnumber of field lines is after all, a matter of how many field lines we choose to\ndraw What is physically significant is the relative number of field lines crossing\na given area at different points Rationalised 2023-24\nElectric Charges\nand Fields\n23\nprojection of area normal to E,  or E^ DS, i"}, {"Chapter": "1", "sentence_range": "687-690", "Text": "The\nnumber of field lines is after all, a matter of how many field lines we choose to\ndraw What is physically significant is the relative number of field lines crossing\na given area at different points Rationalised 2023-24\nElectric Charges\nand Fields\n23\nprojection of area normal to E,  or E^ DS, i e"}, {"Chapter": "1", "sentence_range": "688-691", "Text": "What is physically significant is the relative number of field lines crossing\na given area at different points Rationalised 2023-24\nElectric Charges\nand Fields\n23\nprojection of area normal to E,  or E^ DS, i e , component of E along the\nnormal to the area element times the magnitude of the area element"}, {"Chapter": "1", "sentence_range": "689-692", "Text": "Rationalised 2023-24\nElectric Charges\nand Fields\n23\nprojection of area normal to E,  or E^ DS, i e , component of E along the\nnormal to the area element times the magnitude of the area element The\nunit of electric flux is N C\u20131 m2"}, {"Chapter": "1", "sentence_range": "690-693", "Text": "e , component of E along the\nnormal to the area element times the magnitude of the area element The\nunit of electric flux is N C\u20131 m2 The basic definition of electric flux given by Eq"}, {"Chapter": "1", "sentence_range": "691-694", "Text": ", component of E along the\nnormal to the area element times the magnitude of the area element The\nunit of electric flux is N C\u20131 m2 The basic definition of electric flux given by Eq (1"}, {"Chapter": "1", "sentence_range": "692-695", "Text": "The\nunit of electric flux is N C\u20131 m2 The basic definition of electric flux given by Eq (1 11) can be used, in\nprinciple, to calculate the total flux through any given surface"}, {"Chapter": "1", "sentence_range": "693-696", "Text": "The basic definition of electric flux given by Eq (1 11) can be used, in\nprinciple, to calculate the total flux through any given surface All we\nhave to do is to divide the surface into small area elements, calculate the\nflux at each element and add them up"}, {"Chapter": "1", "sentence_range": "694-697", "Text": "(1 11) can be used, in\nprinciple, to calculate the total flux through any given surface All we\nhave to do is to divide the surface into small area elements, calculate the\nflux at each element and add them up Thus, the total flux f through a\nsurface S is\nf ~ S E"}, {"Chapter": "1", "sentence_range": "695-698", "Text": "11) can be used, in\nprinciple, to calculate the total flux through any given surface All we\nhave to do is to divide the surface into small area elements, calculate the\nflux at each element and add them up Thus, the total flux f through a\nsurface S is\nf ~ S E DS\n(1"}, {"Chapter": "1", "sentence_range": "696-699", "Text": "All we\nhave to do is to divide the surface into small area elements, calculate the\nflux at each element and add them up Thus, the total flux f through a\nsurface S is\nf ~ S E DS\n(1 12)\nThe approximation sign is put because the electric field E is taken to\nbe constant over the small area element"}, {"Chapter": "1", "sentence_range": "697-700", "Text": "Thus, the total flux f through a\nsurface S is\nf ~ S E DS\n(1 12)\nThe approximation sign is put because the electric field E is taken to\nbe constant over the small area element This is mathematically exact\nonly when you take the limit DS \u00ae 0  and the sum in Eq"}, {"Chapter": "1", "sentence_range": "698-701", "Text": "DS\n(1 12)\nThe approximation sign is put because the electric field E is taken to\nbe constant over the small area element This is mathematically exact\nonly when you take the limit DS \u00ae 0  and the sum in Eq (1"}, {"Chapter": "1", "sentence_range": "699-702", "Text": "12)\nThe approximation sign is put because the electric field E is taken to\nbe constant over the small area element This is mathematically exact\nonly when you take the limit DS \u00ae 0  and the sum in Eq (1 12) is written\nas an integral"}, {"Chapter": "1", "sentence_range": "700-703", "Text": "This is mathematically exact\nonly when you take the limit DS \u00ae 0  and the sum in Eq (1 12) is written\nas an integral 1"}, {"Chapter": "1", "sentence_range": "701-704", "Text": "(1 12) is written\nas an integral 1 10  ELECTRIC DIPOLE\nAn electric dipole is a pair of equal and opposite point charges q and \u2013q,\nseparated by a distance 2a"}, {"Chapter": "1", "sentence_range": "702-705", "Text": "12) is written\nas an integral 1 10  ELECTRIC DIPOLE\nAn electric dipole is a pair of equal and opposite point charges q and \u2013q,\nseparated by a distance 2a The line connecting the two charges defines\na direction in space"}, {"Chapter": "1", "sentence_range": "703-706", "Text": "1 10  ELECTRIC DIPOLE\nAn electric dipole is a pair of equal and opposite point charges q and \u2013q,\nseparated by a distance 2a The line connecting the two charges defines\na direction in space By convention, the direction from \u2013q to q is said to\nbe the direction of the dipole"}, {"Chapter": "1", "sentence_range": "704-707", "Text": "10  ELECTRIC DIPOLE\nAn electric dipole is a pair of equal and opposite point charges q and \u2013q,\nseparated by a distance 2a The line connecting the two charges defines\na direction in space By convention, the direction from \u2013q to q is said to\nbe the direction of the dipole The mid-point of locations of \u2013q and q is\ncalled the centre of the dipole"}, {"Chapter": "1", "sentence_range": "705-708", "Text": "The line connecting the two charges defines\na direction in space By convention, the direction from \u2013q to q is said to\nbe the direction of the dipole The mid-point of locations of \u2013q and q is\ncalled the centre of the dipole The total charge of the electric dipole is obviously zero"}, {"Chapter": "1", "sentence_range": "706-709", "Text": "By convention, the direction from \u2013q to q is said to\nbe the direction of the dipole The mid-point of locations of \u2013q and q is\ncalled the centre of the dipole The total charge of the electric dipole is obviously zero This does not\nmean that the field of the electric dipole is zero"}, {"Chapter": "1", "sentence_range": "707-710", "Text": "The mid-point of locations of \u2013q and q is\ncalled the centre of the dipole The total charge of the electric dipole is obviously zero This does not\nmean that the field of the electric dipole is zero Since the charge q and\n\u2013q are separated by some distance, the electric fields due to them, when\nadded, do not exactly cancel out"}, {"Chapter": "1", "sentence_range": "708-711", "Text": "The total charge of the electric dipole is obviously zero This does not\nmean that the field of the electric dipole is zero Since the charge q and\n\u2013q are separated by some distance, the electric fields due to them, when\nadded, do not exactly cancel out However, at distances much larger than\nthe separation of the two charges forming a dipole (r >> 2a), the fields\ndue to q and \u2013q nearly cancel out"}, {"Chapter": "1", "sentence_range": "709-712", "Text": "This does not\nmean that the field of the electric dipole is zero Since the charge q and\n\u2013q are separated by some distance, the electric fields due to them, when\nadded, do not exactly cancel out However, at distances much larger than\nthe separation of the two charges forming a dipole (r >> 2a), the fields\ndue to q and \u2013q nearly cancel out The electric field due to a dipole\ntherefore falls off, at large distance, faster than like 1/r 2 (the dependence\non r of the field due to a single charge q)"}, {"Chapter": "1", "sentence_range": "710-713", "Text": "Since the charge q and\n\u2013q are separated by some distance, the electric fields due to them, when\nadded, do not exactly cancel out However, at distances much larger than\nthe separation of the two charges forming a dipole (r >> 2a), the fields\ndue to q and \u2013q nearly cancel out The electric field due to a dipole\ntherefore falls off, at large distance, faster than like 1/r 2 (the dependence\non r of the field due to a single charge q) These qualitative ideas are\nborne out by the explicit calculation as follows:\n1"}, {"Chapter": "1", "sentence_range": "711-714", "Text": "However, at distances much larger than\nthe separation of the two charges forming a dipole (r >> 2a), the fields\ndue to q and \u2013q nearly cancel out The electric field due to a dipole\ntherefore falls off, at large distance, faster than like 1/r 2 (the dependence\non r of the field due to a single charge q) These qualitative ideas are\nborne out by the explicit calculation as follows:\n1 10"}, {"Chapter": "1", "sentence_range": "712-715", "Text": "The electric field due to a dipole\ntherefore falls off, at large distance, faster than like 1/r 2 (the dependence\non r of the field due to a single charge q) These qualitative ideas are\nborne out by the explicit calculation as follows:\n1 10 1  The field of an electric dipole\nThe electric field of the pair of charges (\u2013q and q) at any point in space\ncan be found out from Coulomb\u2019s law and the superposition principle"}, {"Chapter": "1", "sentence_range": "713-716", "Text": "These qualitative ideas are\nborne out by the explicit calculation as follows:\n1 10 1  The field of an electric dipole\nThe electric field of the pair of charges (\u2013q and q) at any point in space\ncan be found out from Coulomb\u2019s law and the superposition principle The results are simple for the following two cases: (i) when the point is on\nthe dipole axis, and (ii) when it is in the equatorial plane of the dipole,\ni"}, {"Chapter": "1", "sentence_range": "714-717", "Text": "10 1  The field of an electric dipole\nThe electric field of the pair of charges (\u2013q and q) at any point in space\ncan be found out from Coulomb\u2019s law and the superposition principle The results are simple for the following two cases: (i) when the point is on\nthe dipole axis, and (ii) when it is in the equatorial plane of the dipole,\ni e"}, {"Chapter": "1", "sentence_range": "715-718", "Text": "1  The field of an electric dipole\nThe electric field of the pair of charges (\u2013q and q) at any point in space\ncan be found out from Coulomb\u2019s law and the superposition principle The results are simple for the following two cases: (i) when the point is on\nthe dipole axis, and (ii) when it is in the equatorial plane of the dipole,\ni e ,  on a plane perpendicular to the dipole axis through its centre"}, {"Chapter": "1", "sentence_range": "716-719", "Text": "The results are simple for the following two cases: (i) when the point is on\nthe dipole axis, and (ii) when it is in the equatorial plane of the dipole,\ni e ,  on a plane perpendicular to the dipole axis through its centre The\nelectric field at any general point P is obtained by adding the electric\nfields E\u2013q due to the charge \u2013q and E+q due to the charge q, by the\nparallelogram law of vectors"}, {"Chapter": "1", "sentence_range": "717-720", "Text": "e ,  on a plane perpendicular to the dipole axis through its centre The\nelectric field at any general point P is obtained by adding the electric\nfields E\u2013q due to the charge \u2013q and E+q due to the charge q, by the\nparallelogram law of vectors (i) For points on the axis\nLet the point P be at distance r from the centre of the dipole on the side of\nthe charge q,  as shown in Fig"}, {"Chapter": "1", "sentence_range": "718-721", "Text": ",  on a plane perpendicular to the dipole axis through its centre The\nelectric field at any general point P is obtained by adding the electric\nfields E\u2013q due to the charge \u2013q and E+q due to the charge q, by the\nparallelogram law of vectors (i) For points on the axis\nLet the point P be at distance r from the centre of the dipole on the side of\nthe charge q,  as shown in Fig 1"}, {"Chapter": "1", "sentence_range": "719-722", "Text": "The\nelectric field at any general point P is obtained by adding the electric\nfields E\u2013q due to the charge \u2013q and E+q due to the charge q, by the\nparallelogram law of vectors (i) For points on the axis\nLet the point P be at distance r from the centre of the dipole on the side of\nthe charge q,  as shown in Fig 1 17(a)"}, {"Chapter": "1", "sentence_range": "720-723", "Text": "(i) For points on the axis\nLet the point P be at distance r from the centre of the dipole on the side of\nthe charge q,  as shown in Fig 1 17(a) Then\nE\np\n\u2212\n= \u2212\n+\nq\nrq\na\n4\n0\n2\n\u03c0\u03b5 (\n)\n\ufffd\n[1"}, {"Chapter": "1", "sentence_range": "721-724", "Text": "1 17(a) Then\nE\np\n\u2212\n= \u2212\n+\nq\nrq\na\n4\n0\n2\n\u03c0\u03b5 (\n)\n\ufffd\n[1 13(a)]\nwhere \u02c6p  is the unit vector along the dipole axis (from \u2013q to q)"}, {"Chapter": "1", "sentence_range": "722-725", "Text": "17(a) Then\nE\np\n\u2212\n= \u2212\n+\nq\nrq\na\n4\n0\n2\n\u03c0\u03b5 (\n)\n\ufffd\n[1 13(a)]\nwhere \u02c6p  is the unit vector along the dipole axis (from \u2013q to q) Also\nE\np\n+\n=\n\u2212\nq\nrq\na\n4\n0\n2\n\u03c0 \u03b5 (\n)\n\ufffd\n[1"}, {"Chapter": "1", "sentence_range": "723-726", "Text": "Then\nE\np\n\u2212\n= \u2212\n+\nq\nrq\na\n4\n0\n2\n\u03c0\u03b5 (\n)\n\ufffd\n[1 13(a)]\nwhere \u02c6p  is the unit vector along the dipole axis (from \u2013q to q) Also\nE\np\n+\n=\n\u2212\nq\nrq\na\n4\n0\n2\n\u03c0 \u03b5 (\n)\n\ufffd\n[1 13(b)]\nRationalised 2023-24\n24\nPhysics\nThe total field at P is\nE\nE\nE\np\n=\n+\n=\n\u2212\n\u2212\n+\n\uf8ee\n\uf8ef\uf8f0\n\uf8f9\n\uf8fa\uf8fb\n+\n\u2212\nq\nq\nq\nr\na\nr\na\n4\n1\n1\n0\n2\n2\n\u03c0 \u03b5\n(\n)\n(\n)\n\ufffd\n    =\n\u2212\nq\na r\nr\na\no\n4\n4\n2\n2 2\n\u03c0 \u03b5\n(\n)\n\ufffdp\n(1"}, {"Chapter": "1", "sentence_range": "724-727", "Text": "13(a)]\nwhere \u02c6p  is the unit vector along the dipole axis (from \u2013q to q) Also\nE\np\n+\n=\n\u2212\nq\nrq\na\n4\n0\n2\n\u03c0 \u03b5 (\n)\n\ufffd\n[1 13(b)]\nRationalised 2023-24\n24\nPhysics\nThe total field at P is\nE\nE\nE\np\n=\n+\n=\n\u2212\n\u2212\n+\n\uf8ee\n\uf8ef\uf8f0\n\uf8f9\n\uf8fa\uf8fb\n+\n\u2212\nq\nq\nq\nr\na\nr\na\n4\n1\n1\n0\n2\n2\n\u03c0 \u03b5\n(\n)\n(\n)\n\ufffd\n    =\n\u2212\nq\na r\nr\na\no\n4\n4\n2\n2 2\n\u03c0 \u03b5\n(\n)\n\ufffdp\n(1 14)\nFor r >> a\nE\np\n=\n44\n0\nq a3\n\u03c0\u03b5r\n\u02c6                (r >> a)\n(1"}, {"Chapter": "1", "sentence_range": "725-728", "Text": "Also\nE\np\n+\n=\n\u2212\nq\nrq\na\n4\n0\n2\n\u03c0 \u03b5 (\n)\n\ufffd\n[1 13(b)]\nRationalised 2023-24\n24\nPhysics\nThe total field at P is\nE\nE\nE\np\n=\n+\n=\n\u2212\n\u2212\n+\n\uf8ee\n\uf8ef\uf8f0\n\uf8f9\n\uf8fa\uf8fb\n+\n\u2212\nq\nq\nq\nr\na\nr\na\n4\n1\n1\n0\n2\n2\n\u03c0 \u03b5\n(\n)\n(\n)\n\ufffd\n    =\n\u2212\nq\na r\nr\na\no\n4\n4\n2\n2 2\n\u03c0 \u03b5\n(\n)\n\ufffdp\n(1 14)\nFor r >> a\nE\np\n=\n44\n0\nq a3\n\u03c0\u03b5r\n\u02c6                (r >> a)\n(1 15)\n(ii) For points on the equatorial plane\nThe magnitudes of the electric fields due to the two\ncharges +q and \u2013q are given by\nE\nq\nr\na\n+q\n=\n+\n4\n1\n0\n2\n2\n\u03c0\u03b5\n[1"}, {"Chapter": "1", "sentence_range": "726-729", "Text": "13(b)]\nRationalised 2023-24\n24\nPhysics\nThe total field at P is\nE\nE\nE\np\n=\n+\n=\n\u2212\n\u2212\n+\n\uf8ee\n\uf8ef\uf8f0\n\uf8f9\n\uf8fa\uf8fb\n+\n\u2212\nq\nq\nq\nr\na\nr\na\n4\n1\n1\n0\n2\n2\n\u03c0 \u03b5\n(\n)\n(\n)\n\ufffd\n    =\n\u2212\nq\na r\nr\na\no\n4\n4\n2\n2 2\n\u03c0 \u03b5\n(\n)\n\ufffdp\n(1 14)\nFor r >> a\nE\np\n=\n44\n0\nq a3\n\u03c0\u03b5r\n\u02c6                (r >> a)\n(1 15)\n(ii) For points on the equatorial plane\nThe magnitudes of the electric fields due to the two\ncharges +q and \u2013q are given by\nE\nq\nr\na\n+q\n=\n+\n4\n1\n0\n2\n2\n\u03c0\u03b5\n[1 16(a)]\nE\nq\nr\na\n\u2013q\n=\n+\n4\n1\n0\n2\n2\n\u03c0\u03b5\n[1"}, {"Chapter": "1", "sentence_range": "727-730", "Text": "14)\nFor r >> a\nE\np\n=\n44\n0\nq a3\n\u03c0\u03b5r\n\u02c6                (r >> a)\n(1 15)\n(ii) For points on the equatorial plane\nThe magnitudes of the electric fields due to the two\ncharges +q and \u2013q are given by\nE\nq\nr\na\n+q\n=\n+\n4\n1\n0\n2\n2\n\u03c0\u03b5\n[1 16(a)]\nE\nq\nr\na\n\u2013q\n=\n+\n4\n1\n0\n2\n2\n\u03c0\u03b5\n[1 16(b)]\nand are equal"}, {"Chapter": "1", "sentence_range": "728-731", "Text": "15)\n(ii) For points on the equatorial plane\nThe magnitudes of the electric fields due to the two\ncharges +q and \u2013q are given by\nE\nq\nr\na\n+q\n=\n+\n4\n1\n0\n2\n2\n\u03c0\u03b5\n[1 16(a)]\nE\nq\nr\na\n\u2013q\n=\n+\n4\n1\n0\n2\n2\n\u03c0\u03b5\n[1 16(b)]\nand are equal The directions of  E+q and E\u2013q are as shown in\nFig"}, {"Chapter": "1", "sentence_range": "729-732", "Text": "16(a)]\nE\nq\nr\na\n\u2013q\n=\n+\n4\n1\n0\n2\n2\n\u03c0\u03b5\n[1 16(b)]\nand are equal The directions of  E+q and E\u2013q are as shown in\nFig 1"}, {"Chapter": "1", "sentence_range": "730-733", "Text": "16(b)]\nand are equal The directions of  E+q and E\u2013q are as shown in\nFig 1 17(b)"}, {"Chapter": "1", "sentence_range": "731-734", "Text": "The directions of  E+q and E\u2013q are as shown in\nFig 1 17(b) Clearly, the components normal to the dipole\naxis cancel away"}, {"Chapter": "1", "sentence_range": "732-735", "Text": "1 17(b) Clearly, the components normal to the dipole\naxis cancel away The components along the dipole axis\nadd up"}, {"Chapter": "1", "sentence_range": "733-736", "Text": "17(b) Clearly, the components normal to the dipole\naxis cancel away The components along the dipole axis\nadd up The total electric field is opposite to \u02c6p"}, {"Chapter": "1", "sentence_range": "734-737", "Text": "Clearly, the components normal to the dipole\naxis cancel away The components along the dipole axis\nadd up The total electric field is opposite to \u02c6p We have\nE = \u2013 (E +q + E \u2013q ) cosq  \u02c6p\n= \u2212\n2+\n4\n2\n2 3\n2\nrq a\na\no\n\u03c0 \u03b5 (\n)\n\ufffd\n/\np\n(1"}, {"Chapter": "1", "sentence_range": "735-738", "Text": "The components along the dipole axis\nadd up The total electric field is opposite to \u02c6p We have\nE = \u2013 (E +q + E \u2013q ) cosq  \u02c6p\n= \u2212\n2+\n4\n2\n2 3\n2\nrq a\na\no\n\u03c0 \u03b5 (\n)\n\ufffd\n/\np\n(1 17)\nAt large distances (r >> a), this reduces to\nE\np\n= \u2212\n>>\n42\nq a3\nr\nr\na\n\u03c0 \u03b5o\n\u02c6\n(\n)\n(1"}, {"Chapter": "1", "sentence_range": "736-739", "Text": "The total electric field is opposite to \u02c6p We have\nE = \u2013 (E +q + E \u2013q ) cosq  \u02c6p\n= \u2212\n2+\n4\n2\n2 3\n2\nrq a\na\no\n\u03c0 \u03b5 (\n)\n\ufffd\n/\np\n(1 17)\nAt large distances (r >> a), this reduces to\nE\np\n= \u2212\n>>\n42\nq a3\nr\nr\na\n\u03c0 \u03b5o\n\u02c6\n(\n)\n(1 18)\nFrom Eqs"}, {"Chapter": "1", "sentence_range": "737-740", "Text": "We have\nE = \u2013 (E +q + E \u2013q ) cosq  \u02c6p\n= \u2212\n2+\n4\n2\n2 3\n2\nrq a\na\no\n\u03c0 \u03b5 (\n)\n\ufffd\n/\np\n(1 17)\nAt large distances (r >> a), this reduces to\nE\np\n= \u2212\n>>\n42\nq a3\nr\nr\na\n\u03c0 \u03b5o\n\u02c6\n(\n)\n(1 18)\nFrom Eqs (1"}, {"Chapter": "1", "sentence_range": "738-741", "Text": "17)\nAt large distances (r >> a), this reduces to\nE\np\n= \u2212\n>>\n42\nq a3\nr\nr\na\n\u03c0 \u03b5o\n\u02c6\n(\n)\n(1 18)\nFrom Eqs (1 15) and (1"}, {"Chapter": "1", "sentence_range": "739-742", "Text": "18)\nFrom Eqs (1 15) and (1 18), it is clear that the dipole field at large\ndistances does not involve q and a separately; it depends on the product\nqa"}, {"Chapter": "1", "sentence_range": "740-743", "Text": "(1 15) and (1 18), it is clear that the dipole field at large\ndistances does not involve q and a separately; it depends on the product\nqa This suggests the definition of dipole moment"}, {"Chapter": "1", "sentence_range": "741-744", "Text": "15) and (1 18), it is clear that the dipole field at large\ndistances does not involve q and a separately; it depends on the product\nqa This suggests the definition of dipole moment The dipole moment\nvector p of an electric dipole is defined by\np  = q \u00d7 2a \u02c6p\n(1"}, {"Chapter": "1", "sentence_range": "742-745", "Text": "18), it is clear that the dipole field at large\ndistances does not involve q and a separately; it depends on the product\nqa This suggests the definition of dipole moment The dipole moment\nvector p of an electric dipole is defined by\np  = q \u00d7 2a \u02c6p\n(1 19)\nthat is, it is a vector whose magnitude is charge q times the separation\n2a (between the pair of charges q, \u2013q) and the direction is along the line\nfrom \u2013q to q"}, {"Chapter": "1", "sentence_range": "743-746", "Text": "This suggests the definition of dipole moment The dipole moment\nvector p of an electric dipole is defined by\np  = q \u00d7 2a \u02c6p\n(1 19)\nthat is, it is a vector whose magnitude is charge q times the separation\n2a (between the pair of charges q, \u2013q) and the direction is along the line\nfrom \u2013q to q In terms of p, the electric field of a dipole at large distances\ntakes simple forms:\nAt a point on the dipole axis\nE\np\n=\n42\n\u03c0\u03b5or3\n(r >> a)\n(1"}, {"Chapter": "1", "sentence_range": "744-747", "Text": "The dipole moment\nvector p of an electric dipole is defined by\np  = q \u00d7 2a \u02c6p\n(1 19)\nthat is, it is a vector whose magnitude is charge q times the separation\n2a (between the pair of charges q, \u2013q) and the direction is along the line\nfrom \u2013q to q In terms of p, the electric field of a dipole at large distances\ntakes simple forms:\nAt a point on the dipole axis\nE\np\n=\n42\n\u03c0\u03b5or3\n(r >> a)\n(1 20)\nAt a point on the equatorial plane\nE\n= \u2212 4p\n\u03c0\u03b5or3\n(r >> a)\n(1"}, {"Chapter": "1", "sentence_range": "745-748", "Text": "19)\nthat is, it is a vector whose magnitude is charge q times the separation\n2a (between the pair of charges q, \u2013q) and the direction is along the line\nfrom \u2013q to q In terms of p, the electric field of a dipole at large distances\ntakes simple forms:\nAt a point on the dipole axis\nE\np\n=\n42\n\u03c0\u03b5or3\n(r >> a)\n(1 20)\nAt a point on the equatorial plane\nE\n= \u2212 4p\n\u03c0\u03b5or3\n(r >> a)\n(1 21)\nFIGURE 1"}, {"Chapter": "1", "sentence_range": "746-749", "Text": "In terms of p, the electric field of a dipole at large distances\ntakes simple forms:\nAt a point on the dipole axis\nE\np\n=\n42\n\u03c0\u03b5or3\n(r >> a)\n(1 20)\nAt a point on the equatorial plane\nE\n= \u2212 4p\n\u03c0\u03b5or3\n(r >> a)\n(1 21)\nFIGURE 1 17 Electric field of a dipole\nat (a) a point on the axis, (b) a point\non the equatorial plane of the dipole"}, {"Chapter": "1", "sentence_range": "747-750", "Text": "20)\nAt a point on the equatorial plane\nE\n= \u2212 4p\n\u03c0\u03b5or3\n(r >> a)\n(1 21)\nFIGURE 1 17 Electric field of a dipole\nat (a) a point on the axis, (b) a point\non the equatorial plane of the dipole p is the dipole moment vector of\nmagnitude p = q \u00d7 2a  and\ndirected from \u2013q to q"}, {"Chapter": "1", "sentence_range": "748-751", "Text": "21)\nFIGURE 1 17 Electric field of a dipole\nat (a) a point on the axis, (b) a point\non the equatorial plane of the dipole p is the dipole moment vector of\nmagnitude p = q \u00d7 2a  and\ndirected from \u2013q to q Rationalised 2023-24\nElectric Charges\nand Fields\n25\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "749-752", "Text": "17 Electric field of a dipole\nat (a) a point on the axis, (b) a point\non the equatorial plane of the dipole p is the dipole moment vector of\nmagnitude p = q \u00d7 2a  and\ndirected from \u2013q to q Rationalised 2023-24\nElectric Charges\nand Fields\n25\n EXAMPLE 1 9\n Notice the important point that the dipole field at large distances\nfalls off not as 1/r 2 but as1/r 3"}, {"Chapter": "1", "sentence_range": "750-753", "Text": "p is the dipole moment vector of\nmagnitude p = q \u00d7 2a  and\ndirected from \u2013q to q Rationalised 2023-24\nElectric Charges\nand Fields\n25\n EXAMPLE 1 9\n Notice the important point that the dipole field at large distances\nfalls off not as 1/r 2 but as1/r 3 Further,  the magnitude and the direction\nof the dipole field depends not only on the distance r but also on the\nangle between the position vector r and the dipole moment p"}, {"Chapter": "1", "sentence_range": "751-754", "Text": "Rationalised 2023-24\nElectric Charges\nand Fields\n25\n EXAMPLE 1 9\n Notice the important point that the dipole field at large distances\nfalls off not as 1/r 2 but as1/r 3 Further,  the magnitude and the direction\nof the dipole field depends not only on the distance r but also on the\nangle between the position vector r and the dipole moment p We can think of the limit when the dipole size 2a approaches zero,\nthe charge q approaches infinity in such a way that the product\np = q \u00d7 2a is finite"}, {"Chapter": "1", "sentence_range": "752-755", "Text": "9\n Notice the important point that the dipole field at large distances\nfalls off not as 1/r 2 but as1/r 3 Further,  the magnitude and the direction\nof the dipole field depends not only on the distance r but also on the\nangle between the position vector r and the dipole moment p We can think of the limit when the dipole size 2a approaches zero,\nthe charge q approaches infinity in such a way that the product\np = q \u00d7 2a is finite Such a dipole is referred to as a point dipole"}, {"Chapter": "1", "sentence_range": "753-756", "Text": "Further,  the magnitude and the direction\nof the dipole field depends not only on the distance r but also on the\nangle between the position vector r and the dipole moment p We can think of the limit when the dipole size 2a approaches zero,\nthe charge q approaches infinity in such a way that the product\np = q \u00d7 2a is finite Such a dipole is referred to as a point dipole For a\npoint dipole, Eqs"}, {"Chapter": "1", "sentence_range": "754-757", "Text": "We can think of the limit when the dipole size 2a approaches zero,\nthe charge q approaches infinity in such a way that the product\np = q \u00d7 2a is finite Such a dipole is referred to as a point dipole For a\npoint dipole, Eqs (1"}, {"Chapter": "1", "sentence_range": "755-758", "Text": "Such a dipole is referred to as a point dipole For a\npoint dipole, Eqs (1 20) and (1"}, {"Chapter": "1", "sentence_range": "756-759", "Text": "For a\npoint dipole, Eqs (1 20) and (1 21) are exact, true for any r"}, {"Chapter": "1", "sentence_range": "757-760", "Text": "(1 20) and (1 21) are exact, true for any r 1"}, {"Chapter": "1", "sentence_range": "758-761", "Text": "20) and (1 21) are exact, true for any r 1 10"}, {"Chapter": "1", "sentence_range": "759-762", "Text": "21) are exact, true for any r 1 10 2  Physical significance of dipoles\nIn most molecules, the centres of positive charges and of negative charges*\nlie at the same place"}, {"Chapter": "1", "sentence_range": "760-763", "Text": "1 10 2  Physical significance of dipoles\nIn most molecules, the centres of positive charges and of negative charges*\nlie at the same place Therefore, their dipole moment is zero"}, {"Chapter": "1", "sentence_range": "761-764", "Text": "10 2  Physical significance of dipoles\nIn most molecules, the centres of positive charges and of negative charges*\nlie at the same place Therefore, their dipole moment is zero CO2 and\nCH4 are of this type of molecules"}, {"Chapter": "1", "sentence_range": "762-765", "Text": "2  Physical significance of dipoles\nIn most molecules, the centres of positive charges and of negative charges*\nlie at the same place Therefore, their dipole moment is zero CO2 and\nCH4 are of this type of molecules However, they develop a dipole moment\nwhen an electric field is applied"}, {"Chapter": "1", "sentence_range": "763-766", "Text": "Therefore, their dipole moment is zero CO2 and\nCH4 are of this type of molecules However, they develop a dipole moment\nwhen an electric field is applied But in some molecules, the centres of\nnegative charges and of positive charges do not coincide"}, {"Chapter": "1", "sentence_range": "764-767", "Text": "CO2 and\nCH4 are of this type of molecules However, they develop a dipole moment\nwhen an electric field is applied But in some molecules, the centres of\nnegative charges and of positive charges do not coincide Therefore they\nhave a permanent electric dipole moment, even in the absence of an electric\nfield"}, {"Chapter": "1", "sentence_range": "765-768", "Text": "However, they develop a dipole moment\nwhen an electric field is applied But in some molecules, the centres of\nnegative charges and of positive charges do not coincide Therefore they\nhave a permanent electric dipole moment, even in the absence of an electric\nfield Such molecules are called polar molecules"}, {"Chapter": "1", "sentence_range": "766-769", "Text": "But in some molecules, the centres of\nnegative charges and of positive charges do not coincide Therefore they\nhave a permanent electric dipole moment, even in the absence of an electric\nfield Such molecules are called polar molecules Water molecules, H2O,\nis an example of this type"}, {"Chapter": "1", "sentence_range": "767-770", "Text": "Therefore they\nhave a permanent electric dipole moment, even in the absence of an electric\nfield Such molecules are called polar molecules Water molecules, H2O,\nis an example of this type Various materials give rise to interesting\nproperties and important applications in the presence or  absence of\nelectric field"}, {"Chapter": "1", "sentence_range": "768-771", "Text": "Such molecules are called polar molecules Water molecules, H2O,\nis an example of this type Various materials give rise to interesting\nproperties and important applications in the presence or  absence of\nelectric field Example 1"}, {"Chapter": "1", "sentence_range": "769-772", "Text": "Water molecules, H2O,\nis an example of this type Various materials give rise to interesting\nproperties and important applications in the presence or  absence of\nelectric field Example 1 9 Two charges \u00b110 mC are placed 5"}, {"Chapter": "1", "sentence_range": "770-773", "Text": "Various materials give rise to interesting\nproperties and important applications in the presence or  absence of\nelectric field Example 1 9 Two charges \u00b110 mC are placed 5 0 mm apart"}, {"Chapter": "1", "sentence_range": "771-774", "Text": "Example 1 9 Two charges \u00b110 mC are placed 5 0 mm apart Determine\nthe electric field at (a) a point P on the axis of the dipole 15 cm away\nfrom its centre O on the side of the positive charge, as shown in Fig"}, {"Chapter": "1", "sentence_range": "772-775", "Text": "9 Two charges \u00b110 mC are placed 5 0 mm apart Determine\nthe electric field at (a) a point P on the axis of the dipole 15 cm away\nfrom its centre O on the side of the positive charge, as shown in Fig 1"}, {"Chapter": "1", "sentence_range": "773-776", "Text": "0 mm apart Determine\nthe electric field at (a) a point P on the axis of the dipole 15 cm away\nfrom its centre O on the side of the positive charge, as shown in Fig 1 18(a), and (b) a point Q, 15 cm away from O on a line passing through\nO and normal to the axis of the dipole, as shown in Fig"}, {"Chapter": "1", "sentence_range": "774-777", "Text": "Determine\nthe electric field at (a) a point P on the axis of the dipole 15 cm away\nfrom its centre O on the side of the positive charge, as shown in Fig 1 18(a), and (b) a point Q, 15 cm away from O on a line passing through\nO and normal to the axis of the dipole, as shown in Fig 1"}, {"Chapter": "1", "sentence_range": "775-778", "Text": "1 18(a), and (b) a point Q, 15 cm away from O on a line passing through\nO and normal to the axis of the dipole, as shown in Fig 1 18(b)"}, {"Chapter": "1", "sentence_range": "776-779", "Text": "18(a), and (b) a point Q, 15 cm away from O on a line passing through\nO and normal to the axis of the dipole, as shown in Fig 1 18(b) FIGURE 1"}, {"Chapter": "1", "sentence_range": "777-780", "Text": "1 18(b) FIGURE 1 18\n*\nCentre of a collection of positive point charges is defined much the same way\nas the centre of mass: r\nr\ncm =\n\u2211\n\u2211\nq\nq\ni i\ni\ni\ni"}, {"Chapter": "1", "sentence_range": "778-781", "Text": "18(b) FIGURE 1 18\n*\nCentre of a collection of positive point charges is defined much the same way\nas the centre of mass: r\nr\ncm =\n\u2211\n\u2211\nq\nq\ni i\ni\ni\ni Rationalised 2023-24\n26\nPhysics\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "779-782", "Text": "FIGURE 1 18\n*\nCentre of a collection of positive point charges is defined much the same way\nas the centre of mass: r\nr\ncm =\n\u2211\n\u2211\nq\nq\ni i\ni\ni\ni Rationalised 2023-24\n26\nPhysics\n EXAMPLE 1 9\nSolution  (a) Field at P due to charge +10 mC\n= \n5\n12\n2\n1\n2\n10\nC\n4 (8"}, {"Chapter": "1", "sentence_range": "780-783", "Text": "18\n*\nCentre of a collection of positive point charges is defined much the same way\nas the centre of mass: r\nr\ncm =\n\u2211\n\u2211\nq\nq\ni i\ni\ni\ni Rationalised 2023-24\n26\nPhysics\n EXAMPLE 1 9\nSolution  (a) Field at P due to charge +10 mC\n= \n5\n12\n2\n1\n2\n10\nC\n4 (8 854 10\nC N\nm\n)\n\u2212\n\u2212\n\u2212\n\u2212\n\u03c0\n\u00d7\n2\n4\n2\n1\n(15\n0"}, {"Chapter": "1", "sentence_range": "781-784", "Text": "Rationalised 2023-24\n26\nPhysics\n EXAMPLE 1 9\nSolution  (a) Field at P due to charge +10 mC\n= \n5\n12\n2\n1\n2\n10\nC\n4 (8 854 10\nC N\nm\n)\n\u2212\n\u2212\n\u2212\n\u2212\n\u03c0\n\u00d7\n2\n4\n2\n1\n(15\n0 25)\n10\nm\n\u2212\n\u00d7\n\u2212\n\u00d7\n=  4"}, {"Chapter": "1", "sentence_range": "782-785", "Text": "9\nSolution  (a) Field at P due to charge +10 mC\n= \n5\n12\n2\n1\n2\n10\nC\n4 (8 854 10\nC N\nm\n)\n\u2212\n\u2212\n\u2212\n\u2212\n\u03c0\n\u00d7\n2\n4\n2\n1\n(15\n0 25)\n10\nm\n\u2212\n\u00d7\n\u2212\n\u00d7\n=  4 13 \u00d7 106  N C\u20131  along BP\nField at P due to charge \u201310 mC\n\u20135\n12\n2\n1\n2\n10\nC\n4 (8"}, {"Chapter": "1", "sentence_range": "783-786", "Text": "854 10\nC N\nm\n)\n\u2212\n\u2212\n\u2212\n\u2212\n\u03c0\n\u00d7\n2\n4\n2\n1\n(15\n0 25)\n10\nm\n\u2212\n\u00d7\n\u2212\n\u00d7\n=  4 13 \u00d7 106  N C\u20131  along BP\nField at P due to charge \u201310 mC\n\u20135\n12\n2\n1\n2\n10\nC\n4 (8 854 10\nC N\nm\n)\n\u2212\n\u2212\n\u2212\n=\n\u03c0\n\u00d7\n \n2\n4\n2\n1\n(15\n0"}, {"Chapter": "1", "sentence_range": "784-787", "Text": "25)\n10\nm\n\u2212\n\u00d7\n\u2212\n\u00d7\n=  4 13 \u00d7 106  N C\u20131  along BP\nField at P due to charge \u201310 mC\n\u20135\n12\n2\n1\n2\n10\nC\n4 (8 854 10\nC N\nm\n)\n\u2212\n\u2212\n\u2212\n=\n\u03c0\n\u00d7\n \n2\n4\n2\n1\n(15\n0 25)\n10\nm\n\u2212\n\u00d7\n+\n\u00d7\n=  3"}, {"Chapter": "1", "sentence_range": "785-788", "Text": "13 \u00d7 106  N C\u20131  along BP\nField at P due to charge \u201310 mC\n\u20135\n12\n2\n1\n2\n10\nC\n4 (8 854 10\nC N\nm\n)\n\u2212\n\u2212\n\u2212\n=\n\u03c0\n\u00d7\n \n2\n4\n2\n1\n(15\n0 25)\n10\nm\n\u2212\n\u00d7\n+\n\u00d7\n=  3 86 \u00d7 106  N C\u20131 along PA\nThe resultant electric field at P due to the two charges at A and B is\n=  2"}, {"Chapter": "1", "sentence_range": "786-789", "Text": "854 10\nC N\nm\n)\n\u2212\n\u2212\n\u2212\n=\n\u03c0\n\u00d7\n \n2\n4\n2\n1\n(15\n0 25)\n10\nm\n\u2212\n\u00d7\n+\n\u00d7\n=  3 86 \u00d7 106  N C\u20131 along PA\nThe resultant electric field at P due to the two charges at A and B is\n=  2 7 \u00d7 105  N C\u20131  along BP"}, {"Chapter": "1", "sentence_range": "787-790", "Text": "25)\n10\nm\n\u2212\n\u00d7\n+\n\u00d7\n=  3 86 \u00d7 106  N C\u20131 along PA\nThe resultant electric field at P due to the two charges at A and B is\n=  2 7 \u00d7 105  N C\u20131  along BP In this example, the ratio OP/OB is quite large (= 60)"}, {"Chapter": "1", "sentence_range": "788-791", "Text": "86 \u00d7 106  N C\u20131 along PA\nThe resultant electric field at P due to the two charges at A and B is\n=  2 7 \u00d7 105  N C\u20131  along BP In this example, the ratio OP/OB is quite large (= 60) Thus, we can\nexpect to get approximately the same result as above by directly using\nthe formula for electric field at a far-away point on the axis of a dipole"}, {"Chapter": "1", "sentence_range": "789-792", "Text": "7 \u00d7 105  N C\u20131  along BP In this example, the ratio OP/OB is quite large (= 60) Thus, we can\nexpect to get approximately the same result as above by directly using\nthe formula for electric field at a far-away point on the axis of a dipole For a dipole consisting of charges \u00b1 q, 2a distance apart, the electric\nfield at a distance r from the centre on the axis of the dipole has a\nmagnitude\nE\np\nr\n=\n2\n4\n0\n3\n\u03c0\u03b5\n(r/a >> 1)\nwhere p = 2a q is the magnitude of the dipole moment"}, {"Chapter": "1", "sentence_range": "790-793", "Text": "In this example, the ratio OP/OB is quite large (= 60) Thus, we can\nexpect to get approximately the same result as above by directly using\nthe formula for electric field at a far-away point on the axis of a dipole For a dipole consisting of charges \u00b1 q, 2a distance apart, the electric\nfield at a distance r from the centre on the axis of the dipole has a\nmagnitude\nE\np\nr\n=\n2\n4\n0\n3\n\u03c0\u03b5\n(r/a >> 1)\nwhere p = 2a q is the magnitude of the dipole moment The direction of electric field on the dipole axis is always along the\ndirection of the dipole moment vector (i"}, {"Chapter": "1", "sentence_range": "791-794", "Text": "Thus, we can\nexpect to get approximately the same result as above by directly using\nthe formula for electric field at a far-away point on the axis of a dipole For a dipole consisting of charges \u00b1 q, 2a distance apart, the electric\nfield at a distance r from the centre on the axis of the dipole has a\nmagnitude\nE\np\nr\n=\n2\n4\n0\n3\n\u03c0\u03b5\n(r/a >> 1)\nwhere p = 2a q is the magnitude of the dipole moment The direction of electric field on the dipole axis is always along the\ndirection of the dipole moment vector (i e"}, {"Chapter": "1", "sentence_range": "792-795", "Text": "For a dipole consisting of charges \u00b1 q, 2a distance apart, the electric\nfield at a distance r from the centre on the axis of the dipole has a\nmagnitude\nE\np\nr\n=\n2\n4\n0\n3\n\u03c0\u03b5\n(r/a >> 1)\nwhere p = 2a q is the magnitude of the dipole moment The direction of electric field on the dipole axis is always along the\ndirection of the dipole moment vector (i e , from \u2013q to q)"}, {"Chapter": "1", "sentence_range": "793-796", "Text": "The direction of electric field on the dipole axis is always along the\ndirection of the dipole moment vector (i e , from \u2013q to q) Here,\np =10\u20135 C \u00d7 5 \u00d7 10\u20133 m  = 5 \u00d7 10\u20138 C m\nTherefore,\nE  =\n8\n12\n2\n1\n2\n2\n5 10\nC m\n4 (8"}, {"Chapter": "1", "sentence_range": "794-797", "Text": "e , from \u2013q to q) Here,\np =10\u20135 C \u00d7 5 \u00d7 10\u20133 m  = 5 \u00d7 10\u20138 C m\nTherefore,\nE  =\n8\n12\n2\n1\n2\n2\n5 10\nC m\n4 (8 854 10\nC N\nm\n)\n\u2212\n\u2212\n\u2212\n\u2212\n\u00d7\n\u00d7\n\u03c0\n\u00d7\n3\n6\n3\n1\n(15)\n10\nm\n\u2212\n\u00d7\n\u00d7\n = 2"}, {"Chapter": "1", "sentence_range": "795-798", "Text": ", from \u2013q to q) Here,\np =10\u20135 C \u00d7 5 \u00d7 10\u20133 m  = 5 \u00d7 10\u20138 C m\nTherefore,\nE  =\n8\n12\n2\n1\n2\n2\n5 10\nC m\n4 (8 854 10\nC N\nm\n)\n\u2212\n\u2212\n\u2212\n\u2212\n\u00d7\n\u00d7\n\u03c0\n\u00d7\n3\n6\n3\n1\n(15)\n10\nm\n\u2212\n\u00d7\n\u00d7\n = 2 6 \u00d7 105  N C\u20131\nalong the dipole moment direction AB, which is close to the result\nobtained earlier"}, {"Chapter": "1", "sentence_range": "796-799", "Text": "Here,\np =10\u20135 C \u00d7 5 \u00d7 10\u20133 m  = 5 \u00d7 10\u20138 C m\nTherefore,\nE  =\n8\n12\n2\n1\n2\n2\n5 10\nC m\n4 (8 854 10\nC N\nm\n)\n\u2212\n\u2212\n\u2212\n\u2212\n\u00d7\n\u00d7\n\u03c0\n\u00d7\n3\n6\n3\n1\n(15)\n10\nm\n\u2212\n\u00d7\n\u00d7\n = 2 6 \u00d7 105  N C\u20131\nalong the dipole moment direction AB, which is close to the result\nobtained earlier (b) Field at Q due to charge + 10 mC at B\n=\n5\n12\n2\n1\n2\n10\nC\n4\n(8"}, {"Chapter": "1", "sentence_range": "797-800", "Text": "854 10\nC N\nm\n)\n\u2212\n\u2212\n\u2212\n\u2212\n\u00d7\n\u00d7\n\u03c0\n\u00d7\n3\n6\n3\n1\n(15)\n10\nm\n\u2212\n\u00d7\n\u00d7\n = 2 6 \u00d7 105  N C\u20131\nalong the dipole moment direction AB, which is close to the result\nobtained earlier (b) Field at Q due to charge + 10 mC at B\n=\n5\n12\n2\n1\n2\n10\nC\n4\n(8 854\n10\nC\nN\nm\n)\n\u2212\n\u2212\n\u2212\n\u2212\n\u03c0\n\u00d7\n2\n2\n4\n2\n1\n[15\n(0"}, {"Chapter": "1", "sentence_range": "798-801", "Text": "6 \u00d7 105  N C\u20131\nalong the dipole moment direction AB, which is close to the result\nobtained earlier (b) Field at Q due to charge + 10 mC at B\n=\n5\n12\n2\n1\n2\n10\nC\n4\n(8 854\n10\nC\nN\nm\n)\n\u2212\n\u2212\n\u2212\n\u2212\n\u03c0\n\u00d7\n2\n2\n4\n2\n1\n[15\n(0 25) ]\n10\nm\n\u2212\n+\n\u00d7\n\u00d7\n=  3"}, {"Chapter": "1", "sentence_range": "799-802", "Text": "(b) Field at Q due to charge + 10 mC at B\n=\n5\n12\n2\n1\n2\n10\nC\n4\n(8 854\n10\nC\nN\nm\n)\n\u2212\n\u2212\n\u2212\n\u2212\n\u03c0\n\u00d7\n2\n2\n4\n2\n1\n[15\n(0 25) ]\n10\nm\n\u2212\n+\n\u00d7\n\u00d7\n=  3 99 \u00d7 106  N C\u20131 along BQ\nField at Q due to charge \u201310 mC at A\n=\n5\n12\n2\n1\n2\n10\nC\n4\n(8"}, {"Chapter": "1", "sentence_range": "800-803", "Text": "854\n10\nC\nN\nm\n)\n\u2212\n\u2212\n\u2212\n\u2212\n\u03c0\n\u00d7\n2\n2\n4\n2\n1\n[15\n(0 25) ]\n10\nm\n\u2212\n+\n\u00d7\n\u00d7\n=  3 99 \u00d7 106  N C\u20131 along BQ\nField at Q due to charge \u201310 mC at A\n=\n5\n12\n2\n1\n2\n10\nC\n4\n(8 854 10\nC N\nm\n)\n\u2212\n\u2212\n\u2212\n\u2212\n\u03c0\n\u00d7\n2\n2\n4\n2\n1\n[15\n(0"}, {"Chapter": "1", "sentence_range": "801-804", "Text": "25) ]\n10\nm\n\u2212\n+\n\u00d7\n\u00d7\n=  3 99 \u00d7 106  N C\u20131 along BQ\nField at Q due to charge \u201310 mC at A\n=\n5\n12\n2\n1\n2\n10\nC\n4\n(8 854 10\nC N\nm\n)\n\u2212\n\u2212\n\u2212\n\u2212\n\u03c0\n\u00d7\n2\n2\n4\n2\n1\n[15\n(0 25) ] 10\nm\n\u2212\n+\n\u00d7\n\u00d7\n=  3"}, {"Chapter": "1", "sentence_range": "802-805", "Text": "99 \u00d7 106  N C\u20131 along BQ\nField at Q due to charge \u201310 mC at A\n=\n5\n12\n2\n1\n2\n10\nC\n4\n(8 854 10\nC N\nm\n)\n\u2212\n\u2212\n\u2212\n\u2212\n\u03c0\n\u00d7\n2\n2\n4\n2\n1\n[15\n(0 25) ] 10\nm\n\u2212\n+\n\u00d7\n\u00d7\n=  3 99 \u00d7 106  N C\u20131 along QA"}, {"Chapter": "1", "sentence_range": "803-806", "Text": "854 10\nC N\nm\n)\n\u2212\n\u2212\n\u2212\n\u2212\n\u03c0\n\u00d7\n2\n2\n4\n2\n1\n[15\n(0 25) ] 10\nm\n\u2212\n+\n\u00d7\n\u00d7\n=  3 99 \u00d7 106  N C\u20131 along QA Clearly, the components of these two forces with equal magnitudes\ncancel along the direction OQ but add up along the direction parallel\nto BA"}, {"Chapter": "1", "sentence_range": "804-807", "Text": "25) ] 10\nm\n\u2212\n+\n\u00d7\n\u00d7\n=  3 99 \u00d7 106  N C\u20131 along QA Clearly, the components of these two forces with equal magnitudes\ncancel along the direction OQ but add up along the direction parallel\nto BA Therefore, the resultant electric field at Q due to the two\ncharges at A and B is\n= 2 \u00d7 \n6\n\u20131\n2\n2\n0"}, {"Chapter": "1", "sentence_range": "805-808", "Text": "99 \u00d7 106  N C\u20131 along QA Clearly, the components of these two forces with equal magnitudes\ncancel along the direction OQ but add up along the direction parallel\nto BA Therefore, the resultant electric field at Q due to the two\ncharges at A and B is\n= 2 \u00d7 \n6\n\u20131\n2\n2\n0 25\n3"}, {"Chapter": "1", "sentence_range": "806-809", "Text": "Clearly, the components of these two forces with equal magnitudes\ncancel along the direction OQ but add up along the direction parallel\nto BA Therefore, the resultant electric field at Q due to the two\ncharges at A and B is\n= 2 \u00d7 \n6\n\u20131\n2\n2\n0 25\n3 99\n10 N C\n15\n(0"}, {"Chapter": "1", "sentence_range": "807-810", "Text": "Therefore, the resultant electric field at Q due to the two\ncharges at A and B is\n= 2 \u00d7 \n6\n\u20131\n2\n2\n0 25\n3 99\n10 N C\n15\n(0 25)\n\u00d7\n\u00d7\n+\nalong BA\n=  1"}, {"Chapter": "1", "sentence_range": "808-811", "Text": "25\n3 99\n10 N C\n15\n(0 25)\n\u00d7\n\u00d7\n+\nalong BA\n=  1 33 \u00d7 105  N C\u20131 along BA"}, {"Chapter": "1", "sentence_range": "809-812", "Text": "99\n10 N C\n15\n(0 25)\n\u00d7\n\u00d7\n+\nalong BA\n=  1 33 \u00d7 105  N C\u20131 along BA As in (a), we can expect to get approximately the same result by\ndirectly using the formula for dipole field at a point on the normal to\nthe axis of the dipole:\nRationalised 2023-24\nElectric Charges\nand Fields\n27\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "810-813", "Text": "25)\n\u00d7\n\u00d7\n+\nalong BA\n=  1 33 \u00d7 105  N C\u20131 along BA As in (a), we can expect to get approximately the same result by\ndirectly using the formula for dipole field at a point on the normal to\nthe axis of the dipole:\nRationalised 2023-24\nElectric Charges\nand Fields\n27\n EXAMPLE 1 9\nE\np\nr\n= 4\n3\n\u03c0\n0\u03b5\n(r/a >> 1)\n8\n12\n2\n\u20131\n\u20132\n5 10\nCm\n4\n(8"}, {"Chapter": "1", "sentence_range": "811-814", "Text": "33 \u00d7 105  N C\u20131 along BA As in (a), we can expect to get approximately the same result by\ndirectly using the formula for dipole field at a point on the normal to\nthe axis of the dipole:\nRationalised 2023-24\nElectric Charges\nand Fields\n27\n EXAMPLE 1 9\nE\np\nr\n= 4\n3\n\u03c0\n0\u03b5\n(r/a >> 1)\n8\n12\n2\n\u20131\n\u20132\n5 10\nCm\n4\n(8 854 10\nC N\nm\n)\n\u2212\n\u2212\n\u00d7\n=\n\u03c0\n\u00d7\n3\n6\n3\n1\n(15)\n10\nm\n\u2212\n\u00d7\n\u00d7\n=  1"}, {"Chapter": "1", "sentence_range": "812-815", "Text": "As in (a), we can expect to get approximately the same result by\ndirectly using the formula for dipole field at a point on the normal to\nthe axis of the dipole:\nRationalised 2023-24\nElectric Charges\nand Fields\n27\n EXAMPLE 1 9\nE\np\nr\n= 4\n3\n\u03c0\n0\u03b5\n(r/a >> 1)\n8\n12\n2\n\u20131\n\u20132\n5 10\nCm\n4\n(8 854 10\nC N\nm\n)\n\u2212\n\u2212\n\u00d7\n=\n\u03c0\n\u00d7\n3\n6\n3\n1\n(15)\n10\nm\n\u2212\n\u00d7\n\u00d7\n=  1 33 \u00d7 105  N C\u20131"}, {"Chapter": "1", "sentence_range": "813-816", "Text": "9\nE\np\nr\n= 4\n3\n\u03c0\n0\u03b5\n(r/a >> 1)\n8\n12\n2\n\u20131\n\u20132\n5 10\nCm\n4\n(8 854 10\nC N\nm\n)\n\u2212\n\u2212\n\u00d7\n=\n\u03c0\n\u00d7\n3\n6\n3\n1\n(15)\n10\nm\n\u2212\n\u00d7\n\u00d7\n=  1 33 \u00d7 105  N C\u20131 The direction of electric field in this case is opposite to the direction\nof the dipole moment vector"}, {"Chapter": "1", "sentence_range": "814-817", "Text": "854 10\nC N\nm\n)\n\u2212\n\u2212\n\u00d7\n=\n\u03c0\n\u00d7\n3\n6\n3\n1\n(15)\n10\nm\n\u2212\n\u00d7\n\u00d7\n=  1 33 \u00d7 105  N C\u20131 The direction of electric field in this case is opposite to the direction\nof the dipole moment vector Again, the result agrees with that obtained\nbefore"}, {"Chapter": "1", "sentence_range": "815-818", "Text": "33 \u00d7 105  N C\u20131 The direction of electric field in this case is opposite to the direction\nof the dipole moment vector Again, the result agrees with that obtained\nbefore 1"}, {"Chapter": "1", "sentence_range": "816-819", "Text": "The direction of electric field in this case is opposite to the direction\nof the dipole moment vector Again, the result agrees with that obtained\nbefore 1 11  DIPOLE IN A UNIFORM EXTERNAL FIELD\nConsider a permanent dipole of dipole moment p in a uniform\nexternal field E, as shown in Fig"}, {"Chapter": "1", "sentence_range": "817-820", "Text": "Again, the result agrees with that obtained\nbefore 1 11  DIPOLE IN A UNIFORM EXTERNAL FIELD\nConsider a permanent dipole of dipole moment p in a uniform\nexternal field E, as shown in Fig 1"}, {"Chapter": "1", "sentence_range": "818-821", "Text": "1 11  DIPOLE IN A UNIFORM EXTERNAL FIELD\nConsider a permanent dipole of dipole moment p in a uniform\nexternal field E, as shown in Fig 1 19"}, {"Chapter": "1", "sentence_range": "819-822", "Text": "11  DIPOLE IN A UNIFORM EXTERNAL FIELD\nConsider a permanent dipole of dipole moment p in a uniform\nexternal field E, as shown in Fig 1 19 (By permanent dipole, we\nmean that p exists  irrespective of E; it has not been induced by E"}, {"Chapter": "1", "sentence_range": "820-823", "Text": "1 19 (By permanent dipole, we\nmean that p exists  irrespective of E; it has not been induced by E )\nThere is a force qE on q and a force \u2013qE on \u2013q"}, {"Chapter": "1", "sentence_range": "821-824", "Text": "19 (By permanent dipole, we\nmean that p exists  irrespective of E; it has not been induced by E )\nThere is a force qE on q and a force \u2013qE on \u2013q The net force on\nthe dipole is zero, since E is uniform"}, {"Chapter": "1", "sentence_range": "822-825", "Text": "(By permanent dipole, we\nmean that p exists  irrespective of E; it has not been induced by E )\nThere is a force qE on q and a force \u2013qE on \u2013q The net force on\nthe dipole is zero, since E is uniform However, the charges are\nseparated, so the forces act at different points, resulting in a torque\non the dipole"}, {"Chapter": "1", "sentence_range": "823-826", "Text": ")\nThere is a force qE on q and a force \u2013qE on \u2013q The net force on\nthe dipole is zero, since E is uniform However, the charges are\nseparated, so the forces act at different points, resulting in a torque\non the dipole When the net force is zero, the torque (couple) is\nindependent of the origin"}, {"Chapter": "1", "sentence_range": "824-827", "Text": "The net force on\nthe dipole is zero, since E is uniform However, the charges are\nseparated, so the forces act at different points, resulting in a torque\non the dipole When the net force is zero, the torque (couple) is\nindependent of the origin Its magnitude equals the magnitude of\neach force multiplied by the arm of the couple (perpendicular\ndistance between the two antiparallel forces)"}, {"Chapter": "1", "sentence_range": "825-828", "Text": "However, the charges are\nseparated, so the forces act at different points, resulting in a torque\non the dipole When the net force is zero, the torque (couple) is\nindependent of the origin Its magnitude equals the magnitude of\neach force multiplied by the arm of the couple (perpendicular\ndistance between the two antiparallel forces) Magnitude of torque = q E \u00d7 2 a sinq\n       = 2 q a E sinq\nIts direction is normal to the plane of the paper, coming out of it"}, {"Chapter": "1", "sentence_range": "826-829", "Text": "When the net force is zero, the torque (couple) is\nindependent of the origin Its magnitude equals the magnitude of\neach force multiplied by the arm of the couple (perpendicular\ndistance between the two antiparallel forces) Magnitude of torque = q E \u00d7 2 a sinq\n       = 2 q a E sinq\nIts direction is normal to the plane of the paper, coming out of it The magnitude of p \u00d7 E is also p E sinq and its direction\nis normal to the paper, coming out of it"}, {"Chapter": "1", "sentence_range": "827-830", "Text": "Its magnitude equals the magnitude of\neach force multiplied by the arm of the couple (perpendicular\ndistance between the two antiparallel forces) Magnitude of torque = q E \u00d7 2 a sinq\n       = 2 q a E sinq\nIts direction is normal to the plane of the paper, coming out of it The magnitude of p \u00d7 E is also p E sinq and its direction\nis normal to the paper, coming out of it Thus,\nttttt  = p \u00d7 E\n(1"}, {"Chapter": "1", "sentence_range": "828-831", "Text": "Magnitude of torque = q E \u00d7 2 a sinq\n       = 2 q a E sinq\nIts direction is normal to the plane of the paper, coming out of it The magnitude of p \u00d7 E is also p E sinq and its direction\nis normal to the paper, coming out of it Thus,\nttttt  = p \u00d7 E\n(1 22)\nThis torque will tend to align the dipole with the field\nE"}, {"Chapter": "1", "sentence_range": "829-832", "Text": "The magnitude of p \u00d7 E is also p E sinq and its direction\nis normal to the paper, coming out of it Thus,\nttttt  = p \u00d7 E\n(1 22)\nThis torque will tend to align the dipole with the field\nE When p is aligned with E, the torque is zero"}, {"Chapter": "1", "sentence_range": "830-833", "Text": "Thus,\nttttt  = p \u00d7 E\n(1 22)\nThis torque will tend to align the dipole with the field\nE When p is aligned with E, the torque is zero What happens if the field is not uniform"}, {"Chapter": "1", "sentence_range": "831-834", "Text": "22)\nThis torque will tend to align the dipole with the field\nE When p is aligned with E, the torque is zero What happens if the field is not uniform In that case,\nthe net force will evidently be non-zero"}, {"Chapter": "1", "sentence_range": "832-835", "Text": "When p is aligned with E, the torque is zero What happens if the field is not uniform In that case,\nthe net force will evidently be non-zero In addition there\nwill, in general, be a torque on the system as before"}, {"Chapter": "1", "sentence_range": "833-836", "Text": "What happens if the field is not uniform In that case,\nthe net force will evidently be non-zero In addition there\nwill, in general, be a torque on the system as before The\ngeneral case is involved, so let us consider the simpler\nsituations when p is parallel to E or antiparallel to E"}, {"Chapter": "1", "sentence_range": "834-837", "Text": "In that case,\nthe net force will evidently be non-zero In addition there\nwill, in general, be a torque on the system as before The\ngeneral case is involved, so let us consider the simpler\nsituations when p is parallel to E or antiparallel to E In\neither case, the net torque is zero, but there is a net force\non the dipole if E is not uniform"}, {"Chapter": "1", "sentence_range": "835-838", "Text": "In addition there\nwill, in general, be a torque on the system as before The\ngeneral case is involved, so let us consider the simpler\nsituations when p is parallel to E or antiparallel to E In\neither case, the net torque is zero, but there is a net force\non the dipole if E is not uniform Figure 1"}, {"Chapter": "1", "sentence_range": "836-839", "Text": "The\ngeneral case is involved, so let us consider the simpler\nsituations when p is parallel to E or antiparallel to E In\neither case, the net torque is zero, but there is a net force\non the dipole if E is not uniform Figure 1 20 is self-explanatory"}, {"Chapter": "1", "sentence_range": "837-840", "Text": "In\neither case, the net torque is zero, but there is a net force\non the dipole if E is not uniform Figure 1 20 is self-explanatory It is easily seen that\nwhen p is parallel to E, the dipole has a net force in the\ndirection of increasing field"}, {"Chapter": "1", "sentence_range": "838-841", "Text": "Figure 1 20 is self-explanatory It is easily seen that\nwhen p is parallel to E, the dipole has a net force in the\ndirection of increasing field When p is antiparallel to E,\nthe net force on the dipole is in the direction of decreasing\nfield"}, {"Chapter": "1", "sentence_range": "839-842", "Text": "20 is self-explanatory It is easily seen that\nwhen p is parallel to E, the dipole has a net force in the\ndirection of increasing field When p is antiparallel to E,\nthe net force on the dipole is in the direction of decreasing\nfield In general, the force depends on the orientation of p\nwith respect to E"}, {"Chapter": "1", "sentence_range": "840-843", "Text": "It is easily seen that\nwhen p is parallel to E, the dipole has a net force in the\ndirection of increasing field When p is antiparallel to E,\nthe net force on the dipole is in the direction of decreasing\nfield In general, the force depends on the orientation of p\nwith respect to E This brings us to a common observation in frictional\nelectricity"}, {"Chapter": "1", "sentence_range": "841-844", "Text": "When p is antiparallel to E,\nthe net force on the dipole is in the direction of decreasing\nfield In general, the force depends on the orientation of p\nwith respect to E This brings us to a common observation in frictional\nelectricity A comb run through dry hair attracts pieces of\npaper"}, {"Chapter": "1", "sentence_range": "842-845", "Text": "In general, the force depends on the orientation of p\nwith respect to E This brings us to a common observation in frictional\nelectricity A comb run through dry hair attracts pieces of\npaper The comb, as we know, acquires charge through\nfriction"}, {"Chapter": "1", "sentence_range": "843-846", "Text": "This brings us to a common observation in frictional\nelectricity A comb run through dry hair attracts pieces of\npaper The comb, as we know, acquires charge through\nfriction But the paper is not charged"}, {"Chapter": "1", "sentence_range": "844-847", "Text": "A comb run through dry hair attracts pieces of\npaper The comb, as we know, acquires charge through\nfriction But the paper is not charged What then explains\nthe attractive force"}, {"Chapter": "1", "sentence_range": "845-848", "Text": "The comb, as we know, acquires charge through\nfriction But the paper is not charged What then explains\nthe attractive force Taking the clue from the preceding\nFIGURE 1"}, {"Chapter": "1", "sentence_range": "846-849", "Text": "But the paper is not charged What then explains\nthe attractive force Taking the clue from the preceding\nFIGURE 1 19 Dipole in a\nuniform electric field"}, {"Chapter": "1", "sentence_range": "847-850", "Text": "What then explains\nthe attractive force Taking the clue from the preceding\nFIGURE 1 19 Dipole in a\nuniform electric field FIGURE 1"}, {"Chapter": "1", "sentence_range": "848-851", "Text": "Taking the clue from the preceding\nFIGURE 1 19 Dipole in a\nuniform electric field FIGURE 1 20 Electric force on a\ndipole: (a) E parallel to p, (b)  E\nantiparallel to p"}, {"Chapter": "1", "sentence_range": "849-852", "Text": "19 Dipole in a\nuniform electric field FIGURE 1 20 Electric force on a\ndipole: (a) E parallel to p, (b)  E\nantiparallel to p Rationalised 2023-24\n28\nPhysics\ndiscussion, the charged comb \u2018polarises\u2019 the piece of paper, i"}, {"Chapter": "1", "sentence_range": "850-853", "Text": "FIGURE 1 20 Electric force on a\ndipole: (a) E parallel to p, (b)  E\nantiparallel to p Rationalised 2023-24\n28\nPhysics\ndiscussion, the charged comb \u2018polarises\u2019 the piece of paper, i e"}, {"Chapter": "1", "sentence_range": "851-854", "Text": "20 Electric force on a\ndipole: (a) E parallel to p, (b)  E\nantiparallel to p Rationalised 2023-24\n28\nPhysics\ndiscussion, the charged comb \u2018polarises\u2019 the piece of paper, i e , induces\na net dipole moment in the direction of field"}, {"Chapter": "1", "sentence_range": "852-855", "Text": "Rationalised 2023-24\n28\nPhysics\ndiscussion, the charged comb \u2018polarises\u2019 the piece of paper, i e , induces\na net dipole moment in the direction of field Further, the electric field\ndue to the comb is not uniform"}, {"Chapter": "1", "sentence_range": "853-856", "Text": "e , induces\na net dipole moment in the direction of field Further, the electric field\ndue to the comb is not uniform This non-uniformity of the field makes a\ndipole to experience a net force on it"}, {"Chapter": "1", "sentence_range": "854-857", "Text": ", induces\na net dipole moment in the direction of field Further, the electric field\ndue to the comb is not uniform This non-uniformity of the field makes a\ndipole to experience a net force on it In this situation, it is easily seen\nthat the paper should move in the direction of the comb"}, {"Chapter": "1", "sentence_range": "855-858", "Text": "Further, the electric field\ndue to the comb is not uniform This non-uniformity of the field makes a\ndipole to experience a net force on it In this situation, it is easily seen\nthat the paper should move in the direction of the comb 1"}, {"Chapter": "1", "sentence_range": "856-859", "Text": "This non-uniformity of the field makes a\ndipole to experience a net force on it In this situation, it is easily seen\nthat the paper should move in the direction of the comb 1 12  CONTINUOUS CHARGE DISTRIBUTION\nWe have so far dealt with charge configurations involving discrete charges\nq1, q2,"}, {"Chapter": "1", "sentence_range": "857-860", "Text": "In this situation, it is easily seen\nthat the paper should move in the direction of the comb 1 12  CONTINUOUS CHARGE DISTRIBUTION\nWe have so far dealt with charge configurations involving discrete charges\nq1, q2, , qn"}, {"Chapter": "1", "sentence_range": "858-861", "Text": "1 12  CONTINUOUS CHARGE DISTRIBUTION\nWe have so far dealt with charge configurations involving discrete charges\nq1, q2, , qn One reason why we restricted to discrete charges is that the\nmathematical treatment is simpler and does not involve calculus"}, {"Chapter": "1", "sentence_range": "859-862", "Text": "12  CONTINUOUS CHARGE DISTRIBUTION\nWe have so far dealt with charge configurations involving discrete charges\nq1, q2, , qn One reason why we restricted to discrete charges is that the\nmathematical treatment is simpler and does not involve calculus For\nmany purposes, however, it is impractical to work in terms of discrete\ncharges and we need to work with continuous charge distributions"}, {"Chapter": "1", "sentence_range": "860-863", "Text": ", qn One reason why we restricted to discrete charges is that the\nmathematical treatment is simpler and does not involve calculus For\nmany purposes, however, it is impractical to work in terms of discrete\ncharges and we need to work with continuous charge distributions For\nexample, on the surface of a charged conductor, it is impractical to specify\nthe charge distribution in terms of the locations of the microscopic charged\nconstituents"}, {"Chapter": "1", "sentence_range": "861-864", "Text": "One reason why we restricted to discrete charges is that the\nmathematical treatment is simpler and does not involve calculus For\nmany purposes, however, it is impractical to work in terms of discrete\ncharges and we need to work with continuous charge distributions For\nexample, on the surface of a charged conductor, it is impractical to specify\nthe charge distribution in terms of the locations of the microscopic charged\nconstituents It is more feasible to consider an area element DS (Fig"}, {"Chapter": "1", "sentence_range": "862-865", "Text": "For\nmany purposes, however, it is impractical to work in terms of discrete\ncharges and we need to work with continuous charge distributions For\nexample, on the surface of a charged conductor, it is impractical to specify\nthe charge distribution in terms of the locations of the microscopic charged\nconstituents It is more feasible to consider an area element DS (Fig 1"}, {"Chapter": "1", "sentence_range": "863-866", "Text": "For\nexample, on the surface of a charged conductor, it is impractical to specify\nthe charge distribution in terms of the locations of the microscopic charged\nconstituents It is more feasible to consider an area element DS (Fig 1 21)\non the surface of the conductor (which is very small on the macroscopic\nscale but big enough to include a very large number of electrons) and\nspecify the charge DQ on that element"}, {"Chapter": "1", "sentence_range": "864-867", "Text": "It is more feasible to consider an area element DS (Fig 1 21)\non the surface of the conductor (which is very small on the macroscopic\nscale but big enough to include a very large number of electrons) and\nspecify the charge DQ on that element We then define a surface charge\ndensity s at the area element by\nSQ\n\u03c3\n= \u2206\u2206\n(1"}, {"Chapter": "1", "sentence_range": "865-868", "Text": "1 21)\non the surface of the conductor (which is very small on the macroscopic\nscale but big enough to include a very large number of electrons) and\nspecify the charge DQ on that element We then define a surface charge\ndensity s at the area element by\nSQ\n\u03c3\n= \u2206\u2206\n(1 23)\nWe can do this at different points on the conductor and thus arrive at\na continuous function s, called the surface charge density"}, {"Chapter": "1", "sentence_range": "866-869", "Text": "21)\non the surface of the conductor (which is very small on the macroscopic\nscale but big enough to include a very large number of electrons) and\nspecify the charge DQ on that element We then define a surface charge\ndensity s at the area element by\nSQ\n\u03c3\n= \u2206\u2206\n(1 23)\nWe can do this at different points on the conductor and thus arrive at\na continuous function s, called the surface charge density The surface\ncharge density s  so defined ignores the quantisation of charge and the\ndiscontinuity in charge distribution at the microscopic level*"}, {"Chapter": "1", "sentence_range": "867-870", "Text": "We then define a surface charge\ndensity s at the area element by\nSQ\n\u03c3\n= \u2206\u2206\n(1 23)\nWe can do this at different points on the conductor and thus arrive at\na continuous function s, called the surface charge density The surface\ncharge density s  so defined ignores the quantisation of charge and the\ndiscontinuity in charge distribution at the microscopic level* s  represents\nmacroscopic surface charge density, which in a sense, is a smoothed out\naverage of the microscopic charge density over an area element DS which,\nas said before, is large microscopically but small macroscopically"}, {"Chapter": "1", "sentence_range": "868-871", "Text": "23)\nWe can do this at different points on the conductor and thus arrive at\na continuous function s, called the surface charge density The surface\ncharge density s  so defined ignores the quantisation of charge and the\ndiscontinuity in charge distribution at the microscopic level* s  represents\nmacroscopic surface charge density, which in a sense, is a smoothed out\naverage of the microscopic charge density over an area element DS which,\nas said before, is large microscopically but small macroscopically The\nunits for s are C/m2"}, {"Chapter": "1", "sentence_range": "869-872", "Text": "The surface\ncharge density s  so defined ignores the quantisation of charge and the\ndiscontinuity in charge distribution at the microscopic level* s  represents\nmacroscopic surface charge density, which in a sense, is a smoothed out\naverage of the microscopic charge density over an area element DS which,\nas said before, is large microscopically but small macroscopically The\nunits for s are C/m2 Similar considerations apply for a line charge distribution and a volume\ncharge distribution"}, {"Chapter": "1", "sentence_range": "870-873", "Text": "s  represents\nmacroscopic surface charge density, which in a sense, is a smoothed out\naverage of the microscopic charge density over an area element DS which,\nas said before, is large microscopically but small macroscopically The\nunits for s are C/m2 Similar considerations apply for a line charge distribution and a volume\ncharge distribution The linear charge density l of a wire is defined by\nlQ\n\u03bb\n=\u2206\n\u2206\n(1"}, {"Chapter": "1", "sentence_range": "871-874", "Text": "The\nunits for s are C/m2 Similar considerations apply for a line charge distribution and a volume\ncharge distribution The linear charge density l of a wire is defined by\nlQ\n\u03bb\n=\u2206\n\u2206\n(1 24)\nwhere Dl is a small line element of wire on the macroscopic scale that,\nhowever, includes a large number of microscopic charged constituents,\nand DQ is the charge contained in that line element"}, {"Chapter": "1", "sentence_range": "872-875", "Text": "Similar considerations apply for a line charge distribution and a volume\ncharge distribution The linear charge density l of a wire is defined by\nlQ\n\u03bb\n=\u2206\n\u2206\n(1 24)\nwhere Dl is a small line element of wire on the macroscopic scale that,\nhowever, includes a large number of microscopic charged constituents,\nand DQ is the charge contained in that line element The units for l are\nC/m"}, {"Chapter": "1", "sentence_range": "873-876", "Text": "The linear charge density l of a wire is defined by\nlQ\n\u03bb\n=\u2206\n\u2206\n(1 24)\nwhere Dl is a small line element of wire on the macroscopic scale that,\nhowever, includes a large number of microscopic charged constituents,\nand DQ is the charge contained in that line element The units for l are\nC/m The volume charge density (sometimes simply called charge density)\nis defined in a similar manner:\nVQ\n\u03c1\n= \u2206\u2206\n(1"}, {"Chapter": "1", "sentence_range": "874-877", "Text": "24)\nwhere Dl is a small line element of wire on the macroscopic scale that,\nhowever, includes a large number of microscopic charged constituents,\nand DQ is the charge contained in that line element The units for l are\nC/m The volume charge density (sometimes simply called charge density)\nis defined in a similar manner:\nVQ\n\u03c1\n= \u2206\u2206\n(1 25)\nwhere DQ is the charge included in the macroscopically small volume\nelement DV that includes a large number of microscopic charged\nconstituents"}, {"Chapter": "1", "sentence_range": "875-878", "Text": "The units for l are\nC/m The volume charge density (sometimes simply called charge density)\nis defined in a similar manner:\nVQ\n\u03c1\n= \u2206\u2206\n(1 25)\nwhere DQ is the charge included in the macroscopically small volume\nelement DV that includes a large number of microscopic charged\nconstituents The units for r are C/m3"}, {"Chapter": "1", "sentence_range": "876-879", "Text": "The volume charge density (sometimes simply called charge density)\nis defined in a similar manner:\nVQ\n\u03c1\n= \u2206\u2206\n(1 25)\nwhere DQ is the charge included in the macroscopically small volume\nelement DV that includes a large number of microscopic charged\nconstituents The units for r are C/m3 The notion of continuous charge distribution is similar to that we\nadopt for continuous mass distribution in mechanics"}, {"Chapter": "1", "sentence_range": "877-880", "Text": "25)\nwhere DQ is the charge included in the macroscopically small volume\nelement DV that includes a large number of microscopic charged\nconstituents The units for r are C/m3 The notion of continuous charge distribution is similar to that we\nadopt for continuous mass distribution in mechanics When we refer to\nFIGURE 1"}, {"Chapter": "1", "sentence_range": "878-881", "Text": "The units for r are C/m3 The notion of continuous charge distribution is similar to that we\nadopt for continuous mass distribution in mechanics When we refer to\nFIGURE 1 21\nDefinition of linear,\nsurface and volume\ncharge densities"}, {"Chapter": "1", "sentence_range": "879-882", "Text": "The notion of continuous charge distribution is similar to that we\nadopt for continuous mass distribution in mechanics When we refer to\nFIGURE 1 21\nDefinition of linear,\nsurface and volume\ncharge densities In each case, the\nelement (Dl, DS, DV)\nchosen is small on\nthe macroscopic\nscale but contains\na very large number\nof microscopic\nconstituents"}, {"Chapter": "1", "sentence_range": "880-883", "Text": "When we refer to\nFIGURE 1 21\nDefinition of linear,\nsurface and volume\ncharge densities In each case, the\nelement (Dl, DS, DV)\nchosen is small on\nthe macroscopic\nscale but contains\na very large number\nof microscopic\nconstituents *\nAt the microscopic level, charge distribution is discontinuous, because they are\ndiscrete charges separated by intervening space where there is no charge"}, {"Chapter": "1", "sentence_range": "881-884", "Text": "21\nDefinition of linear,\nsurface and volume\ncharge densities In each case, the\nelement (Dl, DS, DV)\nchosen is small on\nthe macroscopic\nscale but contains\na very large number\nof microscopic\nconstituents *\nAt the microscopic level, charge distribution is discontinuous, because they are\ndiscrete charges separated by intervening space where there is no charge Rationalised 2023-24\nElectric Charges\nand Fields\n29\nthe density of a liquid, we are referring to its macroscopic density"}, {"Chapter": "1", "sentence_range": "882-885", "Text": "In each case, the\nelement (Dl, DS, DV)\nchosen is small on\nthe macroscopic\nscale but contains\na very large number\nof microscopic\nconstituents *\nAt the microscopic level, charge distribution is discontinuous, because they are\ndiscrete charges separated by intervening space where there is no charge Rationalised 2023-24\nElectric Charges\nand Fields\n29\nthe density of a liquid, we are referring to its macroscopic density We\nregard it as a continuous fluid and ignore its discrete molecular\nconstitution"}, {"Chapter": "1", "sentence_range": "883-886", "Text": "*\nAt the microscopic level, charge distribution is discontinuous, because they are\ndiscrete charges separated by intervening space where there is no charge Rationalised 2023-24\nElectric Charges\nand Fields\n29\nthe density of a liquid, we are referring to its macroscopic density We\nregard it as a continuous fluid and ignore its discrete molecular\nconstitution The field due to a continuous charge distribution can be obtained in\nmuch the same way as for a system of discrete charges, Eq"}, {"Chapter": "1", "sentence_range": "884-887", "Text": "Rationalised 2023-24\nElectric Charges\nand Fields\n29\nthe density of a liquid, we are referring to its macroscopic density We\nregard it as a continuous fluid and ignore its discrete molecular\nconstitution The field due to a continuous charge distribution can be obtained in\nmuch the same way as for a system of discrete charges, Eq (1"}, {"Chapter": "1", "sentence_range": "885-888", "Text": "We\nregard it as a continuous fluid and ignore its discrete molecular\nconstitution The field due to a continuous charge distribution can be obtained in\nmuch the same way as for a system of discrete charges, Eq (1 10)"}, {"Chapter": "1", "sentence_range": "886-889", "Text": "The field due to a continuous charge distribution can be obtained in\nmuch the same way as for a system of discrete charges, Eq (1 10) Suppose\na continuous charge distribution in space has a charge density r"}, {"Chapter": "1", "sentence_range": "887-890", "Text": "(1 10) Suppose\na continuous charge distribution in space has a charge density r Choose\nany convenient origin O and let the position vector of any point in the\ncharge distribution be r"}, {"Chapter": "1", "sentence_range": "888-891", "Text": "10) Suppose\na continuous charge distribution in space has a charge density r Choose\nany convenient origin O and let the position vector of any point in the\ncharge distribution be r The charge density r may vary from point to\npoint, i"}, {"Chapter": "1", "sentence_range": "889-892", "Text": "Suppose\na continuous charge distribution in space has a charge density r Choose\nany convenient origin O and let the position vector of any point in the\ncharge distribution be r The charge density r may vary from point to\npoint, i e"}, {"Chapter": "1", "sentence_range": "890-893", "Text": "Choose\nany convenient origin O and let the position vector of any point in the\ncharge distribution be r The charge density r may vary from point to\npoint, i e , it is a function of r"}, {"Chapter": "1", "sentence_range": "891-894", "Text": "The charge density r may vary from point to\npoint, i e , it is a function of r Divide the charge distribution into small\nvolume elements of size DV"}, {"Chapter": "1", "sentence_range": "892-895", "Text": "e , it is a function of r Divide the charge distribution into small\nvolume elements of size DV The charge in a volume element DV is rDV"}, {"Chapter": "1", "sentence_range": "893-896", "Text": ", it is a function of r Divide the charge distribution into small\nvolume elements of size DV The charge in a volume element DV is rDV Now, consider any general point P (inside or outside the distribution)\nwith position vector R (Fig"}, {"Chapter": "1", "sentence_range": "894-897", "Text": "Divide the charge distribution into small\nvolume elements of size DV The charge in a volume element DV is rDV Now, consider any general point P (inside or outside the distribution)\nwith position vector R (Fig 1"}, {"Chapter": "1", "sentence_range": "895-898", "Text": "The charge in a volume element DV is rDV Now, consider any general point P (inside or outside the distribution)\nwith position vector R (Fig 1 21)"}, {"Chapter": "1", "sentence_range": "896-899", "Text": "Now, consider any general point P (inside or outside the distribution)\nwith position vector R (Fig 1 21) Electric field due to the charge rDV is\ngiven by Coulomb\u2019s law:\n2\n0\n1\n\u02c6\n4\nV\n'\nr'\n\u03c1\n\u03b5\n\u2206\n\u2206\n=\n\u03c0\nE\nr\n(1"}, {"Chapter": "1", "sentence_range": "897-900", "Text": "1 21) Electric field due to the charge rDV is\ngiven by Coulomb\u2019s law:\n2\n0\n1\n\u02c6\n4\nV\n'\nr'\n\u03c1\n\u03b5\n\u2206\n\u2206\n=\n\u03c0\nE\nr\n(1 26)\nwhere r\u00a2 is the distance between the charge element and P, and  \u02c6r\u00a2 is a\nunit vector in the direction from the charge element to P"}, {"Chapter": "1", "sentence_range": "898-901", "Text": "21) Electric field due to the charge rDV is\ngiven by Coulomb\u2019s law:\n2\n0\n1\n\u02c6\n4\nV\n'\nr'\n\u03c1\n\u03b5\n\u2206\n\u2206\n=\n\u03c0\nE\nr\n(1 26)\nwhere r\u00a2 is the distance between the charge element and P, and  \u02c6r\u00a2 is a\nunit vector in the direction from the charge element to P By the\nsuperposition principle, the total electric field due to the charge\ndistribution is obtained by summing over electric fields due to different\nvolume elements:\n2\n0\n1\n\u02c6\n4\nall\nV\nV\n'\nr'\n\u03c1\n\u03b5\n\u2206\n\u2206\n\u2245\n\u03a3\n\u03c0\nE\nr\n(1"}, {"Chapter": "1", "sentence_range": "899-902", "Text": "Electric field due to the charge rDV is\ngiven by Coulomb\u2019s law:\n2\n0\n1\n\u02c6\n4\nV\n'\nr'\n\u03c1\n\u03b5\n\u2206\n\u2206\n=\n\u03c0\nE\nr\n(1 26)\nwhere r\u00a2 is the distance between the charge element and P, and  \u02c6r\u00a2 is a\nunit vector in the direction from the charge element to P By the\nsuperposition principle, the total electric field due to the charge\ndistribution is obtained by summing over electric fields due to different\nvolume elements:\n2\n0\n1\n\u02c6\n4\nall\nV\nV\n'\nr'\n\u03c1\n\u03b5\n\u2206\n\u2206\n\u2245\n\u03a3\n\u03c0\nE\nr\n(1 27)\nNote that r, r\u00a2,  \u02c6\u2032r  all can vary from point to point"}, {"Chapter": "1", "sentence_range": "900-903", "Text": "26)\nwhere r\u00a2 is the distance between the charge element and P, and  \u02c6r\u00a2 is a\nunit vector in the direction from the charge element to P By the\nsuperposition principle, the total electric field due to the charge\ndistribution is obtained by summing over electric fields due to different\nvolume elements:\n2\n0\n1\n\u02c6\n4\nall\nV\nV\n'\nr'\n\u03c1\n\u03b5\n\u2206\n\u2206\n\u2245\n\u03a3\n\u03c0\nE\nr\n(1 27)\nNote that r, r\u00a2,  \u02c6\u2032r  all can vary from point to point In a strict\nmathematical method, we should let DV\u00ae0 and the sum then becomes\nan integral; but we omit that discussion here, for simplicity"}, {"Chapter": "1", "sentence_range": "901-904", "Text": "By the\nsuperposition principle, the total electric field due to the charge\ndistribution is obtained by summing over electric fields due to different\nvolume elements:\n2\n0\n1\n\u02c6\n4\nall\nV\nV\n'\nr'\n\u03c1\n\u03b5\n\u2206\n\u2206\n\u2245\n\u03a3\n\u03c0\nE\nr\n(1 27)\nNote that r, r\u00a2,  \u02c6\u2032r  all can vary from point to point In a strict\nmathematical method, we should let DV\u00ae0 and the sum then becomes\nan integral; but we omit that discussion here, for simplicity In short,\nusing Coulomb\u2019s law and the superposition principle, electric field can\nbe determined for any charge distribution, discrete or continuous or part\ndiscrete and part continuous"}, {"Chapter": "1", "sentence_range": "902-905", "Text": "27)\nNote that r, r\u00a2,  \u02c6\u2032r  all can vary from point to point In a strict\nmathematical method, we should let DV\u00ae0 and the sum then becomes\nan integral; but we omit that discussion here, for simplicity In short,\nusing Coulomb\u2019s law and the superposition principle, electric field can\nbe determined for any charge distribution, discrete or continuous or part\ndiscrete and part continuous 1"}, {"Chapter": "1", "sentence_range": "903-906", "Text": "In a strict\nmathematical method, we should let DV\u00ae0 and the sum then becomes\nan integral; but we omit that discussion here, for simplicity In short,\nusing Coulomb\u2019s law and the superposition principle, electric field can\nbe determined for any charge distribution, discrete or continuous or part\ndiscrete and part continuous 1 13  GAUSS\u2019S LAW\nAs a simple application of the notion of electric flux, let us consider the\ntotal flux through a sphere of radius r, which encloses a point charge q\nat its centre"}, {"Chapter": "1", "sentence_range": "904-907", "Text": "In short,\nusing Coulomb\u2019s law and the superposition principle, electric field can\nbe determined for any charge distribution, discrete or continuous or part\ndiscrete and part continuous 1 13  GAUSS\u2019S LAW\nAs a simple application of the notion of electric flux, let us consider the\ntotal flux through a sphere of radius r, which encloses a point charge q\nat its centre Divide the sphere into small area elements, as shown in\nFig"}, {"Chapter": "1", "sentence_range": "905-908", "Text": "1 13  GAUSS\u2019S LAW\nAs a simple application of the notion of electric flux, let us consider the\ntotal flux through a sphere of radius r, which encloses a point charge q\nat its centre Divide the sphere into small area elements, as shown in\nFig 1"}, {"Chapter": "1", "sentence_range": "906-909", "Text": "13  GAUSS\u2019S LAW\nAs a simple application of the notion of electric flux, let us consider the\ntotal flux through a sphere of radius r, which encloses a point charge q\nat its centre Divide the sphere into small area elements, as shown in\nFig 1 22"}, {"Chapter": "1", "sentence_range": "907-910", "Text": "Divide the sphere into small area elements, as shown in\nFig 1 22 The flux through an area element DS is\n2\n0\n\u02c6\n4\nq\nr\n\u03c6\n\u03b5\n\u2206\n=\n\u2206\n=\n\u2206\n\u03c0\nE\nS\nr\nS\ni\ni\n(1"}, {"Chapter": "1", "sentence_range": "908-911", "Text": "1 22 The flux through an area element DS is\n2\n0\n\u02c6\n4\nq\nr\n\u03c6\n\u03b5\n\u2206\n=\n\u2206\n=\n\u2206\n\u03c0\nE\nS\nr\nS\ni\ni\n(1 28)\nwhere we have used Coulomb\u2019s law for the electric field due to a single\ncharge q"}, {"Chapter": "1", "sentence_range": "909-912", "Text": "22 The flux through an area element DS is\n2\n0\n\u02c6\n4\nq\nr\n\u03c6\n\u03b5\n\u2206\n=\n\u2206\n=\n\u2206\n\u03c0\nE\nS\nr\nS\ni\ni\n(1 28)\nwhere we have used Coulomb\u2019s law for the electric field due to a single\ncharge q The unit vector \u02c6r is along the radius vector from the centre to\nthe area element"}, {"Chapter": "1", "sentence_range": "910-913", "Text": "The flux through an area element DS is\n2\n0\n\u02c6\n4\nq\nr\n\u03c6\n\u03b5\n\u2206\n=\n\u2206\n=\n\u2206\n\u03c0\nE\nS\nr\nS\ni\ni\n(1 28)\nwhere we have used Coulomb\u2019s law for the electric field due to a single\ncharge q The unit vector \u02c6r is along the radius vector from the centre to\nthe area element Now, since the normal to a sphere at every point is\nalong the radius vector at that point, the area element DS and \u02c6r  have\nthe same direction"}, {"Chapter": "1", "sentence_range": "911-914", "Text": "28)\nwhere we have used Coulomb\u2019s law for the electric field due to a single\ncharge q The unit vector \u02c6r is along the radius vector from the centre to\nthe area element Now, since the normal to a sphere at every point is\nalong the radius vector at that point, the area element DS and \u02c6r  have\nthe same direction Therefore,\n2\n0\n4\nq\nS\nr\n\u03c6\n\u03b5\n\u2206\n=\n\u2206\n\u03c0\n(1"}, {"Chapter": "1", "sentence_range": "912-915", "Text": "The unit vector \u02c6r is along the radius vector from the centre to\nthe area element Now, since the normal to a sphere at every point is\nalong the radius vector at that point, the area element DS and \u02c6r  have\nthe same direction Therefore,\n2\n0\n4\nq\nS\nr\n\u03c6\n\u03b5\n\u2206\n=\n\u2206\n\u03c0\n(1 29)\nsince the magnitude of a unit vector is 1"}, {"Chapter": "1", "sentence_range": "913-916", "Text": "Now, since the normal to a sphere at every point is\nalong the radius vector at that point, the area element DS and \u02c6r  have\nthe same direction Therefore,\n2\n0\n4\nq\nS\nr\n\u03c6\n\u03b5\n\u2206\n=\n\u2206\n\u03c0\n(1 29)\nsince the magnitude of a unit vector is 1 The total flux through the sphere is obtained by adding up flux\nthrough all the different area elements:\nFIGURE 1"}, {"Chapter": "1", "sentence_range": "914-917", "Text": "Therefore,\n2\n0\n4\nq\nS\nr\n\u03c6\n\u03b5\n\u2206\n=\n\u2206\n\u03c0\n(1 29)\nsince the magnitude of a unit vector is 1 The total flux through the sphere is obtained by adding up flux\nthrough all the different area elements:\nFIGURE 1 22 Flux\nthrough a sphere\nenclosing a point\ncharge q at its centre"}, {"Chapter": "1", "sentence_range": "915-918", "Text": "29)\nsince the magnitude of a unit vector is 1 The total flux through the sphere is obtained by adding up flux\nthrough all the different area elements:\nFIGURE 1 22 Flux\nthrough a sphere\nenclosing a point\ncharge q at its centre Rationalised 2023-24\n30\nPhysics\n2\n0\n4\nall\nS\nq\nS\nr\n\u03c6\n\u03b5\n\u2206\n=\n\u03a3\n\u2206\n\u03c0\nSince each area element of the sphere is at the same\ndistance r from the charge,\n2\n2\n0\n4\n4\nall\nS\no\nq\nq\nS\nS\nr\nr\n\u03c6\n\u03b5\n\u03b5\n\u2206\n=\n\u03a3 \u2206\n=\n\u03c0\n\u03c0\nNow S, the total area of the sphere, equals 4pr 2"}, {"Chapter": "1", "sentence_range": "916-919", "Text": "The total flux through the sphere is obtained by adding up flux\nthrough all the different area elements:\nFIGURE 1 22 Flux\nthrough a sphere\nenclosing a point\ncharge q at its centre Rationalised 2023-24\n30\nPhysics\n2\n0\n4\nall\nS\nq\nS\nr\n\u03c6\n\u03b5\n\u2206\n=\n\u03a3\n\u2206\n\u03c0\nSince each area element of the sphere is at the same\ndistance r from the charge,\n2\n2\n0\n4\n4\nall\nS\no\nq\nq\nS\nS\nr\nr\n\u03c6\n\u03b5\n\u03b5\n\u2206\n=\n\u03a3 \u2206\n=\n\u03c0\n\u03c0\nNow S, the total area of the sphere, equals 4pr 2 Thus,\n2\n2\n0\n0\n4\n4\nq\nq\nr\nr\n\u03c6\n\u03b5\n\u03b5\n=\n\u00d7\n\u03c0\n=\n\u03c0\n(1"}, {"Chapter": "1", "sentence_range": "917-920", "Text": "22 Flux\nthrough a sphere\nenclosing a point\ncharge q at its centre Rationalised 2023-24\n30\nPhysics\n2\n0\n4\nall\nS\nq\nS\nr\n\u03c6\n\u03b5\n\u2206\n=\n\u03a3\n\u2206\n\u03c0\nSince each area element of the sphere is at the same\ndistance r from the charge,\n2\n2\n0\n4\n4\nall\nS\no\nq\nq\nS\nS\nr\nr\n\u03c6\n\u03b5\n\u03b5\n\u2206\n=\n\u03a3 \u2206\n=\n\u03c0\n\u03c0\nNow S, the total area of the sphere, equals 4pr 2 Thus,\n2\n2\n0\n0\n4\n4\nq\nq\nr\nr\n\u03c6\n\u03b5\n\u03b5\n=\n\u00d7\n\u03c0\n=\n\u03c0\n(1 30)\nEquation (1"}, {"Chapter": "1", "sentence_range": "918-921", "Text": "Rationalised 2023-24\n30\nPhysics\n2\n0\n4\nall\nS\nq\nS\nr\n\u03c6\n\u03b5\n\u2206\n=\n\u03a3\n\u2206\n\u03c0\nSince each area element of the sphere is at the same\ndistance r from the charge,\n2\n2\n0\n4\n4\nall\nS\no\nq\nq\nS\nS\nr\nr\n\u03c6\n\u03b5\n\u03b5\n\u2206\n=\n\u03a3 \u2206\n=\n\u03c0\n\u03c0\nNow S, the total area of the sphere, equals 4pr 2 Thus,\n2\n2\n0\n0\n4\n4\nq\nq\nr\nr\n\u03c6\n\u03b5\n\u03b5\n=\n\u00d7\n\u03c0\n=\n\u03c0\n(1 30)\nEquation (1 30) is a simple illustration of a general result of\nelectrostatics called Gauss\u2019s law"}, {"Chapter": "1", "sentence_range": "919-922", "Text": "Thus,\n2\n2\n0\n0\n4\n4\nq\nq\nr\nr\n\u03c6\n\u03b5\n\u03b5\n=\n\u00d7\n\u03c0\n=\n\u03c0\n(1 30)\nEquation (1 30) is a simple illustration of a general result of\nelectrostatics called Gauss\u2019s law We state Gauss\u2019s law without proof:\nElectric flux through a closed surface S\n= q/e0\n(1"}, {"Chapter": "1", "sentence_range": "920-923", "Text": "30)\nEquation (1 30) is a simple illustration of a general result of\nelectrostatics called Gauss\u2019s law We state Gauss\u2019s law without proof:\nElectric flux through a closed surface S\n= q/e0\n(1 31)\nq = total charge enclosed by S"}, {"Chapter": "1", "sentence_range": "921-924", "Text": "30) is a simple illustration of a general result of\nelectrostatics called Gauss\u2019s law We state Gauss\u2019s law without proof:\nElectric flux through a closed surface S\n= q/e0\n(1 31)\nq = total charge enclosed by S The law implies that the total electric flux through a closed surface is\nzero if no charge is enclosed by the surface"}, {"Chapter": "1", "sentence_range": "922-925", "Text": "We state Gauss\u2019s law without proof:\nElectric flux through a closed surface S\n= q/e0\n(1 31)\nq = total charge enclosed by S The law implies that the total electric flux through a closed surface is\nzero if no charge is enclosed by the surface We can see that explicitly in\nthe simple situation of Fig"}, {"Chapter": "1", "sentence_range": "923-926", "Text": "31)\nq = total charge enclosed by S The law implies that the total electric flux through a closed surface is\nzero if no charge is enclosed by the surface We can see that explicitly in\nthe simple situation of Fig 1"}, {"Chapter": "1", "sentence_range": "924-927", "Text": "The law implies that the total electric flux through a closed surface is\nzero if no charge is enclosed by the surface We can see that explicitly in\nthe simple situation of Fig 1 23"}, {"Chapter": "1", "sentence_range": "925-928", "Text": "We can see that explicitly in\nthe simple situation of Fig 1 23 Here the electric field is uniform and we are considering a closed\ncylindrical surface, with its axis parallel to the uniform field E"}, {"Chapter": "1", "sentence_range": "926-929", "Text": "1 23 Here the electric field is uniform and we are considering a closed\ncylindrical surface, with its axis parallel to the uniform field E The total\nflux f  through the surface is  f = f1 + f2 + f3, where f1 and f2 represent\nthe flux through the surfaces 1 and 2 (of circular cross-section) of the\ncylinder and f3 is the flux through the curved cylindrical part of the\nclosed surface"}, {"Chapter": "1", "sentence_range": "927-930", "Text": "23 Here the electric field is uniform and we are considering a closed\ncylindrical surface, with its axis parallel to the uniform field E The total\nflux f  through the surface is  f = f1 + f2 + f3, where f1 and f2 represent\nthe flux through the surfaces 1 and 2 (of circular cross-section) of the\ncylinder and f3 is the flux through the curved cylindrical part of the\nclosed surface Now the normal to the surface 3 at every point is\nperpendicular to E, so by definition of flux, f3 = 0"}, {"Chapter": "1", "sentence_range": "928-931", "Text": "Here the electric field is uniform and we are considering a closed\ncylindrical surface, with its axis parallel to the uniform field E The total\nflux f  through the surface is  f = f1 + f2 + f3, where f1 and f2 represent\nthe flux through the surfaces 1 and 2 (of circular cross-section) of the\ncylinder and f3 is the flux through the curved cylindrical part of the\nclosed surface Now the normal to the surface 3 at every point is\nperpendicular to E, so by definition of flux, f3 = 0 Further, the outward\nnormal to 2 is along E while the outward normal to 1 is opposite to E"}, {"Chapter": "1", "sentence_range": "929-932", "Text": "The total\nflux f  through the surface is  f = f1 + f2 + f3, where f1 and f2 represent\nthe flux through the surfaces 1 and 2 (of circular cross-section) of the\ncylinder and f3 is the flux through the curved cylindrical part of the\nclosed surface Now the normal to the surface 3 at every point is\nperpendicular to E, so by definition of flux, f3 = 0 Further, the outward\nnormal to 2 is along E while the outward normal to 1 is opposite to E Therefore,\nf1 = \u2013E S1,     f2 = +E S2\nS1 = S2 = S\nwhere S is the area of circular cross-section"}, {"Chapter": "1", "sentence_range": "930-933", "Text": "Now the normal to the surface 3 at every point is\nperpendicular to E, so by definition of flux, f3 = 0 Further, the outward\nnormal to 2 is along E while the outward normal to 1 is opposite to E Therefore,\nf1 = \u2013E S1,     f2 = +E S2\nS1 = S2 = S\nwhere S is the area of circular cross-section Thus, the total flux is zero,\nas expected by Gauss\u2019s law"}, {"Chapter": "1", "sentence_range": "931-934", "Text": "Further, the outward\nnormal to 2 is along E while the outward normal to 1 is opposite to E Therefore,\nf1 = \u2013E S1,     f2 = +E S2\nS1 = S2 = S\nwhere S is the area of circular cross-section Thus, the total flux is zero,\nas expected by Gauss\u2019s law Thus, whenever you find that the net electric\nflux through a closed surface is zero, we conclude that the total charge\ncontained in the closed surface is zero"}, {"Chapter": "1", "sentence_range": "932-935", "Text": "Therefore,\nf1 = \u2013E S1,     f2 = +E S2\nS1 = S2 = S\nwhere S is the area of circular cross-section Thus, the total flux is zero,\nas expected by Gauss\u2019s law Thus, whenever you find that the net electric\nflux through a closed surface is zero, we conclude that the total charge\ncontained in the closed surface is zero The great significance of Gauss\u2019s law Eq"}, {"Chapter": "1", "sentence_range": "933-936", "Text": "Thus, the total flux is zero,\nas expected by Gauss\u2019s law Thus, whenever you find that the net electric\nflux through a closed surface is zero, we conclude that the total charge\ncontained in the closed surface is zero The great significance of Gauss\u2019s law Eq (1"}, {"Chapter": "1", "sentence_range": "934-937", "Text": "Thus, whenever you find that the net electric\nflux through a closed surface is zero, we conclude that the total charge\ncontained in the closed surface is zero The great significance of Gauss\u2019s law Eq (1 31), is that it is true in\ngeneral, and not only for the simple cases we have considered above"}, {"Chapter": "1", "sentence_range": "935-938", "Text": "The great significance of Gauss\u2019s law Eq (1 31), is that it is true in\ngeneral, and not only for the simple cases we have considered above Let\nus note some important points regarding this law:\n(i)\nGauss\u2019s law is true for any closed surface, no matter what its shape\nor size"}, {"Chapter": "1", "sentence_range": "936-939", "Text": "(1 31), is that it is true in\ngeneral, and not only for the simple cases we have considered above Let\nus note some important points regarding this law:\n(i)\nGauss\u2019s law is true for any closed surface, no matter what its shape\nor size (ii) The term q on the right side of Gauss\u2019s law, Eq"}, {"Chapter": "1", "sentence_range": "937-940", "Text": "31), is that it is true in\ngeneral, and not only for the simple cases we have considered above Let\nus note some important points regarding this law:\n(i)\nGauss\u2019s law is true for any closed surface, no matter what its shape\nor size (ii) The term q on the right side of Gauss\u2019s law, Eq (1"}, {"Chapter": "1", "sentence_range": "938-941", "Text": "Let\nus note some important points regarding this law:\n(i)\nGauss\u2019s law is true for any closed surface, no matter what its shape\nor size (ii) The term q on the right side of Gauss\u2019s law, Eq (1 31), includes the\nsum of all charges enclosed by the surface"}, {"Chapter": "1", "sentence_range": "939-942", "Text": "(ii) The term q on the right side of Gauss\u2019s law, Eq (1 31), includes the\nsum of all charges enclosed by the surface The charges may be located\nanywhere inside the surface"}, {"Chapter": "1", "sentence_range": "940-943", "Text": "(1 31), includes the\nsum of all charges enclosed by the surface The charges may be located\nanywhere inside the surface (iii) In the situation when the surface is so chosen that there are some\ncharges inside and some outside, the electric field [whose flux appears\non the left side of Eq"}, {"Chapter": "1", "sentence_range": "941-944", "Text": "31), includes the\nsum of all charges enclosed by the surface The charges may be located\nanywhere inside the surface (iii) In the situation when the surface is so chosen that there are some\ncharges inside and some outside, the electric field [whose flux appears\non the left side of Eq (1"}, {"Chapter": "1", "sentence_range": "942-945", "Text": "The charges may be located\nanywhere inside the surface (iii) In the situation when the surface is so chosen that there are some\ncharges inside and some outside, the electric field [whose flux appears\non the left side of Eq (1 31)] is due to all the charges, both inside and\noutside S"}, {"Chapter": "1", "sentence_range": "943-946", "Text": "(iii) In the situation when the surface is so chosen that there are some\ncharges inside and some outside, the electric field [whose flux appears\non the left side of Eq (1 31)] is due to all the charges, both inside and\noutside S The term q on the right side of Gauss\u2019s law, however,\nrepresents only the total charge inside S"}, {"Chapter": "1", "sentence_range": "944-947", "Text": "(1 31)] is due to all the charges, both inside and\noutside S The term q on the right side of Gauss\u2019s law, however,\nrepresents only the total charge inside S FIGURE 1"}, {"Chapter": "1", "sentence_range": "945-948", "Text": "31)] is due to all the charges, both inside and\noutside S The term q on the right side of Gauss\u2019s law, however,\nrepresents only the total charge inside S FIGURE 1 23 Calculation of the\nflux of uniform electric field\nthrough the surface of a cylinder"}, {"Chapter": "1", "sentence_range": "946-949", "Text": "The term q on the right side of Gauss\u2019s law, however,\nrepresents only the total charge inside S FIGURE 1 23 Calculation of the\nflux of uniform electric field\nthrough the surface of a cylinder Rationalised 2023-24\nElectric Charges\nand Fields\n31\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "947-950", "Text": "FIGURE 1 23 Calculation of the\nflux of uniform electric field\nthrough the surface of a cylinder Rationalised 2023-24\nElectric Charges\nand Fields\n31\n EXAMPLE 1 10\n(iv) The surface that we choose for the application of Gauss\u2019s law is called\nthe Gaussian surface"}, {"Chapter": "1", "sentence_range": "948-951", "Text": "23 Calculation of the\nflux of uniform electric field\nthrough the surface of a cylinder Rationalised 2023-24\nElectric Charges\nand Fields\n31\n EXAMPLE 1 10\n(iv) The surface that we choose for the application of Gauss\u2019s law is called\nthe Gaussian surface You may choose any Gaussian surface and\napply Gauss\u2019s law"}, {"Chapter": "1", "sentence_range": "949-952", "Text": "Rationalised 2023-24\nElectric Charges\nand Fields\n31\n EXAMPLE 1 10\n(iv) The surface that we choose for the application of Gauss\u2019s law is called\nthe Gaussian surface You may choose any Gaussian surface and\napply Gauss\u2019s law However, take care not to let the Gaussian surface\npass through any discrete charge"}, {"Chapter": "1", "sentence_range": "950-953", "Text": "10\n(iv) The surface that we choose for the application of Gauss\u2019s law is called\nthe Gaussian surface You may choose any Gaussian surface and\napply Gauss\u2019s law However, take care not to let the Gaussian surface\npass through any discrete charge This is because electric field due\nto a system of discrete charges is not well defined at the location of\nany charge"}, {"Chapter": "1", "sentence_range": "951-954", "Text": "You may choose any Gaussian surface and\napply Gauss\u2019s law However, take care not to let the Gaussian surface\npass through any discrete charge This is because electric field due\nto a system of discrete charges is not well defined at the location of\nany charge (As you go close to the charge, the field grows without\nany bound"}, {"Chapter": "1", "sentence_range": "952-955", "Text": "However, take care not to let the Gaussian surface\npass through any discrete charge This is because electric field due\nto a system of discrete charges is not well defined at the location of\nany charge (As you go close to the charge, the field grows without\nany bound ) However, the Gaussian surface can pass through a\ncontinuous charge distribution"}, {"Chapter": "1", "sentence_range": "953-956", "Text": "This is because electric field due\nto a system of discrete charges is not well defined at the location of\nany charge (As you go close to the charge, the field grows without\nany bound ) However, the Gaussian surface can pass through a\ncontinuous charge distribution (v) Gauss\u2019s law is often useful towards a much easier calculation of the\nelectrostatic field when the system has some symmetry"}, {"Chapter": "1", "sentence_range": "954-957", "Text": "(As you go close to the charge, the field grows without\nany bound ) However, the Gaussian surface can pass through a\ncontinuous charge distribution (v) Gauss\u2019s law is often useful towards a much easier calculation of the\nelectrostatic field when the system has some symmetry This is\nfacilitated by the choice of a suitable Gaussian surface"}, {"Chapter": "1", "sentence_range": "955-958", "Text": ") However, the Gaussian surface can pass through a\ncontinuous charge distribution (v) Gauss\u2019s law is often useful towards a much easier calculation of the\nelectrostatic field when the system has some symmetry This is\nfacilitated by the choice of a suitable Gaussian surface (vi) Finally, Gauss\u2019s law is based on the inverse square dependence on\ndistance contained in the Coulomb\u2019s law"}, {"Chapter": "1", "sentence_range": "956-959", "Text": "(v) Gauss\u2019s law is often useful towards a much easier calculation of the\nelectrostatic field when the system has some symmetry This is\nfacilitated by the choice of a suitable Gaussian surface (vi) Finally, Gauss\u2019s law is based on the inverse square dependence on\ndistance contained in the Coulomb\u2019s law Any violation of Gauss\u2019s\nlaw will indicate departure from the inverse square law"}, {"Chapter": "1", "sentence_range": "957-960", "Text": "This is\nfacilitated by the choice of a suitable Gaussian surface (vi) Finally, Gauss\u2019s law is based on the inverse square dependence on\ndistance contained in the Coulomb\u2019s law Any violation of Gauss\u2019s\nlaw will indicate departure from the inverse square law Example 1"}, {"Chapter": "1", "sentence_range": "958-961", "Text": "(vi) Finally, Gauss\u2019s law is based on the inverse square dependence on\ndistance contained in the Coulomb\u2019s law Any violation of Gauss\u2019s\nlaw will indicate departure from the inverse square law Example 1 10 The electric field components in Fig"}, {"Chapter": "1", "sentence_range": "959-962", "Text": "Any violation of Gauss\u2019s\nlaw will indicate departure from the inverse square law Example 1 10 The electric field components in Fig 1"}, {"Chapter": "1", "sentence_range": "960-963", "Text": "Example 1 10 The electric field components in Fig 1 24 are\nEx = ax1/2, Ey = Ez = 0, in which a = 800 N/C m1/2"}, {"Chapter": "1", "sentence_range": "961-964", "Text": "10 The electric field components in Fig 1 24 are\nEx = ax1/2, Ey = Ez = 0, in which a = 800 N/C m1/2 Calculate (a) the\nflux through the cube, and (b) the charge within the cube"}, {"Chapter": "1", "sentence_range": "962-965", "Text": "1 24 are\nEx = ax1/2, Ey = Ez = 0, in which a = 800 N/C m1/2 Calculate (a) the\nflux through the cube, and (b) the charge within the cube Assume\nthat a = 0"}, {"Chapter": "1", "sentence_range": "963-966", "Text": "24 are\nEx = ax1/2, Ey = Ez = 0, in which a = 800 N/C m1/2 Calculate (a) the\nflux through the cube, and (b) the charge within the cube Assume\nthat a = 0 1 m"}, {"Chapter": "1", "sentence_range": "964-967", "Text": "Calculate (a) the\nflux through the cube, and (b) the charge within the cube Assume\nthat a = 0 1 m FIGURE 1"}, {"Chapter": "1", "sentence_range": "965-968", "Text": "Assume\nthat a = 0 1 m FIGURE 1 24\nSolution\n(a) Since the electric field has only an x component, for faces\nperpendicular to x direction, the angle between E and DS is\n\u00b1 p/2"}, {"Chapter": "1", "sentence_range": "966-969", "Text": "1 m FIGURE 1 24\nSolution\n(a) Since the electric field has only an x component, for faces\nperpendicular to x direction, the angle between E and DS is\n\u00b1 p/2 Therefore, the flux  f = E"}, {"Chapter": "1", "sentence_range": "967-970", "Text": "FIGURE 1 24\nSolution\n(a) Since the electric field has only an x component, for faces\nperpendicular to x direction, the angle between E and DS is\n\u00b1 p/2 Therefore, the flux  f = E DS is separately zero for each face\nof the cube except the two shaded ones"}, {"Chapter": "1", "sentence_range": "968-971", "Text": "24\nSolution\n(a) Since the electric field has only an x component, for faces\nperpendicular to x direction, the angle between E and DS is\n\u00b1 p/2 Therefore, the flux  f = E DS is separately zero for each face\nof the cube except the two shaded ones Now the magnitude of\nthe electric field at the left face is\nEL = ax1/2 = aa1/2\n(x = a at the left face)"}, {"Chapter": "1", "sentence_range": "969-972", "Text": "Therefore, the flux  f = E DS is separately zero for each face\nof the cube except the two shaded ones Now the magnitude of\nthe electric field at the left face is\nEL = ax1/2 = aa1/2\n(x = a at the left face) The magnitude of electric field at the right face is\nER = a x1/2 = a (2a)1/2\n(x = 2a at the right face)"}, {"Chapter": "1", "sentence_range": "970-973", "Text": "DS is separately zero for each face\nof the cube except the two shaded ones Now the magnitude of\nthe electric field at the left face is\nEL = ax1/2 = aa1/2\n(x = a at the left face) The magnitude of electric field at the right face is\nER = a x1/2 = a (2a)1/2\n(x = 2a at the right face) The corresponding fluxes are\nfL= EL"}, {"Chapter": "1", "sentence_range": "971-974", "Text": "Now the magnitude of\nthe electric field at the left face is\nEL = ax1/2 = aa1/2\n(x = a at the left face) The magnitude of electric field at the right face is\nER = a x1/2 = a (2a)1/2\n(x = 2a at the right face) The corresponding fluxes are\nfL= EL DS = \n\u02c6\nL\nL\n\u2206S\nE\n\u22c5n\n=EL DS cosq = \u2013EL DS, since q = 180\u00b0\n   = \u2013ELa2\nfR= ER"}, {"Chapter": "1", "sentence_range": "972-975", "Text": "The magnitude of electric field at the right face is\nER = a x1/2 = a (2a)1/2\n(x = 2a at the right face) The corresponding fluxes are\nfL= EL DS = \n\u02c6\nL\nL\n\u2206S\nE\n\u22c5n\n=EL DS cosq = \u2013EL DS, since q = 180\u00b0\n   = \u2013ELa2\nfR= ER DS = ER DS cosq  = ER DS,   since q = 0\u00b0\n   = ERa2\nNet flux through the cube\nRationalised 2023-24\n32\nPhysics\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "973-976", "Text": "The corresponding fluxes are\nfL= EL DS = \n\u02c6\nL\nL\n\u2206S\nE\n\u22c5n\n=EL DS cosq = \u2013EL DS, since q = 180\u00b0\n   = \u2013ELa2\nfR= ER DS = ER DS cosq  = ER DS,   since q = 0\u00b0\n   = ERa2\nNet flux through the cube\nRationalised 2023-24\n32\nPhysics\n EXAMPLE 1 11\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "974-977", "Text": "DS = \n\u02c6\nL\nL\n\u2206S\nE\n\u22c5n\n=EL DS cosq = \u2013EL DS, since q = 180\u00b0\n   = \u2013ELa2\nfR= ER DS = ER DS cosq  = ER DS,   since q = 0\u00b0\n   = ERa2\nNet flux through the cube\nRationalised 2023-24\n32\nPhysics\n EXAMPLE 1 11\n EXAMPLE 1 10\n= fR + fL = ERa2 \u2013 ELa2 = a2 (ER \u2013 EL) = aa2 [(2a)1/2 \u2013 a1/2]\n=  aa5/2 (\n)\n2 \u2013 1\n=  800 (0"}, {"Chapter": "1", "sentence_range": "975-978", "Text": "DS = ER DS cosq  = ER DS,   since q = 0\u00b0\n   = ERa2\nNet flux through the cube\nRationalised 2023-24\n32\nPhysics\n EXAMPLE 1 11\n EXAMPLE 1 10\n= fR + fL = ERa2 \u2013 ELa2 = a2 (ER \u2013 EL) = aa2 [(2a)1/2 \u2013 a1/2]\n=  aa5/2 (\n)\n2 \u2013 1\n=  800 (0 1)5/2 (\n)\n2 \u2013 1\n= 1"}, {"Chapter": "1", "sentence_range": "976-979", "Text": "11\n EXAMPLE 1 10\n= fR + fL = ERa2 \u2013 ELa2 = a2 (ER \u2013 EL) = aa2 [(2a)1/2 \u2013 a1/2]\n=  aa5/2 (\n)\n2 \u2013 1\n=  800 (0 1)5/2 (\n)\n2 \u2013 1\n= 1 05 N m2 C\u20131\n(b) We can use Gauss\u2019s law to find the total charge q inside the cube"}, {"Chapter": "1", "sentence_range": "977-980", "Text": "10\n= fR + fL = ERa2 \u2013 ELa2 = a2 (ER \u2013 EL) = aa2 [(2a)1/2 \u2013 a1/2]\n=  aa5/2 (\n)\n2 \u2013 1\n=  800 (0 1)5/2 (\n)\n2 \u2013 1\n= 1 05 N m2 C\u20131\n(b) We can use Gauss\u2019s law to find the total charge q inside the cube We have f = q/e0 or q = fe0"}, {"Chapter": "1", "sentence_range": "978-981", "Text": "1)5/2 (\n)\n2 \u2013 1\n= 1 05 N m2 C\u20131\n(b) We can use Gauss\u2019s law to find the total charge q inside the cube We have f = q/e0 or q = fe0 Therefore,\n  q = 1"}, {"Chapter": "1", "sentence_range": "979-982", "Text": "05 N m2 C\u20131\n(b) We can use Gauss\u2019s law to find the total charge q inside the cube We have f = q/e0 or q = fe0 Therefore,\n  q = 1 05 \u00d7 8"}, {"Chapter": "1", "sentence_range": "980-983", "Text": "We have f = q/e0 or q = fe0 Therefore,\n  q = 1 05 \u00d7 8 854 \u00d7 10\u201312 C = 9"}, {"Chapter": "1", "sentence_range": "981-984", "Text": "Therefore,\n  q = 1 05 \u00d7 8 854 \u00d7 10\u201312 C = 9 27 \u00d7 10\u201312 C"}, {"Chapter": "1", "sentence_range": "982-985", "Text": "05 \u00d7 8 854 \u00d7 10\u201312 C = 9 27 \u00d7 10\u201312 C Example 1"}, {"Chapter": "1", "sentence_range": "983-986", "Text": "854 \u00d7 10\u201312 C = 9 27 \u00d7 10\u201312 C Example 1 11 An electric field is uniform, and in the positive x\ndirection for positive x, and uniform with the same magnitude but in\nthe negative x direction for negative x"}, {"Chapter": "1", "sentence_range": "984-987", "Text": "27 \u00d7 10\u201312 C Example 1 11 An electric field is uniform, and in the positive x\ndirection for positive x, and uniform with the same magnitude but in\nthe negative x direction for negative x It is given that E = 200 \u02c6i   N/C\nfor x > 0 and E = \u2013200 \u02c6i  N/C for x < 0"}, {"Chapter": "1", "sentence_range": "985-988", "Text": "Example 1 11 An electric field is uniform, and in the positive x\ndirection for positive x, and uniform with the same magnitude but in\nthe negative x direction for negative x It is given that E = 200 \u02c6i   N/C\nfor x > 0 and E = \u2013200 \u02c6i  N/C for x < 0 A right circular cylinder of\nlength 20 cm and radius 5 cm has its centre at the origin and its axis\nalong the x-axis so that one face is at x = +10 cm and the other is at\nx = \u201310 cm (Fig"}, {"Chapter": "1", "sentence_range": "986-989", "Text": "11 An electric field is uniform, and in the positive x\ndirection for positive x, and uniform with the same magnitude but in\nthe negative x direction for negative x It is given that E = 200 \u02c6i   N/C\nfor x > 0 and E = \u2013200 \u02c6i  N/C for x < 0 A right circular cylinder of\nlength 20 cm and radius 5 cm has its centre at the origin and its axis\nalong the x-axis so that one face is at x = +10 cm and the other is at\nx = \u201310 cm (Fig 1"}, {"Chapter": "1", "sentence_range": "987-990", "Text": "It is given that E = 200 \u02c6i   N/C\nfor x > 0 and E = \u2013200 \u02c6i  N/C for x < 0 A right circular cylinder of\nlength 20 cm and radius 5 cm has its centre at the origin and its axis\nalong the x-axis so that one face is at x = +10 cm and the other is at\nx = \u201310 cm (Fig 1 25)"}, {"Chapter": "1", "sentence_range": "988-991", "Text": "A right circular cylinder of\nlength 20 cm and radius 5 cm has its centre at the origin and its axis\nalong the x-axis so that one face is at x = +10 cm and the other is at\nx = \u201310 cm (Fig 1 25) (a) What is the net outward flux through each\nflat face"}, {"Chapter": "1", "sentence_range": "989-992", "Text": "1 25) (a) What is the net outward flux through each\nflat face (b) What is the flux through the side of the cylinder"}, {"Chapter": "1", "sentence_range": "990-993", "Text": "25) (a) What is the net outward flux through each\nflat face (b) What is the flux through the side of the cylinder (c) What is the net outward flux through the cylinder"}, {"Chapter": "1", "sentence_range": "991-994", "Text": "(a) What is the net outward flux through each\nflat face (b) What is the flux through the side of the cylinder (c) What is the net outward flux through the cylinder (d) What is the\nnet charge inside the cylinder"}, {"Chapter": "1", "sentence_range": "992-995", "Text": "(b) What is the flux through the side of the cylinder (c) What is the net outward flux through the cylinder (d) What is the\nnet charge inside the cylinder Solution\n(a)\nWe can see from the figure that on the left face E and DS are\nparallel"}, {"Chapter": "1", "sentence_range": "993-996", "Text": "(c) What is the net outward flux through the cylinder (d) What is the\nnet charge inside the cylinder Solution\n(a)\nWe can see from the figure that on the left face E and DS are\nparallel Therefore, the outward flux is\nfL= E"}, {"Chapter": "1", "sentence_range": "994-997", "Text": "(d) What is the\nnet charge inside the cylinder Solution\n(a)\nWe can see from the figure that on the left face E and DS are\nparallel Therefore, the outward flux is\nfL= E DS = \u2013 200 \u02c6 \u2206\ni\nS\ni\n=  + 200 DS, since  \u02c6 \u2206\ni\niS\n= \u2013 DS\n=  + 200 \u00d7 p (0"}, {"Chapter": "1", "sentence_range": "995-998", "Text": "Solution\n(a)\nWe can see from the figure that on the left face E and DS are\nparallel Therefore, the outward flux is\nfL= E DS = \u2013 200 \u02c6 \u2206\ni\nS\ni\n=  + 200 DS, since  \u02c6 \u2206\ni\niS\n= \u2013 DS\n=  + 200 \u00d7 p (0 05)2 = + 1"}, {"Chapter": "1", "sentence_range": "996-999", "Text": "Therefore, the outward flux is\nfL= E DS = \u2013 200 \u02c6 \u2206\ni\nS\ni\n=  + 200 DS, since  \u02c6 \u2206\ni\niS\n= \u2013 DS\n=  + 200 \u00d7 p (0 05)2 = + 1 57 N m2 C\u20131\nOn the right face, E and DS are parallel and therefore\nfR =  E"}, {"Chapter": "1", "sentence_range": "997-1000", "Text": "DS = \u2013 200 \u02c6 \u2206\ni\nS\ni\n=  + 200 DS, since  \u02c6 \u2206\ni\niS\n= \u2013 DS\n=  + 200 \u00d7 p (0 05)2 = + 1 57 N m2 C\u20131\nOn the right face, E and DS are parallel and therefore\nfR =  E DS =  + 1"}, {"Chapter": "1", "sentence_range": "998-1001", "Text": "05)2 = + 1 57 N m2 C\u20131\nOn the right face, E and DS are parallel and therefore\nfR =  E DS =  + 1 57 N m2 C\u20131"}, {"Chapter": "1", "sentence_range": "999-1002", "Text": "57 N m2 C\u20131\nOn the right face, E and DS are parallel and therefore\nfR =  E DS =  + 1 57 N m2 C\u20131 (b)\nFor any point on the side of the cylinder E is perpendicular to\nDS and hence E"}, {"Chapter": "1", "sentence_range": "1000-1003", "Text": "DS =  + 1 57 N m2 C\u20131 (b)\nFor any point on the side of the cylinder E is perpendicular to\nDS and hence E DS = 0"}, {"Chapter": "1", "sentence_range": "1001-1004", "Text": "57 N m2 C\u20131 (b)\nFor any point on the side of the cylinder E is perpendicular to\nDS and hence E DS = 0 Therefore, the flux out of the side of the\ncylinder is zero"}, {"Chapter": "1", "sentence_range": "1002-1005", "Text": "(b)\nFor any point on the side of the cylinder E is perpendicular to\nDS and hence E DS = 0 Therefore, the flux out of the side of the\ncylinder is zero (c)\nNet outward flux through the cylinder\nf = 1"}, {"Chapter": "1", "sentence_range": "1003-1006", "Text": "DS = 0 Therefore, the flux out of the side of the\ncylinder is zero (c)\nNet outward flux through the cylinder\nf = 1 57 + 1"}, {"Chapter": "1", "sentence_range": "1004-1007", "Text": "Therefore, the flux out of the side of the\ncylinder is zero (c)\nNet outward flux through the cylinder\nf = 1 57 + 1 57 + 0 = 3"}, {"Chapter": "1", "sentence_range": "1005-1008", "Text": "(c)\nNet outward flux through the cylinder\nf = 1 57 + 1 57 + 0 = 3 14 N m2 C\u20131\nFIGURE 1"}, {"Chapter": "1", "sentence_range": "1006-1009", "Text": "57 + 1 57 + 0 = 3 14 N m2 C\u20131\nFIGURE 1 25\n (d) The net charge within the cylinder can be found by using Gauss\u2019s\nlaw which gives\nq =  e0f\n   =  3"}, {"Chapter": "1", "sentence_range": "1007-1010", "Text": "57 + 0 = 3 14 N m2 C\u20131\nFIGURE 1 25\n (d) The net charge within the cylinder can be found by using Gauss\u2019s\nlaw which gives\nq =  e0f\n   =  3 14 \u00d7 8"}, {"Chapter": "1", "sentence_range": "1008-1011", "Text": "14 N m2 C\u20131\nFIGURE 1 25\n (d) The net charge within the cylinder can be found by using Gauss\u2019s\nlaw which gives\nq =  e0f\n   =  3 14 \u00d7 8 854 \u00d7 10\u201312  C\n   =  2"}, {"Chapter": "1", "sentence_range": "1009-1012", "Text": "25\n (d) The net charge within the cylinder can be found by using Gauss\u2019s\nlaw which gives\nq =  e0f\n   =  3 14 \u00d7 8 854 \u00d7 10\u201312  C\n   =  2 78 \u00d7 10\u201311 C\nRationalised 2023-24\nElectric Charges\nand Fields\n33\n1"}, {"Chapter": "1", "sentence_range": "1010-1013", "Text": "14 \u00d7 8 854 \u00d7 10\u201312  C\n   =  2 78 \u00d7 10\u201311 C\nRationalised 2023-24\nElectric Charges\nand Fields\n33\n1 14  APPLICATIONS OF GAUSS\u2019S LAW\nThe electric field due to a general charge distribution is, as seen above,\ngiven by Eq"}, {"Chapter": "1", "sentence_range": "1011-1014", "Text": "854 \u00d7 10\u201312  C\n   =  2 78 \u00d7 10\u201311 C\nRationalised 2023-24\nElectric Charges\nand Fields\n33\n1 14  APPLICATIONS OF GAUSS\u2019S LAW\nThe electric field due to a general charge distribution is, as seen above,\ngiven by Eq (1"}, {"Chapter": "1", "sentence_range": "1012-1015", "Text": "78 \u00d7 10\u201311 C\nRationalised 2023-24\nElectric Charges\nand Fields\n33\n1 14  APPLICATIONS OF GAUSS\u2019S LAW\nThe electric field due to a general charge distribution is, as seen above,\ngiven by Eq (1 27)"}, {"Chapter": "1", "sentence_range": "1013-1016", "Text": "14  APPLICATIONS OF GAUSS\u2019S LAW\nThe electric field due to a general charge distribution is, as seen above,\ngiven by Eq (1 27) In practice, except for some special cases, the\nsummation (or integration) involved in this equation cannot be carried\nout to give electric field at every point in\nspace"}, {"Chapter": "1", "sentence_range": "1014-1017", "Text": "(1 27) In practice, except for some special cases, the\nsummation (or integration) involved in this equation cannot be carried\nout to give electric field at every point in\nspace For some symmetric charge\nconfigurations, however, it is possible to\nobtain the electric field in a simple way using\nthe Gauss\u2019s law"}, {"Chapter": "1", "sentence_range": "1015-1018", "Text": "27) In practice, except for some special cases, the\nsummation (or integration) involved in this equation cannot be carried\nout to give electric field at every point in\nspace For some symmetric charge\nconfigurations, however, it is possible to\nobtain the electric field in a simple way using\nthe Gauss\u2019s law This is best understood by\nsome examples"}, {"Chapter": "1", "sentence_range": "1016-1019", "Text": "In practice, except for some special cases, the\nsummation (or integration) involved in this equation cannot be carried\nout to give electric field at every point in\nspace For some symmetric charge\nconfigurations, however, it is possible to\nobtain the electric field in a simple way using\nthe Gauss\u2019s law This is best understood by\nsome examples 1"}, {"Chapter": "1", "sentence_range": "1017-1020", "Text": "For some symmetric charge\nconfigurations, however, it is possible to\nobtain the electric field in a simple way using\nthe Gauss\u2019s law This is best understood by\nsome examples 1 14"}, {"Chapter": "1", "sentence_range": "1018-1021", "Text": "This is best understood by\nsome examples 1 14 1\nField due to an infinitely\nlong straight uniformly\ncharged wire\nConsider an infinitely long thin straight wire\nwith uniform linear charge density l"}, {"Chapter": "1", "sentence_range": "1019-1022", "Text": "1 14 1\nField due to an infinitely\nlong straight uniformly\ncharged wire\nConsider an infinitely long thin straight wire\nwith uniform linear charge density l The wire\nis obviously an axis of symmetry"}, {"Chapter": "1", "sentence_range": "1020-1023", "Text": "14 1\nField due to an infinitely\nlong straight uniformly\ncharged wire\nConsider an infinitely long thin straight wire\nwith uniform linear charge density l The wire\nis obviously an axis of symmetry Suppose we\ntake the radial vector from O to P and rotate it\naround the wire"}, {"Chapter": "1", "sentence_range": "1021-1024", "Text": "1\nField due to an infinitely\nlong straight uniformly\ncharged wire\nConsider an infinitely long thin straight wire\nwith uniform linear charge density l The wire\nis obviously an axis of symmetry Suppose we\ntake the radial vector from O to P and rotate it\naround the wire The points P, P\u00a2, P\u00a2\u00a2 so\nobtained are completely equivalent with\nrespect to the charged wire"}, {"Chapter": "1", "sentence_range": "1022-1025", "Text": "The wire\nis obviously an axis of symmetry Suppose we\ntake the radial vector from O to P and rotate it\naround the wire The points P, P\u00a2, P\u00a2\u00a2 so\nobtained are completely equivalent with\nrespect to the charged wire This implies that\nthe electric field must have the same magnitude\nat these points"}, {"Chapter": "1", "sentence_range": "1023-1026", "Text": "Suppose we\ntake the radial vector from O to P and rotate it\naround the wire The points P, P\u00a2, P\u00a2\u00a2 so\nobtained are completely equivalent with\nrespect to the charged wire This implies that\nthe electric field must have the same magnitude\nat these points The direction of electric field at\nevery point must be radial (outward if l > 0,\ninward if l < 0)"}, {"Chapter": "1", "sentence_range": "1024-1027", "Text": "The points P, P\u00a2, P\u00a2\u00a2 so\nobtained are completely equivalent with\nrespect to the charged wire This implies that\nthe electric field must have the same magnitude\nat these points The direction of electric field at\nevery point must be radial (outward if l > 0,\ninward if l < 0) This is clear from Fig"}, {"Chapter": "1", "sentence_range": "1025-1028", "Text": "This implies that\nthe electric field must have the same magnitude\nat these points The direction of electric field at\nevery point must be radial (outward if l > 0,\ninward if l < 0) This is clear from Fig 1"}, {"Chapter": "1", "sentence_range": "1026-1029", "Text": "The direction of electric field at\nevery point must be radial (outward if l > 0,\ninward if l < 0) This is clear from Fig 1 26"}, {"Chapter": "1", "sentence_range": "1027-1030", "Text": "This is clear from Fig 1 26 Consider a pair of line elements P1 and P2\nof the wire, as shown"}, {"Chapter": "1", "sentence_range": "1028-1031", "Text": "1 26 Consider a pair of line elements P1 and P2\nof the wire, as shown The electric fields\nproduced by the two elements of the pair when\nsummed give a resultant electric field which\nis radial (the components normal to the radial\nvector cancel)"}, {"Chapter": "1", "sentence_range": "1029-1032", "Text": "26 Consider a pair of line elements P1 and P2\nof the wire, as shown The electric fields\nproduced by the two elements of the pair when\nsummed give a resultant electric field which\nis radial (the components normal to the radial\nvector cancel) This is true for any such pair\nand hence the total field at any point P is\nradial"}, {"Chapter": "1", "sentence_range": "1030-1033", "Text": "Consider a pair of line elements P1 and P2\nof the wire, as shown The electric fields\nproduced by the two elements of the pair when\nsummed give a resultant electric field which\nis radial (the components normal to the radial\nvector cancel) This is true for any such pair\nand hence the total field at any point P is\nradial Finally, since the wire is infinite,\nelectric field does not depend on the position\nof P along the length of the wire"}, {"Chapter": "1", "sentence_range": "1031-1034", "Text": "The electric fields\nproduced by the two elements of the pair when\nsummed give a resultant electric field which\nis radial (the components normal to the radial\nvector cancel) This is true for any such pair\nand hence the total field at any point P is\nradial Finally, since the wire is infinite,\nelectric field does not depend on the position\nof P along the length of the wire In short, the\nelectric field is everywhere radial in the plane\ncutting the wire normally, and its magnitude\ndepends only on the radial distance r"}, {"Chapter": "1", "sentence_range": "1032-1035", "Text": "This is true for any such pair\nand hence the total field at any point P is\nradial Finally, since the wire is infinite,\nelectric field does not depend on the position\nof P along the length of the wire In short, the\nelectric field is everywhere radial in the plane\ncutting the wire normally, and its magnitude\ndepends only on the radial distance r To calculate the field, imagine a cylindrical\nGaussian surface, as shown in the Fig"}, {"Chapter": "1", "sentence_range": "1033-1036", "Text": "Finally, since the wire is infinite,\nelectric field does not depend on the position\nof P along the length of the wire In short, the\nelectric field is everywhere radial in the plane\ncutting the wire normally, and its magnitude\ndepends only on the radial distance r To calculate the field, imagine a cylindrical\nGaussian surface, as shown in the Fig 1"}, {"Chapter": "1", "sentence_range": "1034-1037", "Text": "In short, the\nelectric field is everywhere radial in the plane\ncutting the wire normally, and its magnitude\ndepends only on the radial distance r To calculate the field, imagine a cylindrical\nGaussian surface, as shown in the Fig 1 26(b)"}, {"Chapter": "1", "sentence_range": "1035-1038", "Text": "To calculate the field, imagine a cylindrical\nGaussian surface, as shown in the Fig 1 26(b) Since the field is everywhere radial, flux\nthrough the two ends of the cylindrical\nGaussian surface is zero"}, {"Chapter": "1", "sentence_range": "1036-1039", "Text": "1 26(b) Since the field is everywhere radial, flux\nthrough the two ends of the cylindrical\nGaussian surface is zero At the cylindrical\npart of the surface, E is normal to the surface\nat every point, and its magnitude is constant,\nsince it depends only on r"}, {"Chapter": "1", "sentence_range": "1037-1040", "Text": "26(b) Since the field is everywhere radial, flux\nthrough the two ends of the cylindrical\nGaussian surface is zero At the cylindrical\npart of the surface, E is normal to the surface\nat every point, and its magnitude is constant,\nsince it depends only on r The surface area\nof the curved part is  2prl, where l is the length\nof the cylinder"}, {"Chapter": "1", "sentence_range": "1038-1041", "Text": "Since the field is everywhere radial, flux\nthrough the two ends of the cylindrical\nGaussian surface is zero At the cylindrical\npart of the surface, E is normal to the surface\nat every point, and its magnitude is constant,\nsince it depends only on r The surface area\nof the curved part is  2prl, where l is the length\nof the cylinder FIGURE 1"}, {"Chapter": "1", "sentence_range": "1039-1042", "Text": "At the cylindrical\npart of the surface, E is normal to the surface\nat every point, and its magnitude is constant,\nsince it depends only on r The surface area\nof the curved part is  2prl, where l is the length\nof the cylinder FIGURE 1 26 (a) Electric field due to an\ninfinitely long thin straight wire is radial,\n(b) The Gaussian surface for a long thin\nwire of uniform linear charge density"}, {"Chapter": "1", "sentence_range": "1040-1043", "Text": "The surface area\nof the curved part is  2prl, where l is the length\nof the cylinder FIGURE 1 26 (a) Electric field due to an\ninfinitely long thin straight wire is radial,\n(b) The Gaussian surface for a long thin\nwire of uniform linear charge density Rationalised 2023-24\n34\nPhysics\nFlux through the Gaussian surface\n=\nflux through the curved cylindrical part of the surface\n=\nE \u00d7 2prl\nThe surface includes charge equal to l l"}, {"Chapter": "1", "sentence_range": "1041-1044", "Text": "FIGURE 1 26 (a) Electric field due to an\ninfinitely long thin straight wire is radial,\n(b) The Gaussian surface for a long thin\nwire of uniform linear charge density Rationalised 2023-24\n34\nPhysics\nFlux through the Gaussian surface\n=\nflux through the curved cylindrical part of the surface\n=\nE \u00d7 2prl\nThe surface includes charge equal to l l Gauss\u2019s law then gives\nE \u00d7 2prl = ll/e0\ni"}, {"Chapter": "1", "sentence_range": "1042-1045", "Text": "26 (a) Electric field due to an\ninfinitely long thin straight wire is radial,\n(b) The Gaussian surface for a long thin\nwire of uniform linear charge density Rationalised 2023-24\n34\nPhysics\nFlux through the Gaussian surface\n=\nflux through the curved cylindrical part of the surface\n=\nE \u00d7 2prl\nThe surface includes charge equal to l l Gauss\u2019s law then gives\nE \u00d7 2prl = ll/e0\ni e"}, {"Chapter": "1", "sentence_range": "1043-1046", "Text": "Rationalised 2023-24\n34\nPhysics\nFlux through the Gaussian surface\n=\nflux through the curved cylindrical part of the surface\n=\nE \u00d7 2prl\nThe surface includes charge equal to l l Gauss\u2019s law then gives\nE \u00d7 2prl = ll/e0\ni e ,E  = \n0\n2\nr\n\u03b5\u03bb\n\u03c0\nVectorially, E at any point is given by\n0\n\u02c6\n2\nr\n\u03b5\u03bb\n=\n\u03c0\nE\nn\n(1"}, {"Chapter": "1", "sentence_range": "1044-1047", "Text": "Gauss\u2019s law then gives\nE \u00d7 2prl = ll/e0\ni e ,E  = \n0\n2\nr\n\u03b5\u03bb\n\u03c0\nVectorially, E at any point is given by\n0\n\u02c6\n2\nr\n\u03b5\u03bb\n=\n\u03c0\nE\nn\n(1 32)\nwhere \u02c6n  is the radial unit vector in the plane normal to the wire passing\nthrough the point"}, {"Chapter": "1", "sentence_range": "1045-1048", "Text": "e ,E  = \n0\n2\nr\n\u03b5\u03bb\n\u03c0\nVectorially, E at any point is given by\n0\n\u02c6\n2\nr\n\u03b5\u03bb\n=\n\u03c0\nE\nn\n(1 32)\nwhere \u02c6n  is the radial unit vector in the plane normal to the wire passing\nthrough the point E is directed outward if l is positive and inward if l is\nnegative"}, {"Chapter": "1", "sentence_range": "1046-1049", "Text": ",E  = \n0\n2\nr\n\u03b5\u03bb\n\u03c0\nVectorially, E at any point is given by\n0\n\u02c6\n2\nr\n\u03b5\u03bb\n=\n\u03c0\nE\nn\n(1 32)\nwhere \u02c6n  is the radial unit vector in the plane normal to the wire passing\nthrough the point E is directed outward if l is positive and inward if l is\nnegative Note that when we write a vector A as a scalar multiplied by a unit\nvector, i"}, {"Chapter": "1", "sentence_range": "1047-1050", "Text": "32)\nwhere \u02c6n  is the radial unit vector in the plane normal to the wire passing\nthrough the point E is directed outward if l is positive and inward if l is\nnegative Note that when we write a vector A as a scalar multiplied by a unit\nvector, i e"}, {"Chapter": "1", "sentence_range": "1048-1051", "Text": "E is directed outward if l is positive and inward if l is\nnegative Note that when we write a vector A as a scalar multiplied by a unit\nvector, i e , as A = A \u02c6a , the scalar A is  an algebraic number"}, {"Chapter": "1", "sentence_range": "1049-1052", "Text": "Note that when we write a vector A as a scalar multiplied by a unit\nvector, i e , as A = A \u02c6a , the scalar A is  an algebraic number It can be\nnegative or positive"}, {"Chapter": "1", "sentence_range": "1050-1053", "Text": "e , as A = A \u02c6a , the scalar A is  an algebraic number It can be\nnegative or positive The direction of A will be the same as that of the unit\nvector \u02c6a if A > 0 and opposite to \u02c6a  if A < 0"}, {"Chapter": "1", "sentence_range": "1051-1054", "Text": ", as A = A \u02c6a , the scalar A is  an algebraic number It can be\nnegative or positive The direction of A will be the same as that of the unit\nvector \u02c6a if A > 0 and opposite to \u02c6a  if A < 0 When we want to restrict to\nnon-negative values, we use the symbol A and call it the modulus of A"}, {"Chapter": "1", "sentence_range": "1052-1055", "Text": "It can be\nnegative or positive The direction of A will be the same as that of the unit\nvector \u02c6a if A > 0 and opposite to \u02c6a  if A < 0 When we want to restrict to\nnon-negative values, we use the symbol A and call it the modulus of A Thus, \nA\u22650"}, {"Chapter": "1", "sentence_range": "1053-1056", "Text": "The direction of A will be the same as that of the unit\nvector \u02c6a if A > 0 and opposite to \u02c6a  if A < 0 When we want to restrict to\nnon-negative values, we use the symbol A and call it the modulus of A Thus, \nA\u22650 Also note that though only  the charge enclosed by the surface (ll)\nwas included above, the electric field E is due to the charge on the entire\nwire"}, {"Chapter": "1", "sentence_range": "1054-1057", "Text": "When we want to restrict to\nnon-negative values, we use the symbol A and call it the modulus of A Thus, \nA\u22650 Also note that though only  the charge enclosed by the surface (ll)\nwas included above, the electric field E is due to the charge on the entire\nwire Further, the assumption that the wire is infinitely long is crucial"}, {"Chapter": "1", "sentence_range": "1055-1058", "Text": "Thus, \nA\u22650 Also note that though only  the charge enclosed by the surface (ll)\nwas included above, the electric field E is due to the charge on the entire\nwire Further, the assumption that the wire is infinitely long is crucial Without this assumption, we cannot take E to be normal to the curved\npart of the cylindrical Gaussian surface"}, {"Chapter": "1", "sentence_range": "1056-1059", "Text": "Also note that though only  the charge enclosed by the surface (ll)\nwas included above, the electric field E is due to the charge on the entire\nwire Further, the assumption that the wire is infinitely long is crucial Without this assumption, we cannot take E to be normal to the curved\npart of the cylindrical Gaussian surface However, Eq"}, {"Chapter": "1", "sentence_range": "1057-1060", "Text": "Further, the assumption that the wire is infinitely long is crucial Without this assumption, we cannot take E to be normal to the curved\npart of the cylindrical Gaussian surface However, Eq (1"}, {"Chapter": "1", "sentence_range": "1058-1061", "Text": "Without this assumption, we cannot take E to be normal to the curved\npart of the cylindrical Gaussian surface However, Eq (1 32) is\napproximately true for electric field around the central portions of a long\nwire, where the end effects may be ignored"}, {"Chapter": "1", "sentence_range": "1059-1062", "Text": "However, Eq (1 32) is\napproximately true for electric field around the central portions of a long\nwire, where the end effects may be ignored 1"}, {"Chapter": "1", "sentence_range": "1060-1063", "Text": "(1 32) is\napproximately true for electric field around the central portions of a long\nwire, where the end effects may be ignored 1 14"}, {"Chapter": "1", "sentence_range": "1061-1064", "Text": "32) is\napproximately true for electric field around the central portions of a long\nwire, where the end effects may be ignored 1 14 2  Field due to a uniformly charged infinite plane sheet\nLet s be the uniform surface charge density of an infinite plane sheet\n(Fig"}, {"Chapter": "1", "sentence_range": "1062-1065", "Text": "1 14 2  Field due to a uniformly charged infinite plane sheet\nLet s be the uniform surface charge density of an infinite plane sheet\n(Fig 1"}, {"Chapter": "1", "sentence_range": "1063-1066", "Text": "14 2  Field due to a uniformly charged infinite plane sheet\nLet s be the uniform surface charge density of an infinite plane sheet\n(Fig 1 27)"}, {"Chapter": "1", "sentence_range": "1064-1067", "Text": "2  Field due to a uniformly charged infinite plane sheet\nLet s be the uniform surface charge density of an infinite plane sheet\n(Fig 1 27) We take the x-axis normal to the given plane"}, {"Chapter": "1", "sentence_range": "1065-1068", "Text": "1 27) We take the x-axis normal to the given plane By symmetry,\nthe electric field will not depend on y and z coordinates and its direction\nat every point must be parallel to the x-direction"}, {"Chapter": "1", "sentence_range": "1066-1069", "Text": "27) We take the x-axis normal to the given plane By symmetry,\nthe electric field will not depend on y and z coordinates and its direction\nat every point must be parallel to the x-direction We can take the Gaussian surface to be a\nrectangular parallelepiped of cross-sectional area\nA, as shown"}, {"Chapter": "1", "sentence_range": "1067-1070", "Text": "We take the x-axis normal to the given plane By symmetry,\nthe electric field will not depend on y and z coordinates and its direction\nat every point must be parallel to the x-direction We can take the Gaussian surface to be a\nrectangular parallelepiped of cross-sectional area\nA, as shown (A cylindrical surface will also do"}, {"Chapter": "1", "sentence_range": "1068-1071", "Text": "By symmetry,\nthe electric field will not depend on y and z coordinates and its direction\nat every point must be parallel to the x-direction We can take the Gaussian surface to be a\nrectangular parallelepiped of cross-sectional area\nA, as shown (A cylindrical surface will also do ) As\nseen from the figure, only the two faces 1 and 2 will\ncontribute to the flux; electric field lines are parallel\nto the other faces and they, therefore, do not\ncontribute to the total flux"}, {"Chapter": "1", "sentence_range": "1069-1072", "Text": "We can take the Gaussian surface to be a\nrectangular parallelepiped of cross-sectional area\nA, as shown (A cylindrical surface will also do ) As\nseen from the figure, only the two faces 1 and 2 will\ncontribute to the flux; electric field lines are parallel\nto the other faces and they, therefore, do not\ncontribute to the total flux The unit vector normal to surface 1 is in \u2013x\ndirection while  the unit  vector normal to surface 2\nis in the +x direction"}, {"Chapter": "1", "sentence_range": "1070-1073", "Text": "(A cylindrical surface will also do ) As\nseen from the figure, only the two faces 1 and 2 will\ncontribute to the flux; electric field lines are parallel\nto the other faces and they, therefore, do not\ncontribute to the total flux The unit vector normal to surface 1 is in \u2013x\ndirection while  the unit  vector normal to surface 2\nis in the +x direction Therefore, flux  E"}, {"Chapter": "1", "sentence_range": "1071-1074", "Text": ") As\nseen from the figure, only the two faces 1 and 2 will\ncontribute to the flux; electric field lines are parallel\nto the other faces and they, therefore, do not\ncontribute to the total flux The unit vector normal to surface 1 is in \u2013x\ndirection while  the unit  vector normal to surface 2\nis in the +x direction Therefore, flux  E DS through\nboth the surfaces are equal and add up"}, {"Chapter": "1", "sentence_range": "1072-1075", "Text": "The unit vector normal to surface 1 is in \u2013x\ndirection while  the unit  vector normal to surface 2\nis in the +x direction Therefore, flux  E DS through\nboth the surfaces are equal and add up Therefore\nthe net flux through the Gaussian surface is 2 EA"}, {"Chapter": "1", "sentence_range": "1073-1076", "Text": "Therefore, flux  E DS through\nboth the surfaces are equal and add up Therefore\nthe net flux through the Gaussian surface is 2 EA The charge enclosed by the closed surface is sA"}, {"Chapter": "1", "sentence_range": "1074-1077", "Text": "DS through\nboth the surfaces are equal and add up Therefore\nthe net flux through the Gaussian surface is 2 EA The charge enclosed by the closed surface is sA Therefore by Gauss\u2019s law,\nFIGURE 1"}, {"Chapter": "1", "sentence_range": "1075-1078", "Text": "Therefore\nthe net flux through the Gaussian surface is 2 EA The charge enclosed by the closed surface is sA Therefore by Gauss\u2019s law,\nFIGURE 1 27 Gaussian surface for a\nuniformly charged infinite plane sheet"}, {"Chapter": "1", "sentence_range": "1076-1079", "Text": "The charge enclosed by the closed surface is sA Therefore by Gauss\u2019s law,\nFIGURE 1 27 Gaussian surface for a\nuniformly charged infinite plane sheet Rationalised 2023-24\nElectric Charges\nand Fields\n35\n2 EA = sA/e0\nor,  E = s/2e0\nVectorically,\n0\n\u02c6\n2\n\u03b5\u03c3\nE=\nn\n(1"}, {"Chapter": "1", "sentence_range": "1077-1080", "Text": "Therefore by Gauss\u2019s law,\nFIGURE 1 27 Gaussian surface for a\nuniformly charged infinite plane sheet Rationalised 2023-24\nElectric Charges\nand Fields\n35\n2 EA = sA/e0\nor,  E = s/2e0\nVectorically,\n0\n\u02c6\n2\n\u03b5\u03c3\nE=\nn\n(1 33)\nwhere \u02c6n  is a unit vector normal to the plane and going away from it"}, {"Chapter": "1", "sentence_range": "1078-1081", "Text": "27 Gaussian surface for a\nuniformly charged infinite plane sheet Rationalised 2023-24\nElectric Charges\nand Fields\n35\n2 EA = sA/e0\nor,  E = s/2e0\nVectorically,\n0\n\u02c6\n2\n\u03b5\u03c3\nE=\nn\n(1 33)\nwhere \u02c6n  is a unit vector normal to the plane and going away from it E is directed away from the plate if s  is positive and toward the plate\nif s is negative"}, {"Chapter": "1", "sentence_range": "1079-1082", "Text": "Rationalised 2023-24\nElectric Charges\nand Fields\n35\n2 EA = sA/e0\nor,  E = s/2e0\nVectorically,\n0\n\u02c6\n2\n\u03b5\u03c3\nE=\nn\n(1 33)\nwhere \u02c6n  is a unit vector normal to the plane and going away from it E is directed away from the plate if s  is positive and toward the plate\nif s is negative Note that the above application of the Gauss\u2019 law has\nbrought out an additional fact: E is independent of x also"}, {"Chapter": "1", "sentence_range": "1080-1083", "Text": "33)\nwhere \u02c6n  is a unit vector normal to the plane and going away from it E is directed away from the plate if s  is positive and toward the plate\nif s is negative Note that the above application of the Gauss\u2019 law has\nbrought out an additional fact: E is independent of x also For a finite large planar sheet, Eq"}, {"Chapter": "1", "sentence_range": "1081-1084", "Text": "E is directed away from the plate if s  is positive and toward the plate\nif s is negative Note that the above application of the Gauss\u2019 law has\nbrought out an additional fact: E is independent of x also For a finite large planar sheet, Eq (1"}, {"Chapter": "1", "sentence_range": "1082-1085", "Text": "Note that the above application of the Gauss\u2019 law has\nbrought out an additional fact: E is independent of x also For a finite large planar sheet, Eq (1 33) is approximately true in the\nmiddle regions of the planar sheet, away from the ends"}, {"Chapter": "1", "sentence_range": "1083-1086", "Text": "For a finite large planar sheet, Eq (1 33) is approximately true in the\nmiddle regions of the planar sheet, away from the ends 1"}, {"Chapter": "1", "sentence_range": "1084-1087", "Text": "(1 33) is approximately true in the\nmiddle regions of the planar sheet, away from the ends 1 14"}, {"Chapter": "1", "sentence_range": "1085-1088", "Text": "33) is approximately true in the\nmiddle regions of the planar sheet, away from the ends 1 14 3  Field due to a uniformly charged thin spherical shell\nLet s be the uniform surface charge density of a thin spherical shell of\nradius R (Fig"}, {"Chapter": "1", "sentence_range": "1086-1089", "Text": "1 14 3  Field due to a uniformly charged thin spherical shell\nLet s be the uniform surface charge density of a thin spherical shell of\nradius R (Fig 1"}, {"Chapter": "1", "sentence_range": "1087-1090", "Text": "14 3  Field due to a uniformly charged thin spherical shell\nLet s be the uniform surface charge density of a thin spherical shell of\nradius R (Fig 1 28)"}, {"Chapter": "1", "sentence_range": "1088-1091", "Text": "3  Field due to a uniformly charged thin spherical shell\nLet s be the uniform surface charge density of a thin spherical shell of\nradius R (Fig 1 28) The situation has obvious spherical symmetry"}, {"Chapter": "1", "sentence_range": "1089-1092", "Text": "1 28) The situation has obvious spherical symmetry The\nfield at any point P, outside or inside, can depend only on r (the radial\ndistance from the centre of the shell to the point) and must be radial (i"}, {"Chapter": "1", "sentence_range": "1090-1093", "Text": "28) The situation has obvious spherical symmetry The\nfield at any point P, outside or inside, can depend only on r (the radial\ndistance from the centre of the shell to the point) and must be radial (i e"}, {"Chapter": "1", "sentence_range": "1091-1094", "Text": "The situation has obvious spherical symmetry The\nfield at any point P, outside or inside, can depend only on r (the radial\ndistance from the centre of the shell to the point) and must be radial (i e ,\nalong the radius vector)"}, {"Chapter": "1", "sentence_range": "1092-1095", "Text": "The\nfield at any point P, outside or inside, can depend only on r (the radial\ndistance from the centre of the shell to the point) and must be radial (i e ,\nalong the radius vector) (i) Field outside the shell: Consider a point P outside the\nshell with radius vector r"}, {"Chapter": "1", "sentence_range": "1093-1096", "Text": "e ,\nalong the radius vector) (i) Field outside the shell: Consider a point P outside the\nshell with radius vector r To calculate E at P, we take the\nGaussian surface to be a sphere of radius r and with centre\nO, passing through P"}, {"Chapter": "1", "sentence_range": "1094-1097", "Text": ",\nalong the radius vector) (i) Field outside the shell: Consider a point P outside the\nshell with radius vector r To calculate E at P, we take the\nGaussian surface to be a sphere of radius r and with centre\nO, passing through P All points on this sphere are equivalent\nrelative to the given charged configuration"}, {"Chapter": "1", "sentence_range": "1095-1098", "Text": "(i) Field outside the shell: Consider a point P outside the\nshell with radius vector r To calculate E at P, we take the\nGaussian surface to be a sphere of radius r and with centre\nO, passing through P All points on this sphere are equivalent\nrelative to the given charged configuration (That is what we\nmean by spherical symmetry"}, {"Chapter": "1", "sentence_range": "1096-1099", "Text": "To calculate E at P, we take the\nGaussian surface to be a sphere of radius r and with centre\nO, passing through P All points on this sphere are equivalent\nrelative to the given charged configuration (That is what we\nmean by spherical symmetry ) The electric field at each point\nof the Gaussian surface, therefore, has the same magnitude\nE and is along the radius vector at each point"}, {"Chapter": "1", "sentence_range": "1097-1100", "Text": "All points on this sphere are equivalent\nrelative to the given charged configuration (That is what we\nmean by spherical symmetry ) The electric field at each point\nof the Gaussian surface, therefore, has the same magnitude\nE and is along the radius vector at each point Thus, E and\nDS at every point are parallel and the flux through each\nelement is E DS"}, {"Chapter": "1", "sentence_range": "1098-1101", "Text": "(That is what we\nmean by spherical symmetry ) The electric field at each point\nof the Gaussian surface, therefore, has the same magnitude\nE and is along the radius vector at each point Thus, E and\nDS at every point are parallel and the flux through each\nelement is E DS Summing over all DS, the flux through the\nGaussian surface is E \u00d7 4 p r 2"}, {"Chapter": "1", "sentence_range": "1099-1102", "Text": ") The electric field at each point\nof the Gaussian surface, therefore, has the same magnitude\nE and is along the radius vector at each point Thus, E and\nDS at every point are parallel and the flux through each\nelement is E DS Summing over all DS, the flux through the\nGaussian surface is E \u00d7 4 p r 2 The charge enclosed is\ns \u00d7 4 p R 2"}, {"Chapter": "1", "sentence_range": "1100-1103", "Text": "Thus, E and\nDS at every point are parallel and the flux through each\nelement is E DS Summing over all DS, the flux through the\nGaussian surface is E \u00d7 4 p r 2 The charge enclosed is\ns \u00d7 4 p R 2 By Gauss\u2019s law\nE \u00d7 4 p r 2 = \n2\n0\n4\nR\n\u03b5\u03c3\n\u03c0\nOr,  \n2\n2\n2\n0\n0\n4\nR\nq\nE\nr\nr\n\u03b5\u03c3\n\u03b5\n=\n=\n\u03c0\nwhere q  =  4 p R2 s is the total charge on the spherical shell"}, {"Chapter": "1", "sentence_range": "1101-1104", "Text": "Summing over all DS, the flux through the\nGaussian surface is E \u00d7 4 p r 2 The charge enclosed is\ns \u00d7 4 p R 2 By Gauss\u2019s law\nE \u00d7 4 p r 2 = \n2\n0\n4\nR\n\u03b5\u03c3\n\u03c0\nOr,  \n2\n2\n2\n0\n0\n4\nR\nq\nE\nr\nr\n\u03b5\u03c3\n\u03b5\n=\n=\n\u03c0\nwhere q  =  4 p R2 s is the total charge on the spherical shell Vectorially,\n2\n0\n\u02c6\n4\nq\n\u03b5r\n=\n\u03c0\nE\nr\n(1"}, {"Chapter": "1", "sentence_range": "1102-1105", "Text": "The charge enclosed is\ns \u00d7 4 p R 2 By Gauss\u2019s law\nE \u00d7 4 p r 2 = \n2\n0\n4\nR\n\u03b5\u03c3\n\u03c0\nOr,  \n2\n2\n2\n0\n0\n4\nR\nq\nE\nr\nr\n\u03b5\u03c3\n\u03b5\n=\n=\n\u03c0\nwhere q  =  4 p R2 s is the total charge on the spherical shell Vectorially,\n2\n0\n\u02c6\n4\nq\n\u03b5r\n=\n\u03c0\nE\nr\n(1 34)\nThe electric field is directed outward if q > 0 and inward if\nq < 0"}, {"Chapter": "1", "sentence_range": "1103-1106", "Text": "By Gauss\u2019s law\nE \u00d7 4 p r 2 = \n2\n0\n4\nR\n\u03b5\u03c3\n\u03c0\nOr,  \n2\n2\n2\n0\n0\n4\nR\nq\nE\nr\nr\n\u03b5\u03c3\n\u03b5\n=\n=\n\u03c0\nwhere q  =  4 p R2 s is the total charge on the spherical shell Vectorially,\n2\n0\n\u02c6\n4\nq\n\u03b5r\n=\n\u03c0\nE\nr\n(1 34)\nThe electric field is directed outward if q > 0 and inward if\nq < 0 This, however, is exactly the field produced by a charge\nq placed at the centre O"}, {"Chapter": "1", "sentence_range": "1104-1107", "Text": "Vectorially,\n2\n0\n\u02c6\n4\nq\n\u03b5r\n=\n\u03c0\nE\nr\n(1 34)\nThe electric field is directed outward if q > 0 and inward if\nq < 0 This, however, is exactly the field produced by a charge\nq placed at the centre O Thus for points outside the shell, the field due\nto a uniformly charged shell is as if  the entire charge of the shell is\nconcentrated at its centre"}, {"Chapter": "1", "sentence_range": "1105-1108", "Text": "34)\nThe electric field is directed outward if q > 0 and inward if\nq < 0 This, however, is exactly the field produced by a charge\nq placed at the centre O Thus for points outside the shell, the field due\nto a uniformly charged shell is as if  the entire charge of the shell is\nconcentrated at its centre (ii) Field inside the shell: In Fig"}, {"Chapter": "1", "sentence_range": "1106-1109", "Text": "This, however, is exactly the field produced by a charge\nq placed at the centre O Thus for points outside the shell, the field due\nto a uniformly charged shell is as if  the entire charge of the shell is\nconcentrated at its centre (ii) Field inside the shell: In Fig 1"}, {"Chapter": "1", "sentence_range": "1107-1110", "Text": "Thus for points outside the shell, the field due\nto a uniformly charged shell is as if  the entire charge of the shell is\nconcentrated at its centre (ii) Field inside the shell: In Fig 1 28(b), the point P is inside the\nshell"}, {"Chapter": "1", "sentence_range": "1108-1111", "Text": "(ii) Field inside the shell: In Fig 1 28(b), the point P is inside the\nshell The Gaussian surface is again a sphere through P centred at O"}, {"Chapter": "1", "sentence_range": "1109-1112", "Text": "1 28(b), the point P is inside the\nshell The Gaussian surface is again a sphere through P centred at O FIGURE 1"}, {"Chapter": "1", "sentence_range": "1110-1113", "Text": "28(b), the point P is inside the\nshell The Gaussian surface is again a sphere through P centred at O FIGURE 1 28 Gaussian\nsurfaces for a point with\n(a) r > R, (b) r < R"}, {"Chapter": "1", "sentence_range": "1111-1114", "Text": "The Gaussian surface is again a sphere through P centred at O FIGURE 1 28 Gaussian\nsurfaces for a point with\n(a) r > R, (b) r < R Rationalised 2023-24\n36\nPhysics\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "1112-1115", "Text": "FIGURE 1 28 Gaussian\nsurfaces for a point with\n(a) r > R, (b) r < R Rationalised 2023-24\n36\nPhysics\n EXAMPLE 1 12\nThe flux through the Gaussian surface, calculated as before, is\nE \u00d7 4 p r2"}, {"Chapter": "1", "sentence_range": "1113-1116", "Text": "28 Gaussian\nsurfaces for a point with\n(a) r > R, (b) r < R Rationalised 2023-24\n36\nPhysics\n EXAMPLE 1 12\nThe flux through the Gaussian surface, calculated as before, is\nE \u00d7 4 p r2 However, in this case, the Gaussian surface encloses no\ncharge"}, {"Chapter": "1", "sentence_range": "1114-1117", "Text": "Rationalised 2023-24\n36\nPhysics\n EXAMPLE 1 12\nThe flux through the Gaussian surface, calculated as before, is\nE \u00d7 4 p r2 However, in this case, the Gaussian surface encloses no\ncharge Gauss\u2019s law then gives\nE \u00d7 4 p r2 =  0\ni"}, {"Chapter": "1", "sentence_range": "1115-1118", "Text": "12\nThe flux through the Gaussian surface, calculated as before, is\nE \u00d7 4 p r2 However, in this case, the Gaussian surface encloses no\ncharge Gauss\u2019s law then gives\nE \u00d7 4 p r2 =  0\ni e"}, {"Chapter": "1", "sentence_range": "1116-1119", "Text": "However, in this case, the Gaussian surface encloses no\ncharge Gauss\u2019s law then gives\nE \u00d7 4 p r2 =  0\ni e , E = 0          (r < R )\n(1"}, {"Chapter": "1", "sentence_range": "1117-1120", "Text": "Gauss\u2019s law then gives\nE \u00d7 4 p r2 =  0\ni e , E = 0          (r < R )\n(1 35)\nthat is, the field due to a uniformly charged thin shell is zero at all points\ninside the shell*"}, {"Chapter": "1", "sentence_range": "1118-1121", "Text": "e , E = 0          (r < R )\n(1 35)\nthat is, the field due to a uniformly charged thin shell is zero at all points\ninside the shell* This important result is a direct consequence of Gauss\u2019s\nlaw which follows from Coulomb\u2019s law"}, {"Chapter": "1", "sentence_range": "1119-1122", "Text": ", E = 0          (r < R )\n(1 35)\nthat is, the field due to a uniformly charged thin shell is zero at all points\ninside the shell* This important result is a direct consequence of Gauss\u2019s\nlaw which follows from Coulomb\u2019s law The experimental verification of\nthis result confirms the 1/r2 dependence in Coulomb\u2019s law"}, {"Chapter": "1", "sentence_range": "1120-1123", "Text": "35)\nthat is, the field due to a uniformly charged thin shell is zero at all points\ninside the shell* This important result is a direct consequence of Gauss\u2019s\nlaw which follows from Coulomb\u2019s law The experimental verification of\nthis result confirms the 1/r2 dependence in Coulomb\u2019s law Example 1"}, {"Chapter": "1", "sentence_range": "1121-1124", "Text": "This important result is a direct consequence of Gauss\u2019s\nlaw which follows from Coulomb\u2019s law The experimental verification of\nthis result confirms the 1/r2 dependence in Coulomb\u2019s law Example 1 12 An early model for an atom considered it to have a\npositively charged point nucleus of charge Ze, surrounded by a\nuniform density of negative charge up to a radius R"}, {"Chapter": "1", "sentence_range": "1122-1125", "Text": "The experimental verification of\nthis result confirms the 1/r2 dependence in Coulomb\u2019s law Example 1 12 An early model for an atom considered it to have a\npositively charged point nucleus of charge Ze, surrounded by a\nuniform density of negative charge up to a radius R The atom as a\nwhole is neutral"}, {"Chapter": "1", "sentence_range": "1123-1126", "Text": "Example 1 12 An early model for an atom considered it to have a\npositively charged point nucleus of charge Ze, surrounded by a\nuniform density of negative charge up to a radius R The atom as a\nwhole is neutral For this model, what is the electric field at a distance\nr from the nucleus"}, {"Chapter": "1", "sentence_range": "1124-1127", "Text": "12 An early model for an atom considered it to have a\npositively charged point nucleus of charge Ze, surrounded by a\nuniform density of negative charge up to a radius R The atom as a\nwhole is neutral For this model, what is the electric field at a distance\nr from the nucleus FIGURE 1"}, {"Chapter": "1", "sentence_range": "1125-1128", "Text": "The atom as a\nwhole is neutral For this model, what is the electric field at a distance\nr from the nucleus FIGURE 1 29\nSolution  The charge distribution for this model of the atom is as\nshown in Fig"}, {"Chapter": "1", "sentence_range": "1126-1129", "Text": "For this model, what is the electric field at a distance\nr from the nucleus FIGURE 1 29\nSolution  The charge distribution for this model of the atom is as\nshown in Fig 1"}, {"Chapter": "1", "sentence_range": "1127-1130", "Text": "FIGURE 1 29\nSolution  The charge distribution for this model of the atom is as\nshown in Fig 1 29"}, {"Chapter": "1", "sentence_range": "1128-1131", "Text": "29\nSolution  The charge distribution for this model of the atom is as\nshown in Fig 1 29 The total negative charge in the uniform spherical\ncharge distribution of radius R must be \u2013Z e, since the atom (nucleus\nof charge Z e + negative charge) is neutral"}, {"Chapter": "1", "sentence_range": "1129-1132", "Text": "1 29 The total negative charge in the uniform spherical\ncharge distribution of radius R must be \u2013Z e, since the atom (nucleus\nof charge Z e + negative charge) is neutral This immediately gives us\nthe negative charge density r, since we must have\n3\n4\n0\u2013\n3\nR\nZe\n\u03c1\n\u03c0\n=\nor\n3\n3\n4\nZe\nR\n\u03c1 = \u2212\n\u03c0\nTo find the electric field E(r) at a point P which is a distance r away\nfrom the nucleus, we use Gauss\u2019s law"}, {"Chapter": "1", "sentence_range": "1130-1133", "Text": "29 The total negative charge in the uniform spherical\ncharge distribution of radius R must be \u2013Z e, since the atom (nucleus\nof charge Z e + negative charge) is neutral This immediately gives us\nthe negative charge density r, since we must have\n3\n4\n0\u2013\n3\nR\nZe\n\u03c1\n\u03c0\n=\nor\n3\n3\n4\nZe\nR\n\u03c1 = \u2212\n\u03c0\nTo find the electric field E(r) at a point P which is a distance r away\nfrom the nucleus, we use Gauss\u2019s law Because of the spherical\nsymmetry of the charge distribution, the magnitude of the electric\nfield E(r) depends only on the radial distance, no matter what the\ndirection of r"}, {"Chapter": "1", "sentence_range": "1131-1134", "Text": "The total negative charge in the uniform spherical\ncharge distribution of radius R must be \u2013Z e, since the atom (nucleus\nof charge Z e + negative charge) is neutral This immediately gives us\nthe negative charge density r, since we must have\n3\n4\n0\u2013\n3\nR\nZe\n\u03c1\n\u03c0\n=\nor\n3\n3\n4\nZe\nR\n\u03c1 = \u2212\n\u03c0\nTo find the electric field E(r) at a point P which is a distance r away\nfrom the nucleus, we use Gauss\u2019s law Because of the spherical\nsymmetry of the charge distribution, the magnitude of the electric\nfield E(r) depends only on the radial distance, no matter what the\ndirection of r Its direction is along (or opposite to) the radius vector r\nfrom the origin to the point P"}, {"Chapter": "1", "sentence_range": "1132-1135", "Text": "This immediately gives us\nthe negative charge density r, since we must have\n3\n4\n0\u2013\n3\nR\nZe\n\u03c1\n\u03c0\n=\nor\n3\n3\n4\nZe\nR\n\u03c1 = \u2212\n\u03c0\nTo find the electric field E(r) at a point P which is a distance r away\nfrom the nucleus, we use Gauss\u2019s law Because of the spherical\nsymmetry of the charge distribution, the magnitude of the electric\nfield E(r) depends only on the radial distance, no matter what the\ndirection of r Its direction is along (or opposite to) the radius vector r\nfrom the origin to the point P The obvious Gaussian surface is a\nspherical surface centred at the nucleus"}, {"Chapter": "1", "sentence_range": "1133-1136", "Text": "Because of the spherical\nsymmetry of the charge distribution, the magnitude of the electric\nfield E(r) depends only on the radial distance, no matter what the\ndirection of r Its direction is along (or opposite to) the radius vector r\nfrom the origin to the point P The obvious Gaussian surface is a\nspherical surface centred at the nucleus We consider two situations,\nnamely, r < R and r > R"}, {"Chapter": "1", "sentence_range": "1134-1137", "Text": "Its direction is along (or opposite to) the radius vector r\nfrom the origin to the point P The obvious Gaussian surface is a\nspherical surface centred at the nucleus We consider two situations,\nnamely, r < R and r > R (i) r < R : The electric flux f enclosed by the spherical surface is\n    f  =  E (r) \u00d7 4 p r 2\n*\nCompare this with a uniform mass shell discussed in Section 7"}, {"Chapter": "1", "sentence_range": "1135-1138", "Text": "The obvious Gaussian surface is a\nspherical surface centred at the nucleus We consider two situations,\nnamely, r < R and r > R (i) r < R : The electric flux f enclosed by the spherical surface is\n    f  =  E (r) \u00d7 4 p r 2\n*\nCompare this with a uniform mass shell discussed in Section 7 5 of Class XI\nTextbook of Physics"}, {"Chapter": "1", "sentence_range": "1136-1139", "Text": "We consider two situations,\nnamely, r < R and r > R (i) r < R : The electric flux f enclosed by the spherical surface is\n    f  =  E (r) \u00d7 4 p r 2\n*\nCompare this with a uniform mass shell discussed in Section 7 5 of Class XI\nTextbook of Physics Rationalised 2023-24\nElectric Charges\nand Fields\n37\n EXAMPLE 1"}, {"Chapter": "1", "sentence_range": "1137-1140", "Text": "(i) r < R : The electric flux f enclosed by the spherical surface is\n    f  =  E (r) \u00d7 4 p r 2\n*\nCompare this with a uniform mass shell discussed in Section 7 5 of Class XI\nTextbook of Physics Rationalised 2023-24\nElectric Charges\nand Fields\n37\n EXAMPLE 1 12\nwhere E (r)  is the magnitude of the electric field at r"}, {"Chapter": "1", "sentence_range": "1138-1141", "Text": "5 of Class XI\nTextbook of Physics Rationalised 2023-24\nElectric Charges\nand Fields\n37\n EXAMPLE 1 12\nwhere E (r)  is the magnitude of the electric field at r This is because\nthe field at any point on the spherical Gaussian surface has the\nsame direction as the normal to the surface there,  and has the same\nmagnitude at all points on the surface"}, {"Chapter": "1", "sentence_range": "1139-1142", "Text": "Rationalised 2023-24\nElectric Charges\nand Fields\n37\n EXAMPLE 1 12\nwhere E (r)  is the magnitude of the electric field at r This is because\nthe field at any point on the spherical Gaussian surface has the\nsame direction as the normal to the surface there,  and has the same\nmagnitude at all points on the surface The charge q enclosed by the Gaussian surface is the positive nuclear\ncharge and the negative charge within the sphere of radius r,\ni"}, {"Chapter": "1", "sentence_range": "1140-1143", "Text": "12\nwhere E (r)  is the magnitude of the electric field at r This is because\nthe field at any point on the spherical Gaussian surface has the\nsame direction as the normal to the surface there,  and has the same\nmagnitude at all points on the surface The charge q enclosed by the Gaussian surface is the positive nuclear\ncharge and the negative charge within the sphere of radius r,\ni e"}, {"Chapter": "1", "sentence_range": "1141-1144", "Text": "This is because\nthe field at any point on the spherical Gaussian surface has the\nsame direction as the normal to the surface there,  and has the same\nmagnitude at all points on the surface The charge q enclosed by the Gaussian surface is the positive nuclear\ncharge and the negative charge within the sphere of radius r,\ni e , \n3\n4\n3\nr\nq\nZ e\n\u03c1\n\u03c0\n=\n+\nSubstituting for the charge density r  obtained earlier, we have\n3\n3\nr\nq\nZ e\nZ e\nR\n=\n\u2212\nGauss\u2019s law then gives,\n2\n3\n0\n1\n( )\n;\n4\nZ e\nr\nE r\nr\nR\nr\nR\n\u03b5\n \n \n=\n\u2212\n<\n \n \n \n \n\u03c0\nThe electric field is directed radially outward"}, {"Chapter": "1", "sentence_range": "1142-1145", "Text": "The charge q enclosed by the Gaussian surface is the positive nuclear\ncharge and the negative charge within the sphere of radius r,\ni e , \n3\n4\n3\nr\nq\nZ e\n\u03c1\n\u03c0\n=\n+\nSubstituting for the charge density r  obtained earlier, we have\n3\n3\nr\nq\nZ e\nZ e\nR\n=\n\u2212\nGauss\u2019s law then gives,\n2\n3\n0\n1\n( )\n;\n4\nZ e\nr\nE r\nr\nR\nr\nR\n\u03b5\n \n \n=\n\u2212\n<\n \n \n \n \n\u03c0\nThe electric field is directed radially outward (ii) r > R:  In this case, the total charge enclosed by the Gaussian\nspherical surface is zero since the atom is neutral"}, {"Chapter": "1", "sentence_range": "1143-1146", "Text": "e , \n3\n4\n3\nr\nq\nZ e\n\u03c1\n\u03c0\n=\n+\nSubstituting for the charge density r  obtained earlier, we have\n3\n3\nr\nq\nZ e\nZ e\nR\n=\n\u2212\nGauss\u2019s law then gives,\n2\n3\n0\n1\n( )\n;\n4\nZ e\nr\nE r\nr\nR\nr\nR\n\u03b5\n \n \n=\n\u2212\n<\n \n \n \n \n\u03c0\nThe electric field is directed radially outward (ii) r > R:  In this case, the total charge enclosed by the Gaussian\nspherical surface is zero since the atom is neutral Thus, from Gauss\u2019s\nlaw,\nE (r) \u00d7 4 p  r 2 = 0  or  E (r) = 0;    r > R\nAt r = R, both cases give the same result: E = 0"}, {"Chapter": "1", "sentence_range": "1144-1147", "Text": ", \n3\n4\n3\nr\nq\nZ e\n\u03c1\n\u03c0\n=\n+\nSubstituting for the charge density r  obtained earlier, we have\n3\n3\nr\nq\nZ e\nZ e\nR\n=\n\u2212\nGauss\u2019s law then gives,\n2\n3\n0\n1\n( )\n;\n4\nZ e\nr\nE r\nr\nR\nr\nR\n\u03b5\n \n \n=\n\u2212\n<\n \n \n \n \n\u03c0\nThe electric field is directed radially outward (ii) r > R:  In this case, the total charge enclosed by the Gaussian\nspherical surface is zero since the atom is neutral Thus, from Gauss\u2019s\nlaw,\nE (r) \u00d7 4 p  r 2 = 0  or  E (r) = 0;    r > R\nAt r = R, both cases give the same result: E = 0 SUMMARY\n1"}, {"Chapter": "1", "sentence_range": "1145-1148", "Text": "(ii) r > R:  In this case, the total charge enclosed by the Gaussian\nspherical surface is zero since the atom is neutral Thus, from Gauss\u2019s\nlaw,\nE (r) \u00d7 4 p  r 2 = 0  or  E (r) = 0;    r > R\nAt r = R, both cases give the same result: E = 0 SUMMARY\n1 Electric and magnetic forces determine the properties of atoms,\nmolecules and bulk matter"}, {"Chapter": "1", "sentence_range": "1146-1149", "Text": "Thus, from Gauss\u2019s\nlaw,\nE (r) \u00d7 4 p  r 2 = 0  or  E (r) = 0;    r > R\nAt r = R, both cases give the same result: E = 0 SUMMARY\n1 Electric and magnetic forces determine the properties of atoms,\nmolecules and bulk matter 2"}, {"Chapter": "1", "sentence_range": "1147-1150", "Text": "SUMMARY\n1 Electric and magnetic forces determine the properties of atoms,\nmolecules and bulk matter 2 From simple experiments on frictional electricity, one can infer that\nthere are two types of charges in nature; and that like charges repel\nand unlike charges attract"}, {"Chapter": "1", "sentence_range": "1148-1151", "Text": "Electric and magnetic forces determine the properties of atoms,\nmolecules and bulk matter 2 From simple experiments on frictional electricity, one can infer that\nthere are two types of charges in nature; and that like charges repel\nand unlike charges attract By convention, the charge on a glass rod\nrubbed with silk is positive; that on a plastic rod rubbed with fur is\nthen negative"}, {"Chapter": "1", "sentence_range": "1149-1152", "Text": "2 From simple experiments on frictional electricity, one can infer that\nthere are two types of charges in nature; and that like charges repel\nand unlike charges attract By convention, the charge on a glass rod\nrubbed with silk is positive; that on a plastic rod rubbed with fur is\nthen negative 3"}, {"Chapter": "1", "sentence_range": "1150-1153", "Text": "From simple experiments on frictional electricity, one can infer that\nthere are two types of charges in nature; and that like charges repel\nand unlike charges attract By convention, the charge on a glass rod\nrubbed with silk is positive; that on a plastic rod rubbed with fur is\nthen negative 3 Conductors allow movement of electric charge through them,\ninsulators do not"}, {"Chapter": "1", "sentence_range": "1151-1154", "Text": "By convention, the charge on a glass rod\nrubbed with silk is positive; that on a plastic rod rubbed with fur is\nthen negative 3 Conductors allow movement of electric charge through them,\ninsulators do not In metals, the mobile charges are electrons; in\nelectrolytes both positive and negative ions are mobile"}, {"Chapter": "1", "sentence_range": "1152-1155", "Text": "3 Conductors allow movement of electric charge through them,\ninsulators do not In metals, the mobile charges are electrons; in\nelectrolytes both positive and negative ions are mobile 4"}, {"Chapter": "1", "sentence_range": "1153-1156", "Text": "Conductors allow movement of electric charge through them,\ninsulators do not In metals, the mobile charges are electrons; in\nelectrolytes both positive and negative ions are mobile 4 Electric charge has three basic properties: quantisation, additivity\nand conservation"}, {"Chapter": "1", "sentence_range": "1154-1157", "Text": "In metals, the mobile charges are electrons; in\nelectrolytes both positive and negative ions are mobile 4 Electric charge has three basic properties: quantisation, additivity\nand conservation Quantisation of electric charge means that total charge (q) of a body\nis always an integral multiple of a basic quantum of charge (e) i"}, {"Chapter": "1", "sentence_range": "1155-1158", "Text": "4 Electric charge has three basic properties: quantisation, additivity\nand conservation Quantisation of electric charge means that total charge (q) of a body\nis always an integral multiple of a basic quantum of charge (e) i e"}, {"Chapter": "1", "sentence_range": "1156-1159", "Text": "Electric charge has three basic properties: quantisation, additivity\nand conservation Quantisation of electric charge means that total charge (q) of a body\nis always an integral multiple of a basic quantum of charge (e) i e ,\nq = n e, where n = 0, \u00b11, \u00b12, \u00b13,"}, {"Chapter": "1", "sentence_range": "1157-1160", "Text": "Quantisation of electric charge means that total charge (q) of a body\nis always an integral multiple of a basic quantum of charge (e) i e ,\nq = n e, where n = 0, \u00b11, \u00b12, \u00b13, Proton and electron have charges\n+e, \u2013e, respectively"}, {"Chapter": "1", "sentence_range": "1158-1161", "Text": "e ,\nq = n e, where n = 0, \u00b11, \u00b12, \u00b13, Proton and electron have charges\n+e, \u2013e, respectively For macroscopic charges for which n is a very large\nnumber, quantisation of charge can be ignored"}, {"Chapter": "1", "sentence_range": "1159-1162", "Text": ",\nq = n e, where n = 0, \u00b11, \u00b12, \u00b13, Proton and electron have charges\n+e, \u2013e, respectively For macroscopic charges for which n is a very large\nnumber, quantisation of charge can be ignored Additivity of electric charges means that the total charge of a system\nis the algebraic sum (i"}, {"Chapter": "1", "sentence_range": "1160-1163", "Text": "Proton and electron have charges\n+e, \u2013e, respectively For macroscopic charges for which n is a very large\nnumber, quantisation of charge can be ignored Additivity of electric charges means that the total charge of a system\nis the algebraic sum (i e"}, {"Chapter": "1", "sentence_range": "1161-1164", "Text": "For macroscopic charges for which n is a very large\nnumber, quantisation of charge can be ignored Additivity of electric charges means that the total charge of a system\nis the algebraic sum (i e , the sum taking into account proper signs)\nof all individual charges in the system"}, {"Chapter": "1", "sentence_range": "1162-1165", "Text": "Additivity of electric charges means that the total charge of a system\nis the algebraic sum (i e , the sum taking into account proper signs)\nof all individual charges in the system Conservation of electric charges means that the total charge of an\nisolated system remains unchanged with time"}, {"Chapter": "1", "sentence_range": "1163-1166", "Text": "e , the sum taking into account proper signs)\nof all individual charges in the system Conservation of electric charges means that the total charge of an\nisolated system remains unchanged with time This means that when\nRationalised 2023-24\n38\nPhysics\nbodies are charged through friction, there is a transfer of electric charge\nfrom one body to another, but no creation or destruction\nof charge"}, {"Chapter": "1", "sentence_range": "1164-1167", "Text": ", the sum taking into account proper signs)\nof all individual charges in the system Conservation of electric charges means that the total charge of an\nisolated system remains unchanged with time This means that when\nRationalised 2023-24\n38\nPhysics\nbodies are charged through friction, there is a transfer of electric charge\nfrom one body to another, but no creation or destruction\nof charge 5"}, {"Chapter": "1", "sentence_range": "1165-1168", "Text": "Conservation of electric charges means that the total charge of an\nisolated system remains unchanged with time This means that when\nRationalised 2023-24\n38\nPhysics\nbodies are charged through friction, there is a transfer of electric charge\nfrom one body to another, but no creation or destruction\nof charge 5 Coulomb\u2019s Law: The mutual electrostatic force between two point\ncharges q1 and q2 is proportional to the product q1q2 and inversely\nproportional to the square of the distance r21 separating them"}, {"Chapter": "1", "sentence_range": "1166-1169", "Text": "This means that when\nRationalised 2023-24\n38\nPhysics\nbodies are charged through friction, there is a transfer of electric charge\nfrom one body to another, but no creation or destruction\nof charge 5 Coulomb\u2019s Law: The mutual electrostatic force between two point\ncharges q1 and q2 is proportional to the product q1q2 and inversely\nproportional to the square of the distance r21 separating them Mathematically,\nF21 = force on q2 due to  \n1\n2\n1\n21\n2\n21\n\u02c6\nk\n(q q )\nq\nr\n=\nr\nwhere \n\u02c6r21\n is a unit vector in the direction from q1 to q2 and k =  \n0\n1\n4 \u03b5\n\u03c0\nis the constant of proportionality"}, {"Chapter": "1", "sentence_range": "1167-1170", "Text": "5 Coulomb\u2019s Law: The mutual electrostatic force between two point\ncharges q1 and q2 is proportional to the product q1q2 and inversely\nproportional to the square of the distance r21 separating them Mathematically,\nF21 = force on q2 due to  \n1\n2\n1\n21\n2\n21\n\u02c6\nk\n(q q )\nq\nr\n=\nr\nwhere \n\u02c6r21\n is a unit vector in the direction from q1 to q2 and k =  \n0\n1\n4 \u03b5\n\u03c0\nis the constant of proportionality In SI units, the unit of charge is coulomb"}, {"Chapter": "1", "sentence_range": "1168-1171", "Text": "Coulomb\u2019s Law: The mutual electrostatic force between two point\ncharges q1 and q2 is proportional to the product q1q2 and inversely\nproportional to the square of the distance r21 separating them Mathematically,\nF21 = force on q2 due to  \n1\n2\n1\n21\n2\n21\n\u02c6\nk\n(q q )\nq\nr\n=\nr\nwhere \n\u02c6r21\n is a unit vector in the direction from q1 to q2 and k =  \n0\n1\n4 \u03b5\n\u03c0\nis the constant of proportionality In SI units, the unit of charge is coulomb The experimental value of\nthe constant e0 is\ne0 = 8"}, {"Chapter": "1", "sentence_range": "1169-1172", "Text": "Mathematically,\nF21 = force on q2 due to  \n1\n2\n1\n21\n2\n21\n\u02c6\nk\n(q q )\nq\nr\n=\nr\nwhere \n\u02c6r21\n is a unit vector in the direction from q1 to q2 and k =  \n0\n1\n4 \u03b5\n\u03c0\nis the constant of proportionality In SI units, the unit of charge is coulomb The experimental value of\nthe constant e0 is\ne0 = 8 854 \u00d7 10\u201312 C2 N\u20131 m\u20132\nThe approximate value of k is\nk = 9 \u00d7 109 N m2 C\u20132\n6"}, {"Chapter": "1", "sentence_range": "1170-1173", "Text": "In SI units, the unit of charge is coulomb The experimental value of\nthe constant e0 is\ne0 = 8 854 \u00d7 10\u201312 C2 N\u20131 m\u20132\nThe approximate value of k is\nk = 9 \u00d7 109 N m2 C\u20132\n6 The ratio of electric force and gravitational force between a proton\nand an electron is\n2\n39\n2 4\n10\ne\np\nk e"}, {"Chapter": "1", "sentence_range": "1171-1174", "Text": "The experimental value of\nthe constant e0 is\ne0 = 8 854 \u00d7 10\u201312 C2 N\u20131 m\u20132\nThe approximate value of k is\nk = 9 \u00d7 109 N m2 C\u20132\n6 The ratio of electric force and gravitational force between a proton\nand an electron is\n2\n39\n2 4\n10\ne\np\nk e G\nm m\n\u2245\n\u00d7\n7"}, {"Chapter": "1", "sentence_range": "1172-1175", "Text": "854 \u00d7 10\u201312 C2 N\u20131 m\u20132\nThe approximate value of k is\nk = 9 \u00d7 109 N m2 C\u20132\n6 The ratio of electric force and gravitational force between a proton\nand an electron is\n2\n39\n2 4\n10\ne\np\nk e G\nm m\n\u2245\n\u00d7\n7 Superposition Principle: The principle is based on the property that the\nforces with which two charges attract or repel each other are not\naffected by the presence of a third (or more) additional charge(s)"}, {"Chapter": "1", "sentence_range": "1173-1176", "Text": "The ratio of electric force and gravitational force between a proton\nand an electron is\n2\n39\n2 4\n10\ne\np\nk e G\nm m\n\u2245\n\u00d7\n7 Superposition Principle: The principle is based on the property that the\nforces with which two charges attract or repel each other are not\naffected by the presence of a third (or more) additional charge(s) For\nan assembly of charges q1, q2, q3,"}, {"Chapter": "1", "sentence_range": "1174-1177", "Text": "G\nm m\n\u2245\n\u00d7\n7 Superposition Principle: The principle is based on the property that the\nforces with which two charges attract or repel each other are not\naffected by the presence of a third (or more) additional charge(s) For\nan assembly of charges q1, q2, q3, , the force on any charge, say q1, is\nthe vector sum of the force on  q1 due to q2, the force on q1 due to q3,\nand so on"}, {"Chapter": "1", "sentence_range": "1175-1178", "Text": "Superposition Principle: The principle is based on the property that the\nforces with which two charges attract or repel each other are not\naffected by the presence of a third (or more) additional charge(s) For\nan assembly of charges q1, q2, q3, , the force on any charge, say q1, is\nthe vector sum of the force on  q1 due to q2, the force on q1 due to q3,\nand so on For each pair, the force is given by the Coulomb\u2019s law for\ntwo charges stated earlier"}, {"Chapter": "1", "sentence_range": "1176-1179", "Text": "For\nan assembly of charges q1, q2, q3, , the force on any charge, say q1, is\nthe vector sum of the force on  q1 due to q2, the force on q1 due to q3,\nand so on For each pair, the force is given by the Coulomb\u2019s law for\ntwo charges stated earlier 8"}, {"Chapter": "1", "sentence_range": "1177-1180", "Text": ", the force on any charge, say q1, is\nthe vector sum of the force on  q1 due to q2, the force on q1 due to q3,\nand so on For each pair, the force is given by the Coulomb\u2019s law for\ntwo charges stated earlier 8 The electric field E at a point due to a charge configuration is the\nforce on a small positive test charge q placed at the point divided by\nthe magnitude of the charge"}, {"Chapter": "1", "sentence_range": "1178-1181", "Text": "For each pair, the force is given by the Coulomb\u2019s law for\ntwo charges stated earlier 8 The electric field E at a point due to a charge configuration is the\nforce on a small positive test charge q placed at the point divided by\nthe magnitude of the charge Electric field due to a point charge q has\na magnitude |q|/4pe0r2; it is radially outwards from q, if q is positive,\nand radially inwards if q is negative"}, {"Chapter": "1", "sentence_range": "1179-1182", "Text": "8 The electric field E at a point due to a charge configuration is the\nforce on a small positive test charge q placed at the point divided by\nthe magnitude of the charge Electric field due to a point charge q has\na magnitude |q|/4pe0r2; it is radially outwards from q, if q is positive,\nand radially inwards if q is negative Like Coulomb force, electric field\nalso satisfies superposition principle"}, {"Chapter": "1", "sentence_range": "1180-1183", "Text": "The electric field E at a point due to a charge configuration is the\nforce on a small positive test charge q placed at the point divided by\nthe magnitude of the charge Electric field due to a point charge q has\na magnitude |q|/4pe0r2; it is radially outwards from q, if q is positive,\nand radially inwards if q is negative Like Coulomb force, electric field\nalso satisfies superposition principle 9"}, {"Chapter": "1", "sentence_range": "1181-1184", "Text": "Electric field due to a point charge q has\na magnitude |q|/4pe0r2; it is radially outwards from q, if q is positive,\nand radially inwards if q is negative Like Coulomb force, electric field\nalso satisfies superposition principle 9 An electric field line is a curve drawn in such a way that the tangent\nat each point on the curve gives the direction of electric field at that\npoint"}, {"Chapter": "1", "sentence_range": "1182-1185", "Text": "Like Coulomb force, electric field\nalso satisfies superposition principle 9 An electric field line is a curve drawn in such a way that the tangent\nat each point on the curve gives the direction of electric field at that\npoint The relative closeness of field lines indicates the relative strength\nof electric field at different points; they crowd near each other in regions\nof strong electric field and are far apart where the electric field is\nweak"}, {"Chapter": "1", "sentence_range": "1183-1186", "Text": "9 An electric field line is a curve drawn in such a way that the tangent\nat each point on the curve gives the direction of electric field at that\npoint The relative closeness of field lines indicates the relative strength\nof electric field at different points; they crowd near each other in regions\nof strong electric field and are far apart where the electric field is\nweak In regions of constant electric field, the field lines are uniformly\nspaced parallel straight lines"}, {"Chapter": "1", "sentence_range": "1184-1187", "Text": "An electric field line is a curve drawn in such a way that the tangent\nat each point on the curve gives the direction of electric field at that\npoint The relative closeness of field lines indicates the relative strength\nof electric field at different points; they crowd near each other in regions\nof strong electric field and are far apart where the electric field is\nweak In regions of constant electric field, the field lines are uniformly\nspaced parallel straight lines 10"}, {"Chapter": "1", "sentence_range": "1185-1188", "Text": "The relative closeness of field lines indicates the relative strength\nof electric field at different points; they crowd near each other in regions\nof strong electric field and are far apart where the electric field is\nweak In regions of constant electric field, the field lines are uniformly\nspaced parallel straight lines 10 Some of the important properties of field lines are: (i) Field lines are\ncontinuous curves without any breaks"}, {"Chapter": "1", "sentence_range": "1186-1189", "Text": "In regions of constant electric field, the field lines are uniformly\nspaced parallel straight lines 10 Some of the important properties of field lines are: (i) Field lines are\ncontinuous curves without any breaks (ii) Two field lines cannot cross\neach other"}, {"Chapter": "1", "sentence_range": "1187-1190", "Text": "10 Some of the important properties of field lines are: (i) Field lines are\ncontinuous curves without any breaks (ii) Two field lines cannot cross\neach other (iii) Electrostatic field lines start at positive charges and\nend at negative charges \u2014they cannot form closed loops"}, {"Chapter": "1", "sentence_range": "1188-1191", "Text": "Some of the important properties of field lines are: (i) Field lines are\ncontinuous curves without any breaks (ii) Two field lines cannot cross\neach other (iii) Electrostatic field lines start at positive charges and\nend at negative charges \u2014they cannot form closed loops 11"}, {"Chapter": "1", "sentence_range": "1189-1192", "Text": "(ii) Two field lines cannot cross\neach other (iii) Electrostatic field lines start at positive charges and\nend at negative charges \u2014they cannot form closed loops 11 An electric dipole is a pair of equal and opposite charges q and \u2013q\nseparated by some distance 2a"}, {"Chapter": "1", "sentence_range": "1190-1193", "Text": "(iii) Electrostatic field lines start at positive charges and\nend at negative charges \u2014they cannot form closed loops 11 An electric dipole is a pair of equal and opposite charges q and \u2013q\nseparated by some distance 2a Its dipole moment vector p has\nmagnitude 2qa and is in the direction of the dipole axis from \u2013q to q"}, {"Chapter": "1", "sentence_range": "1191-1194", "Text": "11 An electric dipole is a pair of equal and opposite charges q and \u2013q\nseparated by some distance 2a Its dipole moment vector p has\nmagnitude 2qa and is in the direction of the dipole axis from \u2013q to q Rationalised 2023-24\nElectric Charges\nand Fields\n39\n12"}, {"Chapter": "1", "sentence_range": "1192-1195", "Text": "An electric dipole is a pair of equal and opposite charges q and \u2013q\nseparated by some distance 2a Its dipole moment vector p has\nmagnitude 2qa and is in the direction of the dipole axis from \u2013q to q Rationalised 2023-24\nElectric Charges\nand Fields\n39\n12 Field of an electric dipole in its equatorial plane (i"}, {"Chapter": "1", "sentence_range": "1193-1196", "Text": "Its dipole moment vector p has\nmagnitude 2qa and is in the direction of the dipole axis from \u2013q to q Rationalised 2023-24\nElectric Charges\nand Fields\n39\n12 Field of an electric dipole in its equatorial plane (i e"}, {"Chapter": "1", "sentence_range": "1194-1197", "Text": "Rationalised 2023-24\nElectric Charges\nand Fields\n39\n12 Field of an electric dipole in its equatorial plane (i e , the plane\nperpendicular to its axis and passing through its centre) at a distance\nr from the centre:\n2\n2 3/2\n1\n4\n(\n)\no\na\nr\n\u03b5\n\u2212\n=\n\u03c0\n+\np\nE\n3 ,\n4\no\nfor r\na\n\u2212\u03b5r\n\u2245\n>>\n\u03c0\np\nDipole electric field on the axis at a distance r from the centre:\n2\n2 2\n0\n2\n4\n(\n)\nr\nr\na\n\u03b5\n=\n\u03c0\np\u2212\nE\n3\n0\n42\nfor\nr\na\n\u03b5r\n\u2245\n>>\n\u03c0\np\nThe 1/r3 dependence of dipole electric fields should be noted in contrast\nto the 1/r 2 dependence of electric field due to a point charge"}, {"Chapter": "1", "sentence_range": "1195-1198", "Text": "Field of an electric dipole in its equatorial plane (i e , the plane\nperpendicular to its axis and passing through its centre) at a distance\nr from the centre:\n2\n2 3/2\n1\n4\n(\n)\no\na\nr\n\u03b5\n\u2212\n=\n\u03c0\n+\np\nE\n3 ,\n4\no\nfor r\na\n\u2212\u03b5r\n\u2245\n>>\n\u03c0\np\nDipole electric field on the axis at a distance r from the centre:\n2\n2 2\n0\n2\n4\n(\n)\nr\nr\na\n\u03b5\n=\n\u03c0\np\u2212\nE\n3\n0\n42\nfor\nr\na\n\u03b5r\n\u2245\n>>\n\u03c0\np\nThe 1/r3 dependence of dipole electric fields should be noted in contrast\nto the 1/r 2 dependence of electric field due to a point charge 13"}, {"Chapter": "1", "sentence_range": "1196-1199", "Text": "e , the plane\nperpendicular to its axis and passing through its centre) at a distance\nr from the centre:\n2\n2 3/2\n1\n4\n(\n)\no\na\nr\n\u03b5\n\u2212\n=\n\u03c0\n+\np\nE\n3 ,\n4\no\nfor r\na\n\u2212\u03b5r\n\u2245\n>>\n\u03c0\np\nDipole electric field on the axis at a distance r from the centre:\n2\n2 2\n0\n2\n4\n(\n)\nr\nr\na\n\u03b5\n=\n\u03c0\np\u2212\nE\n3\n0\n42\nfor\nr\na\n\u03b5r\n\u2245\n>>\n\u03c0\np\nThe 1/r3 dependence of dipole electric fields should be noted in contrast\nto the 1/r 2 dependence of electric field due to a point charge 13 In a uniform electric field E, a dipole experiences a torque \u03c4  given by\n\u03c4  = p \u00d7 E\nbut experiences no net force"}, {"Chapter": "1", "sentence_range": "1197-1200", "Text": ", the plane\nperpendicular to its axis and passing through its centre) at a distance\nr from the centre:\n2\n2 3/2\n1\n4\n(\n)\no\na\nr\n\u03b5\n\u2212\n=\n\u03c0\n+\np\nE\n3 ,\n4\no\nfor r\na\n\u2212\u03b5r\n\u2245\n>>\n\u03c0\np\nDipole electric field on the axis at a distance r from the centre:\n2\n2 2\n0\n2\n4\n(\n)\nr\nr\na\n\u03b5\n=\n\u03c0\np\u2212\nE\n3\n0\n42\nfor\nr\na\n\u03b5r\n\u2245\n>>\n\u03c0\np\nThe 1/r3 dependence of dipole electric fields should be noted in contrast\nto the 1/r 2 dependence of electric field due to a point charge 13 In a uniform electric field E, a dipole experiences a torque \u03c4  given by\n\u03c4  = p \u00d7 E\nbut experiences no net force 14"}, {"Chapter": "1", "sentence_range": "1198-1201", "Text": "13 In a uniform electric field E, a dipole experiences a torque \u03c4  given by\n\u03c4  = p \u00d7 E\nbut experiences no net force 14 The flux Df of electric field E through a small area element DS is\ngiven by\n Df  = E"}, {"Chapter": "1", "sentence_range": "1199-1202", "Text": "In a uniform electric field E, a dipole experiences a torque \u03c4  given by\n\u03c4  = p \u00d7 E\nbut experiences no net force 14 The flux Df of electric field E through a small area element DS is\ngiven by\n Df  = E DS\nThe vector area element DS is\nDS = DS \u02c6n\nwhere DS is the magnitude of the area element and \u02c6n  is normal to the\narea element, which can be considered planar for sufficiently small DS"}, {"Chapter": "1", "sentence_range": "1200-1203", "Text": "14 The flux Df of electric field E through a small area element DS is\ngiven by\n Df  = E DS\nThe vector area element DS is\nDS = DS \u02c6n\nwhere DS is the magnitude of the area element and \u02c6n  is normal to the\narea element, which can be considered planar for sufficiently small DS For an area element of a closed surface, \u02c6n  is taken to be the direction\nof outward normal, by convention"}, {"Chapter": "1", "sentence_range": "1201-1204", "Text": "The flux Df of electric field E through a small area element DS is\ngiven by\n Df  = E DS\nThe vector area element DS is\nDS = DS \u02c6n\nwhere DS is the magnitude of the area element and \u02c6n  is normal to the\narea element, which can be considered planar for sufficiently small DS For an area element of a closed surface, \u02c6n  is taken to be the direction\nof outward normal, by convention 15"}, {"Chapter": "1", "sentence_range": "1202-1205", "Text": "DS\nThe vector area element DS is\nDS = DS \u02c6n\nwhere DS is the magnitude of the area element and \u02c6n  is normal to the\narea element, which can be considered planar for sufficiently small DS For an area element of a closed surface, \u02c6n  is taken to be the direction\nof outward normal, by convention 15 Gauss\u2019s law: The flux of electric field through any closed surface S is\n1/e0 times the total charge enclosed by S"}, {"Chapter": "1", "sentence_range": "1203-1206", "Text": "For an area element of a closed surface, \u02c6n  is taken to be the direction\nof outward normal, by convention 15 Gauss\u2019s law: The flux of electric field through any closed surface S is\n1/e0 times the total charge enclosed by S The law is especially useful\nin determining electric field E, when the source distribution has simple\nsymmetry:\n(i) Thin infinitely long straight wire of uniform linear charge density l\n0\n\u02c6\n2\nr\n\u03b5\u03bb\n=\n\u03c0\nE\nn\nwhere r is the perpendicular distance of the point from the wire and\n\u02c6n is the radial unit vector in the plane normal to the wire passing\nthrough the point"}, {"Chapter": "1", "sentence_range": "1204-1207", "Text": "15 Gauss\u2019s law: The flux of electric field through any closed surface S is\n1/e0 times the total charge enclosed by S The law is especially useful\nin determining electric field E, when the source distribution has simple\nsymmetry:\n(i) Thin infinitely long straight wire of uniform linear charge density l\n0\n\u02c6\n2\nr\n\u03b5\u03bb\n=\n\u03c0\nE\nn\nwhere r is the perpendicular distance of the point from the wire and\n\u02c6n is the radial unit vector in the plane normal to the wire passing\nthrough the point (ii) Infinite thin plane sheet of uniform surface charge density s\n0\n\u02c6\n2\n\u03c3\n\u03b5\nE=\nn\nwhere \u02c6n  is a unit vector normal to the plane, outward on either side"}, {"Chapter": "1", "sentence_range": "1205-1208", "Text": "Gauss\u2019s law: The flux of electric field through any closed surface S is\n1/e0 times the total charge enclosed by S The law is especially useful\nin determining electric field E, when the source distribution has simple\nsymmetry:\n(i) Thin infinitely long straight wire of uniform linear charge density l\n0\n\u02c6\n2\nr\n\u03b5\u03bb\n=\n\u03c0\nE\nn\nwhere r is the perpendicular distance of the point from the wire and\n\u02c6n is the radial unit vector in the plane normal to the wire passing\nthrough the point (ii) Infinite thin plane sheet of uniform surface charge density s\n0\n\u02c6\n2\n\u03c3\n\u03b5\nE=\nn\nwhere \u02c6n  is a unit vector normal to the plane, outward on either side Rationalised 2023-24\n40\nPhysics\n(iii) Thin spherical shell of uniform surface charge density s\n2\n0\n\u02c6\n(\n)\n4\nq\nr\nR\nr\n\u03b5\n=\n\u2265\n\u03c0\nE\nr\nE = 0\n     (r   <   R)\nwhere r is the distance of the point from the centre of the shell and R\nthe radius of the shell"}, {"Chapter": "1", "sentence_range": "1206-1209", "Text": "The law is especially useful\nin determining electric field E, when the source distribution has simple\nsymmetry:\n(i) Thin infinitely long straight wire of uniform linear charge density l\n0\n\u02c6\n2\nr\n\u03b5\u03bb\n=\n\u03c0\nE\nn\nwhere r is the perpendicular distance of the point from the wire and\n\u02c6n is the radial unit vector in the plane normal to the wire passing\nthrough the point (ii) Infinite thin plane sheet of uniform surface charge density s\n0\n\u02c6\n2\n\u03c3\n\u03b5\nE=\nn\nwhere \u02c6n  is a unit vector normal to the plane, outward on either side Rationalised 2023-24\n40\nPhysics\n(iii) Thin spherical shell of uniform surface charge density s\n2\n0\n\u02c6\n(\n)\n4\nq\nr\nR\nr\n\u03b5\n=\n\u2265\n\u03c0\nE\nr\nE = 0\n     (r   <   R)\nwhere r is the distance of the point from the centre of the shell and R\nthe radius of the shell q is the total charge of the shell:  q = 4pR2s"}, {"Chapter": "1", "sentence_range": "1207-1210", "Text": "(ii) Infinite thin plane sheet of uniform surface charge density s\n0\n\u02c6\n2\n\u03c3\n\u03b5\nE=\nn\nwhere \u02c6n  is a unit vector normal to the plane, outward on either side Rationalised 2023-24\n40\nPhysics\n(iii) Thin spherical shell of uniform surface charge density s\n2\n0\n\u02c6\n(\n)\n4\nq\nr\nR\nr\n\u03b5\n=\n\u2265\n\u03c0\nE\nr\nE = 0\n     (r   <   R)\nwhere r is the distance of the point from the centre of the shell and R\nthe radius of the shell q is the total charge of the shell:  q = 4pR2s The electric field outside the shell is as though the total charge is\nconcentrated at the centre"}, {"Chapter": "1", "sentence_range": "1208-1211", "Text": "Rationalised 2023-24\n40\nPhysics\n(iii) Thin spherical shell of uniform surface charge density s\n2\n0\n\u02c6\n(\n)\n4\nq\nr\nR\nr\n\u03b5\n=\n\u2265\n\u03c0\nE\nr\nE = 0\n     (r   <   R)\nwhere r is the distance of the point from the centre of the shell and R\nthe radius of the shell q is the total charge of the shell:  q = 4pR2s The electric field outside the shell is as though the total charge is\nconcentrated at the centre The same result is true for a solid sphere\nof uniform volume charge density"}, {"Chapter": "1", "sentence_range": "1209-1212", "Text": "q is the total charge of the shell:  q = 4pR2s The electric field outside the shell is as though the total charge is\nconcentrated at the centre The same result is true for a solid sphere\nof uniform volume charge density The field is zero at all points inside\nthe shell"}, {"Chapter": "1", "sentence_range": "1210-1213", "Text": "The electric field outside the shell is as though the total charge is\nconcentrated at the centre The same result is true for a solid sphere\nof uniform volume charge density The field is zero at all points inside\nthe shell Physical quantity\nSymbol\nDimensions\nUnit\nRemarks\nVector area element\nD S\n[L2]\nm2\nDS = DS \u02c6n\nElectric field\nE\n[MLT\u20133A\u20131]\nV m\u20131\nElectric flux\nf\n[ML3 T\u20133A\u20131]\nV m\nDf  =  E"}, {"Chapter": "1", "sentence_range": "1211-1214", "Text": "The same result is true for a solid sphere\nof uniform volume charge density The field is zero at all points inside\nthe shell Physical quantity\nSymbol\nDimensions\nUnit\nRemarks\nVector area element\nD S\n[L2]\nm2\nDS = DS \u02c6n\nElectric field\nE\n[MLT\u20133A\u20131]\nV m\u20131\nElectric flux\nf\n[ML3 T\u20133A\u20131]\nV m\nDf  =  E DS\nDipole moment\np\n[LTA]\nC m\nVector directed\nfrom negative to\npositive charge\nCharge density:\nlinear\nl\n[L\u20131 TA]\nC m\u20131\n Charge/length\nsurface\ns\n[L\u20132 TA]\nC m\u20132\nCharge/area\nvolume\nr\n[L\u20133 TA]\nC m\u20133\nCharge/volume\nPOINTS TO PONDER\n1"}, {"Chapter": "1", "sentence_range": "1212-1215", "Text": "The field is zero at all points inside\nthe shell Physical quantity\nSymbol\nDimensions\nUnit\nRemarks\nVector area element\nD S\n[L2]\nm2\nDS = DS \u02c6n\nElectric field\nE\n[MLT\u20133A\u20131]\nV m\u20131\nElectric flux\nf\n[ML3 T\u20133A\u20131]\nV m\nDf  =  E DS\nDipole moment\np\n[LTA]\nC m\nVector directed\nfrom negative to\npositive charge\nCharge density:\nlinear\nl\n[L\u20131 TA]\nC m\u20131\n Charge/length\nsurface\ns\n[L\u20132 TA]\nC m\u20132\nCharge/area\nvolume\nr\n[L\u20133 TA]\nC m\u20133\nCharge/volume\nPOINTS TO PONDER\n1 You might wonder why the protons, all carrying positive charges,\nare compactly residing inside the nucleus"}, {"Chapter": "1", "sentence_range": "1213-1216", "Text": "Physical quantity\nSymbol\nDimensions\nUnit\nRemarks\nVector area element\nD S\n[L2]\nm2\nDS = DS \u02c6n\nElectric field\nE\n[MLT\u20133A\u20131]\nV m\u20131\nElectric flux\nf\n[ML3 T\u20133A\u20131]\nV m\nDf  =  E DS\nDipole moment\np\n[LTA]\nC m\nVector directed\nfrom negative to\npositive charge\nCharge density:\nlinear\nl\n[L\u20131 TA]\nC m\u20131\n Charge/length\nsurface\ns\n[L\u20132 TA]\nC m\u20132\nCharge/area\nvolume\nr\n[L\u20133 TA]\nC m\u20133\nCharge/volume\nPOINTS TO PONDER\n1 You might wonder why the protons, all carrying positive charges,\nare compactly residing inside the nucleus Why do they not fly away"}, {"Chapter": "1", "sentence_range": "1214-1217", "Text": "DS\nDipole moment\np\n[LTA]\nC m\nVector directed\nfrom negative to\npositive charge\nCharge density:\nlinear\nl\n[L\u20131 TA]\nC m\u20131\n Charge/length\nsurface\ns\n[L\u20132 TA]\nC m\u20132\nCharge/area\nvolume\nr\n[L\u20133 TA]\nC m\u20133\nCharge/volume\nPOINTS TO PONDER\n1 You might wonder why the protons, all carrying positive charges,\nare compactly residing inside the nucleus Why do they not fly away You will learn that there is a third kind of a fundamental force,\ncalled the strong force which holds them together"}, {"Chapter": "1", "sentence_range": "1215-1218", "Text": "You might wonder why the protons, all carrying positive charges,\nare compactly residing inside the nucleus Why do they not fly away You will learn that there is a third kind of a fundamental force,\ncalled the strong force which holds them together The range of\ndistance where this force is effective is, however, very small ~10-14\nm"}, {"Chapter": "1", "sentence_range": "1216-1219", "Text": "Why do they not fly away You will learn that there is a third kind of a fundamental force,\ncalled the strong force which holds them together The range of\ndistance where this force is effective is, however, very small ~10-14\nm This is precisely the size of the nucleus"}, {"Chapter": "1", "sentence_range": "1217-1220", "Text": "You will learn that there is a third kind of a fundamental force,\ncalled the strong force which holds them together The range of\ndistance where this force is effective is, however, very small ~10-14\nm This is precisely the size of the nucleus Also the electrons are\nnot allowed to sit on top of the protons, i"}, {"Chapter": "1", "sentence_range": "1218-1221", "Text": "The range of\ndistance where this force is effective is, however, very small ~10-14\nm This is precisely the size of the nucleus Also the electrons are\nnot allowed to sit on top of the protons, i e"}, {"Chapter": "1", "sentence_range": "1219-1222", "Text": "This is precisely the size of the nucleus Also the electrons are\nnot allowed to sit on top of the protons, i e inside the nucleus,\ndue to the laws of quantum mechanics"}, {"Chapter": "1", "sentence_range": "1220-1223", "Text": "Also the electrons are\nnot allowed to sit on top of the protons, i e inside the nucleus,\ndue to the laws of quantum mechanics This gives the atoms their\nstructure as they exist in nature"}, {"Chapter": "1", "sentence_range": "1221-1224", "Text": "e inside the nucleus,\ndue to the laws of quantum mechanics This gives the atoms their\nstructure as they exist in nature 2"}, {"Chapter": "1", "sentence_range": "1222-1225", "Text": "inside the nucleus,\ndue to the laws of quantum mechanics This gives the atoms their\nstructure as they exist in nature 2 Coulomb force and gravitational force follow the same inverse-square\nlaw"}, {"Chapter": "1", "sentence_range": "1223-1226", "Text": "This gives the atoms their\nstructure as they exist in nature 2 Coulomb force and gravitational force follow the same inverse-square\nlaw But gravitational force has only one sign (always attractive), while\nRationalised 2023-24\nElectric Charges\nand Fields\n41\nCoulomb force can be of both signs (attractive and repulsive), allowing\npossibility of cancellation of electric forces"}, {"Chapter": "1", "sentence_range": "1224-1227", "Text": "2 Coulomb force and gravitational force follow the same inverse-square\nlaw But gravitational force has only one sign (always attractive), while\nRationalised 2023-24\nElectric Charges\nand Fields\n41\nCoulomb force can be of both signs (attractive and repulsive), allowing\npossibility of cancellation of electric forces This is how gravity, despite\nbeing a much weaker force, can be a dominating and more pervasive\nforce in nature"}, {"Chapter": "1", "sentence_range": "1225-1228", "Text": "Coulomb force and gravitational force follow the same inverse-square\nlaw But gravitational force has only one sign (always attractive), while\nRationalised 2023-24\nElectric Charges\nand Fields\n41\nCoulomb force can be of both signs (attractive and repulsive), allowing\npossibility of cancellation of electric forces This is how gravity, despite\nbeing a much weaker force, can be a dominating and more pervasive\nforce in nature 3"}, {"Chapter": "1", "sentence_range": "1226-1229", "Text": "But gravitational force has only one sign (always attractive), while\nRationalised 2023-24\nElectric Charges\nand Fields\n41\nCoulomb force can be of both signs (attractive and repulsive), allowing\npossibility of cancellation of electric forces This is how gravity, despite\nbeing a much weaker force, can be a dominating and more pervasive\nforce in nature 3 The constant of proportionality k in Coulomb\u2019s law is a matter of\nchoice if the unit of charge is to be defined using Coulomb\u2019s law"}, {"Chapter": "1", "sentence_range": "1227-1230", "Text": "This is how gravity, despite\nbeing a much weaker force, can be a dominating and more pervasive\nforce in nature 3 The constant of proportionality k in Coulomb\u2019s law is a matter of\nchoice if the unit of charge is to be defined using Coulomb\u2019s law In SI\nunits, however, what is defined is the unit of current (A) via its magnetic\neffect (Ampere\u2019s law) and the unit of charge (coulomb) is simply defined\nby (1C = 1 A s)"}, {"Chapter": "1", "sentence_range": "1228-1231", "Text": "3 The constant of proportionality k in Coulomb\u2019s law is a matter of\nchoice if the unit of charge is to be defined using Coulomb\u2019s law In SI\nunits, however, what is defined is the unit of current (A) via its magnetic\neffect (Ampere\u2019s law) and the unit of charge (coulomb) is simply defined\nby (1C = 1 A s) In this case, the value of k is no longer arbitrary; it is\napproximately 9 \u00d7 109 N m2 C\u20132"}, {"Chapter": "1", "sentence_range": "1229-1232", "Text": "The constant of proportionality k in Coulomb\u2019s law is a matter of\nchoice if the unit of charge is to be defined using Coulomb\u2019s law In SI\nunits, however, what is defined is the unit of current (A) via its magnetic\neffect (Ampere\u2019s law) and the unit of charge (coulomb) is simply defined\nby (1C = 1 A s) In this case, the value of k is no longer arbitrary; it is\napproximately 9 \u00d7 109 N m2 C\u20132 4"}, {"Chapter": "1", "sentence_range": "1230-1233", "Text": "In SI\nunits, however, what is defined is the unit of current (A) via its magnetic\neffect (Ampere\u2019s law) and the unit of charge (coulomb) is simply defined\nby (1C = 1 A s) In this case, the value of k is no longer arbitrary; it is\napproximately 9 \u00d7 109 N m2 C\u20132 4 The rather large value of k, i"}, {"Chapter": "1", "sentence_range": "1231-1234", "Text": "In this case, the value of k is no longer arbitrary; it is\napproximately 9 \u00d7 109 N m2 C\u20132 4 The rather large value of k, i e"}, {"Chapter": "1", "sentence_range": "1232-1235", "Text": "4 The rather large value of k, i e , the large size of the unit of charge\n(1C) from the point of view of electric effects arises because (as\nmentioned in point 3 already) the unit of charge is defined in terms of\nmagnetic forces (forces on  current\u2013carrying wires) which are generally\nmuch weaker than the electric forces"}, {"Chapter": "1", "sentence_range": "1233-1236", "Text": "The rather large value of k, i e , the large size of the unit of charge\n(1C) from the point of view of electric effects arises because (as\nmentioned in point 3 already) the unit of charge is defined in terms of\nmagnetic forces (forces on  current\u2013carrying wires) which are generally\nmuch weaker than the electric forces Thus while 1 ampere is a unit\nof reasonable size for magnetic effects, 1 C = 1 A s, is too big a unit for\nelectric effects"}, {"Chapter": "1", "sentence_range": "1234-1237", "Text": "e , the large size of the unit of charge\n(1C) from the point of view of electric effects arises because (as\nmentioned in point 3 already) the unit of charge is defined in terms of\nmagnetic forces (forces on  current\u2013carrying wires) which are generally\nmuch weaker than the electric forces Thus while 1 ampere is a unit\nof reasonable size for magnetic effects, 1 C = 1 A s, is too big a unit for\nelectric effects 5"}, {"Chapter": "1", "sentence_range": "1235-1238", "Text": ", the large size of the unit of charge\n(1C) from the point of view of electric effects arises because (as\nmentioned in point 3 already) the unit of charge is defined in terms of\nmagnetic forces (forces on  current\u2013carrying wires) which are generally\nmuch weaker than the electric forces Thus while 1 ampere is a unit\nof reasonable size for magnetic effects, 1 C = 1 A s, is too big a unit for\nelectric effects 5 The additive property of charge is not an \u2018obvious\u2019 property"}, {"Chapter": "1", "sentence_range": "1236-1239", "Text": "Thus while 1 ampere is a unit\nof reasonable size for magnetic effects, 1 C = 1 A s, is too big a unit for\nelectric effects 5 The additive property of charge is not an \u2018obvious\u2019 property It is related\nto the fact that electric charge has no direction associated with it;\ncharge is a scalar"}, {"Chapter": "1", "sentence_range": "1237-1240", "Text": "5 The additive property of charge is not an \u2018obvious\u2019 property It is related\nto the fact that electric charge has no direction associated with it;\ncharge is a scalar 6"}, {"Chapter": "1", "sentence_range": "1238-1241", "Text": "The additive property of charge is not an \u2018obvious\u2019 property It is related\nto the fact that electric charge has no direction associated with it;\ncharge is a scalar 6 Charge is not only a scalar (or invariant) under rotation; it is also\ninvariant for frames of reference in relative motion"}, {"Chapter": "1", "sentence_range": "1239-1242", "Text": "It is related\nto the fact that electric charge has no direction associated with it;\ncharge is a scalar 6 Charge is not only a scalar (or invariant) under rotation; it is also\ninvariant for frames of reference in relative motion This is not always\ntrue for every scalar"}, {"Chapter": "1", "sentence_range": "1240-1243", "Text": "6 Charge is not only a scalar (or invariant) under rotation; it is also\ninvariant for frames of reference in relative motion This is not always\ntrue for every scalar For example, kinetic energy is a scalar under\nrotation, but is not invariant for frames of reference in relative\nmotion"}, {"Chapter": "1", "sentence_range": "1241-1244", "Text": "Charge is not only a scalar (or invariant) under rotation; it is also\ninvariant for frames of reference in relative motion This is not always\ntrue for every scalar For example, kinetic energy is a scalar under\nrotation, but is not invariant for frames of reference in relative\nmotion 7"}, {"Chapter": "1", "sentence_range": "1242-1245", "Text": "This is not always\ntrue for every scalar For example, kinetic energy is a scalar under\nrotation, but is not invariant for frames of reference in relative\nmotion 7 Conservation of total charge of an isolated system is a property\nindependent of the scalar nature of charge noted in point 6"}, {"Chapter": "1", "sentence_range": "1243-1246", "Text": "For example, kinetic energy is a scalar under\nrotation, but is not invariant for frames of reference in relative\nmotion 7 Conservation of total charge of an isolated system is a property\nindependent of the scalar nature of charge noted in point 6 Conservation refers to invariance in time in a given frame of reference"}, {"Chapter": "1", "sentence_range": "1244-1247", "Text": "7 Conservation of total charge of an isolated system is a property\nindependent of the scalar nature of charge noted in point 6 Conservation refers to invariance in time in a given frame of reference A quantity may be scalar but not conserved (like kinetic energy in an\ninelastic collision)"}, {"Chapter": "1", "sentence_range": "1245-1248", "Text": "Conservation of total charge of an isolated system is a property\nindependent of the scalar nature of charge noted in point 6 Conservation refers to invariance in time in a given frame of reference A quantity may be scalar but not conserved (like kinetic energy in an\ninelastic collision) On the other hand, one can have conserved vector\nquantity (e"}, {"Chapter": "1", "sentence_range": "1246-1249", "Text": "Conservation refers to invariance in time in a given frame of reference A quantity may be scalar but not conserved (like kinetic energy in an\ninelastic collision) On the other hand, one can have conserved vector\nquantity (e g"}, {"Chapter": "1", "sentence_range": "1247-1250", "Text": "A quantity may be scalar but not conserved (like kinetic energy in an\ninelastic collision) On the other hand, one can have conserved vector\nquantity (e g , angular momentum of an isolated system)"}, {"Chapter": "1", "sentence_range": "1248-1251", "Text": "On the other hand, one can have conserved vector\nquantity (e g , angular momentum of an isolated system) 8"}, {"Chapter": "1", "sentence_range": "1249-1252", "Text": "g , angular momentum of an isolated system) 8 Quantisation of electric charge is a basic (unexplained) law of nature;\ninterestingly, there is no analogous law on quantisation of mass"}, {"Chapter": "1", "sentence_range": "1250-1253", "Text": ", angular momentum of an isolated system) 8 Quantisation of electric charge is a basic (unexplained) law of nature;\ninterestingly, there is no analogous law on quantisation of mass 9"}, {"Chapter": "1", "sentence_range": "1251-1254", "Text": "8 Quantisation of electric charge is a basic (unexplained) law of nature;\ninterestingly, there is no analogous law on quantisation of mass 9 Superposition principle should not be regarded as \u2018obvious\u2019, or\nequated with the law of addition of vectors"}, {"Chapter": "1", "sentence_range": "1252-1255", "Text": "Quantisation of electric charge is a basic (unexplained) law of nature;\ninterestingly, there is no analogous law on quantisation of mass 9 Superposition principle should not be regarded as \u2018obvious\u2019, or\nequated with the law of addition of vectors It says two things:\nforce on one charge due to another charge is unaffected by the\npresence of other charges, and there are no additional three-body,\nfour-body, etc"}, {"Chapter": "1", "sentence_range": "1253-1256", "Text": "9 Superposition principle should not be regarded as \u2018obvious\u2019, or\nequated with the law of addition of vectors It says two things:\nforce on one charge due to another charge is unaffected by the\npresence of other charges, and there are no additional three-body,\nfour-body, etc , forces which arise only when there are more than\ntwo charges"}, {"Chapter": "1", "sentence_range": "1254-1257", "Text": "Superposition principle should not be regarded as \u2018obvious\u2019, or\nequated with the law of addition of vectors It says two things:\nforce on one charge due to another charge is unaffected by the\npresence of other charges, and there are no additional three-body,\nfour-body, etc , forces which arise only when there are more than\ntwo charges 10"}, {"Chapter": "1", "sentence_range": "1255-1258", "Text": "It says two things:\nforce on one charge due to another charge is unaffected by the\npresence of other charges, and there are no additional three-body,\nfour-body, etc , forces which arise only when there are more than\ntwo charges 10 The electric field due to a discrete charge configuration is not defined\nat the locations of the discrete charges"}, {"Chapter": "1", "sentence_range": "1256-1259", "Text": ", forces which arise only when there are more than\ntwo charges 10 The electric field due to a discrete charge configuration is not defined\nat the locations of the discrete charges For continuous volume\ncharge distribution, it is defined at any point in the distribution"}, {"Chapter": "1", "sentence_range": "1257-1260", "Text": "10 The electric field due to a discrete charge configuration is not defined\nat the locations of the discrete charges For continuous volume\ncharge distribution, it is defined at any point in the distribution For a surface charge distribution, electric field is discontinuous\nacross the surface"}, {"Chapter": "1", "sentence_range": "1258-1261", "Text": "The electric field due to a discrete charge configuration is not defined\nat the locations of the discrete charges For continuous volume\ncharge distribution, it is defined at any point in the distribution For a surface charge distribution, electric field is discontinuous\nacross the surface 11"}, {"Chapter": "1", "sentence_range": "1259-1262", "Text": "For continuous volume\ncharge distribution, it is defined at any point in the distribution For a surface charge distribution, electric field is discontinuous\nacross the surface 11 The electric field due to a charge configuration with total charge zero\nis not zero; but for distances large compared to the size of\nthe configuration, its field falls off faster than 1/r 2, typical of field\ndue to a single charge"}, {"Chapter": "1", "sentence_range": "1260-1263", "Text": "For a surface charge distribution, electric field is discontinuous\nacross the surface 11 The electric field due to a charge configuration with total charge zero\nis not zero; but for distances large compared to the size of\nthe configuration, its field falls off faster than 1/r 2, typical of field\ndue to a single charge An electric dipole is the simplest example of\nthis fact"}, {"Chapter": "1", "sentence_range": "1261-1264", "Text": "11 The electric field due to a charge configuration with total charge zero\nis not zero; but for distances large compared to the size of\nthe configuration, its field falls off faster than 1/r 2, typical of field\ndue to a single charge An electric dipole is the simplest example of\nthis fact Rationalised 2023-24\n42\nPhysics\nEXERCISES\n1"}, {"Chapter": "1", "sentence_range": "1262-1265", "Text": "The electric field due to a charge configuration with total charge zero\nis not zero; but for distances large compared to the size of\nthe configuration, its field falls off faster than 1/r 2, typical of field\ndue to a single charge An electric dipole is the simplest example of\nthis fact Rationalised 2023-24\n42\nPhysics\nEXERCISES\n1 1\nWhat is the force between  two small charged spheres having\ncharges of 2 \u00d7 10\u20137C and 3 \u00d7 10\u20137C placed 30 cm apart in air"}, {"Chapter": "1", "sentence_range": "1263-1266", "Text": "An electric dipole is the simplest example of\nthis fact Rationalised 2023-24\n42\nPhysics\nEXERCISES\n1 1\nWhat is the force between  two small charged spheres having\ncharges of 2 \u00d7 10\u20137C and 3 \u00d7 10\u20137C placed 30 cm apart in air 1"}, {"Chapter": "1", "sentence_range": "1264-1267", "Text": "Rationalised 2023-24\n42\nPhysics\nEXERCISES\n1 1\nWhat is the force between  two small charged spheres having\ncharges of 2 \u00d7 10\u20137C and 3 \u00d7 10\u20137C placed 30 cm apart in air 1 2\nThe electrostatic force on a small sphere of charge 0"}, {"Chapter": "1", "sentence_range": "1265-1268", "Text": "1\nWhat is the force between  two small charged spheres having\ncharges of 2 \u00d7 10\u20137C and 3 \u00d7 10\u20137C placed 30 cm apart in air 1 2\nThe electrostatic force on a small sphere of charge 0 4 mC due to\nanother small sphere of charge \u20130"}, {"Chapter": "1", "sentence_range": "1266-1269", "Text": "1 2\nThe electrostatic force on a small sphere of charge 0 4 mC due to\nanother small sphere of charge \u20130 8 mC in air is 0"}, {"Chapter": "1", "sentence_range": "1267-1270", "Text": "2\nThe electrostatic force on a small sphere of charge 0 4 mC due to\nanother small sphere of charge \u20130 8 mC in air is 0 2 N"}, {"Chapter": "1", "sentence_range": "1268-1271", "Text": "4 mC due to\nanother small sphere of charge \u20130 8 mC in air is 0 2 N (a) What is\nthe distance between the two spheres"}, {"Chapter": "1", "sentence_range": "1269-1272", "Text": "8 mC in air is 0 2 N (a) What is\nthe distance between the two spheres (b) What is the force on the\nsecond sphere due to the first"}, {"Chapter": "1", "sentence_range": "1270-1273", "Text": "2 N (a) What is\nthe distance between the two spheres (b) What is the force on the\nsecond sphere due to the first 1"}, {"Chapter": "1", "sentence_range": "1271-1274", "Text": "(a) What is\nthe distance between the two spheres (b) What is the force on the\nsecond sphere due to the first 1 3\nCheck that the ratio ke2/G memp is dimensionless"}, {"Chapter": "1", "sentence_range": "1272-1275", "Text": "(b) What is the force on the\nsecond sphere due to the first 1 3\nCheck that the ratio ke2/G memp is dimensionless Look up a Table\nof Physical Constants and determine the value of this ratio"}, {"Chapter": "1", "sentence_range": "1273-1276", "Text": "1 3\nCheck that the ratio ke2/G memp is dimensionless Look up a Table\nof Physical Constants and determine the value of this ratio What\ndoes the ratio signify"}, {"Chapter": "1", "sentence_range": "1274-1277", "Text": "3\nCheck that the ratio ke2/G memp is dimensionless Look up a Table\nof Physical Constants and determine the value of this ratio What\ndoes the ratio signify 1"}, {"Chapter": "1", "sentence_range": "1275-1278", "Text": "Look up a Table\nof Physical Constants and determine the value of this ratio What\ndoes the ratio signify 1 4\n(a) Explain the meaning of the statement \u2018electric charge of a body\nis quantised\u2019"}, {"Chapter": "1", "sentence_range": "1276-1279", "Text": "What\ndoes the ratio signify 1 4\n(a) Explain the meaning of the statement \u2018electric charge of a body\nis quantised\u2019 (b) Why can one ignore quantisation of electric charge when dealing\nwith macroscopic i"}, {"Chapter": "1", "sentence_range": "1277-1280", "Text": "1 4\n(a) Explain the meaning of the statement \u2018electric charge of a body\nis quantised\u2019 (b) Why can one ignore quantisation of electric charge when dealing\nwith macroscopic i e"}, {"Chapter": "1", "sentence_range": "1278-1281", "Text": "4\n(a) Explain the meaning of the statement \u2018electric charge of a body\nis quantised\u2019 (b) Why can one ignore quantisation of electric charge when dealing\nwith macroscopic i e , large scale charges"}, {"Chapter": "1", "sentence_range": "1279-1282", "Text": "(b) Why can one ignore quantisation of electric charge when dealing\nwith macroscopic i e , large scale charges 1"}, {"Chapter": "1", "sentence_range": "1280-1283", "Text": "e , large scale charges 1 5\nWhen a glass rod is rubbed with a silk cloth, charges appear on\nboth"}, {"Chapter": "1", "sentence_range": "1281-1284", "Text": ", large scale charges 1 5\nWhen a glass rod is rubbed with a silk cloth, charges appear on\nboth A similar phenomenon is observed with many other pairs of\nbodies"}, {"Chapter": "1", "sentence_range": "1282-1285", "Text": "1 5\nWhen a glass rod is rubbed with a silk cloth, charges appear on\nboth A similar phenomenon is observed with many other pairs of\nbodies Explain how this observation is consistent with the law of\nconservation of charge"}, {"Chapter": "1", "sentence_range": "1283-1286", "Text": "5\nWhen a glass rod is rubbed with a silk cloth, charges appear on\nboth A similar phenomenon is observed with many other pairs of\nbodies Explain how this observation is consistent with the law of\nconservation of charge 1"}, {"Chapter": "1", "sentence_range": "1284-1287", "Text": "A similar phenomenon is observed with many other pairs of\nbodies Explain how this observation is consistent with the law of\nconservation of charge 1 6\nFour point charges qA = 2 mC, qB = \u20135 mC, qC = 2 mC, and qD = \u20135 mC are\nlocated at the corners of a square ABCD of side 10 cm"}, {"Chapter": "1", "sentence_range": "1285-1288", "Text": "Explain how this observation is consistent with the law of\nconservation of charge 1 6\nFour point charges qA = 2 mC, qB = \u20135 mC, qC = 2 mC, and qD = \u20135 mC are\nlocated at the corners of a square ABCD of side 10 cm What is the\nforce on a charge of 1 mC placed at the centre of the square"}, {"Chapter": "1", "sentence_range": "1286-1289", "Text": "1 6\nFour point charges qA = 2 mC, qB = \u20135 mC, qC = 2 mC, and qD = \u20135 mC are\nlocated at the corners of a square ABCD of side 10 cm What is the\nforce on a charge of 1 mC placed at the centre of the square 1"}, {"Chapter": "1", "sentence_range": "1287-1290", "Text": "6\nFour point charges qA = 2 mC, qB = \u20135 mC, qC = 2 mC, and qD = \u20135 mC are\nlocated at the corners of a square ABCD of side 10 cm What is the\nforce on a charge of 1 mC placed at the centre of the square 1 7\n(a) An electrostatic field line is a continuous curve"}, {"Chapter": "1", "sentence_range": "1288-1291", "Text": "What is the\nforce on a charge of 1 mC placed at the centre of the square 1 7\n(a) An electrostatic field line is a continuous curve That is, a field\nline cannot have sudden breaks"}, {"Chapter": "1", "sentence_range": "1289-1292", "Text": "1 7\n(a) An electrostatic field line is a continuous curve That is, a field\nline cannot have sudden breaks Why not"}, {"Chapter": "1", "sentence_range": "1290-1293", "Text": "7\n(a) An electrostatic field line is a continuous curve That is, a field\nline cannot have sudden breaks Why not (b) Explain why two field lines never cross each other at any point"}, {"Chapter": "1", "sentence_range": "1291-1294", "Text": "That is, a field\nline cannot have sudden breaks Why not (b) Explain why two field lines never cross each other at any point 1"}, {"Chapter": "1", "sentence_range": "1292-1295", "Text": "Why not (b) Explain why two field lines never cross each other at any point 1 8\nTwo point charges qA = 3 mC and qB = \u20133 mC are located 20 cm apart\nin vacuum"}, {"Chapter": "1", "sentence_range": "1293-1296", "Text": "(b) Explain why two field lines never cross each other at any point 1 8\nTwo point charges qA = 3 mC and qB = \u20133 mC are located 20 cm apart\nin vacuum (a) What is the electric field at the midpoint O of the line AB joining\nthe two charges"}, {"Chapter": "1", "sentence_range": "1294-1297", "Text": "1 8\nTwo point charges qA = 3 mC and qB = \u20133 mC are located 20 cm apart\nin vacuum (a) What is the electric field at the midpoint O of the line AB joining\nthe two charges (b) If a negative test charge of magnitude 1"}, {"Chapter": "1", "sentence_range": "1295-1298", "Text": "8\nTwo point charges qA = 3 mC and qB = \u20133 mC are located 20 cm apart\nin vacuum (a) What is the electric field at the midpoint O of the line AB joining\nthe two charges (b) If a negative test charge of magnitude 1 5 \u00d7 10\u20139 C is placed at\nthis point, what is the force experienced by the test charge"}, {"Chapter": "1", "sentence_range": "1296-1299", "Text": "(a) What is the electric field at the midpoint O of the line AB joining\nthe two charges (b) If a negative test charge of magnitude 1 5 \u00d7 10\u20139 C is placed at\nthis point, what is the force experienced by the test charge 1"}, {"Chapter": "1", "sentence_range": "1297-1300", "Text": "(b) If a negative test charge of magnitude 1 5 \u00d7 10\u20139 C is placed at\nthis point, what is the force experienced by the test charge 1 9\nA system has two charges qA = 2"}, {"Chapter": "1", "sentence_range": "1298-1301", "Text": "5 \u00d7 10\u20139 C is placed at\nthis point, what is the force experienced by the test charge 1 9\nA system has two charges qA = 2 5 \u00d7 10\u20137 C and qB  =  \u20132"}, {"Chapter": "1", "sentence_range": "1299-1302", "Text": "1 9\nA system has two charges qA = 2 5 \u00d7 10\u20137 C and qB  =  \u20132 5 \u00d7 10\u20137 C\nlocated at points A: (0, 0, \u201315 cm) and B: (0,0, +15 cm), respectively"}, {"Chapter": "1", "sentence_range": "1300-1303", "Text": "9\nA system has two charges qA = 2 5 \u00d7 10\u20137 C and qB  =  \u20132 5 \u00d7 10\u20137 C\nlocated at points A: (0, 0, \u201315 cm) and B: (0,0, +15 cm), respectively What are the total charge and electric dipole moment of the system"}, {"Chapter": "1", "sentence_range": "1301-1304", "Text": "5 \u00d7 10\u20137 C and qB  =  \u20132 5 \u00d7 10\u20137 C\nlocated at points A: (0, 0, \u201315 cm) and B: (0,0, +15 cm), respectively What are the total charge and electric dipole moment of the system 1"}, {"Chapter": "1", "sentence_range": "1302-1305", "Text": "5 \u00d7 10\u20137 C\nlocated at points A: (0, 0, \u201315 cm) and B: (0,0, +15 cm), respectively What are the total charge and electric dipole moment of the system 1 10\nAn electric dipole with dipole moment 4 \u00d7 10\u20139 C m is aligned at 30\u00b0\nwith the direction of a uniform electric field of magnitude 5 \u00d7 104 NC\u20131"}, {"Chapter": "1", "sentence_range": "1303-1306", "Text": "What are the total charge and electric dipole moment of the system 1 10\nAn electric dipole with dipole moment 4 \u00d7 10\u20139 C m is aligned at 30\u00b0\nwith the direction of a uniform electric field of magnitude 5 \u00d7 104 NC\u20131 Calculate the magnitude of the torque acting on the dipole"}, {"Chapter": "1", "sentence_range": "1304-1307", "Text": "1 10\nAn electric dipole with dipole moment 4 \u00d7 10\u20139 C m is aligned at 30\u00b0\nwith the direction of a uniform electric field of magnitude 5 \u00d7 104 NC\u20131 Calculate the magnitude of the torque acting on the dipole 1"}, {"Chapter": "1", "sentence_range": "1305-1308", "Text": "10\nAn electric dipole with dipole moment 4 \u00d7 10\u20139 C m is aligned at 30\u00b0\nwith the direction of a uniform electric field of magnitude 5 \u00d7 104 NC\u20131 Calculate the magnitude of the torque acting on the dipole 1 11\nA polythene piece rubbed with wool is found to have a negative\ncharge of 3 \u00d7 10\u20137 C"}, {"Chapter": "1", "sentence_range": "1306-1309", "Text": "Calculate the magnitude of the torque acting on the dipole 1 11\nA polythene piece rubbed with wool is found to have a negative\ncharge of 3 \u00d7 10\u20137 C (a) Estimate the number of electrons transferred (from which to\nwhich"}, {"Chapter": "1", "sentence_range": "1307-1310", "Text": "1 11\nA polythene piece rubbed with wool is found to have a negative\ncharge of 3 \u00d7 10\u20137 C (a) Estimate the number of electrons transferred (from which to\nwhich )\n(b) Is there a transfer of mass from wool to polythene"}, {"Chapter": "1", "sentence_range": "1308-1311", "Text": "11\nA polythene piece rubbed with wool is found to have a negative\ncharge of 3 \u00d7 10\u20137 C (a) Estimate the number of electrons transferred (from which to\nwhich )\n(b) Is there a transfer of mass from wool to polythene 1"}, {"Chapter": "1", "sentence_range": "1309-1312", "Text": "(a) Estimate the number of electrons transferred (from which to\nwhich )\n(b) Is there a transfer of mass from wool to polythene 1 12\n(a) Two insulated charged copper spheres A and B have their centres\nseparated by a distance of 50 cm"}, {"Chapter": "1", "sentence_range": "1310-1313", "Text": ")\n(b) Is there a transfer of mass from wool to polythene 1 12\n(a) Two insulated charged copper spheres A and B have their centres\nseparated by a distance of 50 cm What is the mutual force of\nelectrostatic repulsion if the charge on each is 6"}, {"Chapter": "1", "sentence_range": "1311-1314", "Text": "1 12\n(a) Two insulated charged copper spheres A and B have their centres\nseparated by a distance of 50 cm What is the mutual force of\nelectrostatic repulsion if the charge on each is 6 5 \u00d7 10\u20137 C"}, {"Chapter": "1", "sentence_range": "1312-1315", "Text": "12\n(a) Two insulated charged copper spheres A and B have their centres\nseparated by a distance of 50 cm What is the mutual force of\nelectrostatic repulsion if the charge on each is 6 5 \u00d7 10\u20137 C The\nradii of A and B are negligible compared to the distance of\nseparation"}, {"Chapter": "1", "sentence_range": "1313-1316", "Text": "What is the mutual force of\nelectrostatic repulsion if the charge on each is 6 5 \u00d7 10\u20137 C The\nradii of A and B are negligible compared to the distance of\nseparation (b) What is the force of repulsion if each sphere is charged double\nthe above amount, and the distance between them is halved"}, {"Chapter": "1", "sentence_range": "1314-1317", "Text": "5 \u00d7 10\u20137 C The\nradii of A and B are negligible compared to the distance of\nseparation (b) What is the force of repulsion if each sphere is charged double\nthe above amount, and the distance between them is halved 1"}, {"Chapter": "1", "sentence_range": "1315-1318", "Text": "The\nradii of A and B are negligible compared to the distance of\nseparation (b) What is the force of repulsion if each sphere is charged double\nthe above amount, and the distance between them is halved 1 13\nFigure 1"}, {"Chapter": "1", "sentence_range": "1316-1319", "Text": "(b) What is the force of repulsion if each sphere is charged double\nthe above amount, and the distance between them is halved 1 13\nFigure 1 30 shows tracks of three charged particles in a uniform\nelectrostatic field"}, {"Chapter": "1", "sentence_range": "1317-1320", "Text": "1 13\nFigure 1 30 shows tracks of three charged particles in a uniform\nelectrostatic field Give the signs of the three charges"}, {"Chapter": "1", "sentence_range": "1318-1321", "Text": "13\nFigure 1 30 shows tracks of three charged particles in a uniform\nelectrostatic field Give the signs of the three charges Which particle\nhas the highest charge to mass ratio"}, {"Chapter": "1", "sentence_range": "1319-1322", "Text": "30 shows tracks of three charged particles in a uniform\nelectrostatic field Give the signs of the three charges Which particle\nhas the highest charge to mass ratio Rationalised 2023-24\nElectric Charges\nand Fields\n43\nFIGURE 1"}, {"Chapter": "1", "sentence_range": "1320-1323", "Text": "Give the signs of the three charges Which particle\nhas the highest charge to mass ratio Rationalised 2023-24\nElectric Charges\nand Fields\n43\nFIGURE 1 30\n1"}, {"Chapter": "1", "sentence_range": "1321-1324", "Text": "Which particle\nhas the highest charge to mass ratio Rationalised 2023-24\nElectric Charges\nand Fields\n43\nFIGURE 1 30\n1 14\nConsider a uniform electric field E = 3 \u00d7 103 \u00ee N/C"}, {"Chapter": "1", "sentence_range": "1322-1325", "Text": "Rationalised 2023-24\nElectric Charges\nand Fields\n43\nFIGURE 1 30\n1 14\nConsider a uniform electric field E = 3 \u00d7 103 \u00ee N/C (a)  What is  the\nflux of this field through a square of 10 cm on a side whose plane is\nparallel to the yz plane"}, {"Chapter": "1", "sentence_range": "1323-1326", "Text": "30\n1 14\nConsider a uniform electric field E = 3 \u00d7 103 \u00ee N/C (a)  What is  the\nflux of this field through a square of 10 cm on a side whose plane is\nparallel to the yz plane (b) What is the  flux through the same\nsquare if the normal  to its plane makes a 60\u00b0 angle with the x-axis"}, {"Chapter": "1", "sentence_range": "1324-1327", "Text": "14\nConsider a uniform electric field E = 3 \u00d7 103 \u00ee N/C (a)  What is  the\nflux of this field through a square of 10 cm on a side whose plane is\nparallel to the yz plane (b) What is the  flux through the same\nsquare if the normal  to its plane makes a 60\u00b0 angle with the x-axis 1"}, {"Chapter": "1", "sentence_range": "1325-1328", "Text": "(a)  What is  the\nflux of this field through a square of 10 cm on a side whose plane is\nparallel to the yz plane (b) What is the  flux through the same\nsquare if the normal  to its plane makes a 60\u00b0 angle with the x-axis 1 15\nWhat is the net flux of the uniform electric field of Exercise 1"}, {"Chapter": "1", "sentence_range": "1326-1329", "Text": "(b) What is the  flux through the same\nsquare if the normal  to its plane makes a 60\u00b0 angle with the x-axis 1 15\nWhat is the net flux of the uniform electric field of Exercise 1 14\nthrough a cube of side 20 cm oriented so that its faces are parallel\nto the coordinate planes"}, {"Chapter": "1", "sentence_range": "1327-1330", "Text": "1 15\nWhat is the net flux of the uniform electric field of Exercise 1 14\nthrough a cube of side 20 cm oriented so that its faces are parallel\nto the coordinate planes 1"}, {"Chapter": "1", "sentence_range": "1328-1331", "Text": "15\nWhat is the net flux of the uniform electric field of Exercise 1 14\nthrough a cube of side 20 cm oriented so that its faces are parallel\nto the coordinate planes 1 16\nCareful measurement of the electric field at the surface of a black\nbox indicates that the net outward flux through the surface of the\nbox is 8"}, {"Chapter": "1", "sentence_range": "1329-1332", "Text": "14\nthrough a cube of side 20 cm oriented so that its faces are parallel\nto the coordinate planes 1 16\nCareful measurement of the electric field at the surface of a black\nbox indicates that the net outward flux through the surface of the\nbox is 8 0 \u00d7 103 Nm2/C"}, {"Chapter": "1", "sentence_range": "1330-1333", "Text": "1 16\nCareful measurement of the electric field at the surface of a black\nbox indicates that the net outward flux through the surface of the\nbox is 8 0 \u00d7 103 Nm2/C (a) What is the net charge inside the box"}, {"Chapter": "1", "sentence_range": "1331-1334", "Text": "16\nCareful measurement of the electric field at the surface of a black\nbox indicates that the net outward flux through the surface of the\nbox is 8 0 \u00d7 103 Nm2/C (a) What is the net charge inside the box (b) If the net outward flux through the surface of the box were zero,\ncould you conclude that there were no charges inside the box"}, {"Chapter": "1", "sentence_range": "1332-1335", "Text": "0 \u00d7 103 Nm2/C (a) What is the net charge inside the box (b) If the net outward flux through the surface of the box were zero,\ncould you conclude that there were no charges inside the box Why\nor Why not"}, {"Chapter": "1", "sentence_range": "1333-1336", "Text": "(a) What is the net charge inside the box (b) If the net outward flux through the surface of the box were zero,\ncould you conclude that there were no charges inside the box Why\nor Why not 1"}, {"Chapter": "1", "sentence_range": "1334-1337", "Text": "(b) If the net outward flux through the surface of the box were zero,\ncould you conclude that there were no charges inside the box Why\nor Why not 1 17\nA point charge +10 mC is a distance 5 cm directly above the centre\nof a square of side 10 cm, as shown in Fig"}, {"Chapter": "1", "sentence_range": "1335-1338", "Text": "Why\nor Why not 1 17\nA point charge +10 mC is a distance 5 cm directly above the centre\nof a square of side 10 cm, as shown in Fig 1"}, {"Chapter": "1", "sentence_range": "1336-1339", "Text": "1 17\nA point charge +10 mC is a distance 5 cm directly above the centre\nof a square of side 10 cm, as shown in Fig 1 31"}, {"Chapter": "1", "sentence_range": "1337-1340", "Text": "17\nA point charge +10 mC is a distance 5 cm directly above the centre\nof a square of side 10 cm, as shown in Fig 1 31 What is the\nmagnitude of the electric flux through the square"}, {"Chapter": "1", "sentence_range": "1338-1341", "Text": "1 31 What is the\nmagnitude of the electric flux through the square (Hint: Think of\nthe square as one face of a cube with edge 10 cm"}, {"Chapter": "1", "sentence_range": "1339-1342", "Text": "31 What is the\nmagnitude of the electric flux through the square (Hint: Think of\nthe square as one face of a cube with edge 10 cm )\nFIGURE 1"}, {"Chapter": "1", "sentence_range": "1340-1343", "Text": "What is the\nmagnitude of the electric flux through the square (Hint: Think of\nthe square as one face of a cube with edge 10 cm )\nFIGURE 1 31\n1"}, {"Chapter": "1", "sentence_range": "1341-1344", "Text": "(Hint: Think of\nthe square as one face of a cube with edge 10 cm )\nFIGURE 1 31\n1 18\nA point charge of 2"}, {"Chapter": "1", "sentence_range": "1342-1345", "Text": ")\nFIGURE 1 31\n1 18\nA point charge of 2 0 mC is at the centre of a  cubic Gaussian\nsurface 9"}, {"Chapter": "1", "sentence_range": "1343-1346", "Text": "31\n1 18\nA point charge of 2 0 mC is at the centre of a  cubic Gaussian\nsurface 9 0 cm on edge"}, {"Chapter": "1", "sentence_range": "1344-1347", "Text": "18\nA point charge of 2 0 mC is at the centre of a  cubic Gaussian\nsurface 9 0 cm on edge What is the net electric flux through the\nsurface"}, {"Chapter": "1", "sentence_range": "1345-1348", "Text": "0 mC is at the centre of a  cubic Gaussian\nsurface 9 0 cm on edge What is the net electric flux through the\nsurface 1"}, {"Chapter": "1", "sentence_range": "1346-1349", "Text": "0 cm on edge What is the net electric flux through the\nsurface 1 19\nA point charge causes an electric flux of \u20131"}, {"Chapter": "1", "sentence_range": "1347-1350", "Text": "What is the net electric flux through the\nsurface 1 19\nA point charge causes an electric flux of \u20131 0 \u00d7 103 Nm2/C to pass\nthrough a spherical Gaussian surface of 10"}, {"Chapter": "1", "sentence_range": "1348-1351", "Text": "1 19\nA point charge causes an electric flux of \u20131 0 \u00d7 103 Nm2/C to pass\nthrough a spherical Gaussian surface of 10 0 cm radius centred on\nthe charge"}, {"Chapter": "1", "sentence_range": "1349-1352", "Text": "19\nA point charge causes an electric flux of \u20131 0 \u00d7 103 Nm2/C to pass\nthrough a spherical Gaussian surface of 10 0 cm radius centred on\nthe charge (a)  If the radius of the Gaussian surface were doubled,\nhow much flux would  pass through the surface"}, {"Chapter": "1", "sentence_range": "1350-1353", "Text": "0 \u00d7 103 Nm2/C to pass\nthrough a spherical Gaussian surface of 10 0 cm radius centred on\nthe charge (a)  If the radius of the Gaussian surface were doubled,\nhow much flux would  pass through the surface (b) What is the\nvalue of the point charge"}, {"Chapter": "1", "sentence_range": "1351-1354", "Text": "0 cm radius centred on\nthe charge (a)  If the radius of the Gaussian surface were doubled,\nhow much flux would  pass through the surface (b) What is the\nvalue of the point charge 1"}, {"Chapter": "1", "sentence_range": "1352-1355", "Text": "(a)  If the radius of the Gaussian surface were doubled,\nhow much flux would  pass through the surface (b) What is the\nvalue of the point charge 1 20\nA conducting sphere of radius 10 cm has an unknown charge"}, {"Chapter": "1", "sentence_range": "1353-1356", "Text": "(b) What is the\nvalue of the point charge 1 20\nA conducting sphere of radius 10 cm has an unknown charge If\nthe electric field 20 cm from the centre of the sphere is 1"}, {"Chapter": "1", "sentence_range": "1354-1357", "Text": "1 20\nA conducting sphere of radius 10 cm has an unknown charge If\nthe electric field 20 cm from the centre of the sphere is 1 5 \u00d7 103 N/C\nand points radially inward, what is the net charge on the sphere"}, {"Chapter": "1", "sentence_range": "1355-1358", "Text": "20\nA conducting sphere of radius 10 cm has an unknown charge If\nthe electric field 20 cm from the centre of the sphere is 1 5 \u00d7 103 N/C\nand points radially inward, what is the net charge on the sphere Rationalised 2023-24\n44\nPhysics\n1"}, {"Chapter": "1", "sentence_range": "1356-1359", "Text": "If\nthe electric field 20 cm from the centre of the sphere is 1 5 \u00d7 103 N/C\nand points radially inward, what is the net charge on the sphere Rationalised 2023-24\n44\nPhysics\n1 21\nA uniformly charged conducting sphere of 2"}, {"Chapter": "1", "sentence_range": "1357-1360", "Text": "5 \u00d7 103 N/C\nand points radially inward, what is the net charge on the sphere Rationalised 2023-24\n44\nPhysics\n1 21\nA uniformly charged conducting sphere of 2 4 m diameter has a\nsurface charge density of 80"}, {"Chapter": "1", "sentence_range": "1358-1361", "Text": "Rationalised 2023-24\n44\nPhysics\n1 21\nA uniformly charged conducting sphere of 2 4 m diameter has a\nsurface charge density of 80 0 mC/m2"}, {"Chapter": "1", "sentence_range": "1359-1362", "Text": "21\nA uniformly charged conducting sphere of 2 4 m diameter has a\nsurface charge density of 80 0 mC/m2 (a) Find the charge on the\nsphere"}, {"Chapter": "1", "sentence_range": "1360-1363", "Text": "4 m diameter has a\nsurface charge density of 80 0 mC/m2 (a) Find the charge on the\nsphere (b) What is the total electric flux leaving the surface of the\nsphere"}, {"Chapter": "1", "sentence_range": "1361-1364", "Text": "0 mC/m2 (a) Find the charge on the\nsphere (b) What is the total electric flux leaving the surface of the\nsphere 1"}, {"Chapter": "1", "sentence_range": "1362-1365", "Text": "(a) Find the charge on the\nsphere (b) What is the total electric flux leaving the surface of the\nsphere 1 22\nAn infinite line charge produces a field of 9 \u00d7 104 N/C at a distance\nof 2 cm"}, {"Chapter": "1", "sentence_range": "1363-1366", "Text": "(b) What is the total electric flux leaving the surface of the\nsphere 1 22\nAn infinite line charge produces a field of 9 \u00d7 104 N/C at a distance\nof 2 cm Calculate the linear charge density"}, {"Chapter": "1", "sentence_range": "1364-1367", "Text": "1 22\nAn infinite line charge produces a field of 9 \u00d7 104 N/C at a distance\nof 2 cm Calculate the linear charge density 1"}, {"Chapter": "1", "sentence_range": "1365-1368", "Text": "22\nAn infinite line charge produces a field of 9 \u00d7 104 N/C at a distance\nof 2 cm Calculate the linear charge density 1 23\nTwo large, thin metal plates are parallel and close to each other"}, {"Chapter": "1", "sentence_range": "1366-1369", "Text": "Calculate the linear charge density 1 23\nTwo large, thin metal plates are parallel and close to each other On\ntheir inner faces, the plates have surface charge densities of opposite\nsigns and of magnitude 17"}, {"Chapter": "1", "sentence_range": "1367-1370", "Text": "1 23\nTwo large, thin metal plates are parallel and close to each other On\ntheir inner faces, the plates have surface charge densities of opposite\nsigns and of magnitude 17 0 \u00d7 10\u201322 C/m2"}, {"Chapter": "1", "sentence_range": "1368-1371", "Text": "23\nTwo large, thin metal plates are parallel and close to each other On\ntheir inner faces, the plates have surface charge densities of opposite\nsigns and of magnitude 17 0 \u00d7 10\u201322 C/m2 What is E: (a) in the outer\nregion of the first plate, (b) in the outer region of the second plate,\nand (c) between the plates"}, {"Chapter": "1", "sentence_range": "1369-1372", "Text": "On\ntheir inner faces, the plates have surface charge densities of opposite\nsigns and of magnitude 17 0 \u00d7 10\u201322 C/m2 What is E: (a) in the outer\nregion of the first plate, (b) in the outer region of the second plate,\nand (c) between the plates Rationalised 2023-24\n2"}, {"Chapter": "1", "sentence_range": "1370-1373", "Text": "0 \u00d7 10\u201322 C/m2 What is E: (a) in the outer\nregion of the first plate, (b) in the outer region of the second plate,\nand (c) between the plates Rationalised 2023-24\n2 1  INTRODUCTION\nIn Chapters 6 and 8 (Class XI), the notion of potential energy was\nintroduced"}, {"Chapter": "1", "sentence_range": "1371-1374", "Text": "What is E: (a) in the outer\nregion of the first plate, (b) in the outer region of the second plate,\nand (c) between the plates Rationalised 2023-24\n2 1  INTRODUCTION\nIn Chapters 6 and 8 (Class XI), the notion of potential energy was\nintroduced When an external force does work in taking a body from a\npoint to another against a force like spring force or gravitational force,\nthat work gets stored as potential energy of the body"}, {"Chapter": "1", "sentence_range": "1372-1375", "Text": "Rationalised 2023-24\n2 1  INTRODUCTION\nIn Chapters 6 and 8 (Class XI), the notion of potential energy was\nintroduced When an external force does work in taking a body from a\npoint to another against a force like spring force or gravitational force,\nthat work gets stored as potential energy of the body When the external\nforce is removed, the body moves, gaining kinetic energy and losing\nan equal amount of potential energy"}, {"Chapter": "1", "sentence_range": "1373-1376", "Text": "1  INTRODUCTION\nIn Chapters 6 and 8 (Class XI), the notion of potential energy was\nintroduced When an external force does work in taking a body from a\npoint to another against a force like spring force or gravitational force,\nthat work gets stored as potential energy of the body When the external\nforce is removed, the body moves, gaining kinetic energy and losing\nan equal amount of potential energy The sum of kinetic and\npotential energies is thus conserved"}, {"Chapter": "1", "sentence_range": "1374-1377", "Text": "When an external force does work in taking a body from a\npoint to another against a force like spring force or gravitational force,\nthat work gets stored as potential energy of the body When the external\nforce is removed, the body moves, gaining kinetic energy and losing\nan equal amount of potential energy The sum of kinetic and\npotential energies is thus conserved Forces of this kind are called\nconservative forces"}, {"Chapter": "1", "sentence_range": "1375-1378", "Text": "When the external\nforce is removed, the body moves, gaining kinetic energy and losing\nan equal amount of potential energy The sum of kinetic and\npotential energies is thus conserved Forces of this kind are called\nconservative forces Spring force and gravitational force are examples of\nconservative forces"}, {"Chapter": "1", "sentence_range": "1376-1379", "Text": "The sum of kinetic and\npotential energies is thus conserved Forces of this kind are called\nconservative forces Spring force and gravitational force are examples of\nconservative forces Coulomb force between two (stationary) charges is also a conservative\nforce"}, {"Chapter": "1", "sentence_range": "1377-1380", "Text": "Forces of this kind are called\nconservative forces Spring force and gravitational force are examples of\nconservative forces Coulomb force between two (stationary) charges is also a conservative\nforce This is not surprising, since both have inverse-square dependence\non distance and differ mainly in the proportionality constants \u2013 the\nmasses in the gravitational law are replaced by charges in Coulomb\u2019s\nlaw"}, {"Chapter": "1", "sentence_range": "1378-1381", "Text": "Spring force and gravitational force are examples of\nconservative forces Coulomb force between two (stationary) charges is also a conservative\nforce This is not surprising, since both have inverse-square dependence\non distance and differ mainly in the proportionality constants \u2013 the\nmasses in the gravitational law are replaced by charges in Coulomb\u2019s\nlaw Thus, like the potential energy of a mass in a gravitational\nfield, we can define electrostatic potential energy of a charge in an\nelectrostatic field"}, {"Chapter": "1", "sentence_range": "1379-1382", "Text": "Coulomb force between two (stationary) charges is also a conservative\nforce This is not surprising, since both have inverse-square dependence\non distance and differ mainly in the proportionality constants \u2013 the\nmasses in the gravitational law are replaced by charges in Coulomb\u2019s\nlaw Thus, like the potential energy of a mass in a gravitational\nfield, we can define electrostatic potential energy of a charge in an\nelectrostatic field Consider an electrostatic field E due to some charge configuration"}, {"Chapter": "1", "sentence_range": "1380-1383", "Text": "This is not surprising, since both have inverse-square dependence\non distance and differ mainly in the proportionality constants \u2013 the\nmasses in the gravitational law are replaced by charges in Coulomb\u2019s\nlaw Thus, like the potential energy of a mass in a gravitational\nfield, we can define electrostatic potential energy of a charge in an\nelectrostatic field Consider an electrostatic field E due to some charge configuration First, for simplicity, consider the field E due to a charge Q placed at the\norigin"}, {"Chapter": "1", "sentence_range": "1381-1384", "Text": "Thus, like the potential energy of a mass in a gravitational\nfield, we can define electrostatic potential energy of a charge in an\nelectrostatic field Consider an electrostatic field E due to some charge configuration First, for simplicity, consider the field E due to a charge Q placed at the\norigin Now, imagine that we bring a test charge q from a point R to a\npoint P against the repulsive force on it due to the charge Q"}, {"Chapter": "1", "sentence_range": "1382-1385", "Text": "Consider an electrostatic field E due to some charge configuration First, for simplicity, consider the field E due to a charge Q placed at the\norigin Now, imagine that we bring a test charge q from a point R to a\npoint P against the repulsive force on it due to the charge Q With reference\nChapter Two\nELECTROSTATIC\nPOTENTIAL AND\nCAPACITANCE\nRationalised 2023-24\nPhysics\n46\nto Fig"}, {"Chapter": "1", "sentence_range": "1383-1386", "Text": "First, for simplicity, consider the field E due to a charge Q placed at the\norigin Now, imagine that we bring a test charge q from a point R to a\npoint P against the repulsive force on it due to the charge Q With reference\nChapter Two\nELECTROSTATIC\nPOTENTIAL AND\nCAPACITANCE\nRationalised 2023-24\nPhysics\n46\nto Fig 2"}, {"Chapter": "1", "sentence_range": "1384-1387", "Text": "Now, imagine that we bring a test charge q from a point R to a\npoint P against the repulsive force on it due to the charge Q With reference\nChapter Two\nELECTROSTATIC\nPOTENTIAL AND\nCAPACITANCE\nRationalised 2023-24\nPhysics\n46\nto Fig 2 1, this will happen if Q and q are both positive\nor both negative"}, {"Chapter": "1", "sentence_range": "1385-1388", "Text": "With reference\nChapter Two\nELECTROSTATIC\nPOTENTIAL AND\nCAPACITANCE\nRationalised 2023-24\nPhysics\n46\nto Fig 2 1, this will happen if Q and q are both positive\nor both negative For definiteness, let us take Q, q > 0"}, {"Chapter": "1", "sentence_range": "1386-1389", "Text": "2 1, this will happen if Q and q are both positive\nor both negative For definiteness, let us take Q, q > 0 Two remarks may be made here"}, {"Chapter": "1", "sentence_range": "1387-1390", "Text": "1, this will happen if Q and q are both positive\nor both negative For definiteness, let us take Q, q > 0 Two remarks may be made here First, we assume\nthat the test charge q is so small that it does not disturb\nthe original configuration, namely the charge Q at the\norigin (or else, we keep Q fixed at the origin by some\nunspecified force)"}, {"Chapter": "1", "sentence_range": "1388-1391", "Text": "For definiteness, let us take Q, q > 0 Two remarks may be made here First, we assume\nthat the test charge q is so small that it does not disturb\nthe original configuration, namely the charge Q at the\norigin (or else, we keep Q fixed at the origin by some\nunspecified force) Second, in bringing the charge q from\nR to P, we apply an external force Fext just enough to\ncounter the repulsive electric force FE (i"}, {"Chapter": "1", "sentence_range": "1389-1392", "Text": "Two remarks may be made here First, we assume\nthat the test charge q is so small that it does not disturb\nthe original configuration, namely the charge Q at the\norigin (or else, we keep Q fixed at the origin by some\nunspecified force) Second, in bringing the charge q from\nR to P, we apply an external force Fext just enough to\ncounter the repulsive electric force FE (i e, Fext= \u2013FE)"}, {"Chapter": "1", "sentence_range": "1390-1393", "Text": "First, we assume\nthat the test charge q is so small that it does not disturb\nthe original configuration, namely the charge Q at the\norigin (or else, we keep Q fixed at the origin by some\nunspecified force) Second, in bringing the charge q from\nR to P, we apply an external force Fext just enough to\ncounter the repulsive electric force FE (i e, Fext= \u2013FE) This means there is no net force on or acceleration of\nthe charge q when it is brought from R to P, i"}, {"Chapter": "1", "sentence_range": "1391-1394", "Text": "Second, in bringing the charge q from\nR to P, we apply an external force Fext just enough to\ncounter the repulsive electric force FE (i e, Fext= \u2013FE) This means there is no net force on or acceleration of\nthe charge q when it is brought from R to P, i e"}, {"Chapter": "1", "sentence_range": "1392-1395", "Text": "e, Fext= \u2013FE) This means there is no net force on or acceleration of\nthe charge q when it is brought from R to P, i e , it is\nbrought with infinitesimally slow constant speed"}, {"Chapter": "1", "sentence_range": "1393-1396", "Text": "This means there is no net force on or acceleration of\nthe charge q when it is brought from R to P, i e , it is\nbrought with infinitesimally slow constant speed In\nthis situation, work done by the external force is the negative of the work\ndone by the electric force, and gets fully stored in the form of potential\nenergy of the charge q"}, {"Chapter": "1", "sentence_range": "1394-1397", "Text": "e , it is\nbrought with infinitesimally slow constant speed In\nthis situation, work done by the external force is the negative of the work\ndone by the electric force, and gets fully stored in the form of potential\nenergy of the charge q If the external force is removed on reaching P, the\nelectric force will take the charge away from Q \u2013 the stored energy (potential\nenergy) at P is used to provide kinetic energy to the charge q in such a\nway that the sum of the kinetic and potential energies is conserved"}, {"Chapter": "1", "sentence_range": "1395-1398", "Text": ", it is\nbrought with infinitesimally slow constant speed In\nthis situation, work done by the external force is the negative of the work\ndone by the electric force, and gets fully stored in the form of potential\nenergy of the charge q If the external force is removed on reaching P, the\nelectric force will take the charge away from Q \u2013 the stored energy (potential\nenergy) at P is used to provide kinetic energy to the charge q in such a\nway that the sum of the kinetic and potential energies is conserved Thus, work done by external forces in moving a charge q from R to P is\nWRP =   \n        =    \u2013\n (2"}, {"Chapter": "1", "sentence_range": "1396-1399", "Text": "In\nthis situation, work done by the external force is the negative of the work\ndone by the electric force, and gets fully stored in the form of potential\nenergy of the charge q If the external force is removed on reaching P, the\nelectric force will take the charge away from Q \u2013 the stored energy (potential\nenergy) at P is used to provide kinetic energy to the charge q in such a\nway that the sum of the kinetic and potential energies is conserved Thus, work done by external forces in moving a charge q from R to P is\nWRP =   \n        =    \u2013\n (2 1)\nThis work done is against electrostatic repulsive force and gets stored\nas potential energy"}, {"Chapter": "1", "sentence_range": "1397-1400", "Text": "If the external force is removed on reaching P, the\nelectric force will take the charge away from Q \u2013 the stored energy (potential\nenergy) at P is used to provide kinetic energy to the charge q in such a\nway that the sum of the kinetic and potential energies is conserved Thus, work done by external forces in moving a charge q from R to P is\nWRP =   \n        =    \u2013\n (2 1)\nThis work done is against electrostatic repulsive force and gets stored\nas potential energy At every point in electric field, a particle with charge q possesses a\ncertain electrostatic potential energy, this work done increases its potential\nenergy by an amount equal to potential energy difference between points\nR and P"}, {"Chapter": "1", "sentence_range": "1398-1401", "Text": "Thus, work done by external forces in moving a charge q from R to P is\nWRP =   \n        =    \u2013\n (2 1)\nThis work done is against electrostatic repulsive force and gets stored\nas potential energy At every point in electric field, a particle with charge q possesses a\ncertain electrostatic potential energy, this work done increases its potential\nenergy by an amount equal to potential energy difference between points\nR and P Thus, potential energy difference\nP\nR\nRP\nU\nU\nU\nW\n\u2206\n=\n\u2212\n=\n(2"}, {"Chapter": "1", "sentence_range": "1399-1402", "Text": "1)\nThis work done is against electrostatic repulsive force and gets stored\nas potential energy At every point in electric field, a particle with charge q possesses a\ncertain electrostatic potential energy, this work done increases its potential\nenergy by an amount equal to potential energy difference between points\nR and P Thus, potential energy difference\nP\nR\nRP\nU\nU\nU\nW\n\u2206\n=\n\u2212\n=\n(2 2)\n(Note here that this displacement is in an opposite sense to the electric\nforce and hence work done by electric field is negative, i"}, {"Chapter": "1", "sentence_range": "1400-1403", "Text": "At every point in electric field, a particle with charge q possesses a\ncertain electrostatic potential energy, this work done increases its potential\nenergy by an amount equal to potential energy difference between points\nR and P Thus, potential energy difference\nP\nR\nRP\nU\nU\nU\nW\n\u2206\n=\n\u2212\n=\n(2 2)\n(Note here that this displacement is in an opposite sense to the electric\nforce and hence work done by electric field is negative, i e"}, {"Chapter": "1", "sentence_range": "1401-1404", "Text": "Thus, potential energy difference\nP\nR\nRP\nU\nU\nU\nW\n\u2206\n=\n\u2212\n=\n(2 2)\n(Note here that this displacement is in an opposite sense to the electric\nforce and hence work done by electric field is negative, i e , \u2013WRP"}, {"Chapter": "1", "sentence_range": "1402-1405", "Text": "2)\n(Note here that this displacement is in an opposite sense to the electric\nforce and hence work done by electric field is negative, i e , \u2013WRP )\nTherefore, we can define electric potential energy difference between\ntwo points as the work required to be done by an external force in moving\n(without accelerating) charge q from one point to another for electric field\nof any arbitrary charge configuration"}, {"Chapter": "1", "sentence_range": "1403-1406", "Text": "e , \u2013WRP )\nTherefore, we can define electric potential energy difference between\ntwo points as the work required to be done by an external force in moving\n(without accelerating) charge q from one point to another for electric field\nof any arbitrary charge configuration Two important comments may be made at this stage:\n(i)\nThe right side of Eq"}, {"Chapter": "1", "sentence_range": "1404-1407", "Text": ", \u2013WRP )\nTherefore, we can define electric potential energy difference between\ntwo points as the work required to be done by an external force in moving\n(without accelerating) charge q from one point to another for electric field\nof any arbitrary charge configuration Two important comments may be made at this stage:\n(i)\nThe right side of Eq (2"}, {"Chapter": "1", "sentence_range": "1405-1408", "Text": ")\nTherefore, we can define electric potential energy difference between\ntwo points as the work required to be done by an external force in moving\n(without accelerating) charge q from one point to another for electric field\nof any arbitrary charge configuration Two important comments may be made at this stage:\n(i)\nThe right side of Eq (2 2) depends only on the initial and final positions\nof the charge"}, {"Chapter": "1", "sentence_range": "1406-1409", "Text": "Two important comments may be made at this stage:\n(i)\nThe right side of Eq (2 2) depends only on the initial and final positions\nof the charge It means that the work done by an electrostatic field in\nmoving a charge from one point to another depends only on the initial\nand the final points and is independent of the path taken to go from\none point to the other"}, {"Chapter": "1", "sentence_range": "1407-1410", "Text": "(2 2) depends only on the initial and final positions\nof the charge It means that the work done by an electrostatic field in\nmoving a charge from one point to another depends only on the initial\nand the final points and is independent of the path taken to go from\none point to the other This is the fundamental characteristic of a\nconservative force"}, {"Chapter": "1", "sentence_range": "1408-1411", "Text": "2) depends only on the initial and final positions\nof the charge It means that the work done by an electrostatic field in\nmoving a charge from one point to another depends only on the initial\nand the final points and is independent of the path taken to go from\none point to the other This is the fundamental characteristic of a\nconservative force The concept of the potential energy would not be\nmeaningful if the work depended on the path"}, {"Chapter": "1", "sentence_range": "1409-1412", "Text": "It means that the work done by an electrostatic field in\nmoving a charge from one point to another depends only on the initial\nand the final points and is independent of the path taken to go from\none point to the other This is the fundamental characteristic of a\nconservative force The concept of the potential energy would not be\nmeaningful if the work depended on the path The path-independence\nof work done by an electrostatic field can be proved using the\nCoulomb\u2019s law"}, {"Chapter": "1", "sentence_range": "1410-1413", "Text": "This is the fundamental characteristic of a\nconservative force The concept of the potential energy would not be\nmeaningful if the work depended on the path The path-independence\nof work done by an electrostatic field can be proved using the\nCoulomb\u2019s law We omit this proof here"}, {"Chapter": "1", "sentence_range": "1411-1414", "Text": "The concept of the potential energy would not be\nmeaningful if the work depended on the path The path-independence\nof work done by an electrostatic field can be proved using the\nCoulomb\u2019s law We omit this proof here FIGURE 2"}, {"Chapter": "1", "sentence_range": "1412-1415", "Text": "The path-independence\nof work done by an electrostatic field can be proved using the\nCoulomb\u2019s law We omit this proof here FIGURE 2 1 A test charge q (> 0) is\nmoved from the point R to the\npoint P against the repulsive\nforce on it by the charge Q (> 0)\nplaced at the origin"}, {"Chapter": "1", "sentence_range": "1413-1416", "Text": "We omit this proof here FIGURE 2 1 A test charge q (> 0) is\nmoved from the point R to the\npoint P against the repulsive\nforce on it by the charge Q (> 0)\nplaced at the origin Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n47\n(ii) Equation (2"}, {"Chapter": "1", "sentence_range": "1414-1417", "Text": "FIGURE 2 1 A test charge q (> 0) is\nmoved from the point R to the\npoint P against the repulsive\nforce on it by the charge Q (> 0)\nplaced at the origin Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n47\n(ii) Equation (2 2) defines potential energy difference in terms\nof the physically meaningful quantity work"}, {"Chapter": "1", "sentence_range": "1415-1418", "Text": "1 A test charge q (> 0) is\nmoved from the point R to the\npoint P against the repulsive\nforce on it by the charge Q (> 0)\nplaced at the origin Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n47\n(ii) Equation (2 2) defines potential energy difference in terms\nof the physically meaningful quantity work Clearly,\npotential energy so defined is undetermined to within an\nadditive constant"}, {"Chapter": "1", "sentence_range": "1416-1419", "Text": "Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n47\n(ii) Equation (2 2) defines potential energy difference in terms\nof the physically meaningful quantity work Clearly,\npotential energy so defined is undetermined to within an\nadditive constant What this means is that the actual value\nof potential energy is not physically significant; it is only\nthe difference of potential energy that is significant"}, {"Chapter": "1", "sentence_range": "1417-1420", "Text": "2) defines potential energy difference in terms\nof the physically meaningful quantity work Clearly,\npotential energy so defined is undetermined to within an\nadditive constant What this means is that the actual value\nof potential energy is not physically significant; it is only\nthe difference of potential energy that is significant We can\nalways add an arbitrary constant a to potential energy at\nevery point, since this will not change the potential energy\ndifference:\n(\n)\n(\n)\nP\nR\nP\nR\nU\nU\nU\nU\n\u03b1\n\u03b1\n+\n\u2212\n+\n=\n\u2212\nPut it differently, there is a freedom in choosing the point\nwhere potential energy is zero"}, {"Chapter": "1", "sentence_range": "1418-1421", "Text": "Clearly,\npotential energy so defined is undetermined to within an\nadditive constant What this means is that the actual value\nof potential energy is not physically significant; it is only\nthe difference of potential energy that is significant We can\nalways add an arbitrary constant a to potential energy at\nevery point, since this will not change the potential energy\ndifference:\n(\n)\n(\n)\nP\nR\nP\nR\nU\nU\nU\nU\n\u03b1\n\u03b1\n+\n\u2212\n+\n=\n\u2212\nPut it differently, there is a freedom in choosing the point\nwhere potential energy is zero A convenient choice is to have\nelectrostatic potential energy zero at infinity"}, {"Chapter": "1", "sentence_range": "1419-1422", "Text": "What this means is that the actual value\nof potential energy is not physically significant; it is only\nthe difference of potential energy that is significant We can\nalways add an arbitrary constant a to potential energy at\nevery point, since this will not change the potential energy\ndifference:\n(\n)\n(\n)\nP\nR\nP\nR\nU\nU\nU\nU\n\u03b1\n\u03b1\n+\n\u2212\n+\n=\n\u2212\nPut it differently, there is a freedom in choosing the point\nwhere potential energy is zero A convenient choice is to have\nelectrostatic potential energy zero at infinity With this choice,\nif we take the point R at infinity, we get from Eq"}, {"Chapter": "1", "sentence_range": "1420-1423", "Text": "We can\nalways add an arbitrary constant a to potential energy at\nevery point, since this will not change the potential energy\ndifference:\n(\n)\n(\n)\nP\nR\nP\nR\nU\nU\nU\nU\n\u03b1\n\u03b1\n+\n\u2212\n+\n=\n\u2212\nPut it differently, there is a freedom in choosing the point\nwhere potential energy is zero A convenient choice is to have\nelectrostatic potential energy zero at infinity With this choice,\nif we take the point R at infinity, we get from Eq (2"}, {"Chapter": "1", "sentence_range": "1421-1424", "Text": "A convenient choice is to have\nelectrostatic potential energy zero at infinity With this choice,\nif we take the point R at infinity, we get from Eq (2 2)\nP\nP\nP\nW\nU\nU\nU\n\u221e\n\u221e\n=\n\u2212\n=\n(2"}, {"Chapter": "1", "sentence_range": "1422-1425", "Text": "With this choice,\nif we take the point R at infinity, we get from Eq (2 2)\nP\nP\nP\nW\nU\nU\nU\n\u221e\n\u221e\n=\n\u2212\n=\n(2 3)\nSince the point P is arbitrary, Eq"}, {"Chapter": "1", "sentence_range": "1423-1426", "Text": "(2 2)\nP\nP\nP\nW\nU\nU\nU\n\u221e\n\u221e\n=\n\u2212\n=\n(2 3)\nSince the point P is arbitrary, Eq (2"}, {"Chapter": "1", "sentence_range": "1424-1427", "Text": "2)\nP\nP\nP\nW\nU\nU\nU\n\u221e\n\u221e\n=\n\u2212\n=\n(2 3)\nSince the point P is arbitrary, Eq (2 3) provides us with a\ndefinition of potential energy of a charge q at any point"}, {"Chapter": "1", "sentence_range": "1425-1428", "Text": "3)\nSince the point P is arbitrary, Eq (2 3) provides us with a\ndefinition of potential energy of a charge q at any point Potential energy of charge q at a point (in the presence of field\ndue to any charge configuration) is the work done by the\nexternal force (equal and opposite to the electric force) in\nbringing the charge q from infinity to that point"}, {"Chapter": "1", "sentence_range": "1426-1429", "Text": "(2 3) provides us with a\ndefinition of potential energy of a charge q at any point Potential energy of charge q at a point (in the presence of field\ndue to any charge configuration) is the work done by the\nexternal force (equal and opposite to the electric force) in\nbringing the charge q from infinity to that point 2"}, {"Chapter": "1", "sentence_range": "1427-1430", "Text": "3) provides us with a\ndefinition of potential energy of a charge q at any point Potential energy of charge q at a point (in the presence of field\ndue to any charge configuration) is the work done by the\nexternal force (equal and opposite to the electric force) in\nbringing the charge q from infinity to that point 2 2  ELECTROSTATIC POTENTIAL\nConsider any general static charge configuration"}, {"Chapter": "1", "sentence_range": "1428-1431", "Text": "Potential energy of charge q at a point (in the presence of field\ndue to any charge configuration) is the work done by the\nexternal force (equal and opposite to the electric force) in\nbringing the charge q from infinity to that point 2 2  ELECTROSTATIC POTENTIAL\nConsider any general static charge configuration We define\npotential energy of a test charge q in terms of the work done\non the charge q"}, {"Chapter": "1", "sentence_range": "1429-1432", "Text": "2 2  ELECTROSTATIC POTENTIAL\nConsider any general static charge configuration We define\npotential energy of a test charge q in terms of the work done\non the charge q This work is obviously proportional to q, since\nthe force at any point is qE, where E is the electric field at that\npoint due to the given charge configuration"}, {"Chapter": "1", "sentence_range": "1430-1433", "Text": "2  ELECTROSTATIC POTENTIAL\nConsider any general static charge configuration We define\npotential energy of a test charge q in terms of the work done\non the charge q This work is obviously proportional to q, since\nthe force at any point is qE, where E is the electric field at that\npoint due to the given charge configuration It is, therefore,\nconvenient to divide the work by the amount of charge q, so\nthat the resulting quantity is independent of q"}, {"Chapter": "1", "sentence_range": "1431-1434", "Text": "We define\npotential energy of a test charge q in terms of the work done\non the charge q This work is obviously proportional to q, since\nthe force at any point is qE, where E is the electric field at that\npoint due to the given charge configuration It is, therefore,\nconvenient to divide the work by the amount of charge q, so\nthat the resulting quantity is independent of q In other words,\nwork done per unit test charge is characteristic of the electric\nfield associated with the charge configuration"}, {"Chapter": "1", "sentence_range": "1432-1435", "Text": "This work is obviously proportional to q, since\nthe force at any point is qE, where E is the electric field at that\npoint due to the given charge configuration It is, therefore,\nconvenient to divide the work by the amount of charge q, so\nthat the resulting quantity is independent of q In other words,\nwork done per unit test charge is characteristic of the electric\nfield associated with the charge configuration This leads to\nthe idea of electrostatic potential V due to a given charge\nconfiguration"}, {"Chapter": "1", "sentence_range": "1433-1436", "Text": "It is, therefore,\nconvenient to divide the work by the amount of charge q, so\nthat the resulting quantity is independent of q In other words,\nwork done per unit test charge is characteristic of the electric\nfield associated with the charge configuration This leads to\nthe idea of electrostatic potential V due to a given charge\nconfiguration From Eq"}, {"Chapter": "1", "sentence_range": "1434-1437", "Text": "In other words,\nwork done per unit test charge is characteristic of the electric\nfield associated with the charge configuration This leads to\nthe idea of electrostatic potential V due to a given charge\nconfiguration From Eq (2"}, {"Chapter": "1", "sentence_range": "1435-1438", "Text": "This leads to\nthe idea of electrostatic potential V due to a given charge\nconfiguration From Eq (2 1), we get:\nWork done by external force in bringing a unit positive\ncharge from point R to P\n= VP \u2013 VR  =\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nU\nU\nq\nP\nR\n(2"}, {"Chapter": "1", "sentence_range": "1436-1439", "Text": "From Eq (2 1), we get:\nWork done by external force in bringing a unit positive\ncharge from point R to P\n= VP \u2013 VR  =\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nU\nU\nq\nP\nR\n(2 4)\nwhere VP and VR  are the electrostatic potentials at P and R, respectively"}, {"Chapter": "1", "sentence_range": "1437-1440", "Text": "(2 1), we get:\nWork done by external force in bringing a unit positive\ncharge from point R to P\n= VP \u2013 VR  =\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nU\nU\nq\nP\nR\n(2 4)\nwhere VP and VR  are the electrostatic potentials at P and R, respectively Note, as before, that it is not the actual value of potential but the potential\ndifference that is physically significant"}, {"Chapter": "1", "sentence_range": "1438-1441", "Text": "1), we get:\nWork done by external force in bringing a unit positive\ncharge from point R to P\n= VP \u2013 VR  =\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nU\nU\nq\nP\nR\n(2 4)\nwhere VP and VR  are the electrostatic potentials at P and R, respectively Note, as before, that it is not the actual value of potential but the potential\ndifference that is physically significant If, as before, we choose the\npotential to be zero at infinity, Eq"}, {"Chapter": "1", "sentence_range": "1439-1442", "Text": "4)\nwhere VP and VR  are the electrostatic potentials at P and R, respectively Note, as before, that it is not the actual value of potential but the potential\ndifference that is physically significant If, as before, we choose the\npotential to be zero at infinity, Eq (2"}, {"Chapter": "1", "sentence_range": "1440-1443", "Text": "Note, as before, that it is not the actual value of potential but the potential\ndifference that is physically significant If, as before, we choose the\npotential to be zero at infinity, Eq (2 4) implies:\nWork done by an external force in bringing a unit positive charge\nfrom infinity to a point = electrostatic potential (V ) at that point"}, {"Chapter": "1", "sentence_range": "1441-1444", "Text": "If, as before, we choose the\npotential to be zero at infinity, Eq (2 4) implies:\nWork done by an external force in bringing a unit positive charge\nfrom infinity to a point = electrostatic potential (V ) at that point COUNT ALESSANDRO VOLTA (1745 \u20131827)\nCount Alessandro Volta\n(1745 \u2013 1827) Italian\nphysicist, professor at\nPavia"}, {"Chapter": "1", "sentence_range": "1442-1445", "Text": "(2 4) implies:\nWork done by an external force in bringing a unit positive charge\nfrom infinity to a point = electrostatic potential (V ) at that point COUNT ALESSANDRO VOLTA (1745 \u20131827)\nCount Alessandro Volta\n(1745 \u2013 1827) Italian\nphysicist, professor at\nPavia Volta established\nthat the animal electri-\ncity observed by Luigi\nGalvani, 1737\u20131798, in\nexperiments with frog\nmuscle tissue placed in\ncontact with dissimilar\nmetals, was not due to\nany exceptional property\nof animal tissues but\nwas \nalso \ngenerated\nwhenever any wet body\nwas sandwiched between\ndissimilar metals"}, {"Chapter": "1", "sentence_range": "1443-1446", "Text": "4) implies:\nWork done by an external force in bringing a unit positive charge\nfrom infinity to a point = electrostatic potential (V ) at that point COUNT ALESSANDRO VOLTA (1745 \u20131827)\nCount Alessandro Volta\n(1745 \u2013 1827) Italian\nphysicist, professor at\nPavia Volta established\nthat the animal electri-\ncity observed by Luigi\nGalvani, 1737\u20131798, in\nexperiments with frog\nmuscle tissue placed in\ncontact with dissimilar\nmetals, was not due to\nany exceptional property\nof animal tissues but\nwas \nalso \ngenerated\nwhenever any wet body\nwas sandwiched between\ndissimilar metals This\nled him to develop the\nfirst voltaic pile, or\nbattery, consisting of a\nlarge stack of moist disks\nof cardboard (electro-\nlyte) \nsandwiched\nbetween disks of metal\n(electrodes)"}, {"Chapter": "1", "sentence_range": "1444-1447", "Text": "COUNT ALESSANDRO VOLTA (1745 \u20131827)\nCount Alessandro Volta\n(1745 \u2013 1827) Italian\nphysicist, professor at\nPavia Volta established\nthat the animal electri-\ncity observed by Luigi\nGalvani, 1737\u20131798, in\nexperiments with frog\nmuscle tissue placed in\ncontact with dissimilar\nmetals, was not due to\nany exceptional property\nof animal tissues but\nwas \nalso \ngenerated\nwhenever any wet body\nwas sandwiched between\ndissimilar metals This\nled him to develop the\nfirst voltaic pile, or\nbattery, consisting of a\nlarge stack of moist disks\nof cardboard (electro-\nlyte) \nsandwiched\nbetween disks of metal\n(electrodes) Rationalised 2023-24\nPhysics\n48\nIn other words, the electrostatic potential (V )\nat any point in a region with electrostatic field is\nthe work done in bringing a unit positive\ncharge (without acceleration) from infinity to\nthat point"}, {"Chapter": "1", "sentence_range": "1445-1448", "Text": "Volta established\nthat the animal electri-\ncity observed by Luigi\nGalvani, 1737\u20131798, in\nexperiments with frog\nmuscle tissue placed in\ncontact with dissimilar\nmetals, was not due to\nany exceptional property\nof animal tissues but\nwas \nalso \ngenerated\nwhenever any wet body\nwas sandwiched between\ndissimilar metals This\nled him to develop the\nfirst voltaic pile, or\nbattery, consisting of a\nlarge stack of moist disks\nof cardboard (electro-\nlyte) \nsandwiched\nbetween disks of metal\n(electrodes) Rationalised 2023-24\nPhysics\n48\nIn other words, the electrostatic potential (V )\nat any point in a region with electrostatic field is\nthe work done in bringing a unit positive\ncharge (without acceleration) from infinity to\nthat point The qualifying remarks made earlier regarding\npotential energy also apply to the definition of\npotential"}, {"Chapter": "1", "sentence_range": "1446-1449", "Text": "This\nled him to develop the\nfirst voltaic pile, or\nbattery, consisting of a\nlarge stack of moist disks\nof cardboard (electro-\nlyte) \nsandwiched\nbetween disks of metal\n(electrodes) Rationalised 2023-24\nPhysics\n48\nIn other words, the electrostatic potential (V )\nat any point in a region with electrostatic field is\nthe work done in bringing a unit positive\ncharge (without acceleration) from infinity to\nthat point The qualifying remarks made earlier regarding\npotential energy also apply to the definition of\npotential To obtain the work done per unit test\ncharge, we should take an infinitesimal test charge\ndq, obtain the work done dW in bringing it from\ninfinity to the point and determine the ratio\ndW/dq"}, {"Chapter": "1", "sentence_range": "1447-1450", "Text": "Rationalised 2023-24\nPhysics\n48\nIn other words, the electrostatic potential (V )\nat any point in a region with electrostatic field is\nthe work done in bringing a unit positive\ncharge (without acceleration) from infinity to\nthat point The qualifying remarks made earlier regarding\npotential energy also apply to the definition of\npotential To obtain the work done per unit test\ncharge, we should take an infinitesimal test charge\ndq, obtain the work done dW in bringing it from\ninfinity to the point and determine the ratio\ndW/dq Also, the external force at every point of the\npath is to be equal and opposite to the electrostatic\nforce on the test charge at that point"}, {"Chapter": "1", "sentence_range": "1448-1451", "Text": "The qualifying remarks made earlier regarding\npotential energy also apply to the definition of\npotential To obtain the work done per unit test\ncharge, we should take an infinitesimal test charge\ndq, obtain the work done dW in bringing it from\ninfinity to the point and determine the ratio\ndW/dq Also, the external force at every point of the\npath is to be equal and opposite to the electrostatic\nforce on the test charge at that point 2"}, {"Chapter": "1", "sentence_range": "1449-1452", "Text": "To obtain the work done per unit test\ncharge, we should take an infinitesimal test charge\ndq, obtain the work done dW in bringing it from\ninfinity to the point and determine the ratio\ndW/dq Also, the external force at every point of the\npath is to be equal and opposite to the electrostatic\nforce on the test charge at that point 2 3  POTENTIAL DUE TO A POINT CHARGE\nConsider a point charge Q at the origin (Fig"}, {"Chapter": "1", "sentence_range": "1450-1453", "Text": "Also, the external force at every point of the\npath is to be equal and opposite to the electrostatic\nforce on the test charge at that point 2 3  POTENTIAL DUE TO A POINT CHARGE\nConsider a point charge Q at the origin (Fig 2"}, {"Chapter": "1", "sentence_range": "1451-1454", "Text": "2 3  POTENTIAL DUE TO A POINT CHARGE\nConsider a point charge Q at the origin (Fig 2 3)"}, {"Chapter": "1", "sentence_range": "1452-1455", "Text": "3  POTENTIAL DUE TO A POINT CHARGE\nConsider a point charge Q at the origin (Fig 2 3) For definiteness, take Q\nto be positive"}, {"Chapter": "1", "sentence_range": "1453-1456", "Text": "2 3) For definiteness, take Q\nto be positive We wish to determine the potential at any point P with\nposition vector r from the origin"}, {"Chapter": "1", "sentence_range": "1454-1457", "Text": "3) For definiteness, take Q\nto be positive We wish to determine the potential at any point P with\nposition vector r from the origin For that we must\ncalculate the work done in bringing a unit positive\ntest charge from infinity to the point P"}, {"Chapter": "1", "sentence_range": "1455-1458", "Text": "For definiteness, take Q\nto be positive We wish to determine the potential at any point P with\nposition vector r from the origin For that we must\ncalculate the work done in bringing a unit positive\ntest charge from infinity to the point P For Q > 0,\nthe work done against the repulsive force on the\ntest charge is positive"}, {"Chapter": "1", "sentence_range": "1456-1459", "Text": "We wish to determine the potential at any point P with\nposition vector r from the origin For that we must\ncalculate the work done in bringing a unit positive\ntest charge from infinity to the point P For Q > 0,\nthe work done against the repulsive force on the\ntest charge is positive Since work done is\nindependent of the path, we choose a convenient\npath \u2013 along the radial direction from infinity to\nthe point P"}, {"Chapter": "1", "sentence_range": "1457-1460", "Text": "For that we must\ncalculate the work done in bringing a unit positive\ntest charge from infinity to the point P For Q > 0,\nthe work done against the repulsive force on the\ntest charge is positive Since work done is\nindependent of the path, we choose a convenient\npath \u2013 along the radial direction from infinity to\nthe point P At some intermediate point P\u00a2 on the path, the\nelectrostatic force on a unit positive charge is\n2\n0\n1 \u02c6\n4\n'\nQ\n\u03b5r\n\u00d7\n\u2032\n\u03c0\nr\n(2"}, {"Chapter": "1", "sentence_range": "1458-1461", "Text": "For Q > 0,\nthe work done against the repulsive force on the\ntest charge is positive Since work done is\nindependent of the path, we choose a convenient\npath \u2013 along the radial direction from infinity to\nthe point P At some intermediate point P\u00a2 on the path, the\nelectrostatic force on a unit positive charge is\n2\n0\n1 \u02c6\n4\n'\nQ\n\u03b5r\n\u00d7\n\u2032\n\u03c0\nr\n(2 5)\nwhere \u02c6\u2032r is the unit vector along OP\u00a2"}, {"Chapter": "1", "sentence_range": "1459-1462", "Text": "Since work done is\nindependent of the path, we choose a convenient\npath \u2013 along the radial direction from infinity to\nthe point P At some intermediate point P\u00a2 on the path, the\nelectrostatic force on a unit positive charge is\n2\n0\n1 \u02c6\n4\n'\nQ\n\u03b5r\n\u00d7\n\u2032\n\u03c0\nr\n(2 5)\nwhere \u02c6\u2032r is the unit vector along OP\u00a2 Work done\nagainst this force from  r\u00a2 to r\u00a2 + Dr\u00a2 is\n2\n0\n4\nQ'\nW\nr\n\u03b5r\n\u2206\n= \u2212\n\u2206 \u2032\n\u03c0\n(2"}, {"Chapter": "1", "sentence_range": "1460-1463", "Text": "At some intermediate point P\u00a2 on the path, the\nelectrostatic force on a unit positive charge is\n2\n0\n1 \u02c6\n4\n'\nQ\n\u03b5r\n\u00d7\n\u2032\n\u03c0\nr\n(2 5)\nwhere \u02c6\u2032r is the unit vector along OP\u00a2 Work done\nagainst this force from  r\u00a2 to r\u00a2 + Dr\u00a2 is\n2\n0\n4\nQ'\nW\nr\n\u03b5r\n\u2206\n= \u2212\n\u2206 \u2032\n\u03c0\n(2 6)\nThe negative sign appears because for Dr\u00a2 < 0, DW is positive"}, {"Chapter": "1", "sentence_range": "1461-1464", "Text": "5)\nwhere \u02c6\u2032r is the unit vector along OP\u00a2 Work done\nagainst this force from  r\u00a2 to r\u00a2 + Dr\u00a2 is\n2\n0\n4\nQ'\nW\nr\n\u03b5r\n\u2206\n= \u2212\n\u2206 \u2032\n\u03c0\n(2 6)\nThe negative sign appears because for Dr\u00a2 < 0, DW is positive Total\nwork done (W) by the external force is obtained by integrating Eq"}, {"Chapter": "1", "sentence_range": "1462-1465", "Text": "Work done\nagainst this force from  r\u00a2 to r\u00a2 + Dr\u00a2 is\n2\n0\n4\nQ'\nW\nr\n\u03b5r\n\u2206\n= \u2212\n\u2206 \u2032\n\u03c0\n(2 6)\nThe negative sign appears because for Dr\u00a2 < 0, DW is positive Total\nwork done (W) by the external force is obtained by integrating Eq (2"}, {"Chapter": "1", "sentence_range": "1463-1466", "Text": "6)\nThe negative sign appears because for Dr\u00a2 < 0, DW is positive Total\nwork done (W) by the external force is obtained by integrating Eq (2 6)\nfrom r\u00a2 = \u00a5 to r\u00a2 = r,\nW\nQ\nr\ndr\nQ\nr\nQ\nr\nr\nr\n= \u2212\n\u2032\n\u2032 =\n\u2032\n=\n\u221e\n\u221e\n\u222b 4\n4\n4\n0\n2\n0\n0\n\u03c0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n\u03b5\n(2"}, {"Chapter": "1", "sentence_range": "1464-1467", "Text": "Total\nwork done (W) by the external force is obtained by integrating Eq (2 6)\nfrom r\u00a2 = \u00a5 to r\u00a2 = r,\nW\nQ\nr\ndr\nQ\nr\nQ\nr\nr\nr\n= \u2212\n\u2032\n\u2032 =\n\u2032\n=\n\u221e\n\u221e\n\u222b 4\n4\n4\n0\n2\n0\n0\n\u03c0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n\u03b5\n(2 7)\nThis, by definition is the potential at P due to the charge Q\n0\n( )\n4\nQ\nV r\n\u03b5r\n=\n\u03c0\n(2"}, {"Chapter": "1", "sentence_range": "1465-1468", "Text": "(2 6)\nfrom r\u00a2 = \u00a5 to r\u00a2 = r,\nW\nQ\nr\ndr\nQ\nr\nQ\nr\nr\nr\n= \u2212\n\u2032\n\u2032 =\n\u2032\n=\n\u221e\n\u221e\n\u222b 4\n4\n4\n0\n2\n0\n0\n\u03c0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n\u03b5\n(2 7)\nThis, by definition is the potential at P due to the charge Q\n0\n( )\n4\nQ\nV r\n\u03b5r\n=\n\u03c0\n(2 8)\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "1466-1469", "Text": "6)\nfrom r\u00a2 = \u00a5 to r\u00a2 = r,\nW\nQ\nr\ndr\nQ\nr\nQ\nr\nr\nr\n= \u2212\n\u2032\n\u2032 =\n\u2032\n=\n\u221e\n\u221e\n\u222b 4\n4\n4\n0\n2\n0\n0\n\u03c0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n\u03b5\n(2 7)\nThis, by definition is the potential at P due to the charge Q\n0\n( )\n4\nQ\nV r\n\u03b5r\n=\n\u03c0\n(2 8)\nFIGURE 2 2 Work done on a test charge q\nby the electrostatic field due to any given\ncharge configuration is independent\nof the path, and depends only on\nits initial and final positions"}, {"Chapter": "1", "sentence_range": "1467-1470", "Text": "7)\nThis, by definition is the potential at P due to the charge Q\n0\n( )\n4\nQ\nV r\n\u03b5r\n=\n\u03c0\n(2 8)\nFIGURE 2 2 Work done on a test charge q\nby the electrostatic field due to any given\ncharge configuration is independent\nof the path, and depends only on\nits initial and final positions FIGURE 2"}, {"Chapter": "1", "sentence_range": "1468-1471", "Text": "8)\nFIGURE 2 2 Work done on a test charge q\nby the electrostatic field due to any given\ncharge configuration is independent\nof the path, and depends only on\nits initial and final positions FIGURE 2 3 Work done in bringing a unit\npositive test charge from infinity to the\npoint P, against the repulsive force of\ncharge Q (Q > 0), is the potential at P due to\nthe charge Q"}, {"Chapter": "1", "sentence_range": "1469-1472", "Text": "2 Work done on a test charge q\nby the electrostatic field due to any given\ncharge configuration is independent\nof the path, and depends only on\nits initial and final positions FIGURE 2 3 Work done in bringing a unit\npositive test charge from infinity to the\npoint P, against the repulsive force of\ncharge Q (Q > 0), is the potential at P due to\nthe charge Q Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n49\n EXAMPLE 2"}, {"Chapter": "1", "sentence_range": "1470-1473", "Text": "FIGURE 2 3 Work done in bringing a unit\npositive test charge from infinity to the\npoint P, against the repulsive force of\ncharge Q (Q > 0), is the potential at P due to\nthe charge Q Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n49\n EXAMPLE 2 1\nEquation (2"}, {"Chapter": "1", "sentence_range": "1471-1474", "Text": "3 Work done in bringing a unit\npositive test charge from infinity to the\npoint P, against the repulsive force of\ncharge Q (Q > 0), is the potential at P due to\nthe charge Q Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n49\n EXAMPLE 2 1\nEquation (2 8) is true for any\nsign of the charge Q, though we\nconsidered Q > 0 in its derivation"}, {"Chapter": "1", "sentence_range": "1472-1475", "Text": "Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n49\n EXAMPLE 2 1\nEquation (2 8) is true for any\nsign of the charge Q, though we\nconsidered Q > 0 in its derivation For Q < 0, V < 0, i"}, {"Chapter": "1", "sentence_range": "1473-1476", "Text": "1\nEquation (2 8) is true for any\nsign of the charge Q, though we\nconsidered Q > 0 in its derivation For Q < 0, V < 0, i e"}, {"Chapter": "1", "sentence_range": "1474-1477", "Text": "8) is true for any\nsign of the charge Q, though we\nconsidered Q > 0 in its derivation For Q < 0, V < 0, i e , work done (by\nthe external force) per unit positive\ntest charge in bringing it from\ninfinity to the point is negative"}, {"Chapter": "1", "sentence_range": "1475-1478", "Text": "For Q < 0, V < 0, i e , work done (by\nthe external force) per unit positive\ntest charge in bringing it from\ninfinity to the point is negative This\nis equivalent to saying that work\ndone by the electrostatic force in\nbringing the unit positive charge\nform infinity to the point P is\npositive"}, {"Chapter": "1", "sentence_range": "1476-1479", "Text": "e , work done (by\nthe external force) per unit positive\ntest charge in bringing it from\ninfinity to the point is negative This\nis equivalent to saying that work\ndone by the electrostatic force in\nbringing the unit positive charge\nform infinity to the point P is\npositive [This is as it should be,\nsince for Q < 0, the force on a unit\npositive test charge is attractive, so\nthat the electrostatic force and the\ndisplacement (from infinity to P) are\nin the same direction"}, {"Chapter": "1", "sentence_range": "1477-1480", "Text": ", work done (by\nthe external force) per unit positive\ntest charge in bringing it from\ninfinity to the point is negative This\nis equivalent to saying that work\ndone by the electrostatic force in\nbringing the unit positive charge\nform infinity to the point P is\npositive [This is as it should be,\nsince for Q < 0, the force on a unit\npositive test charge is attractive, so\nthat the electrostatic force and the\ndisplacement (from infinity to P) are\nin the same direction ] Finally, we\nnote that Eq"}, {"Chapter": "1", "sentence_range": "1478-1481", "Text": "This\nis equivalent to saying that work\ndone by the electrostatic force in\nbringing the unit positive charge\nform infinity to the point P is\npositive [This is as it should be,\nsince for Q < 0, the force on a unit\npositive test charge is attractive, so\nthat the electrostatic force and the\ndisplacement (from infinity to P) are\nin the same direction ] Finally, we\nnote that Eq (2"}, {"Chapter": "1", "sentence_range": "1479-1482", "Text": "[This is as it should be,\nsince for Q < 0, the force on a unit\npositive test charge is attractive, so\nthat the electrostatic force and the\ndisplacement (from infinity to P) are\nin the same direction ] Finally, we\nnote that Eq (2 8) is consistent with\nthe choice that potential at infinity\nbe zero"}, {"Chapter": "1", "sentence_range": "1480-1483", "Text": "] Finally, we\nnote that Eq (2 8) is consistent with\nthe choice that potential at infinity\nbe zero Figure (2"}, {"Chapter": "1", "sentence_range": "1481-1484", "Text": "(2 8) is consistent with\nthe choice that potential at infinity\nbe zero Figure (2 4) shows how the electrostatic potential ( \uf0b5 1/r) and the\nelectrostatic field (\uf0b5 1/r 2 ) varies with r"}, {"Chapter": "1", "sentence_range": "1482-1485", "Text": "8) is consistent with\nthe choice that potential at infinity\nbe zero Figure (2 4) shows how the electrostatic potential ( \uf0b5 1/r) and the\nelectrostatic field (\uf0b5 1/r 2 ) varies with r Example 2"}, {"Chapter": "1", "sentence_range": "1483-1486", "Text": "Figure (2 4) shows how the electrostatic potential ( \uf0b5 1/r) and the\nelectrostatic field (\uf0b5 1/r 2 ) varies with r Example 2 1\n(a) Calculate the potential at a point P due to a charge of 4 \u00d7 10\u20137C\nlocated 9 cm away"}, {"Chapter": "1", "sentence_range": "1484-1487", "Text": "4) shows how the electrostatic potential ( \uf0b5 1/r) and the\nelectrostatic field (\uf0b5 1/r 2 ) varies with r Example 2 1\n(a) Calculate the potential at a point P due to a charge of 4 \u00d7 10\u20137C\nlocated 9 cm away (b) Hence obtain the work done in bringing a charge of 2 \u00d7 10\u20139 C\nfrom infinity to the point P"}, {"Chapter": "1", "sentence_range": "1485-1488", "Text": "Example 2 1\n(a) Calculate the potential at a point P due to a charge of 4 \u00d7 10\u20137C\nlocated 9 cm away (b) Hence obtain the work done in bringing a charge of 2 \u00d7 10\u20139 C\nfrom infinity to the point P Does the answer depend on the path\nalong which the charge is brought"}, {"Chapter": "1", "sentence_range": "1486-1489", "Text": "1\n(a) Calculate the potential at a point P due to a charge of 4 \u00d7 10\u20137C\nlocated 9 cm away (b) Hence obtain the work done in bringing a charge of 2 \u00d7 10\u20139 C\nfrom infinity to the point P Does the answer depend on the path\nalong which the charge is brought Solution\n(a)  \n          = 4 \u00d7 104 V\n(b) W = qV = 2 \u00d7 10\u20139C \u00d7 4 \u00d7 104V\n     = 8 \u00d7 10\u20135 J\nNo, work done will be path independent"}, {"Chapter": "1", "sentence_range": "1487-1490", "Text": "(b) Hence obtain the work done in bringing a charge of 2 \u00d7 10\u20139 C\nfrom infinity to the point P Does the answer depend on the path\nalong which the charge is brought Solution\n(a)  \n          = 4 \u00d7 104 V\n(b) W = qV = 2 \u00d7 10\u20139C \u00d7 4 \u00d7 104V\n     = 8 \u00d7 10\u20135 J\nNo, work done will be path independent Any arbitrary infinitesimal\npath can be resolved into two perpendicular displacements: One along\nr and another perpendicular to r"}, {"Chapter": "1", "sentence_range": "1488-1491", "Text": "Does the answer depend on the path\nalong which the charge is brought Solution\n(a)  \n          = 4 \u00d7 104 V\n(b) W = qV = 2 \u00d7 10\u20139C \u00d7 4 \u00d7 104V\n     = 8 \u00d7 10\u20135 J\nNo, work done will be path independent Any arbitrary infinitesimal\npath can be resolved into two perpendicular displacements: One along\nr and another perpendicular to r The work done corresponding to\nthe later will be zero"}, {"Chapter": "1", "sentence_range": "1489-1492", "Text": "Solution\n(a)  \n          = 4 \u00d7 104 V\n(b) W = qV = 2 \u00d7 10\u20139C \u00d7 4 \u00d7 104V\n     = 8 \u00d7 10\u20135 J\nNo, work done will be path independent Any arbitrary infinitesimal\npath can be resolved into two perpendicular displacements: One along\nr and another perpendicular to r The work done corresponding to\nthe later will be zero 2"}, {"Chapter": "1", "sentence_range": "1490-1493", "Text": "Any arbitrary infinitesimal\npath can be resolved into two perpendicular displacements: One along\nr and another perpendicular to r The work done corresponding to\nthe later will be zero 2 4  POTENTIAL DUE TO AN ELECTRIC DIPOLE\nAs we learnt in the last chapter, an electric dipole consists of two charges\nq and  \u2013q separated by a (small) distance 2a"}, {"Chapter": "1", "sentence_range": "1491-1494", "Text": "The work done corresponding to\nthe later will be zero 2 4  POTENTIAL DUE TO AN ELECTRIC DIPOLE\nAs we learnt in the last chapter, an electric dipole consists of two charges\nq and  \u2013q separated by a (small) distance 2a Its total charge is zero"}, {"Chapter": "1", "sentence_range": "1492-1495", "Text": "2 4  POTENTIAL DUE TO AN ELECTRIC DIPOLE\nAs we learnt in the last chapter, an electric dipole consists of two charges\nq and  \u2013q separated by a (small) distance 2a Its total charge is zero It is\ncharacterised by a dipole moment vector p whose magnitude is q \u00d7 2a\nand which points in the direction from \u2013q to q (Fig"}, {"Chapter": "1", "sentence_range": "1493-1496", "Text": "4  POTENTIAL DUE TO AN ELECTRIC DIPOLE\nAs we learnt in the last chapter, an electric dipole consists of two charges\nq and  \u2013q separated by a (small) distance 2a Its total charge is zero It is\ncharacterised by a dipole moment vector p whose magnitude is q \u00d7 2a\nand which points in the direction from \u2013q to q (Fig 2"}, {"Chapter": "1", "sentence_range": "1494-1497", "Text": "Its total charge is zero It is\ncharacterised by a dipole moment vector p whose magnitude is q \u00d7 2a\nand which points in the direction from \u2013q to q (Fig 2 5)"}, {"Chapter": "1", "sentence_range": "1495-1498", "Text": "It is\ncharacterised by a dipole moment vector p whose magnitude is q \u00d7 2a\nand which points in the direction from \u2013q to q (Fig 2 5) We also saw that\nthe electric field of a dipole at a point with position vector r depends not\njust on the magnitude r, but also on the angle between r and p"}, {"Chapter": "1", "sentence_range": "1496-1499", "Text": "2 5) We also saw that\nthe electric field of a dipole at a point with position vector r depends not\njust on the magnitude r, but also on the angle between r and p Further,\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "1497-1500", "Text": "5) We also saw that\nthe electric field of a dipole at a point with position vector r depends not\njust on the magnitude r, but also on the angle between r and p Further,\nFIGURE 2 4 Variation of potential V with r [in units of\n(Q/4pe0) m-1] (blue curve) and field with r [in units\nof (Q/4pe0) m-2] (black curve) for a point charge Q"}, {"Chapter": "1", "sentence_range": "1498-1501", "Text": "We also saw that\nthe electric field of a dipole at a point with position vector r depends not\njust on the magnitude r, but also on the angle between r and p Further,\nFIGURE 2 4 Variation of potential V with r [in units of\n(Q/4pe0) m-1] (blue curve) and field with r [in units\nof (Q/4pe0) m-2] (black curve) for a point charge Q Rationalised 2023-24\nPhysics\n50\nthe field falls off, at large distance, not as\n1/r 2 (typical of field due to a single charge)\nbut as 1/r3"}, {"Chapter": "1", "sentence_range": "1499-1502", "Text": "Further,\nFIGURE 2 4 Variation of potential V with r [in units of\n(Q/4pe0) m-1] (blue curve) and field with r [in units\nof (Q/4pe0) m-2] (black curve) for a point charge Q Rationalised 2023-24\nPhysics\n50\nthe field falls off, at large distance, not as\n1/r 2 (typical of field due to a single charge)\nbut as 1/r3 We, now, determine the electric\npotential due to a dipole and contrast it\nwith the potential due to a single charge"}, {"Chapter": "1", "sentence_range": "1500-1503", "Text": "4 Variation of potential V with r [in units of\n(Q/4pe0) m-1] (blue curve) and field with r [in units\nof (Q/4pe0) m-2] (black curve) for a point charge Q Rationalised 2023-24\nPhysics\n50\nthe field falls off, at large distance, not as\n1/r 2 (typical of field due to a single charge)\nbut as 1/r3 We, now, determine the electric\npotential due to a dipole and contrast it\nwith the potential due to a single charge As before, we take the origin at the\ncentre of the dipole"}, {"Chapter": "1", "sentence_range": "1501-1504", "Text": "Rationalised 2023-24\nPhysics\n50\nthe field falls off, at large distance, not as\n1/r 2 (typical of field due to a single charge)\nbut as 1/r3 We, now, determine the electric\npotential due to a dipole and contrast it\nwith the potential due to a single charge As before, we take the origin at the\ncentre of the dipole Now we know that the\nelectric field obeys the superposition\nprinciple"}, {"Chapter": "1", "sentence_range": "1502-1505", "Text": "We, now, determine the electric\npotential due to a dipole and contrast it\nwith the potential due to a single charge As before, we take the origin at the\ncentre of the dipole Now we know that the\nelectric field obeys the superposition\nprinciple Since potential is related to the\nwork done by the field, electrostatic\npotential also follows the superposition\nprinciple"}, {"Chapter": "1", "sentence_range": "1503-1506", "Text": "As before, we take the origin at the\ncentre of the dipole Now we know that the\nelectric field obeys the superposition\nprinciple Since potential is related to the\nwork done by the field, electrostatic\npotential also follows the superposition\nprinciple Thus, the potential due to the\ndipole is the sum of potentials due to the\ncharges q and \u2013q\nV\nrq\nrq\n=\n\u2212\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n41\n0\n1\n2\n\u03c0\u03b5\n(2"}, {"Chapter": "1", "sentence_range": "1504-1507", "Text": "Now we know that the\nelectric field obeys the superposition\nprinciple Since potential is related to the\nwork done by the field, electrostatic\npotential also follows the superposition\nprinciple Thus, the potential due to the\ndipole is the sum of potentials due to the\ncharges q and \u2013q\nV\nrq\nrq\n=\n\u2212\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n41\n0\n1\n2\n\u03c0\u03b5\n(2 9)\nwhere r1 and r2 are the distances of the\npoint P from q and \u2013q, respectively"}, {"Chapter": "1", "sentence_range": "1505-1508", "Text": "Since potential is related to the\nwork done by the field, electrostatic\npotential also follows the superposition\nprinciple Thus, the potential due to the\ndipole is the sum of potentials due to the\ncharges q and \u2013q\nV\nrq\nrq\n=\n\u2212\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n41\n0\n1\n2\n\u03c0\u03b5\n(2 9)\nwhere r1 and r2 are the distances of the\npoint P from q and \u2013q, respectively Now, by geometry,\n2\n2\n2\n1\n2\nr\nr\na\nar\n=\n+\n\u2212\ncosq\n2\n2\n2\n2\n2\nr\nr\na\nar\n=\n+\n+\n cosq\n(2"}, {"Chapter": "1", "sentence_range": "1506-1509", "Text": "Thus, the potential due to the\ndipole is the sum of potentials due to the\ncharges q and \u2013q\nV\nrq\nrq\n=\n\u2212\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n41\n0\n1\n2\n\u03c0\u03b5\n(2 9)\nwhere r1 and r2 are the distances of the\npoint P from q and \u2013q, respectively Now, by geometry,\n2\n2\n2\n1\n2\nr\nr\na\nar\n=\n+\n\u2212\ncosq\n2\n2\n2\n2\n2\nr\nr\na\nar\n=\n+\n+\n cosq\n(2 10)\nWe take r much greater than a (\nr \uf03e\uf03ea\n) and retain terms only upto\nthe first order in a/r\n  r\nr\na\nr\nra\n1\n2\n2\n2\n2\n1\n2\n=\n\u2212\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\ncos\u03b8\n  \u2245\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f8\uf8f7\uf8f6\nr\na\nr\n2\n1\n2\ncos\u03b8\n(2"}, {"Chapter": "1", "sentence_range": "1507-1510", "Text": "9)\nwhere r1 and r2 are the distances of the\npoint P from q and \u2013q, respectively Now, by geometry,\n2\n2\n2\n1\n2\nr\nr\na\nar\n=\n+\n\u2212\ncosq\n2\n2\n2\n2\n2\nr\nr\na\nar\n=\n+\n+\n cosq\n(2 10)\nWe take r much greater than a (\nr \uf03e\uf03ea\n) and retain terms only upto\nthe first order in a/r\n  r\nr\na\nr\nra\n1\n2\n2\n2\n2\n1\n2\n=\n\u2212\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\ncos\u03b8\n  \u2245\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f8\uf8f7\uf8f6\nr\na\nr\n2\n1\n2\ncos\u03b8\n(2 11)\n Similarly,\n  r\nr\na\nr\n2\n2\n2\n1\n2\n\u2245\n\uf8ed\uf8ec\uf8eb+\n\uf8f8\uf8f7\uf8f6\ncos\u03b8\n(2"}, {"Chapter": "1", "sentence_range": "1508-1511", "Text": "Now, by geometry,\n2\n2\n2\n1\n2\nr\nr\na\nar\n=\n+\n\u2212\ncosq\n2\n2\n2\n2\n2\nr\nr\na\nar\n=\n+\n+\n cosq\n(2 10)\nWe take r much greater than a (\nr \uf03e\uf03ea\n) and retain terms only upto\nthe first order in a/r\n  r\nr\na\nr\nra\n1\n2\n2\n2\n2\n1\n2\n=\n\u2212\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\ncos\u03b8\n  \u2245\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f8\uf8f7\uf8f6\nr\na\nr\n2\n1\n2\ncos\u03b8\n(2 11)\n Similarly,\n  r\nr\na\nr\n2\n2\n2\n1\n2\n\u2245\n\uf8ed\uf8ec\uf8eb+\n\uf8f8\uf8f7\uf8f6\ncos\u03b8\n(2 12)\nUsing the Binomial theorem and retaining terms upto the first order\nin a/r ; we obtain,\n1\n1 1\n2\n1 1\n1\n1 2\nr\nr\na\nr\nr\nra\n\u2245\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f8\uf8f7\uf8f6\n\u2245\n\uf8ed\uf8ec\uf8eb+\n\uf8f6\n\uf8f8\uf8f7\n\u2212\ncos\ncos\n/\n\u03b8\n\u03b8\n[2"}, {"Chapter": "1", "sentence_range": "1509-1512", "Text": "10)\nWe take r much greater than a (\nr \uf03e\uf03ea\n) and retain terms only upto\nthe first order in a/r\n  r\nr\na\nr\nra\n1\n2\n2\n2\n2\n1\n2\n=\n\u2212\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\ncos\u03b8\n  \u2245\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f8\uf8f7\uf8f6\nr\na\nr\n2\n1\n2\ncos\u03b8\n(2 11)\n Similarly,\n  r\nr\na\nr\n2\n2\n2\n1\n2\n\u2245\n\uf8ed\uf8ec\uf8eb+\n\uf8f8\uf8f7\uf8f6\ncos\u03b8\n(2 12)\nUsing the Binomial theorem and retaining terms upto the first order\nin a/r ; we obtain,\n1\n1 1\n2\n1 1\n1\n1 2\nr\nr\na\nr\nr\nra\n\u2245\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f8\uf8f7\uf8f6\n\u2245\n\uf8ed\uf8ec\uf8eb+\n\uf8f6\n\uf8f8\uf8f7\n\u2212\ncos\ncos\n/\n\u03b8\n\u03b8\n[2 13(a)]\n1\n1 1\n2\n1 1\n2\n1 2\nr\nr\na\nr\nr\nra\n\u2245\n\uf8ed\uf8ec\uf8eb+\n\uf8f8\uf8f7\uf8f6\n\u2245\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f6\n\uf8f8\uf8f7\n\u2212\ncos\ncos\n/\n\u03b8\n\u03b8\n[2"}, {"Chapter": "1", "sentence_range": "1510-1513", "Text": "11)\n Similarly,\n  r\nr\na\nr\n2\n2\n2\n1\n2\n\u2245\n\uf8ed\uf8ec\uf8eb+\n\uf8f8\uf8f7\uf8f6\ncos\u03b8\n(2 12)\nUsing the Binomial theorem and retaining terms upto the first order\nin a/r ; we obtain,\n1\n1 1\n2\n1 1\n1\n1 2\nr\nr\na\nr\nr\nra\n\u2245\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f8\uf8f7\uf8f6\n\u2245\n\uf8ed\uf8ec\uf8eb+\n\uf8f6\n\uf8f8\uf8f7\n\u2212\ncos\ncos\n/\n\u03b8\n\u03b8\n[2 13(a)]\n1\n1 1\n2\n1 1\n2\n1 2\nr\nr\na\nr\nr\nra\n\u2245\n\uf8ed\uf8ec\uf8eb+\n\uf8f8\uf8f7\uf8f6\n\u2245\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f6\n\uf8f8\uf8f7\n\u2212\ncos\ncos\n/\n\u03b8\n\u03b8\n[2 13(b)]\n Using Eqs"}, {"Chapter": "1", "sentence_range": "1511-1514", "Text": "12)\nUsing the Binomial theorem and retaining terms upto the first order\nin a/r ; we obtain,\n1\n1 1\n2\n1 1\n1\n1 2\nr\nr\na\nr\nr\nra\n\u2245\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f8\uf8f7\uf8f6\n\u2245\n\uf8ed\uf8ec\uf8eb+\n\uf8f6\n\uf8f8\uf8f7\n\u2212\ncos\ncos\n/\n\u03b8\n\u03b8\n[2 13(a)]\n1\n1 1\n2\n1 1\n2\n1 2\nr\nr\na\nr\nr\nra\n\u2245\n\uf8ed\uf8ec\uf8eb+\n\uf8f8\uf8f7\uf8f6\n\u2245\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f6\n\uf8f8\uf8f7\n\u2212\ncos\ncos\n/\n\u03b8\n\u03b8\n[2 13(b)]\n Using Eqs (2"}, {"Chapter": "1", "sentence_range": "1512-1515", "Text": "13(a)]\n1\n1 1\n2\n1 1\n2\n1 2\nr\nr\na\nr\nr\nra\n\u2245\n\uf8ed\uf8ec\uf8eb+\n\uf8f8\uf8f7\uf8f6\n\u2245\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f6\n\uf8f8\uf8f7\n\u2212\ncos\ncos\n/\n\u03b8\n\u03b8\n[2 13(b)]\n Using Eqs (2 9) and (2"}, {"Chapter": "1", "sentence_range": "1513-1516", "Text": "13(b)]\n Using Eqs (2 9) and (2 13) and p = 2qa, we get\nV\nq\na\nr\np\nr\n=\n=\n4\n4\n0\n2\n0\n2\n\u03c0\n\u03c0\n\u03b5\n\u03b8\n\u03b8\n\u03b5\n2 cos\ncos\n(2"}, {"Chapter": "1", "sentence_range": "1514-1517", "Text": "(2 9) and (2 13) and p = 2qa, we get\nV\nq\na\nr\np\nr\n=\n=\n4\n4\n0\n2\n0\n2\n\u03c0\n\u03c0\n\u03b5\n\u03b8\n\u03b8\n\u03b5\n2 cos\ncos\n(2 14)\nNow, p cos q = p"}, {"Chapter": "1", "sentence_range": "1515-1518", "Text": "9) and (2 13) and p = 2qa, we get\nV\nq\na\nr\np\nr\n=\n=\n4\n4\n0\n2\n0\n2\n\u03c0\n\u03c0\n\u03b5\n\u03b8\n\u03b8\n\u03b5\n2 cos\ncos\n(2 14)\nNow, p cos q = p r\u02c6\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "1516-1519", "Text": "13) and p = 2qa, we get\nV\nq\na\nr\np\nr\n=\n=\n4\n4\n0\n2\n0\n2\n\u03c0\n\u03c0\n\u03b5\n\u03b8\n\u03b8\n\u03b5\n2 cos\ncos\n(2 14)\nNow, p cos q = p r\u02c6\nFIGURE 2 5 Quantities involved in the calculation\nof potential due to a dipole"}, {"Chapter": "1", "sentence_range": "1517-1520", "Text": "14)\nNow, p cos q = p r\u02c6\nFIGURE 2 5 Quantities involved in the calculation\nof potential due to a dipole Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n51\nwhere \u02c6r  is the unit vector along the position vector OP"}, {"Chapter": "1", "sentence_range": "1518-1521", "Text": "r\u02c6\nFIGURE 2 5 Quantities involved in the calculation\nof potential due to a dipole Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n51\nwhere \u02c6r  is the unit vector along the position vector OP The electric potential of a dipole is then given by\nV\nr\n=\n41\n0\n2\n\u03c0\u03b5\np"}, {"Chapter": "1", "sentence_range": "1519-1522", "Text": "5 Quantities involved in the calculation\nof potential due to a dipole Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n51\nwhere \u02c6r  is the unit vector along the position vector OP The electric potential of a dipole is then given by\nV\nr\n=\n41\n0\n2\n\u03c0\u03b5\np r\u02c6\n;       (r >> a)\n (2"}, {"Chapter": "1", "sentence_range": "1520-1523", "Text": "Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n51\nwhere \u02c6r  is the unit vector along the position vector OP The electric potential of a dipole is then given by\nV\nr\n=\n41\n0\n2\n\u03c0\u03b5\np r\u02c6\n;       (r >> a)\n (2 15)\nEquation (2"}, {"Chapter": "1", "sentence_range": "1521-1524", "Text": "The electric potential of a dipole is then given by\nV\nr\n=\n41\n0\n2\n\u03c0\u03b5\np r\u02c6\n;       (r >> a)\n (2 15)\nEquation (2 15) is, as indicated, approximately true only for distances\nlarge compared to the size of the dipole, so that higher order terms in\na/r are negligible"}, {"Chapter": "1", "sentence_range": "1522-1525", "Text": "r\u02c6\n;       (r >> a)\n (2 15)\nEquation (2 15) is, as indicated, approximately true only for distances\nlarge compared to the size of the dipole, so that higher order terms in\na/r are negligible For a point dipole p at the origin, Eq"}, {"Chapter": "1", "sentence_range": "1523-1526", "Text": "15)\nEquation (2 15) is, as indicated, approximately true only for distances\nlarge compared to the size of the dipole, so that higher order terms in\na/r are negligible For a point dipole p at the origin, Eq (2"}, {"Chapter": "1", "sentence_range": "1524-1527", "Text": "15) is, as indicated, approximately true only for distances\nlarge compared to the size of the dipole, so that higher order terms in\na/r are negligible For a point dipole p at the origin, Eq (2 15) is, however,\nexact"}, {"Chapter": "1", "sentence_range": "1525-1528", "Text": "For a point dipole p at the origin, Eq (2 15) is, however,\nexact From Eq"}, {"Chapter": "1", "sentence_range": "1526-1529", "Text": "(2 15) is, however,\nexact From Eq (2"}, {"Chapter": "1", "sentence_range": "1527-1530", "Text": "15) is, however,\nexact From Eq (2 15), potential on the dipole axis (q = 0, p ) is given by\n2\n0\n1\n4\np\nV\n\u03b5r\n= \u00b1\n\u03c0\n(2"}, {"Chapter": "1", "sentence_range": "1528-1531", "Text": "From Eq (2 15), potential on the dipole axis (q = 0, p ) is given by\n2\n0\n1\n4\np\nV\n\u03b5r\n= \u00b1\n\u03c0\n(2 16)\n(Positive sign for q = 0, negative sign for q = p"}, {"Chapter": "1", "sentence_range": "1529-1532", "Text": "(2 15), potential on the dipole axis (q = 0, p ) is given by\n2\n0\n1\n4\np\nV\n\u03b5r\n= \u00b1\n\u03c0\n(2 16)\n(Positive sign for q = 0, negative sign for q = p ) The potential in the\nequatorial plane (q = p/2) is zero"}, {"Chapter": "1", "sentence_range": "1530-1533", "Text": "15), potential on the dipole axis (q = 0, p ) is given by\n2\n0\n1\n4\np\nV\n\u03b5r\n= \u00b1\n\u03c0\n(2 16)\n(Positive sign for q = 0, negative sign for q = p ) The potential in the\nequatorial plane (q = p/2) is zero The important contrasting features of electric potential of a dipole\nfrom that due to a single charge are clear from Eqs"}, {"Chapter": "1", "sentence_range": "1531-1534", "Text": "16)\n(Positive sign for q = 0, negative sign for q = p ) The potential in the\nequatorial plane (q = p/2) is zero The important contrasting features of electric potential of a dipole\nfrom that due to a single charge are clear from Eqs (2"}, {"Chapter": "1", "sentence_range": "1532-1535", "Text": ") The potential in the\nequatorial plane (q = p/2) is zero The important contrasting features of electric potential of a dipole\nfrom that due to a single charge are clear from Eqs (2 8) and (2"}, {"Chapter": "1", "sentence_range": "1533-1536", "Text": "The important contrasting features of electric potential of a dipole\nfrom that due to a single charge are clear from Eqs (2 8) and (2 15):\n(i)\nThe potential due to a dipole depends not just on r but also on the\nangle between the position vector  r and the dipole moment vector p"}, {"Chapter": "1", "sentence_range": "1534-1537", "Text": "(2 8) and (2 15):\n(i)\nThe potential due to a dipole depends not just on r but also on the\nangle between the position vector  r and the dipole moment vector p (It is, however, axially symmetric about p"}, {"Chapter": "1", "sentence_range": "1535-1538", "Text": "8) and (2 15):\n(i)\nThe potential due to a dipole depends not just on r but also on the\nangle between the position vector  r and the dipole moment vector p (It is, however, axially symmetric about p That is, if you rotate the\nposition vector r  about  p, keeping q fixed, the points corresponding\nto P on the cone so generated will have the same potential as at P"}, {"Chapter": "1", "sentence_range": "1536-1539", "Text": "15):\n(i)\nThe potential due to a dipole depends not just on r but also on the\nangle between the position vector  r and the dipole moment vector p (It is, however, axially symmetric about p That is, if you rotate the\nposition vector r  about  p, keeping q fixed, the points corresponding\nto P on the cone so generated will have the same potential as at P )\n(ii) The electric dipole potential falls off, at large distance, as 1/r 2, not as\n1/r, characteristic of the potential due to a single charge"}, {"Chapter": "1", "sentence_range": "1537-1540", "Text": "(It is, however, axially symmetric about p That is, if you rotate the\nposition vector r  about  p, keeping q fixed, the points corresponding\nto P on the cone so generated will have the same potential as at P )\n(ii) The electric dipole potential falls off, at large distance, as 1/r 2, not as\n1/r, characteristic of the potential due to a single charge (You can\nrefer to the Fig"}, {"Chapter": "1", "sentence_range": "1538-1541", "Text": "That is, if you rotate the\nposition vector r  about  p, keeping q fixed, the points corresponding\nto P on the cone so generated will have the same potential as at P )\n(ii) The electric dipole potential falls off, at large distance, as 1/r 2, not as\n1/r, characteristic of the potential due to a single charge (You can\nrefer to the Fig 2"}, {"Chapter": "1", "sentence_range": "1539-1542", "Text": ")\n(ii) The electric dipole potential falls off, at large distance, as 1/r 2, not as\n1/r, characteristic of the potential due to a single charge (You can\nrefer to the Fig 2 5 for graphs of 1/r 2 versus r and 1/r versus r,\ndrawn there in another context"}, {"Chapter": "1", "sentence_range": "1540-1543", "Text": "(You can\nrefer to the Fig 2 5 for graphs of 1/r 2 versus r and 1/r versus r,\ndrawn there in another context )\n2"}, {"Chapter": "1", "sentence_range": "1541-1544", "Text": "2 5 for graphs of 1/r 2 versus r and 1/r versus r,\ndrawn there in another context )\n2 5  POTENTIAL DUE TO A SYSTEM OF CHARGES\nConsider a system of charges q1, q2,\u2026, qn with position vectors r1, r2,\u2026,\nrn relative to some origin (Fig"}, {"Chapter": "1", "sentence_range": "1542-1545", "Text": "5 for graphs of 1/r 2 versus r and 1/r versus r,\ndrawn there in another context )\n2 5  POTENTIAL DUE TO A SYSTEM OF CHARGES\nConsider a system of charges q1, q2,\u2026, qn with position vectors r1, r2,\u2026,\nrn relative to some origin (Fig 2"}, {"Chapter": "1", "sentence_range": "1543-1546", "Text": ")\n2 5  POTENTIAL DUE TO A SYSTEM OF CHARGES\nConsider a system of charges q1, q2,\u2026, qn with position vectors r1, r2,\u2026,\nrn relative to some origin (Fig 2 6)"}, {"Chapter": "1", "sentence_range": "1544-1547", "Text": "5  POTENTIAL DUE TO A SYSTEM OF CHARGES\nConsider a system of charges q1, q2,\u2026, qn with position vectors r1, r2,\u2026,\nrn relative to some origin (Fig 2 6) The potential V1 at P due to the charge\nq1 is\n1\n1\n0\n1P\n1\n4\nq\nV\n\u03b5r\n=\n\u03c0\nwhere r1P is the distance between q1 and P"}, {"Chapter": "1", "sentence_range": "1545-1548", "Text": "2 6) The potential V1 at P due to the charge\nq1 is\n1\n1\n0\n1P\n1\n4\nq\nV\n\u03b5r\n=\n\u03c0\nwhere r1P is the distance between q1 and P Similarly, the potential V2 at P due to q2 and\nV3 due to q3 are given by\n2\n2\n0\n2P\n1\n4\nq\nV\n\u03b5r\n=\n\u03c0\n, \n3\n3\n0\n3P\n1\n4\nq\nV\n\u03b5r\n=\n\u03c0\nwhere r2P and r3P are the distances of P from\ncharges q2 and q3, respectively; and so on for the\npotential due to other charges"}, {"Chapter": "1", "sentence_range": "1546-1549", "Text": "6) The potential V1 at P due to the charge\nq1 is\n1\n1\n0\n1P\n1\n4\nq\nV\n\u03b5r\n=\n\u03c0\nwhere r1P is the distance between q1 and P Similarly, the potential V2 at P due to q2 and\nV3 due to q3 are given by\n2\n2\n0\n2P\n1\n4\nq\nV\n\u03b5r\n=\n\u03c0\n, \n3\n3\n0\n3P\n1\n4\nq\nV\n\u03b5r\n=\n\u03c0\nwhere r2P and r3P are the distances of P from\ncharges q2 and q3, respectively; and so on for the\npotential due to other charges By the\nsuperposition principle, the potential V at P due\nto the total charge configuration is the algebraic\nsum of the potentials due to the individual\ncharges\nV = V1 + V2 +"}, {"Chapter": "1", "sentence_range": "1547-1550", "Text": "The potential V1 at P due to the charge\nq1 is\n1\n1\n0\n1P\n1\n4\nq\nV\n\u03b5r\n=\n\u03c0\nwhere r1P is the distance between q1 and P Similarly, the potential V2 at P due to q2 and\nV3 due to q3 are given by\n2\n2\n0\n2P\n1\n4\nq\nV\n\u03b5r\n=\n\u03c0\n, \n3\n3\n0\n3P\n1\n4\nq\nV\n\u03b5r\n=\n\u03c0\nwhere r2P and r3P are the distances of P from\ncharges q2 and q3, respectively; and so on for the\npotential due to other charges By the\nsuperposition principle, the potential V at P due\nto the total charge configuration is the algebraic\nsum of the potentials due to the individual\ncharges\nV = V1 + V2 + + Vn\n(2"}, {"Chapter": "1", "sentence_range": "1548-1551", "Text": "Similarly, the potential V2 at P due to q2 and\nV3 due to q3 are given by\n2\n2\n0\n2P\n1\n4\nq\nV\n\u03b5r\n=\n\u03c0\n, \n3\n3\n0\n3P\n1\n4\nq\nV\n\u03b5r\n=\n\u03c0\nwhere r2P and r3P are the distances of P from\ncharges q2 and q3, respectively; and so on for the\npotential due to other charges By the\nsuperposition principle, the potential V at P due\nto the total charge configuration is the algebraic\nsum of the potentials due to the individual\ncharges\nV = V1 + V2 + + Vn\n(2 17)\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "1549-1552", "Text": "By the\nsuperposition principle, the potential V at P due\nto the total charge configuration is the algebraic\nsum of the potentials due to the individual\ncharges\nV = V1 + V2 + + Vn\n(2 17)\nFIGURE 2 6 Potential at a point due to a\nsystem of charges is the sum of potentials\ndue to individual charges"}, {"Chapter": "1", "sentence_range": "1550-1553", "Text": "+ Vn\n(2 17)\nFIGURE 2 6 Potential at a point due to a\nsystem of charges is the sum of potentials\ndue to individual charges Rationalised 2023-24\nPhysics\n52\n EXAMPLE 2"}, {"Chapter": "1", "sentence_range": "1551-1554", "Text": "17)\nFIGURE 2 6 Potential at a point due to a\nsystem of charges is the sum of potentials\ndue to individual charges Rationalised 2023-24\nPhysics\n52\n EXAMPLE 2 2\n=\n+\n+\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n41\n0\n1\n1\n2\n2\n\u03c0\u03b5\nrq\nrq\nq\nr\nn\nn\nP\nP\nP"}, {"Chapter": "1", "sentence_range": "1552-1555", "Text": "6 Potential at a point due to a\nsystem of charges is the sum of potentials\ndue to individual charges Rationalised 2023-24\nPhysics\n52\n EXAMPLE 2 2\n=\n+\n+\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n41\n0\n1\n1\n2\n2\n\u03c0\u03b5\nrq\nrq\nq\nr\nn\nn\nP\nP\nP (2"}, {"Chapter": "1", "sentence_range": "1553-1556", "Text": "Rationalised 2023-24\nPhysics\n52\n EXAMPLE 2 2\n=\n+\n+\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n41\n0\n1\n1\n2\n2\n\u03c0\u03b5\nrq\nrq\nq\nr\nn\nn\nP\nP\nP (2 18)\nIf we have a continuous charge distribution characterised by a charge\ndensity r (r), we divide it, as before, into small volume elements each of\nsize Dv and carrying a charge rDv"}, {"Chapter": "1", "sentence_range": "1554-1557", "Text": "2\n=\n+\n+\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n41\n0\n1\n1\n2\n2\n\u03c0\u03b5\nrq\nrq\nq\nr\nn\nn\nP\nP\nP (2 18)\nIf we have a continuous charge distribution characterised by a charge\ndensity r (r), we divide it, as before, into small volume elements each of\nsize Dv and carrying a charge rDv We then determine the potential due\nto each volume element and sum (strictly speaking , integrate) over all\nsuch contributions, and thus determine the potential due to the entire\ndistribution"}, {"Chapter": "1", "sentence_range": "1555-1558", "Text": "(2 18)\nIf we have a continuous charge distribution characterised by a charge\ndensity r (r), we divide it, as before, into small volume elements each of\nsize Dv and carrying a charge rDv We then determine the potential due\nto each volume element and sum (strictly speaking , integrate) over all\nsuch contributions, and thus determine the potential due to the entire\ndistribution We have seen in Chapter 1 that for a uniformly charged spherical shell,\nthe electric field outside the shell is as if the entire charge is concentrated\nat the centre"}, {"Chapter": "1", "sentence_range": "1556-1559", "Text": "18)\nIf we have a continuous charge distribution characterised by a charge\ndensity r (r), we divide it, as before, into small volume elements each of\nsize Dv and carrying a charge rDv We then determine the potential due\nto each volume element and sum (strictly speaking , integrate) over all\nsuch contributions, and thus determine the potential due to the entire\ndistribution We have seen in Chapter 1 that for a uniformly charged spherical shell,\nthe electric field outside the shell is as if the entire charge is concentrated\nat the centre Thus, the potential outside the shell is given by\n0\n1\n4\nq\nV\nr\n\u03b5\n=\n\u03c0\n   (\n)\nr\n\u2265R\n[2"}, {"Chapter": "1", "sentence_range": "1557-1560", "Text": "We then determine the potential due\nto each volume element and sum (strictly speaking , integrate) over all\nsuch contributions, and thus determine the potential due to the entire\ndistribution We have seen in Chapter 1 that for a uniformly charged spherical shell,\nthe electric field outside the shell is as if the entire charge is concentrated\nat the centre Thus, the potential outside the shell is given by\n0\n1\n4\nq\nV\nr\n\u03b5\n=\n\u03c0\n   (\n)\nr\n\u2265R\n[2 19(a)]\nwhere q is the total charge on the shell and R its radius"}, {"Chapter": "1", "sentence_range": "1558-1561", "Text": "We have seen in Chapter 1 that for a uniformly charged spherical shell,\nthe electric field outside the shell is as if the entire charge is concentrated\nat the centre Thus, the potential outside the shell is given by\n0\n1\n4\nq\nV\nr\n\u03b5\n=\n\u03c0\n   (\n)\nr\n\u2265R\n[2 19(a)]\nwhere q is the total charge on the shell and R its radius The electric field\ninside the shell is zero"}, {"Chapter": "1", "sentence_range": "1559-1562", "Text": "Thus, the potential outside the shell is given by\n0\n1\n4\nq\nV\nr\n\u03b5\n=\n\u03c0\n   (\n)\nr\n\u2265R\n[2 19(a)]\nwhere q is the total charge on the shell and R its radius The electric field\ninside the shell is zero This implies (Section 2"}, {"Chapter": "1", "sentence_range": "1560-1563", "Text": "19(a)]\nwhere q is the total charge on the shell and R its radius The electric field\ninside the shell is zero This implies (Section 2 6) that potential is constant\ninside the shell (as no work is done in moving a charge inside the shell),\nand, therefore, equals its value at the surface, which is\n0\n1\n4\nq\nV\nR\n\u03b5\n=\n\u03c0\n[2"}, {"Chapter": "1", "sentence_range": "1561-1564", "Text": "The electric field\ninside the shell is zero This implies (Section 2 6) that potential is constant\ninside the shell (as no work is done in moving a charge inside the shell),\nand, therefore, equals its value at the surface, which is\n0\n1\n4\nq\nV\nR\n\u03b5\n=\n\u03c0\n[2 19(b)]\nExample 2"}, {"Chapter": "1", "sentence_range": "1562-1565", "Text": "This implies (Section 2 6) that potential is constant\ninside the shell (as no work is done in moving a charge inside the shell),\nand, therefore, equals its value at the surface, which is\n0\n1\n4\nq\nV\nR\n\u03b5\n=\n\u03c0\n[2 19(b)]\nExample 2 2 Two charges 3 \u00d7 10\u20138 C and \u20132 \u00d7 10\u20138 C are located\n15 cm apart"}, {"Chapter": "1", "sentence_range": "1563-1566", "Text": "6) that potential is constant\ninside the shell (as no work is done in moving a charge inside the shell),\nand, therefore, equals its value at the surface, which is\n0\n1\n4\nq\nV\nR\n\u03b5\n=\n\u03c0\n[2 19(b)]\nExample 2 2 Two charges 3 \u00d7 10\u20138 C and \u20132 \u00d7 10\u20138 C are located\n15 cm apart At what point on the line joining the two charges is the\nelectric potential zero"}, {"Chapter": "1", "sentence_range": "1564-1567", "Text": "19(b)]\nExample 2 2 Two charges 3 \u00d7 10\u20138 C and \u20132 \u00d7 10\u20138 C are located\n15 cm apart At what point on the line joining the two charges is the\nelectric potential zero Take the potential at infinity to be zero"}, {"Chapter": "1", "sentence_range": "1565-1568", "Text": "2 Two charges 3 \u00d7 10\u20138 C and \u20132 \u00d7 10\u20138 C are located\n15 cm apart At what point on the line joining the two charges is the\nelectric potential zero Take the potential at infinity to be zero Solution Let us take the origin O at the location of the positive charge"}, {"Chapter": "1", "sentence_range": "1566-1569", "Text": "At what point on the line joining the two charges is the\nelectric potential zero Take the potential at infinity to be zero Solution Let us take the origin O at the location of the positive charge The line joining the two charges is taken to be the x-axis;  the negative\ncharge is taken to be on the right side of the origin (Fig"}, {"Chapter": "1", "sentence_range": "1567-1570", "Text": "Take the potential at infinity to be zero Solution Let us take the origin O at the location of the positive charge The line joining the two charges is taken to be the x-axis;  the negative\ncharge is taken to be on the right side of the origin (Fig 2"}, {"Chapter": "1", "sentence_range": "1568-1571", "Text": "Solution Let us take the origin O at the location of the positive charge The line joining the two charges is taken to be the x-axis;  the negative\ncharge is taken to be on the right side of the origin (Fig 2 7)"}, {"Chapter": "1", "sentence_range": "1569-1572", "Text": "The line joining the two charges is taken to be the x-axis;  the negative\ncharge is taken to be on the right side of the origin (Fig 2 7) FIGURE 2"}, {"Chapter": "1", "sentence_range": "1570-1573", "Text": "2 7) FIGURE 2 7\nLet P be the required point on the x-axis where the potential is zero"}, {"Chapter": "1", "sentence_range": "1571-1574", "Text": "7) FIGURE 2 7\nLet P be the required point on the x-axis where the potential is zero If x is the x-coordinate of P, obviously x must be positive"}, {"Chapter": "1", "sentence_range": "1572-1575", "Text": "FIGURE 2 7\nLet P be the required point on the x-axis where the potential is zero If x is the x-coordinate of P, obviously x must be positive (There is no\npossibility of potentials due to the two charges adding up to zero for\nx < 0"}, {"Chapter": "1", "sentence_range": "1573-1576", "Text": "7\nLet P be the required point on the x-axis where the potential is zero If x is the x-coordinate of P, obviously x must be positive (There is no\npossibility of potentials due to the two charges adding up to zero for\nx < 0 ) If x lies between O and A, we have\n41\n3 10\n10\n152 10\n10\n0\n0\n8\n2\n8\n2\n\u03c0\u03b5\n\u00d7\u00d7\n\u2212\n\u2212\u00d7\n\u00d7\n=\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n\u2013\n\u2013\n\u2013\n\u2013\n(\n)\nx\nx\nwhere x is in cm"}, {"Chapter": "1", "sentence_range": "1574-1577", "Text": "If x is the x-coordinate of P, obviously x must be positive (There is no\npossibility of potentials due to the two charges adding up to zero for\nx < 0 ) If x lies between O and A, we have\n41\n3 10\n10\n152 10\n10\n0\n0\n8\n2\n8\n2\n\u03c0\u03b5\n\u00d7\u00d7\n\u2212\n\u2212\u00d7\n\u00d7\n=\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n\u2013\n\u2013\n\u2013\n\u2013\n(\n)\nx\nx\nwhere x is in cm That is,\n3\n2\n0\n15\nx\nx\n\u2212\n=\n\u2212\nwhich gives x  =  9 cm"}, {"Chapter": "1", "sentence_range": "1575-1578", "Text": "(There is no\npossibility of potentials due to the two charges adding up to zero for\nx < 0 ) If x lies between O and A, we have\n41\n3 10\n10\n152 10\n10\n0\n0\n8\n2\n8\n2\n\u03c0\u03b5\n\u00d7\u00d7\n\u2212\n\u2212\u00d7\n\u00d7\n=\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n\u2013\n\u2013\n\u2013\n\u2013\n(\n)\nx\nx\nwhere x is in cm That is,\n3\n2\n0\n15\nx\nx\n\u2212\n=\n\u2212\nwhich gives x  =  9 cm If x lies on the extended line OA, the required condition is\n3\n2\n0\n15\nx\n\u2212x\n=\n\u2212\nRationalised 2023-24\nElectrostatic Potential\nand Capacitance\n53\n EXAMPLE 2"}, {"Chapter": "1", "sentence_range": "1576-1579", "Text": ") If x lies between O and A, we have\n41\n3 10\n10\n152 10\n10\n0\n0\n8\n2\n8\n2\n\u03c0\u03b5\n\u00d7\u00d7\n\u2212\n\u2212\u00d7\n\u00d7\n=\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n\u2013\n\u2013\n\u2013\n\u2013\n(\n)\nx\nx\nwhere x is in cm That is,\n3\n2\n0\n15\nx\nx\n\u2212\n=\n\u2212\nwhich gives x  =  9 cm If x lies on the extended line OA, the required condition is\n3\n2\n0\n15\nx\n\u2212x\n=\n\u2212\nRationalised 2023-24\nElectrostatic Potential\nand Capacitance\n53\n EXAMPLE 2 2\nwhich gives\nx  = 45 cm\nThus, electric potential is zero at 9 cm and 45 cm away from the\npositive charge on the side of the negative charge"}, {"Chapter": "1", "sentence_range": "1577-1580", "Text": "That is,\n3\n2\n0\n15\nx\nx\n\u2212\n=\n\u2212\nwhich gives x  =  9 cm If x lies on the extended line OA, the required condition is\n3\n2\n0\n15\nx\n\u2212x\n=\n\u2212\nRationalised 2023-24\nElectrostatic Potential\nand Capacitance\n53\n EXAMPLE 2 2\nwhich gives\nx  = 45 cm\nThus, electric potential is zero at 9 cm and 45 cm away from the\npositive charge on the side of the negative charge Note that the\nformula for potential used in the calculation required choosing\npotential to be zero at infinity"}, {"Chapter": "1", "sentence_range": "1578-1581", "Text": "If x lies on the extended line OA, the required condition is\n3\n2\n0\n15\nx\n\u2212x\n=\n\u2212\nRationalised 2023-24\nElectrostatic Potential\nand Capacitance\n53\n EXAMPLE 2 2\nwhich gives\nx  = 45 cm\nThus, electric potential is zero at 9 cm and 45 cm away from the\npositive charge on the side of the negative charge Note that the\nformula for potential used in the calculation required choosing\npotential to be zero at infinity Example 2"}, {"Chapter": "1", "sentence_range": "1579-1582", "Text": "2\nwhich gives\nx  = 45 cm\nThus, electric potential is zero at 9 cm and 45 cm away from the\npositive charge on the side of the negative charge Note that the\nformula for potential used in the calculation required choosing\npotential to be zero at infinity Example 2 3  Figures 2"}, {"Chapter": "1", "sentence_range": "1580-1583", "Text": "Note that the\nformula for potential used in the calculation required choosing\npotential to be zero at infinity Example 2 3  Figures 2 8 (a) and (b) show the field lines of a positive\nand negative point charge respectively"}, {"Chapter": "1", "sentence_range": "1581-1584", "Text": "Example 2 3  Figures 2 8 (a) and (b) show the field lines of a positive\nand negative point charge respectively FIGURE 2"}, {"Chapter": "1", "sentence_range": "1582-1585", "Text": "3  Figures 2 8 (a) and (b) show the field lines of a positive\nand negative point charge respectively FIGURE 2 8\n(a) Give the signs of the potential difference VP \u2013 VQ; VB \u2013 VA"}, {"Chapter": "1", "sentence_range": "1583-1586", "Text": "8 (a) and (b) show the field lines of a positive\nand negative point charge respectively FIGURE 2 8\n(a) Give the signs of the potential difference VP \u2013 VQ; VB \u2013 VA (b) Give the sign of the potential energy difference of a small negative\ncharge between the points Q and P; A and B"}, {"Chapter": "1", "sentence_range": "1584-1587", "Text": "FIGURE 2 8\n(a) Give the signs of the potential difference VP \u2013 VQ; VB \u2013 VA (b) Give the sign of the potential energy difference of a small negative\ncharge between the points Q and P; A and B (c) Give the sign of the work done by the field in moving a small\npositive charge from Q to P"}, {"Chapter": "1", "sentence_range": "1585-1588", "Text": "8\n(a) Give the signs of the potential difference VP \u2013 VQ; VB \u2013 VA (b) Give the sign of the potential energy difference of a small negative\ncharge between the points Q and P; A and B (c) Give the sign of the work done by the field in moving a small\npositive charge from Q to P (d) Give the sign of the work done by the external agency in moving\na small negative charge from B to A"}, {"Chapter": "1", "sentence_range": "1586-1589", "Text": "(b) Give the sign of the potential energy difference of a small negative\ncharge between the points Q and P; A and B (c) Give the sign of the work done by the field in moving a small\npositive charge from Q to P (d) Give the sign of the work done by the external agency in moving\na small negative charge from B to A (e) Does the kinetic energy of a small negative charge increase or\ndecrease in going from B to A"}, {"Chapter": "1", "sentence_range": "1587-1590", "Text": "(c) Give the sign of the work done by the field in moving a small\npositive charge from Q to P (d) Give the sign of the work done by the external agency in moving\na small negative charge from B to A (e) Does the kinetic energy of a small negative charge increase or\ndecrease in going from B to A Solution\n(a) As \n1\nV\n\u221dr\n, VP > VQ"}, {"Chapter": "1", "sentence_range": "1588-1591", "Text": "(d) Give the sign of the work done by the external agency in moving\na small negative charge from B to A (e) Does the kinetic energy of a small negative charge increase or\ndecrease in going from B to A Solution\n(a) As \n1\nV\n\u221dr\n, VP > VQ Thus, (VP \u2013 VQ) is positive"}, {"Chapter": "1", "sentence_range": "1589-1592", "Text": "(e) Does the kinetic energy of a small negative charge increase or\ndecrease in going from B to A Solution\n(a) As \n1\nV\n\u221dr\n, VP > VQ Thus, (VP \u2013 VQ) is positive Also VB is less negative\nthan VA"}, {"Chapter": "1", "sentence_range": "1590-1593", "Text": "Solution\n(a) As \n1\nV\n\u221dr\n, VP > VQ Thus, (VP \u2013 VQ) is positive Also VB is less negative\nthan VA Thus, VB > VA or (VB \u2013 VA) is positive"}, {"Chapter": "1", "sentence_range": "1591-1594", "Text": "Thus, (VP \u2013 VQ) is positive Also VB is less negative\nthan VA Thus, VB > VA or (VB \u2013 VA) is positive (b) A small negative charge will be attracted towards positive charge"}, {"Chapter": "1", "sentence_range": "1592-1595", "Text": "Also VB is less negative\nthan VA Thus, VB > VA or (VB \u2013 VA) is positive (b) A small negative charge will be attracted towards positive charge The negative charge moves from higher potential energy to lower\npotential energy"}, {"Chapter": "1", "sentence_range": "1593-1596", "Text": "Thus, VB > VA or (VB \u2013 VA) is positive (b) A small negative charge will be attracted towards positive charge The negative charge moves from higher potential energy to lower\npotential energy Therefore the sign of potential energy difference\nof a small negative charge between Q and P is positive"}, {"Chapter": "1", "sentence_range": "1594-1597", "Text": "(b) A small negative charge will be attracted towards positive charge The negative charge moves from higher potential energy to lower\npotential energy Therefore the sign of potential energy difference\nof a small negative charge between Q and P is positive Similarly, (P"}, {"Chapter": "1", "sentence_range": "1595-1598", "Text": "The negative charge moves from higher potential energy to lower\npotential energy Therefore the sign of potential energy difference\nof a small negative charge between Q and P is positive Similarly, (P E"}, {"Chapter": "1", "sentence_range": "1596-1599", "Text": "Therefore the sign of potential energy difference\nof a small negative charge between Q and P is positive Similarly, (P E )A > (P"}, {"Chapter": "1", "sentence_range": "1597-1600", "Text": "Similarly, (P E )A > (P E"}, {"Chapter": "1", "sentence_range": "1598-1601", "Text": "E )A > (P E )B\n and hence sign of potential energy\ndifferences is positive"}, {"Chapter": "1", "sentence_range": "1599-1602", "Text": ")A > (P E )B\n and hence sign of potential energy\ndifferences is positive (c) In moving a small positive charge from Q to P, work has to be\ndone by an external agency against the electric field"}, {"Chapter": "1", "sentence_range": "1600-1603", "Text": "E )B\n and hence sign of potential energy\ndifferences is positive (c) In moving a small positive charge from Q to P, work has to be\ndone by an external agency against the electric field Therefore,\nwork done by the field is negative"}, {"Chapter": "1", "sentence_range": "1601-1604", "Text": ")B\n and hence sign of potential energy\ndifferences is positive (c) In moving a small positive charge from Q to P, work has to be\ndone by an external agency against the electric field Therefore,\nwork done by the field is negative (d) In moving a small negative charge from B to A work has to be\ndone by the external agency"}, {"Chapter": "1", "sentence_range": "1602-1605", "Text": "(c) In moving a small positive charge from Q to P, work has to be\ndone by an external agency against the electric field Therefore,\nwork done by the field is negative (d) In moving a small negative charge from B to A work has to be\ndone by the external agency It is positive"}, {"Chapter": "1", "sentence_range": "1603-1606", "Text": "Therefore,\nwork done by the field is negative (d) In moving a small negative charge from B to A work has to be\ndone by the external agency It is positive (e) Due to force of repulsion on the negative charge, velocity decreases\nand hence the kinetic energy decreases in going from B to A"}, {"Chapter": "1", "sentence_range": "1604-1607", "Text": "(d) In moving a small negative charge from B to A work has to be\ndone by the external agency It is positive (e) Due to force of repulsion on the negative charge, velocity decreases\nand hence the kinetic energy decreases in going from B to A EXAMPLE 2"}, {"Chapter": "1", "sentence_range": "1605-1608", "Text": "It is positive (e) Due to force of repulsion on the negative charge, velocity decreases\nand hence the kinetic energy decreases in going from B to A EXAMPLE 2 3\nElectric potential, equipotential surfaces:\nhttp://video"}, {"Chapter": "1", "sentence_range": "1606-1609", "Text": "(e) Due to force of repulsion on the negative charge, velocity decreases\nand hence the kinetic energy decreases in going from B to A EXAMPLE 2 3\nElectric potential, equipotential surfaces:\nhttp://video mit"}, {"Chapter": "1", "sentence_range": "1607-1610", "Text": "EXAMPLE 2 3\nElectric potential, equipotential surfaces:\nhttp://video mit edu/watch/4-electrostatic-potential-elctric-energy-ev-conservative-field-\nequipotential-sufaces-12584/\nRationalised 2023-24\nPhysics\n54\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "1608-1611", "Text": "3\nElectric potential, equipotential surfaces:\nhttp://video mit edu/watch/4-electrostatic-potential-elctric-energy-ev-conservative-field-\nequipotential-sufaces-12584/\nRationalised 2023-24\nPhysics\n54\nFIGURE 2 10 Equipotential surfaces for a uniform electric field"}, {"Chapter": "1", "sentence_range": "1609-1612", "Text": "mit edu/watch/4-electrostatic-potential-elctric-energy-ev-conservative-field-\nequipotential-sufaces-12584/\nRationalised 2023-24\nPhysics\n54\nFIGURE 2 10 Equipotential surfaces for a uniform electric field 2"}, {"Chapter": "1", "sentence_range": "1610-1613", "Text": "edu/watch/4-electrostatic-potential-elctric-energy-ev-conservative-field-\nequipotential-sufaces-12584/\nRationalised 2023-24\nPhysics\n54\nFIGURE 2 10 Equipotential surfaces for a uniform electric field 2 6  EQUIPOTENTIAL SURFACES\nAn equipotential surface is a surface with a constant value of potential\nat all points on the surface"}, {"Chapter": "1", "sentence_range": "1611-1614", "Text": "10 Equipotential surfaces for a uniform electric field 2 6  EQUIPOTENTIAL SURFACES\nAn equipotential surface is a surface with a constant value of potential\nat all points on the surface For a single charge q, the potential is given\nby Eq"}, {"Chapter": "1", "sentence_range": "1612-1615", "Text": "2 6  EQUIPOTENTIAL SURFACES\nAn equipotential surface is a surface with a constant value of potential\nat all points on the surface For a single charge q, the potential is given\nby Eq (2"}, {"Chapter": "1", "sentence_range": "1613-1616", "Text": "6  EQUIPOTENTIAL SURFACES\nAn equipotential surface is a surface with a constant value of potential\nat all points on the surface For a single charge q, the potential is given\nby Eq (2 8):\n41\no\nq\nV\n\u03b5r\n=\n\u03c0\nThis shows that V is a constant if r is constant"}, {"Chapter": "1", "sentence_range": "1614-1617", "Text": "For a single charge q, the potential is given\nby Eq (2 8):\n41\no\nq\nV\n\u03b5r\n=\n\u03c0\nThis shows that V is a constant if r is constant Thus, equipotential\nsurfaces of a single point charge are concentric spherical surfaces centred\nat the charge"}, {"Chapter": "1", "sentence_range": "1615-1618", "Text": "(2 8):\n41\no\nq\nV\n\u03b5r\n=\n\u03c0\nThis shows that V is a constant if r is constant Thus, equipotential\nsurfaces of a single point charge are concentric spherical surfaces centred\nat the charge Now the electric field lines for a single charge q are radial lines starting\nfrom or ending at the charge, depending on whether q is positive or negative"}, {"Chapter": "1", "sentence_range": "1616-1619", "Text": "8):\n41\no\nq\nV\n\u03b5r\n=\n\u03c0\nThis shows that V is a constant if r is constant Thus, equipotential\nsurfaces of a single point charge are concentric spherical surfaces centred\nat the charge Now the electric field lines for a single charge q are radial lines starting\nfrom or ending at the charge, depending on whether q is positive or negative Clearly, the electric field at every point is normal to the equipotential surface\npassing through that point"}, {"Chapter": "1", "sentence_range": "1617-1620", "Text": "Thus, equipotential\nsurfaces of a single point charge are concentric spherical surfaces centred\nat the charge Now the electric field lines for a single charge q are radial lines starting\nfrom or ending at the charge, depending on whether q is positive or negative Clearly, the electric field at every point is normal to the equipotential surface\npassing through that point This is true in general: for any charge\nconfiguration, equipotential surface through a point is normal to the\nelectric field at that point"}, {"Chapter": "1", "sentence_range": "1618-1621", "Text": "Now the electric field lines for a single charge q are radial lines starting\nfrom or ending at the charge, depending on whether q is positive or negative Clearly, the electric field at every point is normal to the equipotential surface\npassing through that point This is true in general: for any charge\nconfiguration, equipotential surface through a point is normal to the\nelectric field at that point The proof of this statement is simple"}, {"Chapter": "1", "sentence_range": "1619-1622", "Text": "Clearly, the electric field at every point is normal to the equipotential surface\npassing through that point This is true in general: for any charge\nconfiguration, equipotential surface through a point is normal to the\nelectric field at that point The proof of this statement is simple If the field were not normal to the equipotential surface, it would\nhave non-zero component along the surface"}, {"Chapter": "1", "sentence_range": "1620-1623", "Text": "This is true in general: for any charge\nconfiguration, equipotential surface through a point is normal to the\nelectric field at that point The proof of this statement is simple If the field were not normal to the equipotential surface, it would\nhave non-zero component along the surface To move a unit test charge\nagainst the direction of the component of the field, work would have to\nbe done"}, {"Chapter": "1", "sentence_range": "1621-1624", "Text": "The proof of this statement is simple If the field were not normal to the equipotential surface, it would\nhave non-zero component along the surface To move a unit test charge\nagainst the direction of the component of the field, work would have to\nbe done But this is in contradiction to the definition of an equipotential\nsurface: there is no potential difference between any two points on the\nsurface and no work is required to move a test charge on the surface"}, {"Chapter": "1", "sentence_range": "1622-1625", "Text": "If the field were not normal to the equipotential surface, it would\nhave non-zero component along the surface To move a unit test charge\nagainst the direction of the component of the field, work would have to\nbe done But this is in contradiction to the definition of an equipotential\nsurface: there is no potential difference between any two points on the\nsurface and no work is required to move a test charge on the surface The electric field must, therefore, be normal to the equipotential surface\nat every point"}, {"Chapter": "1", "sentence_range": "1623-1626", "Text": "To move a unit test charge\nagainst the direction of the component of the field, work would have to\nbe done But this is in contradiction to the definition of an equipotential\nsurface: there is no potential difference between any two points on the\nsurface and no work is required to move a test charge on the surface The electric field must, therefore, be normal to the equipotential surface\nat every point Equipotential surfaces offer an alternative visual picture\nin addition to the picture of electric field lines around a charge\nconfiguration"}, {"Chapter": "1", "sentence_range": "1624-1627", "Text": "But this is in contradiction to the definition of an equipotential\nsurface: there is no potential difference between any two points on the\nsurface and no work is required to move a test charge on the surface The electric field must, therefore, be normal to the equipotential surface\nat every point Equipotential surfaces offer an alternative visual picture\nin addition to the picture of electric field lines around a charge\nconfiguration FIGURE 2"}, {"Chapter": "1", "sentence_range": "1625-1628", "Text": "The electric field must, therefore, be normal to the equipotential surface\nat every point Equipotential surfaces offer an alternative visual picture\nin addition to the picture of electric field lines around a charge\nconfiguration FIGURE 2 9 For a\nsingle charge q\n(a) equipotential\nsurfaces are\nspherical surfaces\ncentred at the\ncharge, and\n(b) electric field\nlines are radial,\nstarting from the\ncharge if q > 0"}, {"Chapter": "1", "sentence_range": "1626-1629", "Text": "Equipotential surfaces offer an alternative visual picture\nin addition to the picture of electric field lines around a charge\nconfiguration FIGURE 2 9 For a\nsingle charge q\n(a) equipotential\nsurfaces are\nspherical surfaces\ncentred at the\ncharge, and\n(b) electric field\nlines are radial,\nstarting from the\ncharge if q > 0 For a uniform electric field E, say, along the x-axis, the equipotential\nsurfaces are planes normal to the x-axis, i"}, {"Chapter": "1", "sentence_range": "1627-1630", "Text": "FIGURE 2 9 For a\nsingle charge q\n(a) equipotential\nsurfaces are\nspherical surfaces\ncentred at the\ncharge, and\n(b) electric field\nlines are radial,\nstarting from the\ncharge if q > 0 For a uniform electric field E, say, along the x-axis, the equipotential\nsurfaces are planes normal to the x-axis, i e"}, {"Chapter": "1", "sentence_range": "1628-1631", "Text": "9 For a\nsingle charge q\n(a) equipotential\nsurfaces are\nspherical surfaces\ncentred at the\ncharge, and\n(b) electric field\nlines are radial,\nstarting from the\ncharge if q > 0 For a uniform electric field E, say, along the x-axis, the equipotential\nsurfaces are planes normal to the x-axis, i e , planes parallel to the y-z\nplane (Fig"}, {"Chapter": "1", "sentence_range": "1629-1632", "Text": "For a uniform electric field E, say, along the x-axis, the equipotential\nsurfaces are planes normal to the x-axis, i e , planes parallel to the y-z\nplane (Fig 2"}, {"Chapter": "1", "sentence_range": "1630-1633", "Text": "e , planes parallel to the y-z\nplane (Fig 2 10)"}, {"Chapter": "1", "sentence_range": "1631-1634", "Text": ", planes parallel to the y-z\nplane (Fig 2 10) Equipotential surfaces for (a) a dipole and (b) two\nidentical positive charges  are shown in Fig"}, {"Chapter": "1", "sentence_range": "1632-1635", "Text": "2 10) Equipotential surfaces for (a) a dipole and (b) two\nidentical positive charges  are shown in Fig 2"}, {"Chapter": "1", "sentence_range": "1633-1636", "Text": "10) Equipotential surfaces for (a) a dipole and (b) two\nidentical positive charges  are shown in Fig 2 11"}, {"Chapter": "1", "sentence_range": "1634-1637", "Text": "Equipotential surfaces for (a) a dipole and (b) two\nidentical positive charges  are shown in Fig 2 11 FIGURE 2"}, {"Chapter": "1", "sentence_range": "1635-1638", "Text": "2 11 FIGURE 2 11 Some equipotential surfaces for (a) a dipole,\n(b) two identical positive charges"}, {"Chapter": "1", "sentence_range": "1636-1639", "Text": "11 FIGURE 2 11 Some equipotential surfaces for (a) a dipole,\n(b) two identical positive charges Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n55\n2"}, {"Chapter": "1", "sentence_range": "1637-1640", "Text": "FIGURE 2 11 Some equipotential surfaces for (a) a dipole,\n(b) two identical positive charges Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n55\n2 6"}, {"Chapter": "1", "sentence_range": "1638-1641", "Text": "11 Some equipotential surfaces for (a) a dipole,\n(b) two identical positive charges Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n55\n2 6 1  Relation between field and potential\nConsider two closely spaced equipotential surfaces A and B (Fig"}, {"Chapter": "1", "sentence_range": "1639-1642", "Text": "Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n55\n2 6 1  Relation between field and potential\nConsider two closely spaced equipotential surfaces A and B (Fig 2"}, {"Chapter": "1", "sentence_range": "1640-1643", "Text": "6 1  Relation between field and potential\nConsider two closely spaced equipotential surfaces A and B (Fig 2 12)\nwith potential values V and V + d V, where d V is the change in V in the\ndirection of the electric field E"}, {"Chapter": "1", "sentence_range": "1641-1644", "Text": "1  Relation between field and potential\nConsider two closely spaced equipotential surfaces A and B (Fig 2 12)\nwith potential values V and V + d V, where d V is the change in V in the\ndirection of the electric field E Let P be a point on the\nsurface B"}, {"Chapter": "1", "sentence_range": "1642-1645", "Text": "2 12)\nwith potential values V and V + d V, where d V is the change in V in the\ndirection of the electric field E Let P be a point on the\nsurface B d l is the perpendicular distance of the\nsurface A from P"}, {"Chapter": "1", "sentence_range": "1643-1646", "Text": "12)\nwith potential values V and V + d V, where d V is the change in V in the\ndirection of the electric field E Let P be a point on the\nsurface B d l is the perpendicular distance of the\nsurface A from P Imagine that a unit positive charge\nis moved along this perpendicular from the surface B\nto surface A against the electric field"}, {"Chapter": "1", "sentence_range": "1644-1647", "Text": "Let P be a point on the\nsurface B d l is the perpendicular distance of the\nsurface A from P Imagine that a unit positive charge\nis moved along this perpendicular from the surface B\nto surface A against the electric field The work done\nin this process is |E|d l"}, {"Chapter": "1", "sentence_range": "1645-1648", "Text": "d l is the perpendicular distance of the\nsurface A from P Imagine that a unit positive charge\nis moved along this perpendicular from the surface B\nto surface A against the electric field The work done\nin this process is |E|d l This work equals the potential difference\nVA\u2013VB"}, {"Chapter": "1", "sentence_range": "1646-1649", "Text": "Imagine that a unit positive charge\nis moved along this perpendicular from the surface B\nto surface A against the electric field The work done\nin this process is |E|d l This work equals the potential difference\nVA\u2013VB Thus,\n|E|d l = V \u2013 (V + dV)= \u2013 dV\ni"}, {"Chapter": "1", "sentence_range": "1647-1650", "Text": "The work done\nin this process is |E|d l This work equals the potential difference\nVA\u2013VB Thus,\n|E|d l = V \u2013 (V + dV)= \u2013 dV\ni e"}, {"Chapter": "1", "sentence_range": "1648-1651", "Text": "This work equals the potential difference\nVA\u2013VB Thus,\n|E|d l = V \u2013 (V + dV)= \u2013 dV\ni e , |E|= \u2212 \u03b4\n\u03b4\nV\nl\n(2"}, {"Chapter": "1", "sentence_range": "1649-1652", "Text": "Thus,\n|E|d l = V \u2013 (V + dV)= \u2013 dV\ni e , |E|= \u2212 \u03b4\n\u03b4\nV\nl\n(2 20)\nSince dV is negative, dV = \u2013 |dV|"}, {"Chapter": "1", "sentence_range": "1650-1653", "Text": "e , |E|= \u2212 \u03b4\n\u03b4\nV\nl\n(2 20)\nSince dV is negative, dV = \u2013 |dV| we can rewrite\nEq (2"}, {"Chapter": "1", "sentence_range": "1651-1654", "Text": ", |E|= \u2212 \u03b4\n\u03b4\nV\nl\n(2 20)\nSince dV is negative, dV = \u2013 |dV| we can rewrite\nEq (2 20) as\nE = \u2212\n= +\n\u03b4\u03b4\n\u03b4\u03b4\nlV\nlV\n(2"}, {"Chapter": "1", "sentence_range": "1652-1655", "Text": "20)\nSince dV is negative, dV = \u2013 |dV| we can rewrite\nEq (2 20) as\nE = \u2212\n= +\n\u03b4\u03b4\n\u03b4\u03b4\nlV\nlV\n(2 21)\nWe thus arrive at two important conclusions concerning the relation\nbetween electric field and potential:\n(i)\nElectric field is in the direction in which the potential decreases\nsteepest"}, {"Chapter": "1", "sentence_range": "1653-1656", "Text": "we can rewrite\nEq (2 20) as\nE = \u2212\n= +\n\u03b4\u03b4\n\u03b4\u03b4\nlV\nlV\n(2 21)\nWe thus arrive at two important conclusions concerning the relation\nbetween electric field and potential:\n(i)\nElectric field is in the direction in which the potential decreases\nsteepest (ii) Its magnitude is given by the change in the magnitude of potential\nper unit displacement normal to the equipotential surface at the point"}, {"Chapter": "1", "sentence_range": "1654-1657", "Text": "20) as\nE = \u2212\n= +\n\u03b4\u03b4\n\u03b4\u03b4\nlV\nlV\n(2 21)\nWe thus arrive at two important conclusions concerning the relation\nbetween electric field and potential:\n(i)\nElectric field is in the direction in which the potential decreases\nsteepest (ii) Its magnitude is given by the change in the magnitude of potential\nper unit displacement normal to the equipotential surface at the point 2"}, {"Chapter": "1", "sentence_range": "1655-1658", "Text": "21)\nWe thus arrive at two important conclusions concerning the relation\nbetween electric field and potential:\n(i)\nElectric field is in the direction in which the potential decreases\nsteepest (ii) Its magnitude is given by the change in the magnitude of potential\nper unit displacement normal to the equipotential surface at the point 2 7  POTENTIAL ENERGY OF A SYSTEM OF CHARGES\nConsider first the simple case of two charges q1and q2 with position vector\nr1 and r2 relative to some origin"}, {"Chapter": "1", "sentence_range": "1656-1659", "Text": "(ii) Its magnitude is given by the change in the magnitude of potential\nper unit displacement normal to the equipotential surface at the point 2 7  POTENTIAL ENERGY OF A SYSTEM OF CHARGES\nConsider first the simple case of two charges q1and q2 with position vector\nr1 and r2 relative to some origin Let us calculate the work done\n(externally) in building up this configuration"}, {"Chapter": "1", "sentence_range": "1657-1660", "Text": "2 7  POTENTIAL ENERGY OF A SYSTEM OF CHARGES\nConsider first the simple case of two charges q1and q2 with position vector\nr1 and r2 relative to some origin Let us calculate the work done\n(externally) in building up this configuration This means that we consider\nthe charges q1 and q2 initially at infinity and determine the work done by\nan external agency to bring the charges to the given locations"}, {"Chapter": "1", "sentence_range": "1658-1661", "Text": "7  POTENTIAL ENERGY OF A SYSTEM OF CHARGES\nConsider first the simple case of two charges q1and q2 with position vector\nr1 and r2 relative to some origin Let us calculate the work done\n(externally) in building up this configuration This means that we consider\nthe charges q1 and q2 initially at infinity and determine the work done by\nan external agency to bring the charges to the given locations Suppose,\nfirst the charge q1 is brought from infinity to the point r1"}, {"Chapter": "1", "sentence_range": "1659-1662", "Text": "Let us calculate the work done\n(externally) in building up this configuration This means that we consider\nthe charges q1 and q2 initially at infinity and determine the work done by\nan external agency to bring the charges to the given locations Suppose,\nfirst the charge q1 is brought from infinity to the point r1 There is no\nexternal field against which work needs to be done, so work done in\nbringing q1 from infinity to r1 is zero"}, {"Chapter": "1", "sentence_range": "1660-1663", "Text": "This means that we consider\nthe charges q1 and q2 initially at infinity and determine the work done by\nan external agency to bring the charges to the given locations Suppose,\nfirst the charge q1 is brought from infinity to the point r1 There is no\nexternal field against which work needs to be done, so work done in\nbringing q1 from infinity to r1 is zero This charge produces a potential in\nspace given by\nV\nrq\n1\n0\n1\n1\n=41\n\u03c0\u03b5\nP\nwhere r1P is the distance of a point P in space from the location of q1"}, {"Chapter": "1", "sentence_range": "1661-1664", "Text": "Suppose,\nfirst the charge q1 is brought from infinity to the point r1 There is no\nexternal field against which work needs to be done, so work done in\nbringing q1 from infinity to r1 is zero This charge produces a potential in\nspace given by\nV\nrq\n1\n0\n1\n1\n=41\n\u03c0\u03b5\nP\nwhere r1P is the distance of a point P in space from the location of q1 From the definition of potential, work done in bringing charge q2 from\ninfinity to the point r2 is q2 times the potential at r2 due to q1:\nwork done on q2 = \n41\n0\n1\n2\n12\n\u03c0\u03b5\nq q\nr\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "1662-1665", "Text": "There is no\nexternal field against which work needs to be done, so work done in\nbringing q1 from infinity to r1 is zero This charge produces a potential in\nspace given by\nV\nrq\n1\n0\n1\n1\n=41\n\u03c0\u03b5\nP\nwhere r1P is the distance of a point P in space from the location of q1 From the definition of potential, work done in bringing charge q2 from\ninfinity to the point r2 is q2 times the potential at r2 due to q1:\nwork done on q2 = \n41\n0\n1\n2\n12\n\u03c0\u03b5\nq q\nr\nFIGURE 2 12 From the\npotential to the field"}, {"Chapter": "1", "sentence_range": "1663-1666", "Text": "This charge produces a potential in\nspace given by\nV\nrq\n1\n0\n1\n1\n=41\n\u03c0\u03b5\nP\nwhere r1P is the distance of a point P in space from the location of q1 From the definition of potential, work done in bringing charge q2 from\ninfinity to the point r2 is q2 times the potential at r2 due to q1:\nwork done on q2 = \n41\n0\n1\n2\n12\n\u03c0\u03b5\nq q\nr\nFIGURE 2 12 From the\npotential to the field Rationalised 2023-24\nPhysics\n56\nwhere r12 is the distance between points 1 and 2"}, {"Chapter": "1", "sentence_range": "1664-1667", "Text": "From the definition of potential, work done in bringing charge q2 from\ninfinity to the point r2 is q2 times the potential at r2 due to q1:\nwork done on q2 = \n41\n0\n1\n2\n12\n\u03c0\u03b5\nq q\nr\nFIGURE 2 12 From the\npotential to the field Rationalised 2023-24\nPhysics\n56\nwhere r12 is the distance between points 1 and 2 Since electrostatic force is conservative, this work gets\nstored in the form of potential energy of the system"}, {"Chapter": "1", "sentence_range": "1665-1668", "Text": "12 From the\npotential to the field Rationalised 2023-24\nPhysics\n56\nwhere r12 is the distance between points 1 and 2 Since electrostatic force is conservative, this work gets\nstored in the form of potential energy of the system Thus,\nthe potential energy of a system of two charges q1 and q2 is\nU\nrq q\n=\n41\n0\n1\n2\n12\n\u03c0\u03b5\n(2"}, {"Chapter": "1", "sentence_range": "1666-1669", "Text": "Rationalised 2023-24\nPhysics\n56\nwhere r12 is the distance between points 1 and 2 Since electrostatic force is conservative, this work gets\nstored in the form of potential energy of the system Thus,\nthe potential energy of a system of two charges q1 and q2 is\nU\nrq q\n=\n41\n0\n1\n2\n12\n\u03c0\u03b5\n(2 22)\nObviously, if q2 was brought first to its present location and\nq1 brought later, the potential energy U would be the same"}, {"Chapter": "1", "sentence_range": "1667-1670", "Text": "Since electrostatic force is conservative, this work gets\nstored in the form of potential energy of the system Thus,\nthe potential energy of a system of two charges q1 and q2 is\nU\nrq q\n=\n41\n0\n1\n2\n12\n\u03c0\u03b5\n(2 22)\nObviously, if q2 was brought first to its present location and\nq1 brought later, the potential energy U would be the same More generally, the potential energy expression,\nEq"}, {"Chapter": "1", "sentence_range": "1668-1671", "Text": "Thus,\nthe potential energy of a system of two charges q1 and q2 is\nU\nrq q\n=\n41\n0\n1\n2\n12\n\u03c0\u03b5\n(2 22)\nObviously, if q2 was brought first to its present location and\nq1 brought later, the potential energy U would be the same More generally, the potential energy expression,\nEq (2"}, {"Chapter": "1", "sentence_range": "1669-1672", "Text": "22)\nObviously, if q2 was brought first to its present location and\nq1 brought later, the potential energy U would be the same More generally, the potential energy expression,\nEq (2 22), is unaltered whatever way the charges are brought to the specified\nlocations, because of path-independence of work for electrostatic force"}, {"Chapter": "1", "sentence_range": "1670-1673", "Text": "More generally, the potential energy expression,\nEq (2 22), is unaltered whatever way the charges are brought to the specified\nlocations, because of path-independence of work for electrostatic force Equation (2"}, {"Chapter": "1", "sentence_range": "1671-1674", "Text": "(2 22), is unaltered whatever way the charges are brought to the specified\nlocations, because of path-independence of work for electrostatic force Equation (2 22) is true for any sign of  q1and q2"}, {"Chapter": "1", "sentence_range": "1672-1675", "Text": "22), is unaltered whatever way the charges are brought to the specified\nlocations, because of path-independence of work for electrostatic force Equation (2 22) is true for any sign of  q1and q2 If q1q2 > 0, potential\nenergy is positive"}, {"Chapter": "1", "sentence_range": "1673-1676", "Text": "Equation (2 22) is true for any sign of  q1and q2 If q1q2 > 0, potential\nenergy is positive This is as expected, since for like charges (q1q2 > 0),\nelectrostatic force is repulsive and a positive amount of work is needed to\nbe done against this force to bring the charges from infinity to a finite\ndistance apart"}, {"Chapter": "1", "sentence_range": "1674-1677", "Text": "22) is true for any sign of  q1and q2 If q1q2 > 0, potential\nenergy is positive This is as expected, since for like charges (q1q2 > 0),\nelectrostatic force is repulsive and a positive amount of work is needed to\nbe done against this force to bring the charges from infinity to a finite\ndistance apart For unlike charges (q1 q2 < 0), the electrostatic force is\nattractive"}, {"Chapter": "1", "sentence_range": "1675-1678", "Text": "If q1q2 > 0, potential\nenergy is positive This is as expected, since for like charges (q1q2 > 0),\nelectrostatic force is repulsive and a positive amount of work is needed to\nbe done against this force to bring the charges from infinity to a finite\ndistance apart For unlike charges (q1 q2 < 0), the electrostatic force is\nattractive In that case, a positive amount of work is needed against this\nforce to take the charges from the given location to infinity"}, {"Chapter": "1", "sentence_range": "1676-1679", "Text": "This is as expected, since for like charges (q1q2 > 0),\nelectrostatic force is repulsive and a positive amount of work is needed to\nbe done against this force to bring the charges from infinity to a finite\ndistance apart For unlike charges (q1 q2 < 0), the electrostatic force is\nattractive In that case, a positive amount of work is needed against this\nforce to take the charges from the given location to infinity In other words,\na negative amount of work is needed for the reverse path (from infinity to\nthe present locations), so the potential energy is negative"}, {"Chapter": "1", "sentence_range": "1677-1680", "Text": "For unlike charges (q1 q2 < 0), the electrostatic force is\nattractive In that case, a positive amount of work is needed against this\nforce to take the charges from the given location to infinity In other words,\na negative amount of work is needed for the reverse path (from infinity to\nthe present locations), so the potential energy is negative Equation (2"}, {"Chapter": "1", "sentence_range": "1678-1681", "Text": "In that case, a positive amount of work is needed against this\nforce to take the charges from the given location to infinity In other words,\na negative amount of work is needed for the reverse path (from infinity to\nthe present locations), so the potential energy is negative Equation (2 22) is easily generalised for a system of any number of\npoint charges"}, {"Chapter": "1", "sentence_range": "1679-1682", "Text": "In other words,\na negative amount of work is needed for the reverse path (from infinity to\nthe present locations), so the potential energy is negative Equation (2 22) is easily generalised for a system of any number of\npoint charges Let us calculate the potential energy of a system of three\ncharges q1, q2 and q3 located at r1, r2, r3, respectively"}, {"Chapter": "1", "sentence_range": "1680-1683", "Text": "Equation (2 22) is easily generalised for a system of any number of\npoint charges Let us calculate the potential energy of a system of three\ncharges q1, q2 and q3 located at r1, r2, r3, respectively To bring q1 first\nfrom infinity to r1, no work is required"}, {"Chapter": "1", "sentence_range": "1681-1684", "Text": "22) is easily generalised for a system of any number of\npoint charges Let us calculate the potential energy of a system of three\ncharges q1, q2 and q3 located at r1, r2, r3, respectively To bring q1 first\nfrom infinity to r1, no work is required Next we bring q2 from infinity to\nr2"}, {"Chapter": "1", "sentence_range": "1682-1685", "Text": "Let us calculate the potential energy of a system of three\ncharges q1, q2 and q3 located at r1, r2, r3, respectively To bring q1 first\nfrom infinity to r1, no work is required Next we bring q2 from infinity to\nr2 As before, work done in this step is\n1\n2\n2\n1\n2\n0\n12\n1\n(\n)\n4\nq q\nq V\nr\n\u03b5\n=\n\u03c0\nr\n(2"}, {"Chapter": "1", "sentence_range": "1683-1686", "Text": "To bring q1 first\nfrom infinity to r1, no work is required Next we bring q2 from infinity to\nr2 As before, work done in this step is\n1\n2\n2\n1\n2\n0\n12\n1\n(\n)\n4\nq q\nq V\nr\n\u03b5\n=\n\u03c0\nr\n(2 23)\nThe charges q1 and q2 produce a potential, which at any point P is\ngiven by\nV\nrq\nrq\n1 2\n0\n1\n1\n2\n2\n41\n,\n=\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n\u03c0\u03b5\nP\nP\n(2"}, {"Chapter": "1", "sentence_range": "1684-1687", "Text": "Next we bring q2 from infinity to\nr2 As before, work done in this step is\n1\n2\n2\n1\n2\n0\n12\n1\n(\n)\n4\nq q\nq V\nr\n\u03b5\n=\n\u03c0\nr\n(2 23)\nThe charges q1 and q2 produce a potential, which at any point P is\ngiven by\nV\nrq\nrq\n1 2\n0\n1\n1\n2\n2\n41\n,\n=\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n\u03c0\u03b5\nP\nP\n(2 24)\nWork done next in bringing q3 from infinity to the point r3 is  q3 times\nV1, 2 at r3\nq V\nrq q\nq q\nr\n3\n1 2\n3\n0\n1\n3\n13\n2\n3\n23\n41\n, (\nr)\n=\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n\u03c0\u03b5\n(2"}, {"Chapter": "1", "sentence_range": "1685-1688", "Text": "As before, work done in this step is\n1\n2\n2\n1\n2\n0\n12\n1\n(\n)\n4\nq q\nq V\nr\n\u03b5\n=\n\u03c0\nr\n(2 23)\nThe charges q1 and q2 produce a potential, which at any point P is\ngiven by\nV\nrq\nrq\n1 2\n0\n1\n1\n2\n2\n41\n,\n=\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n\u03c0\u03b5\nP\nP\n(2 24)\nWork done next in bringing q3 from infinity to the point r3 is  q3 times\nV1, 2 at r3\nq V\nrq q\nq q\nr\n3\n1 2\n3\n0\n1\n3\n13\n2\n3\n23\n41\n, (\nr)\n=\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n\u03c0\u03b5\n(2 25)\nThe total work done in assembling the charges\nat the given locations is obtained by adding the work\ndone in different steps [Eq"}, {"Chapter": "1", "sentence_range": "1686-1689", "Text": "23)\nThe charges q1 and q2 produce a potential, which at any point P is\ngiven by\nV\nrq\nrq\n1 2\n0\n1\n1\n2\n2\n41\n,\n=\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n\u03c0\u03b5\nP\nP\n(2 24)\nWork done next in bringing q3 from infinity to the point r3 is  q3 times\nV1, 2 at r3\nq V\nrq q\nq q\nr\n3\n1 2\n3\n0\n1\n3\n13\n2\n3\n23\n41\n, (\nr)\n=\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n\u03c0\u03b5\n(2 25)\nThe total work done in assembling the charges\nat the given locations is obtained by adding the work\ndone in different steps [Eq (2"}, {"Chapter": "1", "sentence_range": "1687-1690", "Text": "24)\nWork done next in bringing q3 from infinity to the point r3 is  q3 times\nV1, 2 at r3\nq V\nrq q\nq q\nr\n3\n1 2\n3\n0\n1\n3\n13\n2\n3\n23\n41\n, (\nr)\n=\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n\u03c0\u03b5\n(2 25)\nThe total work done in assembling the charges\nat the given locations is obtained by adding the work\ndone in different steps [Eq (2 23) and Eq"}, {"Chapter": "1", "sentence_range": "1688-1691", "Text": "25)\nThe total work done in assembling the charges\nat the given locations is obtained by adding the work\ndone in different steps [Eq (2 23) and Eq (2"}, {"Chapter": "1", "sentence_range": "1689-1692", "Text": "(2 23) and Eq (2 25)],\nU\nrq q\nrq q\nq q\nr\n=\n+\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n41\n0\n1\n2\n12\n1\n3\n13\n2\n3\n23\n\u03c0\u03b5\n(2"}, {"Chapter": "1", "sentence_range": "1690-1693", "Text": "23) and Eq (2 25)],\nU\nrq q\nrq q\nq q\nr\n=\n+\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n41\n0\n1\n2\n12\n1\n3\n13\n2\n3\n23\n\u03c0\u03b5\n(2 26)\nAgain, because of the conservative nature of the\nelectrostatic force (or equivalently, the path\nindependence of work done), the final expression for\nU, Eq"}, {"Chapter": "1", "sentence_range": "1691-1694", "Text": "(2 25)],\nU\nrq q\nrq q\nq q\nr\n=\n+\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n41\n0\n1\n2\n12\n1\n3\n13\n2\n3\n23\n\u03c0\u03b5\n(2 26)\nAgain, because of the conservative nature of the\nelectrostatic force (or equivalently, the path\nindependence of work done), the final expression for\nU, Eq (2"}, {"Chapter": "1", "sentence_range": "1692-1695", "Text": "25)],\nU\nrq q\nrq q\nq q\nr\n=\n+\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n41\n0\n1\n2\n12\n1\n3\n13\n2\n3\n23\n\u03c0\u03b5\n(2 26)\nAgain, because of the conservative nature of the\nelectrostatic force (or equivalently, the path\nindependence of work done), the final expression for\nU, Eq (2 26), is independent of the manner in which\nthe configuration is assembled"}, {"Chapter": "1", "sentence_range": "1693-1696", "Text": "26)\nAgain, because of the conservative nature of the\nelectrostatic force (or equivalently, the path\nindependence of work done), the final expression for\nU, Eq (2 26), is independent of the manner in which\nthe configuration is assembled The potential energy\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "1694-1697", "Text": "(2 26), is independent of the manner in which\nthe configuration is assembled The potential energy\nFIGURE 2 13 Potential energy of a\nsystem of charges q1 and q2 is\ndirectly proportional to the product\nof charges and inversely to the\ndistance between them"}, {"Chapter": "1", "sentence_range": "1695-1698", "Text": "26), is independent of the manner in which\nthe configuration is assembled The potential energy\nFIGURE 2 13 Potential energy of a\nsystem of charges q1 and q2 is\ndirectly proportional to the product\nof charges and inversely to the\ndistance between them FIGURE 2"}, {"Chapter": "1", "sentence_range": "1696-1699", "Text": "The potential energy\nFIGURE 2 13 Potential energy of a\nsystem of charges q1 and q2 is\ndirectly proportional to the product\nof charges and inversely to the\ndistance between them FIGURE 2 14 Potential energy of a\nsystem of three charges is given by\nEq"}, {"Chapter": "1", "sentence_range": "1697-1700", "Text": "13 Potential energy of a\nsystem of charges q1 and q2 is\ndirectly proportional to the product\nof charges and inversely to the\ndistance between them FIGURE 2 14 Potential energy of a\nsystem of three charges is given by\nEq (2"}, {"Chapter": "1", "sentence_range": "1698-1701", "Text": "FIGURE 2 14 Potential energy of a\nsystem of three charges is given by\nEq (2 26), with the notation given\nin the figure"}, {"Chapter": "1", "sentence_range": "1699-1702", "Text": "14 Potential energy of a\nsystem of three charges is given by\nEq (2 26), with the notation given\nin the figure Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n57\n EXAMPLE 2"}, {"Chapter": "1", "sentence_range": "1700-1703", "Text": "(2 26), with the notation given\nin the figure Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n57\n EXAMPLE 2 4\nis characteristic of the present state of configuration, and not the way\nthe state is achieved"}, {"Chapter": "1", "sentence_range": "1701-1704", "Text": "26), with the notation given\nin the figure Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n57\n EXAMPLE 2 4\nis characteristic of the present state of configuration, and not the way\nthe state is achieved Example 2"}, {"Chapter": "1", "sentence_range": "1702-1705", "Text": "Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n57\n EXAMPLE 2 4\nis characteristic of the present state of configuration, and not the way\nthe state is achieved Example 2 4 Four charges are arranged at the corners of a square\nABCD of side d, as shown in Fig"}, {"Chapter": "1", "sentence_range": "1703-1706", "Text": "4\nis characteristic of the present state of configuration, and not the way\nthe state is achieved Example 2 4 Four charges are arranged at the corners of a square\nABCD of side d, as shown in Fig 2"}, {"Chapter": "1", "sentence_range": "1704-1707", "Text": "Example 2 4 Four charges are arranged at the corners of a square\nABCD of side d, as shown in Fig 2 15"}, {"Chapter": "1", "sentence_range": "1705-1708", "Text": "4 Four charges are arranged at the corners of a square\nABCD of side d, as shown in Fig 2 15 (a) Find the work required to\nput together this arrangement"}, {"Chapter": "1", "sentence_range": "1706-1709", "Text": "2 15 (a) Find the work required to\nput together this arrangement (b) A charge q0 is brought to the centre\nE of the square, the four charges being held fixed at its corners"}, {"Chapter": "1", "sentence_range": "1707-1710", "Text": "15 (a) Find the work required to\nput together this arrangement (b) A charge q0 is brought to the centre\nE of the square, the four charges being held fixed at its corners How\nmuch extra work is needed to do this"}, {"Chapter": "1", "sentence_range": "1708-1711", "Text": "(a) Find the work required to\nput together this arrangement (b) A charge q0 is brought to the centre\nE of the square, the four charges being held fixed at its corners How\nmuch extra work is needed to do this FIGURE 2"}, {"Chapter": "1", "sentence_range": "1709-1712", "Text": "(b) A charge q0 is brought to the centre\nE of the square, the four charges being held fixed at its corners How\nmuch extra work is needed to do this FIGURE 2 15\nSolution\n(a) Since the work done depends on the final arrangement of the\ncharges, and not on how they are put together, we calculate work\nneeded for one way of putting the charges at A, B, C and D"}, {"Chapter": "1", "sentence_range": "1710-1713", "Text": "How\nmuch extra work is needed to do this FIGURE 2 15\nSolution\n(a) Since the work done depends on the final arrangement of the\ncharges, and not on how they are put together, we calculate work\nneeded for one way of putting the charges at A, B, C and D Suppose,\nfirst the charge +q is brought to A, and then the charges \u2013q, +q, and\n\u2013q are brought to B, C and D, respectively"}, {"Chapter": "1", "sentence_range": "1711-1714", "Text": "FIGURE 2 15\nSolution\n(a) Since the work done depends on the final arrangement of the\ncharges, and not on how they are put together, we calculate work\nneeded for one way of putting the charges at A, B, C and D Suppose,\nfirst the charge +q is brought to A, and then the charges \u2013q, +q, and\n\u2013q are brought to B, C and D, respectively The total work needed can\nbe calculated in steps:\n(i)\nWork needed to bring charge +q to A when no charge is present\nelsewhere: this is zero"}, {"Chapter": "1", "sentence_range": "1712-1715", "Text": "15\nSolution\n(a) Since the work done depends on the final arrangement of the\ncharges, and not on how they are put together, we calculate work\nneeded for one way of putting the charges at A, B, C and D Suppose,\nfirst the charge +q is brought to A, and then the charges \u2013q, +q, and\n\u2013q are brought to B, C and D, respectively The total work needed can\nbe calculated in steps:\n(i)\nWork needed to bring charge +q to A when no charge is present\nelsewhere: this is zero (ii) Work needed to bring \u2013q to B when +q is at A"}, {"Chapter": "1", "sentence_range": "1713-1716", "Text": "Suppose,\nfirst the charge +q is brought to A, and then the charges \u2013q, +q, and\n\u2013q are brought to B, C and D, respectively The total work needed can\nbe calculated in steps:\n(i)\nWork needed to bring charge +q to A when no charge is present\nelsewhere: this is zero (ii) Work needed to bring \u2013q to B when +q is at A This is given by\n(charge at B) \u00d7 (electrostatic potential at B due to charge +q at A)\n= \u2212\n\u00d7 \uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7 = \u2212\nq\nq\nd\nq\nd\n4\n4\n0\n2\n0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n(iii) Work needed to bring charge +q to C when +q is at A and \u2013q is at\nB"}, {"Chapter": "1", "sentence_range": "1714-1717", "Text": "The total work needed can\nbe calculated in steps:\n(i)\nWork needed to bring charge +q to A when no charge is present\nelsewhere: this is zero (ii) Work needed to bring \u2013q to B when +q is at A This is given by\n(charge at B) \u00d7 (electrostatic potential at B due to charge +q at A)\n= \u2212\n\u00d7 \uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7 = \u2212\nq\nq\nd\nq\nd\n4\n4\n0\n2\n0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n(iii) Work needed to bring charge +q to C when +q is at A and \u2013q is at\nB This is given by (charge at C) \u00d7 (potential at C due to charges\nat A and B)\n= +\n+\n+\n\u2212\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\nq\nq\nd\nq\nd\n4\n2\n4\n0\n0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n      =\n\u2212\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f8\uf8f7\uf8f6\nq\nd\n2\n0\n4\n1\n1\n2\n\u03c0\u03b5\n(iv) Work needed to bring \u2013q to D when +q at A,\u2013q at B, and +q at C"}, {"Chapter": "1", "sentence_range": "1715-1718", "Text": "(ii) Work needed to bring \u2013q to B when +q is at A This is given by\n(charge at B) \u00d7 (electrostatic potential at B due to charge +q at A)\n= \u2212\n\u00d7 \uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7 = \u2212\nq\nq\nd\nq\nd\n4\n4\n0\n2\n0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n(iii) Work needed to bring charge +q to C when +q is at A and \u2013q is at\nB This is given by (charge at C) \u00d7 (potential at C due to charges\nat A and B)\n= +\n+\n+\n\u2212\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\nq\nq\nd\nq\nd\n4\n2\n4\n0\n0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n      =\n\u2212\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f8\uf8f7\uf8f6\nq\nd\n2\n0\n4\n1\n1\n2\n\u03c0\u03b5\n(iv) Work needed to bring \u2013q to D when +q at A,\u2013q at B, and +q at C This is given by (charge at D) \u00d7 (potential at D due to charges at A,\nB and C)\n      = \u2212\n+\n+\n\u2212\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\nq\nq\nd\nq\nd\nq\nd\n4\n4\n2\n4\n0\n0\n0\n\u03c0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n\u03b5\n=\n\u2212\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nq\nd\n2\n0\n4\n2\n1\n2\n\u03c0\u03b5\nRationalised 2023-24\nPhysics\n58\n EXAMPLE 2"}, {"Chapter": "1", "sentence_range": "1716-1719", "Text": "This is given by\n(charge at B) \u00d7 (electrostatic potential at B due to charge +q at A)\n= \u2212\n\u00d7 \uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7 = \u2212\nq\nq\nd\nq\nd\n4\n4\n0\n2\n0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n(iii) Work needed to bring charge +q to C when +q is at A and \u2013q is at\nB This is given by (charge at C) \u00d7 (potential at C due to charges\nat A and B)\n= +\n+\n+\n\u2212\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\nq\nq\nd\nq\nd\n4\n2\n4\n0\n0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n      =\n\u2212\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f8\uf8f7\uf8f6\nq\nd\n2\n0\n4\n1\n1\n2\n\u03c0\u03b5\n(iv) Work needed to bring \u2013q to D when +q at A,\u2013q at B, and +q at C This is given by (charge at D) \u00d7 (potential at D due to charges at A,\nB and C)\n      = \u2212\n+\n+\n\u2212\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\nq\nq\nd\nq\nd\nq\nd\n4\n4\n2\n4\n0\n0\n0\n\u03c0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n\u03b5\n=\n\u2212\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nq\nd\n2\n0\n4\n2\n1\n2\n\u03c0\u03b5\nRationalised 2023-24\nPhysics\n58\n EXAMPLE 2 4\nAdd the work done in steps (i), (ii), (iii) and (iv)"}, {"Chapter": "1", "sentence_range": "1717-1720", "Text": "This is given by (charge at C) \u00d7 (potential at C due to charges\nat A and B)\n= +\n+\n+\n\u2212\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\nq\nq\nd\nq\nd\n4\n2\n4\n0\n0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n      =\n\u2212\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f8\uf8f7\uf8f6\nq\nd\n2\n0\n4\n1\n1\n2\n\u03c0\u03b5\n(iv) Work needed to bring \u2013q to D when +q at A,\u2013q at B, and +q at C This is given by (charge at D) \u00d7 (potential at D due to charges at A,\nB and C)\n      = \u2212\n+\n+\n\u2212\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\nq\nq\nd\nq\nd\nq\nd\n4\n4\n2\n4\n0\n0\n0\n\u03c0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n\u03b5\n=\n\u2212\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nq\nd\n2\n0\n4\n2\n1\n2\n\u03c0\u03b5\nRationalised 2023-24\nPhysics\n58\n EXAMPLE 2 4\nAdd the work done in steps (i), (ii), (iii) and (iv) The total work\nrequired is\n=\n\u2212\n+\n+\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f6\n\uf8f8\uf8f7 +\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8f2\uf8f1\n\uf8f3\n\uf8fd\uf8fc\n\uf8fe\nq\nd\n2\n0\n4\n0\n1\n1\n1\n2\n2\n1\n2\n\u03c0\u03b5\n( )\n( )\n =\n\u2212\n(\u2212\n)\nq\nd\n2\n0\n4\n4\n2\n\u03c0\u03b5\nThe work done depends only on the arrangement of the charges, and\nnot how they are assembled"}, {"Chapter": "1", "sentence_range": "1718-1721", "Text": "This is given by (charge at D) \u00d7 (potential at D due to charges at A,\nB and C)\n      = \u2212\n+\n+\n\u2212\n+\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\nq\nq\nd\nq\nd\nq\nd\n4\n4\n2\n4\n0\n0\n0\n\u03c0\n\u03c0\n\u03c0\n\u03b5\n\u03b5\n\u03b5\n=\n\u2212\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nq\nd\n2\n0\n4\n2\n1\n2\n\u03c0\u03b5\nRationalised 2023-24\nPhysics\n58\n EXAMPLE 2 4\nAdd the work done in steps (i), (ii), (iii) and (iv) The total work\nrequired is\n=\n\u2212\n+\n+\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f6\n\uf8f8\uf8f7 +\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8f2\uf8f1\n\uf8f3\n\uf8fd\uf8fc\n\uf8fe\nq\nd\n2\n0\n4\n0\n1\n1\n1\n2\n2\n1\n2\n\u03c0\u03b5\n( )\n( )\n =\n\u2212\n(\u2212\n)\nq\nd\n2\n0\n4\n4\n2\n\u03c0\u03b5\nThe work done depends only on the arrangement of the charges, and\nnot how they are assembled By definition, this is the total\nelectrostatic energy of the charges"}, {"Chapter": "1", "sentence_range": "1719-1722", "Text": "4\nAdd the work done in steps (i), (ii), (iii) and (iv) The total work\nrequired is\n=\n\u2212\n+\n+\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f6\n\uf8f8\uf8f7 +\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8f2\uf8f1\n\uf8f3\n\uf8fd\uf8fc\n\uf8fe\nq\nd\n2\n0\n4\n0\n1\n1\n1\n2\n2\n1\n2\n\u03c0\u03b5\n( )\n( )\n =\n\u2212\n(\u2212\n)\nq\nd\n2\n0\n4\n4\n2\n\u03c0\u03b5\nThe work done depends only on the arrangement of the charges, and\nnot how they are assembled By definition, this is the total\nelectrostatic energy of the charges (Students may try calculating same work/energy by taking charges\nin any other order they desire and convince themselves that the energy\nwill remain the same"}, {"Chapter": "1", "sentence_range": "1720-1723", "Text": "The total work\nrequired is\n=\n\u2212\n+\n+\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f6\n\uf8f8\uf8f7 +\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8f2\uf8f1\n\uf8f3\n\uf8fd\uf8fc\n\uf8fe\nq\nd\n2\n0\n4\n0\n1\n1\n1\n2\n2\n1\n2\n\u03c0\u03b5\n( )\n( )\n =\n\u2212\n(\u2212\n)\nq\nd\n2\n0\n4\n4\n2\n\u03c0\u03b5\nThe work done depends only on the arrangement of the charges, and\nnot how they are assembled By definition, this is the total\nelectrostatic energy of the charges (Students may try calculating same work/energy by taking charges\nin any other order they desire and convince themselves that the energy\nwill remain the same )\n(b) The extra work necessary to bring a charge q0 to the point E when\nthe four charges are at A, B, C and D is q0 \u00d7 (electrostatic potential at\nE due to the charges at A, B, C and D)"}, {"Chapter": "1", "sentence_range": "1721-1724", "Text": "By definition, this is the total\nelectrostatic energy of the charges (Students may try calculating same work/energy by taking charges\nin any other order they desire and convince themselves that the energy\nwill remain the same )\n(b) The extra work necessary to bring a charge q0 to the point E when\nthe four charges are at A, B, C and D is q0 \u00d7 (electrostatic potential at\nE due to the charges at A, B, C and D) The electrostatic potential at\nE is clearly zero since potential due to A and C is cancelled by that\ndue to B and D"}, {"Chapter": "1", "sentence_range": "1722-1725", "Text": "(Students may try calculating same work/energy by taking charges\nin any other order they desire and convince themselves that the energy\nwill remain the same )\n(b) The extra work necessary to bring a charge q0 to the point E when\nthe four charges are at A, B, C and D is q0 \u00d7 (electrostatic potential at\nE due to the charges at A, B, C and D) The electrostatic potential at\nE is clearly zero since potential due to A and C is cancelled by that\ndue to B and D Hence, no work is required to bring any charge to\npoint E"}, {"Chapter": "1", "sentence_range": "1723-1726", "Text": ")\n(b) The extra work necessary to bring a charge q0 to the point E when\nthe four charges are at A, B, C and D is q0 \u00d7 (electrostatic potential at\nE due to the charges at A, B, C and D) The electrostatic potential at\nE is clearly zero since potential due to A and C is cancelled by that\ndue to B and D Hence, no work is required to bring any charge to\npoint E 2"}, {"Chapter": "1", "sentence_range": "1724-1727", "Text": "The electrostatic potential at\nE is clearly zero since potential due to A and C is cancelled by that\ndue to B and D Hence, no work is required to bring any charge to\npoint E 2 8  POTENTIAL ENERGY IN AN EXTERNAL FIELD\n2"}, {"Chapter": "1", "sentence_range": "1725-1728", "Text": "Hence, no work is required to bring any charge to\npoint E 2 8  POTENTIAL ENERGY IN AN EXTERNAL FIELD\n2 8"}, {"Chapter": "1", "sentence_range": "1726-1729", "Text": "2 8  POTENTIAL ENERGY IN AN EXTERNAL FIELD\n2 8 1  Potential energy of a single charge\nIn Section 2"}, {"Chapter": "1", "sentence_range": "1727-1730", "Text": "8  POTENTIAL ENERGY IN AN EXTERNAL FIELD\n2 8 1  Potential energy of a single charge\nIn Section 2 7, the source of the electric field was specified \u2013 the charges\nand their locations - and the potential energy of the system of those charges\nwas  determined"}, {"Chapter": "1", "sentence_range": "1728-1731", "Text": "8 1  Potential energy of a single charge\nIn Section 2 7, the source of the electric field was specified \u2013 the charges\nand their locations - and the potential energy of the system of those charges\nwas  determined In this section, we ask a related but a distinct question"}, {"Chapter": "1", "sentence_range": "1729-1732", "Text": "1  Potential energy of a single charge\nIn Section 2 7, the source of the electric field was specified \u2013 the charges\nand their locations - and the potential energy of the system of those charges\nwas  determined In this section, we ask a related but a distinct question What is the potential energy of a charge q in a given field"}, {"Chapter": "1", "sentence_range": "1730-1733", "Text": "7, the source of the electric field was specified \u2013 the charges\nand their locations - and the potential energy of the system of those charges\nwas  determined In this section, we ask a related but a distinct question What is the potential energy of a charge q in a given field This question\nwas, in fact, the starting point that led us to the notion of the electrostatic\npotential (Sections 2"}, {"Chapter": "1", "sentence_range": "1731-1734", "Text": "In this section, we ask a related but a distinct question What is the potential energy of a charge q in a given field This question\nwas, in fact, the starting point that led us to the notion of the electrostatic\npotential (Sections 2 1 and 2"}, {"Chapter": "1", "sentence_range": "1732-1735", "Text": "What is the potential energy of a charge q in a given field This question\nwas, in fact, the starting point that led us to the notion of the electrostatic\npotential (Sections 2 1 and 2 2)"}, {"Chapter": "1", "sentence_range": "1733-1736", "Text": "This question\nwas, in fact, the starting point that led us to the notion of the electrostatic\npotential (Sections 2 1 and 2 2) But here we address this question again\nto clarify in what way it is different from the discussion in Section 2"}, {"Chapter": "1", "sentence_range": "1734-1737", "Text": "1 and 2 2) But here we address this question again\nto clarify in what way it is different from the discussion in Section 2 7"}, {"Chapter": "1", "sentence_range": "1735-1738", "Text": "2) But here we address this question again\nto clarify in what way it is different from the discussion in Section 2 7 The main difference is that we are now concerned with the potential\nenergy of a charge (or charges) in an external field"}, {"Chapter": "1", "sentence_range": "1736-1739", "Text": "But here we address this question again\nto clarify in what way it is different from the discussion in Section 2 7 The main difference is that we are now concerned with the potential\nenergy of a charge (or charges) in an external field The external field E is\nnot produced by the given charge(s) whose potential energy we wish to\ncalculate"}, {"Chapter": "1", "sentence_range": "1737-1740", "Text": "7 The main difference is that we are now concerned with the potential\nenergy of a charge (or charges) in an external field The external field E is\nnot produced by the given charge(s) whose potential energy we wish to\ncalculate E is produced by sources external to the given charge(s)"}, {"Chapter": "1", "sentence_range": "1738-1741", "Text": "The main difference is that we are now concerned with the potential\nenergy of a charge (or charges) in an external field The external field E is\nnot produced by the given charge(s) whose potential energy we wish to\ncalculate E is produced by sources external to the given charge(s) The\nexternal sources may be known, but often they are unknown or\nunspecified; what is specified is the electric field E or the electrostatic\npotential V due to the external sources"}, {"Chapter": "1", "sentence_range": "1739-1742", "Text": "The external field E is\nnot produced by the given charge(s) whose potential energy we wish to\ncalculate E is produced by sources external to the given charge(s) The\nexternal sources may be known, but often they are unknown or\nunspecified; what is specified is the electric field E or the electrostatic\npotential V due to the external sources We assume that the charge q\ndoes not significantly affect the sources producing the external field"}, {"Chapter": "1", "sentence_range": "1740-1743", "Text": "E is produced by sources external to the given charge(s) The\nexternal sources may be known, but often they are unknown or\nunspecified; what is specified is the electric field E or the electrostatic\npotential V due to the external sources We assume that the charge q\ndoes not significantly affect the sources producing the external field This\nis true if q is very small, or the external sources are held fixed by other\nunspecified forces"}, {"Chapter": "1", "sentence_range": "1741-1744", "Text": "The\nexternal sources may be known, but often they are unknown or\nunspecified; what is specified is the electric field E or the electrostatic\npotential V due to the external sources We assume that the charge q\ndoes not significantly affect the sources producing the external field This\nis true if q is very small, or the external sources are held fixed by other\nunspecified forces Even if q is finite, its influence on the external sources\nmay still be ignored in the situation when very strong sources far away\nat infinity produce a finite field E in the region of interest"}, {"Chapter": "1", "sentence_range": "1742-1745", "Text": "We assume that the charge q\ndoes not significantly affect the sources producing the external field This\nis true if q is very small, or the external sources are held fixed by other\nunspecified forces Even if q is finite, its influence on the external sources\nmay still be ignored in the situation when very strong sources far away\nat infinity produce a finite field E in the region of interest Note again that\nwe are interested in determining the potential energy of a given charge q\n(and later, a system of charges) in the external field; we are not interested\nin the potential energy of the sources producing the external electric field"}, {"Chapter": "1", "sentence_range": "1743-1746", "Text": "This\nis true if q is very small, or the external sources are held fixed by other\nunspecified forces Even if q is finite, its influence on the external sources\nmay still be ignored in the situation when very strong sources far away\nat infinity produce a finite field E in the region of interest Note again that\nwe are interested in determining the potential energy of a given charge q\n(and later, a system of charges) in the external field; we are not interested\nin the potential energy of the sources producing the external electric field The external electric field E and the corresponding external potential\nV may vary from point to point"}, {"Chapter": "1", "sentence_range": "1744-1747", "Text": "Even if q is finite, its influence on the external sources\nmay still be ignored in the situation when very strong sources far away\nat infinity produce a finite field E in the region of interest Note again that\nwe are interested in determining the potential energy of a given charge q\n(and later, a system of charges) in the external field; we are not interested\nin the potential energy of the sources producing the external electric field The external electric field E and the corresponding external potential\nV may vary from point to point By definition, V at a point P is the work\ndone in bringing a unit positive charge from infinity to the point P"}, {"Chapter": "1", "sentence_range": "1745-1748", "Text": "Note again that\nwe are interested in determining the potential energy of a given charge q\n(and later, a system of charges) in the external field; we are not interested\nin the potential energy of the sources producing the external electric field The external electric field E and the corresponding external potential\nV may vary from point to point By definition, V at a point P is the work\ndone in bringing a unit positive charge from infinity to the point P Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n59\n EXAMPLE 2"}, {"Chapter": "1", "sentence_range": "1746-1749", "Text": "The external electric field E and the corresponding external potential\nV may vary from point to point By definition, V at a point P is the work\ndone in bringing a unit positive charge from infinity to the point P Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n59\n EXAMPLE 2 5\n(We continue to take potential at infinity to be zero"}, {"Chapter": "1", "sentence_range": "1747-1750", "Text": "By definition, V at a point P is the work\ndone in bringing a unit positive charge from infinity to the point P Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n59\n EXAMPLE 2 5\n(We continue to take potential at infinity to be zero ) Thus, work done in\nbringing a charge q from infinity to the point P in the external field is qV"}, {"Chapter": "1", "sentence_range": "1748-1751", "Text": "Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n59\n EXAMPLE 2 5\n(We continue to take potential at infinity to be zero ) Thus, work done in\nbringing a charge q from infinity to the point P in the external field is qV This work is stored in the form of potential energy of q"}, {"Chapter": "1", "sentence_range": "1749-1752", "Text": "5\n(We continue to take potential at infinity to be zero ) Thus, work done in\nbringing a charge q from infinity to the point P in the external field is qV This work is stored in the form of potential energy of q If the point P has\nposition vector r relative to some origin, we can write:\nPotential energy of q at r in an external field\n= qV(r)\n(2"}, {"Chapter": "1", "sentence_range": "1750-1753", "Text": ") Thus, work done in\nbringing a charge q from infinity to the point P in the external field is qV This work is stored in the form of potential energy of q If the point P has\nposition vector r relative to some origin, we can write:\nPotential energy of q at r in an external field\n= qV(r)\n(2 27)\nwhere V(r) is the external potential at the point r"}, {"Chapter": "1", "sentence_range": "1751-1754", "Text": "This work is stored in the form of potential energy of q If the point P has\nposition vector r relative to some origin, we can write:\nPotential energy of q at r in an external field\n= qV(r)\n(2 27)\nwhere V(r) is the external potential at the point r Thus, if an electron with charge q = e = 1"}, {"Chapter": "1", "sentence_range": "1752-1755", "Text": "If the point P has\nposition vector r relative to some origin, we can write:\nPotential energy of q at r in an external field\n= qV(r)\n(2 27)\nwhere V(r) is the external potential at the point r Thus, if an electron with charge q = e = 1 6\u00d710\u201319 C is accelerated by\na potential difference of DV = 1 volt, it would gain energy of qDV = 1"}, {"Chapter": "1", "sentence_range": "1753-1756", "Text": "27)\nwhere V(r) is the external potential at the point r Thus, if an electron with charge q = e = 1 6\u00d710\u201319 C is accelerated by\na potential difference of DV = 1 volt, it would gain energy of qDV = 1 6 \u00d7\n10\u201319J"}, {"Chapter": "1", "sentence_range": "1754-1757", "Text": "Thus, if an electron with charge q = e = 1 6\u00d710\u201319 C is accelerated by\na potential difference of DV = 1 volt, it would gain energy of qDV = 1 6 \u00d7\n10\u201319J This unit of energy is defined as 1 electron volt or 1eV, i"}, {"Chapter": "1", "sentence_range": "1755-1758", "Text": "6\u00d710\u201319 C is accelerated by\na potential difference of DV = 1 volt, it would gain energy of qDV = 1 6 \u00d7\n10\u201319J This unit of energy is defined as 1 electron volt or 1eV, i e"}, {"Chapter": "1", "sentence_range": "1756-1759", "Text": "6 \u00d7\n10\u201319J This unit of energy is defined as 1 electron volt or 1eV, i e ,\n1 eV=1"}, {"Chapter": "1", "sentence_range": "1757-1760", "Text": "This unit of energy is defined as 1 electron volt or 1eV, i e ,\n1 eV=1 6 \u00d7 10\u201319J"}, {"Chapter": "1", "sentence_range": "1758-1761", "Text": "e ,\n1 eV=1 6 \u00d7 10\u201319J The units based on eV are most commonly used in\natomic, nuclear and particle physics, (1 keV = 103eV = 1"}, {"Chapter": "1", "sentence_range": "1759-1762", "Text": ",\n1 eV=1 6 \u00d7 10\u201319J The units based on eV are most commonly used in\natomic, nuclear and particle physics, (1 keV = 103eV = 1 6 \u00d7 10\u201316J, 1 MeV\n= 106eV = 1"}, {"Chapter": "1", "sentence_range": "1760-1763", "Text": "6 \u00d7 10\u201319J The units based on eV are most commonly used in\natomic, nuclear and particle physics, (1 keV = 103eV = 1 6 \u00d7 10\u201316J, 1 MeV\n= 106eV = 1 6 \u00d7 10\u201313J, 1 GeV = 109eV = 1"}, {"Chapter": "1", "sentence_range": "1761-1764", "Text": "The units based on eV are most commonly used in\natomic, nuclear and particle physics, (1 keV = 103eV = 1 6 \u00d7 10\u201316J, 1 MeV\n= 106eV = 1 6 \u00d7 10\u201313J, 1 GeV = 109eV = 1 6 \u00d7 10\u201310J and 1 TeV = 1012eV\n= 1"}, {"Chapter": "1", "sentence_range": "1762-1765", "Text": "6 \u00d7 10\u201316J, 1 MeV\n= 106eV = 1 6 \u00d7 10\u201313J, 1 GeV = 109eV = 1 6 \u00d7 10\u201310J and 1 TeV = 1012eV\n= 1 6 \u00d7 10\u20137J)"}, {"Chapter": "1", "sentence_range": "1763-1766", "Text": "6 \u00d7 10\u201313J, 1 GeV = 109eV = 1 6 \u00d7 10\u201310J and 1 TeV = 1012eV\n= 1 6 \u00d7 10\u20137J) [This has already been defined on Page 117, XI Physics\nPart I, Table 6"}, {"Chapter": "1", "sentence_range": "1764-1767", "Text": "6 \u00d7 10\u201310J and 1 TeV = 1012eV\n= 1 6 \u00d7 10\u20137J) [This has already been defined on Page 117, XI Physics\nPart I, Table 6 1"}, {"Chapter": "1", "sentence_range": "1765-1768", "Text": "6 \u00d7 10\u20137J) [This has already been defined on Page 117, XI Physics\nPart I, Table 6 1 ]\n2"}, {"Chapter": "1", "sentence_range": "1766-1769", "Text": "[This has already been defined on Page 117, XI Physics\nPart I, Table 6 1 ]\n2 8"}, {"Chapter": "1", "sentence_range": "1767-1770", "Text": "1 ]\n2 8 2\nPotential energy of a system of two charges in an\nexternal field\nNext, we ask: what is the potential energy of a system of two charges q1\nand q2 located at r1and r2, respectively, in an external field"}, {"Chapter": "1", "sentence_range": "1768-1771", "Text": "]\n2 8 2\nPotential energy of a system of two charges in an\nexternal field\nNext, we ask: what is the potential energy of a system of two charges q1\nand q2 located at r1and r2, respectively, in an external field First, we\ncalculate the work done in bringing the charge q1 from infinity to r1"}, {"Chapter": "1", "sentence_range": "1769-1772", "Text": "8 2\nPotential energy of a system of two charges in an\nexternal field\nNext, we ask: what is the potential energy of a system of two charges q1\nand q2 located at r1and r2, respectively, in an external field First, we\ncalculate the work done in bringing the charge q1 from infinity to r1 Work done in this step is q1 V(r1), using Eq"}, {"Chapter": "1", "sentence_range": "1770-1773", "Text": "2\nPotential energy of a system of two charges in an\nexternal field\nNext, we ask: what is the potential energy of a system of two charges q1\nand q2 located at r1and r2, respectively, in an external field First, we\ncalculate the work done in bringing the charge q1 from infinity to r1 Work done in this step is q1 V(r1), using Eq (2"}, {"Chapter": "1", "sentence_range": "1771-1774", "Text": "First, we\ncalculate the work done in bringing the charge q1 from infinity to r1 Work done in this step is q1 V(r1), using Eq (2 27)"}, {"Chapter": "1", "sentence_range": "1772-1775", "Text": "Work done in this step is q1 V(r1), using Eq (2 27) Next, we consider the\nwork done in bringing q2 to r2"}, {"Chapter": "1", "sentence_range": "1773-1776", "Text": "(2 27) Next, we consider the\nwork done in bringing q2 to r2 In this step, work is done not only against\nthe external field E but also against the field due to q1"}, {"Chapter": "1", "sentence_range": "1774-1777", "Text": "27) Next, we consider the\nwork done in bringing q2 to r2 In this step, work is done not only against\nthe external field E but also against the field due to q1 Work done on q2 against the external field\n= q2 V (r2)\nWork done on q2 against the field due to q1\n1\n2\n12\n4\no\nq q\n\u03b5r\n=\n\u03c0\nwhere  r12 is the distance between q1 and q2"}, {"Chapter": "1", "sentence_range": "1775-1778", "Text": "Next, we consider the\nwork done in bringing q2 to r2 In this step, work is done not only against\nthe external field E but also against the field due to q1 Work done on q2 against the external field\n= q2 V (r2)\nWork done on q2 against the field due to q1\n1\n2\n12\n4\no\nq q\n\u03b5r\n=\n\u03c0\nwhere  r12 is the distance between q1 and q2 We have made use of Eqs"}, {"Chapter": "1", "sentence_range": "1776-1779", "Text": "In this step, work is done not only against\nthe external field E but also against the field due to q1 Work done on q2 against the external field\n= q2 V (r2)\nWork done on q2 against the field due to q1\n1\n2\n12\n4\no\nq q\n\u03b5r\n=\n\u03c0\nwhere  r12 is the distance between q1 and q2 We have made use of Eqs (2"}, {"Chapter": "1", "sentence_range": "1777-1780", "Text": "Work done on q2 against the external field\n= q2 V (r2)\nWork done on q2 against the field due to q1\n1\n2\n12\n4\no\nq q\n\u03b5r\n=\n\u03c0\nwhere  r12 is the distance between q1 and q2 We have made use of Eqs (2 27) and (2"}, {"Chapter": "1", "sentence_range": "1778-1781", "Text": "We have made use of Eqs (2 27) and (2 22)"}, {"Chapter": "1", "sentence_range": "1779-1782", "Text": "(2 27) and (2 22) By the superposition principle for fields, we add up\nthe work done on q2 against the two fields (E and that due to q1):\nWork done in bringing q2 to r2\n1\n2\n2\n2\n12\n(\n)\n4\no\nq q\nq V\n\u03b5r\n=\n+\n\u03c0\nr\n(2"}, {"Chapter": "1", "sentence_range": "1780-1783", "Text": "27) and (2 22) By the superposition principle for fields, we add up\nthe work done on q2 against the two fields (E and that due to q1):\nWork done in bringing q2 to r2\n1\n2\n2\n2\n12\n(\n)\n4\no\nq q\nq V\n\u03b5r\n=\n+\n\u03c0\nr\n(2 28)\nThus,\n Potential energy of the system\n= the total work done in assembling the configuration\n1\n2\n1\n1\n2\n2\n0 12\n(\n)\n(\n)\n4\nq q\nq V\nq V\n\u03b5r\n=\n+\n+\n\u03c0\nr\nr\n (2"}, {"Chapter": "1", "sentence_range": "1781-1784", "Text": "22) By the superposition principle for fields, we add up\nthe work done on q2 against the two fields (E and that due to q1):\nWork done in bringing q2 to r2\n1\n2\n2\n2\n12\n(\n)\n4\no\nq q\nq V\n\u03b5r\n=\n+\n\u03c0\nr\n(2 28)\nThus,\n Potential energy of the system\n= the total work done in assembling the configuration\n1\n2\n1\n1\n2\n2\n0 12\n(\n)\n(\n)\n4\nq q\nq V\nq V\n\u03b5r\n=\n+\n+\n\u03c0\nr\nr\n (2 29)\nExample 2"}, {"Chapter": "1", "sentence_range": "1782-1785", "Text": "By the superposition principle for fields, we add up\nthe work done on q2 against the two fields (E and that due to q1):\nWork done in bringing q2 to r2\n1\n2\n2\n2\n12\n(\n)\n4\no\nq q\nq V\n\u03b5r\n=\n+\n\u03c0\nr\n(2 28)\nThus,\n Potential energy of the system\n= the total work done in assembling the configuration\n1\n2\n1\n1\n2\n2\n0 12\n(\n)\n(\n)\n4\nq q\nq V\nq V\n\u03b5r\n=\n+\n+\n\u03c0\nr\nr\n (2 29)\nExample 2 5\n(a) Determine the electrostatic potential energy of a system consisting\nof  two charges 7 mC and \u20132 mC (and with no external field) placed\nat (\u20139 cm, 0, 0) and (9 cm, 0, 0) respectively"}, {"Chapter": "1", "sentence_range": "1783-1786", "Text": "28)\nThus,\n Potential energy of the system\n= the total work done in assembling the configuration\n1\n2\n1\n1\n2\n2\n0 12\n(\n)\n(\n)\n4\nq q\nq V\nq V\n\u03b5r\n=\n+\n+\n\u03c0\nr\nr\n (2 29)\nExample 2 5\n(a) Determine the electrostatic potential energy of a system consisting\nof  two charges 7 mC and \u20132 mC (and with no external field) placed\nat (\u20139 cm, 0, 0) and (9 cm, 0, 0) respectively (b) How much work is required to separate the two charges infinitely\naway from each other"}, {"Chapter": "1", "sentence_range": "1784-1787", "Text": "29)\nExample 2 5\n(a) Determine the electrostatic potential energy of a system consisting\nof  two charges 7 mC and \u20132 mC (and with no external field) placed\nat (\u20139 cm, 0, 0) and (9 cm, 0, 0) respectively (b) How much work is required to separate the two charges infinitely\naway from each other Rationalised 2023-24\nPhysics\n60\n EXAMPLE 2"}, {"Chapter": "1", "sentence_range": "1785-1788", "Text": "5\n(a) Determine the electrostatic potential energy of a system consisting\nof  two charges 7 mC and \u20132 mC (and with no external field) placed\nat (\u20139 cm, 0, 0) and (9 cm, 0, 0) respectively (b) How much work is required to separate the two charges infinitely\naway from each other Rationalised 2023-24\nPhysics\n60\n EXAMPLE 2 5\n(c) Suppose that the same system of charges is now placed in an\nexternal electric field E = A (1/r 2); A = 9 \u00d7 105 NC\u20131 m2"}, {"Chapter": "1", "sentence_range": "1786-1789", "Text": "(b) How much work is required to separate the two charges infinitely\naway from each other Rationalised 2023-24\nPhysics\n60\n EXAMPLE 2 5\n(c) Suppose that the same system of charges is now placed in an\nexternal electric field E = A (1/r 2); A = 9 \u00d7 105 NC\u20131 m2 What would\nthe electrostatic energy of the configuration be"}, {"Chapter": "1", "sentence_range": "1787-1790", "Text": "Rationalised 2023-24\nPhysics\n60\n EXAMPLE 2 5\n(c) Suppose that the same system of charges is now placed in an\nexternal electric field E = A (1/r 2); A = 9 \u00d7 105 NC\u20131 m2 What would\nthe electrostatic energy of the configuration be Solution\n(a)\n12\n9\n1\n2\n0\n1\n7\n( 2)\n10\n9\n10\n4\n0"}, {"Chapter": "1", "sentence_range": "1788-1791", "Text": "5\n(c) Suppose that the same system of charges is now placed in an\nexternal electric field E = A (1/r 2); A = 9 \u00d7 105 NC\u20131 m2 What would\nthe electrostatic energy of the configuration be Solution\n(a)\n12\n9\n1\n2\n0\n1\n7\n( 2)\n10\n9\n10\n4\n0 18\nq q\nU\nr\n\u03b5\n\u2212\n\u00d7 \u2212\n\u00d7\n=\n=\n\u00d7\n\u00d7\n\u03c0\n= \u20130"}, {"Chapter": "1", "sentence_range": "1789-1792", "Text": "What would\nthe electrostatic energy of the configuration be Solution\n(a)\n12\n9\n1\n2\n0\n1\n7\n( 2)\n10\n9\n10\n4\n0 18\nq q\nU\nr\n\u03b5\n\u2212\n\u00d7 \u2212\n\u00d7\n=\n=\n\u00d7\n\u00d7\n\u03c0\n= \u20130 7 J"}, {"Chapter": "1", "sentence_range": "1790-1793", "Text": "Solution\n(a)\n12\n9\n1\n2\n0\n1\n7\n( 2)\n10\n9\n10\n4\n0 18\nq q\nU\nr\n\u03b5\n\u2212\n\u00d7 \u2212\n\u00d7\n=\n=\n\u00d7\n\u00d7\n\u03c0\n= \u20130 7 J (b) W = U2 \u2013 U1  = 0 \u2013 U = 0 \u2013 (\u20130"}, {"Chapter": "1", "sentence_range": "1791-1794", "Text": "18\nq q\nU\nr\n\u03b5\n\u2212\n\u00d7 \u2212\n\u00d7\n=\n=\n\u00d7\n\u00d7\n\u03c0\n= \u20130 7 J (b) W = U2 \u2013 U1  = 0 \u2013 U = 0 \u2013 (\u20130 7) = 0"}, {"Chapter": "1", "sentence_range": "1792-1795", "Text": "7 J (b) W = U2 \u2013 U1  = 0 \u2013 U = 0 \u2013 (\u20130 7) = 0 7 J"}, {"Chapter": "1", "sentence_range": "1793-1796", "Text": "(b) W = U2 \u2013 U1  = 0 \u2013 U = 0 \u2013 (\u20130 7) = 0 7 J (c) The mutual interaction energy of the two charges remains\nunchanged"}, {"Chapter": "1", "sentence_range": "1794-1797", "Text": "7) = 0 7 J (c) The mutual interaction energy of the two charges remains\nunchanged In addition, there is the energy of interaction of the\ntwo charges with the external electric field"}, {"Chapter": "1", "sentence_range": "1795-1798", "Text": "7 J (c) The mutual interaction energy of the two charges remains\nunchanged In addition, there is the energy of interaction of the\ntwo charges with the external electric field We find,\n( )\n(\n)\n1\n1\n2\n2\n7 C\n2 C\n0"}, {"Chapter": "1", "sentence_range": "1796-1799", "Text": "(c) The mutual interaction energy of the two charges remains\nunchanged In addition, there is the energy of interaction of the\ntwo charges with the external electric field We find,\n( )\n(\n)\n1\n1\n2\n2\n7 C\n2 C\n0 09m\n0"}, {"Chapter": "1", "sentence_range": "1797-1800", "Text": "In addition, there is the energy of interaction of the\ntwo charges with the external electric field We find,\n( )\n(\n)\n1\n1\n2\n2\n7 C\n2 C\n0 09m\n0 09m\nq V\nq V\nA\nA\n\u00b5\n\u2212 \u00b5\n+\n=\n+\nr\nr\nand the net electrostatic energy is\n( )\n(\n)\n1\n2\n1\n1\n2\n2\n0 12\n7 C\n2 C\n0"}, {"Chapter": "1", "sentence_range": "1798-1801", "Text": "We find,\n( )\n(\n)\n1\n1\n2\n2\n7 C\n2 C\n0 09m\n0 09m\nq V\nq V\nA\nA\n\u00b5\n\u2212 \u00b5\n+\n=\n+\nr\nr\nand the net electrostatic energy is\n( )\n(\n)\n1\n2\n1\n1\n2\n2\n0 12\n7 C\n2 C\n0 7 J\n4\n0"}, {"Chapter": "1", "sentence_range": "1799-1802", "Text": "09m\n0 09m\nq V\nq V\nA\nA\n\u00b5\n\u2212 \u00b5\n+\n=\n+\nr\nr\nand the net electrostatic energy is\n( )\n(\n)\n1\n2\n1\n1\n2\n2\n0 12\n7 C\n2 C\n0 7 J\n4\n0 09 m\n0"}, {"Chapter": "1", "sentence_range": "1800-1803", "Text": "09m\nq V\nq V\nA\nA\n\u00b5\n\u2212 \u00b5\n+\n=\n+\nr\nr\nand the net electrostatic energy is\n( )\n(\n)\n1\n2\n1\n1\n2\n2\n0 12\n7 C\n2 C\n0 7 J\n4\n0 09 m\n0 09 m\nq q\nq V\nq V\nA\nA\nr\n\u03b5\n\u00b5\n\u2212 \u00b5\n+\n+\n=\n+\n\u2212\n\u03c0\nr\nr\n         \n70\n20\n0"}, {"Chapter": "1", "sentence_range": "1801-1804", "Text": "7 J\n4\n0 09 m\n0 09 m\nq q\nq V\nq V\nA\nA\nr\n\u03b5\n\u00b5\n\u2212 \u00b5\n+\n+\n=\n+\n\u2212\n\u03c0\nr\nr\n         \n70\n20\n0 7\n49"}, {"Chapter": "1", "sentence_range": "1802-1805", "Text": "09 m\n0 09 m\nq q\nq V\nq V\nA\nA\nr\n\u03b5\n\u00b5\n\u2212 \u00b5\n+\n+\n=\n+\n\u2212\n\u03c0\nr\nr\n         \n70\n20\n0 7\n49 3 J\n=\n\u2212\n\u2212\n=\n2"}, {"Chapter": "1", "sentence_range": "1803-1806", "Text": "09 m\nq q\nq V\nq V\nA\nA\nr\n\u03b5\n\u00b5\n\u2212 \u00b5\n+\n+\n=\n+\n\u2212\n\u03c0\nr\nr\n         \n70\n20\n0 7\n49 3 J\n=\n\u2212\n\u2212\n=\n2 8"}, {"Chapter": "1", "sentence_range": "1804-1807", "Text": "7\n49 3 J\n=\n\u2212\n\u2212\n=\n2 8 3  Potential energy of a dipole in an external field\nConsider a dipole with charges q1 = +q and q2 = \u2013q placed in a uniform\nelectric field E, as shown in Fig"}, {"Chapter": "1", "sentence_range": "1805-1808", "Text": "3 J\n=\n\u2212\n\u2212\n=\n2 8 3  Potential energy of a dipole in an external field\nConsider a dipole with charges q1 = +q and q2 = \u2013q placed in a uniform\nelectric field E, as shown in Fig 2"}, {"Chapter": "1", "sentence_range": "1806-1809", "Text": "8 3  Potential energy of a dipole in an external field\nConsider a dipole with charges q1 = +q and q2 = \u2013q placed in a uniform\nelectric field E, as shown in Fig 2 16"}, {"Chapter": "1", "sentence_range": "1807-1810", "Text": "3  Potential energy of a dipole in an external field\nConsider a dipole with charges q1 = +q and q2 = \u2013q placed in a uniform\nelectric field E, as shown in Fig 2 16 As seen in the last chapter, in a uniform electric field,\nthe dipole experiences no net force; but experiences a\ntorque t t t t t given by\nt = \nt = \nt = \nt = \nt = p \u00d7 E\n(2"}, {"Chapter": "1", "sentence_range": "1808-1811", "Text": "2 16 As seen in the last chapter, in a uniform electric field,\nthe dipole experiences no net force; but experiences a\ntorque t t t t t given by\nt = \nt = \nt = \nt = \nt = p \u00d7 E\n(2 30)\nwhich will tend to rotate it (unless p is parallel or\nantiparallel to E)"}, {"Chapter": "1", "sentence_range": "1809-1812", "Text": "16 As seen in the last chapter, in a uniform electric field,\nthe dipole experiences no net force; but experiences a\ntorque t t t t t given by\nt = \nt = \nt = \nt = \nt = p \u00d7 E\n(2 30)\nwhich will tend to rotate it (unless p is parallel or\nantiparallel to E) Suppose an external torque tttttext is\napplied in such a manner that it just neutralises this\ntorque and rotates it in the plane of paper from angle q0\nto angle q1 at an infinitesimal angular speed and without\nangular acceleration"}, {"Chapter": "1", "sentence_range": "1810-1813", "Text": "As seen in the last chapter, in a uniform electric field,\nthe dipole experiences no net force; but experiences a\ntorque t t t t t given by\nt = \nt = \nt = \nt = \nt = p \u00d7 E\n(2 30)\nwhich will tend to rotate it (unless p is parallel or\nantiparallel to E) Suppose an external torque tttttext is\napplied in such a manner that it just neutralises this\ntorque and rotates it in the plane of paper from angle q0\nto angle q1 at an infinitesimal angular speed and without\nangular acceleration The amount of work done by the\nexternal torque will be given by\n(\n)\ncos\ncos\npE\n\u03b8\n\u03b8\n0\n1\n=\n\u2212\n(2"}, {"Chapter": "1", "sentence_range": "1811-1814", "Text": "30)\nwhich will tend to rotate it (unless p is parallel or\nantiparallel to E) Suppose an external torque tttttext is\napplied in such a manner that it just neutralises this\ntorque and rotates it in the plane of paper from angle q0\nto angle q1 at an infinitesimal angular speed and without\nangular acceleration The amount of work done by the\nexternal torque will be given by\n(\n)\ncos\ncos\npE\n\u03b8\n\u03b8\n0\n1\n=\n\u2212\n(2 31)\nThis work is stored as the potential energy of the system"}, {"Chapter": "1", "sentence_range": "1812-1815", "Text": "Suppose an external torque tttttext is\napplied in such a manner that it just neutralises this\ntorque and rotates it in the plane of paper from angle q0\nto angle q1 at an infinitesimal angular speed and without\nangular acceleration The amount of work done by the\nexternal torque will be given by\n(\n)\ncos\ncos\npE\n\u03b8\n\u03b8\n0\n1\n=\n\u2212\n(2 31)\nThis work is stored as the potential energy of the system We can\nthen associate potential energy U(q) with an inclination q  of the dipole"}, {"Chapter": "1", "sentence_range": "1813-1816", "Text": "The amount of work done by the\nexternal torque will be given by\n(\n)\ncos\ncos\npE\n\u03b8\n\u03b8\n0\n1\n=\n\u2212\n(2 31)\nThis work is stored as the potential energy of the system We can\nthen associate potential energy U(q) with an inclination q  of the dipole Similar to other potential energies, there is a freedom in choosing the\nangle where the potential energy U is taken to be zero"}, {"Chapter": "1", "sentence_range": "1814-1817", "Text": "31)\nThis work is stored as the potential energy of the system We can\nthen associate potential energy U(q) with an inclination q  of the dipole Similar to other potential energies, there is a freedom in choosing the\nangle where the potential energy U is taken to be zero A natural choice\nis to take q0 = p / 2"}, {"Chapter": "1", "sentence_range": "1815-1818", "Text": "We can\nthen associate potential energy U(q) with an inclination q  of the dipole Similar to other potential energies, there is a freedom in choosing the\nangle where the potential energy U is taken to be zero A natural choice\nis to take q0 = p / 2 (An explanation for it is provided towards the end of\ndiscussion"}, {"Chapter": "1", "sentence_range": "1816-1819", "Text": "Similar to other potential energies, there is a freedom in choosing the\nangle where the potential energy U is taken to be zero A natural choice\nis to take q0 = p / 2 (An explanation for it is provided towards the end of\ndiscussion )  We can then write,\n(2"}, {"Chapter": "1", "sentence_range": "1817-1820", "Text": "A natural choice\nis to take q0 = p / 2 (An explanation for it is provided towards the end of\ndiscussion )  We can then write,\n(2 32)\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "1818-1821", "Text": "(An explanation for it is provided towards the end of\ndiscussion )  We can then write,\n(2 32)\nFIGURE 2 16 Potential energy of a\ndipole in a uniform external field"}, {"Chapter": "1", "sentence_range": "1819-1822", "Text": ")  We can then write,\n(2 32)\nFIGURE 2 16 Potential energy of a\ndipole in a uniform external field Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n61\n EXAMPLE 2"}, {"Chapter": "1", "sentence_range": "1820-1823", "Text": "32)\nFIGURE 2 16 Potential energy of a\ndipole in a uniform external field Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n61\n EXAMPLE 2 6\nThis expression can alternately be understood also from Eq"}, {"Chapter": "1", "sentence_range": "1821-1824", "Text": "16 Potential energy of a\ndipole in a uniform external field Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n61\n EXAMPLE 2 6\nThis expression can alternately be understood also from Eq (2"}, {"Chapter": "1", "sentence_range": "1822-1825", "Text": "Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n61\n EXAMPLE 2 6\nThis expression can alternately be understood also from Eq (2 29)"}, {"Chapter": "1", "sentence_range": "1823-1826", "Text": "6\nThis expression can alternately be understood also from Eq (2 29) We apply Eq"}, {"Chapter": "1", "sentence_range": "1824-1827", "Text": "(2 29) We apply Eq (2"}, {"Chapter": "1", "sentence_range": "1825-1828", "Text": "29) We apply Eq (2 29) to the present system of two charges +q and \u2013q"}, {"Chapter": "1", "sentence_range": "1826-1829", "Text": "We apply Eq (2 29) to the present system of two charges +q and \u2013q The\npotential energy  expression then reads\n( )\n( )\n(\n)\n2\n1\n2\n[\n]\n4\n2\nq\nU\nq V\nV\na\n\u03b8\n\u03b50\n=\n\u2212\n\u2212\n\u2032\n\u03c0\n\u00d7\nr\nr\n(2"}, {"Chapter": "1", "sentence_range": "1827-1830", "Text": "(2 29) to the present system of two charges +q and \u2013q The\npotential energy  expression then reads\n( )\n( )\n(\n)\n2\n1\n2\n[\n]\n4\n2\nq\nU\nq V\nV\na\n\u03b8\n\u03b50\n=\n\u2212\n\u2212\n\u2032\n\u03c0\n\u00d7\nr\nr\n(2 33)\nHere, r1 and r2 denote the position vectors of +q and \u2013q"}, {"Chapter": "1", "sentence_range": "1828-1831", "Text": "29) to the present system of two charges +q and \u2013q The\npotential energy  expression then reads\n( )\n( )\n(\n)\n2\n1\n2\n[\n]\n4\n2\nq\nU\nq V\nV\na\n\u03b8\n\u03b50\n=\n\u2212\n\u2212\n\u2032\n\u03c0\n\u00d7\nr\nr\n(2 33)\nHere, r1 and r2 denote the position vectors of +q and \u2013q Now, the\npotential difference between positions r1 and r2 equals the work done\nin bringing a unit positive charge against field from r2 to r1"}, {"Chapter": "1", "sentence_range": "1829-1832", "Text": "The\npotential energy  expression then reads\n( )\n( )\n(\n)\n2\n1\n2\n[\n]\n4\n2\nq\nU\nq V\nV\na\n\u03b8\n\u03b50\n=\n\u2212\n\u2212\n\u2032\n\u03c0\n\u00d7\nr\nr\n(2 33)\nHere, r1 and r2 denote the position vectors of +q and \u2013q Now, the\npotential difference between positions r1 and r2 equals the work done\nin bringing a unit positive charge against field from r2 to r1 The\ndisplacement parallel to the force is 2a cosq"}, {"Chapter": "1", "sentence_range": "1830-1833", "Text": "33)\nHere, r1 and r2 denote the position vectors of +q and \u2013q Now, the\npotential difference between positions r1 and r2 equals the work done\nin bringing a unit positive charge against field from r2 to r1 The\ndisplacement parallel to the force is 2a cosq Thus, [V(r1)\u2013V (r2)] =\n\u2013E \u00d7 2a cosq"}, {"Chapter": "1", "sentence_range": "1831-1834", "Text": "Now, the\npotential difference between positions r1 and r2 equals the work done\nin bringing a unit positive charge against field from r2 to r1 The\ndisplacement parallel to the force is 2a cosq Thus, [V(r1)\u2013V (r2)] =\n\u2013E \u00d7 2a cosq We thus obtain,\n( )\n2\n2\ncos\n4\n2\n4\n2\n\u03b8\n\u03b8\n\u03b5\n\u03b5\n0\n0\n= \u2212\n\u2212\n= \u2212\n\u2212\n\u2032\n\u03c0\n\u00d7\n\u03c0\n\u00d7\np"}, {"Chapter": "1", "sentence_range": "1832-1835", "Text": "The\ndisplacement parallel to the force is 2a cosq Thus, [V(r1)\u2013V (r2)] =\n\u2013E \u00d7 2a cosq We thus obtain,\n( )\n2\n2\ncos\n4\n2\n4\n2\n\u03b8\n\u03b8\n\u03b5\n\u03b5\n0\n0\n= \u2212\n\u2212\n= \u2212\n\u2212\n\u2032\n\u03c0\n\u00d7\n\u03c0\n\u00d7\np E\nq\nq\nU\npE\na\na\n(2"}, {"Chapter": "1", "sentence_range": "1833-1836", "Text": "Thus, [V(r1)\u2013V (r2)] =\n\u2013E \u00d7 2a cosq We thus obtain,\n( )\n2\n2\ncos\n4\n2\n4\n2\n\u03b8\n\u03b8\n\u03b5\n\u03b5\n0\n0\n= \u2212\n\u2212\n= \u2212\n\u2212\n\u2032\n\u03c0\n\u00d7\n\u03c0\n\u00d7\np E\nq\nq\nU\npE\na\na\n(2 34)\nWe note that U\u00a2 (q) differs from U(q ) by a quantity which is just a constant\nfor a given dipole"}, {"Chapter": "1", "sentence_range": "1834-1837", "Text": "We thus obtain,\n( )\n2\n2\ncos\n4\n2\n4\n2\n\u03b8\n\u03b8\n\u03b5\n\u03b5\n0\n0\n= \u2212\n\u2212\n= \u2212\n\u2212\n\u2032\n\u03c0\n\u00d7\n\u03c0\n\u00d7\np E\nq\nq\nU\npE\na\na\n(2 34)\nWe note that U\u00a2 (q) differs from U(q ) by a quantity which is just a constant\nfor a given dipole Since a constant is insignificant for potential energy, we\ncan drop the second term in Eq"}, {"Chapter": "1", "sentence_range": "1835-1838", "Text": "E\nq\nq\nU\npE\na\na\n(2 34)\nWe note that U\u00a2 (q) differs from U(q ) by a quantity which is just a constant\nfor a given dipole Since a constant is insignificant for potential energy, we\ncan drop the second term in Eq (2"}, {"Chapter": "1", "sentence_range": "1836-1839", "Text": "34)\nWe note that U\u00a2 (q) differs from U(q ) by a quantity which is just a constant\nfor a given dipole Since a constant is insignificant for potential energy, we\ncan drop the second term in Eq (2 34) and it then reduces to Eq"}, {"Chapter": "1", "sentence_range": "1837-1840", "Text": "Since a constant is insignificant for potential energy, we\ncan drop the second term in Eq (2 34) and it then reduces to Eq (2"}, {"Chapter": "1", "sentence_range": "1838-1841", "Text": "(2 34) and it then reduces to Eq (2 32)"}, {"Chapter": "1", "sentence_range": "1839-1842", "Text": "34) and it then reduces to Eq (2 32) We can now understand why we took q0=p/2"}, {"Chapter": "1", "sentence_range": "1840-1843", "Text": "(2 32) We can now understand why we took q0=p/2 In this case, the work\ndone against the external field E in bringing +q and \u2013 q are equal and\nopposite and cancel out, i"}, {"Chapter": "1", "sentence_range": "1841-1844", "Text": "32) We can now understand why we took q0=p/2 In this case, the work\ndone against the external field E in bringing +q and \u2013 q are equal and\nopposite and cancel out, i e"}, {"Chapter": "1", "sentence_range": "1842-1845", "Text": "We can now understand why we took q0=p/2 In this case, the work\ndone against the external field E in bringing +q and \u2013 q are equal and\nopposite and cancel out, i e , q [V (r1)  \u2013 V (r2)]=0"}, {"Chapter": "1", "sentence_range": "1843-1846", "Text": "In this case, the work\ndone against the external field E in bringing +q and \u2013 q are equal and\nopposite and cancel out, i e , q [V (r1)  \u2013 V (r2)]=0 Example 2"}, {"Chapter": "1", "sentence_range": "1844-1847", "Text": "e , q [V (r1)  \u2013 V (r2)]=0 Example 2 6 A molecule of a substance has a permanent electric\ndipole moment of magnitude 10\u201329 C m"}, {"Chapter": "1", "sentence_range": "1845-1848", "Text": ", q [V (r1)  \u2013 V (r2)]=0 Example 2 6 A molecule of a substance has a permanent electric\ndipole moment of magnitude 10\u201329 C m A mole of this substance is\npolarised (at low temperature) by applying a strong electrostatic field\nof magnitude 106 V m\u20131"}, {"Chapter": "1", "sentence_range": "1846-1849", "Text": "Example 2 6 A molecule of a substance has a permanent electric\ndipole moment of magnitude 10\u201329 C m A mole of this substance is\npolarised (at low temperature) by applying a strong electrostatic field\nof magnitude 106 V m\u20131 The direction of the field is suddenly changed\nby an angle of 60\u00ba"}, {"Chapter": "1", "sentence_range": "1847-1850", "Text": "6 A molecule of a substance has a permanent electric\ndipole moment of magnitude 10\u201329 C m A mole of this substance is\npolarised (at low temperature) by applying a strong electrostatic field\nof magnitude 106 V m\u20131 The direction of the field is suddenly changed\nby an angle of 60\u00ba Estimate the heat released by the substance in\naligning its dipoles along the new direction of the field"}, {"Chapter": "1", "sentence_range": "1848-1851", "Text": "A mole of this substance is\npolarised (at low temperature) by applying a strong electrostatic field\nof magnitude 106 V m\u20131 The direction of the field is suddenly changed\nby an angle of 60\u00ba Estimate the heat released by the substance in\naligning its dipoles along the new direction of the field For simplicity,\nassume 100% polarisation of the sample"}, {"Chapter": "1", "sentence_range": "1849-1852", "Text": "The direction of the field is suddenly changed\nby an angle of 60\u00ba Estimate the heat released by the substance in\naligning its dipoles along the new direction of the field For simplicity,\nassume 100% polarisation of the sample Solution   Here, dipole moment of each molecules = 10\u201329 C m\nAs 1 mole of the substance contains 6 \u00d7 1023 molecules,\ntotal dipole moment of all the molecules, p = 6 \u00d7 1023 \u00d7 10\u201329 C m\n    = 6 \u00d7 10\u20136\n C m\nInitial potential energy, Ui = \u2013pE cos q = \u20136\u00d710\u20136\u00d7106 cos 0\u00b0 = \u20136 J\nFinal potential energy (when q = 60\u00b0), Uf = \u20136 \u00d7 10\u20136 \u00d7 106 cos 60\u00b0 = \u20133 J\nChange in potential energy = \u20133 J \u2013 (\u20136J) = 3 J\nSo, there is loss in potential energy"}, {"Chapter": "1", "sentence_range": "1850-1853", "Text": "Estimate the heat released by the substance in\naligning its dipoles along the new direction of the field For simplicity,\nassume 100% polarisation of the sample Solution   Here, dipole moment of each molecules = 10\u201329 C m\nAs 1 mole of the substance contains 6 \u00d7 1023 molecules,\ntotal dipole moment of all the molecules, p = 6 \u00d7 1023 \u00d7 10\u201329 C m\n    = 6 \u00d7 10\u20136\n C m\nInitial potential energy, Ui = \u2013pE cos q = \u20136\u00d710\u20136\u00d7106 cos 0\u00b0 = \u20136 J\nFinal potential energy (when q = 60\u00b0), Uf = \u20136 \u00d7 10\u20136 \u00d7 106 cos 60\u00b0 = \u20133 J\nChange in potential energy = \u20133 J \u2013 (\u20136J) = 3 J\nSo, there is loss in potential energy This must be the energy released\nby the substance in the form of heat in aligning its dipoles"}, {"Chapter": "1", "sentence_range": "1851-1854", "Text": "For simplicity,\nassume 100% polarisation of the sample Solution   Here, dipole moment of each molecules = 10\u201329 C m\nAs 1 mole of the substance contains 6 \u00d7 1023 molecules,\ntotal dipole moment of all the molecules, p = 6 \u00d7 1023 \u00d7 10\u201329 C m\n    = 6 \u00d7 10\u20136\n C m\nInitial potential energy, Ui = \u2013pE cos q = \u20136\u00d710\u20136\u00d7106 cos 0\u00b0 = \u20136 J\nFinal potential energy (when q = 60\u00b0), Uf = \u20136 \u00d7 10\u20136 \u00d7 106 cos 60\u00b0 = \u20133 J\nChange in potential energy = \u20133 J \u2013 (\u20136J) = 3 J\nSo, there is loss in potential energy This must be the energy released\nby the substance in the form of heat in aligning its dipoles 2"}, {"Chapter": "1", "sentence_range": "1852-1855", "Text": "Solution   Here, dipole moment of each molecules = 10\u201329 C m\nAs 1 mole of the substance contains 6 \u00d7 1023 molecules,\ntotal dipole moment of all the molecules, p = 6 \u00d7 1023 \u00d7 10\u201329 C m\n    = 6 \u00d7 10\u20136\n C m\nInitial potential energy, Ui = \u2013pE cos q = \u20136\u00d710\u20136\u00d7106 cos 0\u00b0 = \u20136 J\nFinal potential energy (when q = 60\u00b0), Uf = \u20136 \u00d7 10\u20136 \u00d7 106 cos 60\u00b0 = \u20133 J\nChange in potential energy = \u20133 J \u2013 (\u20136J) = 3 J\nSo, there is loss in potential energy This must be the energy released\nby the substance in the form of heat in aligning its dipoles 2 9  ELECTROSTATICS OF CONDUCTORS\nConductors and insulators were described briefly in Chapter 1"}, {"Chapter": "1", "sentence_range": "1853-1856", "Text": "This must be the energy released\nby the substance in the form of heat in aligning its dipoles 2 9  ELECTROSTATICS OF CONDUCTORS\nConductors and insulators were described briefly in Chapter 1 Conductors contain mobile charge carriers"}, {"Chapter": "1", "sentence_range": "1854-1857", "Text": "2 9  ELECTROSTATICS OF CONDUCTORS\nConductors and insulators were described briefly in Chapter 1 Conductors contain mobile charge carriers In metallic conductors, these\ncharge carriers are electrons"}, {"Chapter": "1", "sentence_range": "1855-1858", "Text": "9  ELECTROSTATICS OF CONDUCTORS\nConductors and insulators were described briefly in Chapter 1 Conductors contain mobile charge carriers In metallic conductors, these\ncharge carriers are electrons In a metal, the outer (valence) electrons\npart away from their atoms and are free to move"}, {"Chapter": "1", "sentence_range": "1856-1859", "Text": "Conductors contain mobile charge carriers In metallic conductors, these\ncharge carriers are electrons In a metal, the outer (valence) electrons\npart away from their atoms and are free to move These electrons are free\nwithin the metal but not free  to leave the metal"}, {"Chapter": "1", "sentence_range": "1857-1860", "Text": "In metallic conductors, these\ncharge carriers are electrons In a metal, the outer (valence) electrons\npart away from their atoms and are free to move These electrons are free\nwithin the metal but not free  to leave the metal The free electrons form a\nkind of \u2018gas\u2019; they collide with each other and with the ions, and move\nrandomly in different directions"}, {"Chapter": "1", "sentence_range": "1858-1861", "Text": "In a metal, the outer (valence) electrons\npart away from their atoms and are free to move These electrons are free\nwithin the metal but not free  to leave the metal The free electrons form a\nkind of \u2018gas\u2019; they collide with each other and with the ions, and move\nrandomly in different directions In an external electric field, they drift\nagainst the direction of the field"}, {"Chapter": "1", "sentence_range": "1859-1862", "Text": "These electrons are free\nwithin the metal but not free  to leave the metal The free electrons form a\nkind of \u2018gas\u2019; they collide with each other and with the ions, and move\nrandomly in different directions In an external electric field, they drift\nagainst the direction of the field The positive ions made up of the nuclei\nand the bound electrons remain held in their fixed positions"}, {"Chapter": "1", "sentence_range": "1860-1863", "Text": "The free electrons form a\nkind of \u2018gas\u2019; they collide with each other and with the ions, and move\nrandomly in different directions In an external electric field, they drift\nagainst the direction of the field The positive ions made up of the nuclei\nand the bound electrons remain held in their fixed positions In electrolytic\nconductors, the charge carriers are both positive and negative ions; but\nRationalised 2023-24\nPhysics\n62\nthe situation in this case is more involved \u2013 the movement of the charge\ncarriers is affected both by the external electric field as also by the\nso-called chemical forces (see Chapter 3)"}, {"Chapter": "1", "sentence_range": "1861-1864", "Text": "In an external electric field, they drift\nagainst the direction of the field The positive ions made up of the nuclei\nand the bound electrons remain held in their fixed positions In electrolytic\nconductors, the charge carriers are both positive and negative ions; but\nRationalised 2023-24\nPhysics\n62\nthe situation in this case is more involved \u2013 the movement of the charge\ncarriers is affected both by the external electric field as also by the\nso-called chemical forces (see Chapter 3) We shall restrict our discussion\nto metallic solid conductors"}, {"Chapter": "1", "sentence_range": "1862-1865", "Text": "The positive ions made up of the nuclei\nand the bound electrons remain held in their fixed positions In electrolytic\nconductors, the charge carriers are both positive and negative ions; but\nRationalised 2023-24\nPhysics\n62\nthe situation in this case is more involved \u2013 the movement of the charge\ncarriers is affected both by the external electric field as also by the\nso-called chemical forces (see Chapter 3) We shall restrict our discussion\nto metallic solid conductors Let us note important results regarding\nelectrostatics of conductors"}, {"Chapter": "1", "sentence_range": "1863-1866", "Text": "In electrolytic\nconductors, the charge carriers are both positive and negative ions; but\nRationalised 2023-24\nPhysics\n62\nthe situation in this case is more involved \u2013 the movement of the charge\ncarriers is affected both by the external electric field as also by the\nso-called chemical forces (see Chapter 3) We shall restrict our discussion\nto metallic solid conductors Let us note important results regarding\nelectrostatics of conductors 1"}, {"Chapter": "1", "sentence_range": "1864-1867", "Text": "We shall restrict our discussion\nto metallic solid conductors Let us note important results regarding\nelectrostatics of conductors 1 Inside a conductor, electrostatic field is zero\nConsider a conductor, neutral or charged"}, {"Chapter": "1", "sentence_range": "1865-1868", "Text": "Let us note important results regarding\nelectrostatics of conductors 1 Inside a conductor, electrostatic field is zero\nConsider a conductor, neutral or charged There may also be an external\nelectrostatic field"}, {"Chapter": "1", "sentence_range": "1866-1869", "Text": "1 Inside a conductor, electrostatic field is zero\nConsider a conductor, neutral or charged There may also be an external\nelectrostatic field In the static situation, when there is no current inside\nor on the surface of the conductor, the electric field is zero everywhere\ninside the conductor"}, {"Chapter": "1", "sentence_range": "1867-1870", "Text": "Inside a conductor, electrostatic field is zero\nConsider a conductor, neutral or charged There may also be an external\nelectrostatic field In the static situation, when there is no current inside\nor on the surface of the conductor, the electric field is zero everywhere\ninside the conductor This fact can be taken as the defining property of a\nconductor"}, {"Chapter": "1", "sentence_range": "1868-1871", "Text": "There may also be an external\nelectrostatic field In the static situation, when there is no current inside\nor on the surface of the conductor, the electric field is zero everywhere\ninside the conductor This fact can be taken as the defining property of a\nconductor A conductor has free electrons"}, {"Chapter": "1", "sentence_range": "1869-1872", "Text": "In the static situation, when there is no current inside\nor on the surface of the conductor, the electric field is zero everywhere\ninside the conductor This fact can be taken as the defining property of a\nconductor A conductor has free electrons As long as electric field is not\nzero, the free charge carriers would experience force and drift"}, {"Chapter": "1", "sentence_range": "1870-1873", "Text": "This fact can be taken as the defining property of a\nconductor A conductor has free electrons As long as electric field is not\nzero, the free charge carriers would experience force and drift In the\nstatic situation, the free charges have so distributed themselves that the\nelectric field is zero everywhere inside"}, {"Chapter": "1", "sentence_range": "1871-1874", "Text": "A conductor has free electrons As long as electric field is not\nzero, the free charge carriers would experience force and drift In the\nstatic situation, the free charges have so distributed themselves that the\nelectric field is zero everywhere inside Electrostatic field is zero inside a\nconductor"}, {"Chapter": "1", "sentence_range": "1872-1875", "Text": "As long as electric field is not\nzero, the free charge carriers would experience force and drift In the\nstatic situation, the free charges have so distributed themselves that the\nelectric field is zero everywhere inside Electrostatic field is zero inside a\nconductor 2"}, {"Chapter": "1", "sentence_range": "1873-1876", "Text": "In the\nstatic situation, the free charges have so distributed themselves that the\nelectric field is zero everywhere inside Electrostatic field is zero inside a\nconductor 2 At the surface of a charged conductor, electrostatic field\nmust be normal to the surface at every point\nIf E were not normal to the surface, it would have some non-zero\ncomponent along the surface"}, {"Chapter": "1", "sentence_range": "1874-1877", "Text": "Electrostatic field is zero inside a\nconductor 2 At the surface of a charged conductor, electrostatic field\nmust be normal to the surface at every point\nIf E were not normal to the surface, it would have some non-zero\ncomponent along the surface Free charges on the surface of the conductor\nwould then experience force and move"}, {"Chapter": "1", "sentence_range": "1875-1878", "Text": "2 At the surface of a charged conductor, electrostatic field\nmust be normal to the surface at every point\nIf E were not normal to the surface, it would have some non-zero\ncomponent along the surface Free charges on the surface of the conductor\nwould then experience force and move In the static situation, therefore,\nE should have no tangential component"}, {"Chapter": "1", "sentence_range": "1876-1879", "Text": "At the surface of a charged conductor, electrostatic field\nmust be normal to the surface at every point\nIf E were not normal to the surface, it would have some non-zero\ncomponent along the surface Free charges on the surface of the conductor\nwould then experience force and move In the static situation, therefore,\nE should have no tangential component Thus electrostatic field at the\nsurface of a charged conductor must be normal to the surface at every\npoint"}, {"Chapter": "1", "sentence_range": "1877-1880", "Text": "Free charges on the surface of the conductor\nwould then experience force and move In the static situation, therefore,\nE should have no tangential component Thus electrostatic field at the\nsurface of a charged conductor must be normal to the surface at every\npoint (For a conductor without any surface charge density, field is zero\neven at the surface"}, {"Chapter": "1", "sentence_range": "1878-1881", "Text": "In the static situation, therefore,\nE should have no tangential component Thus electrostatic field at the\nsurface of a charged conductor must be normal to the surface at every\npoint (For a conductor without any surface charge density, field is zero\neven at the surface ) See result 5"}, {"Chapter": "1", "sentence_range": "1879-1882", "Text": "Thus electrostatic field at the\nsurface of a charged conductor must be normal to the surface at every\npoint (For a conductor without any surface charge density, field is zero\neven at the surface ) See result 5 3"}, {"Chapter": "1", "sentence_range": "1880-1883", "Text": "(For a conductor without any surface charge density, field is zero\neven at the surface ) See result 5 3 The interior of a conductor can have no excess charge in\nthe static situation\nA neutral conductor has equal amounts of positive and negative charges\nin every small volume or surface element"}, {"Chapter": "1", "sentence_range": "1881-1884", "Text": ") See result 5 3 The interior of a conductor can have no excess charge in\nthe static situation\nA neutral conductor has equal amounts of positive and negative charges\nin every small volume or surface element When the conductor is charged,\nthe excess charge can reside only on the surface in the static situation"}, {"Chapter": "1", "sentence_range": "1882-1885", "Text": "3 The interior of a conductor can have no excess charge in\nthe static situation\nA neutral conductor has equal amounts of positive and negative charges\nin every small volume or surface element When the conductor is charged,\nthe excess charge can reside only on the surface in the static situation This follows from the Gauss\u2019s law"}, {"Chapter": "1", "sentence_range": "1883-1886", "Text": "The interior of a conductor can have no excess charge in\nthe static situation\nA neutral conductor has equal amounts of positive and negative charges\nin every small volume or surface element When the conductor is charged,\nthe excess charge can reside only on the surface in the static situation This follows from the Gauss\u2019s law Consider any arbitrary volume element\nv inside a conductor"}, {"Chapter": "1", "sentence_range": "1884-1887", "Text": "When the conductor is charged,\nthe excess charge can reside only on the surface in the static situation This follows from the Gauss\u2019s law Consider any arbitrary volume element\nv inside a conductor On the closed surface S bounding the volume\nelement v, electrostatic field is zero"}, {"Chapter": "1", "sentence_range": "1885-1888", "Text": "This follows from the Gauss\u2019s law Consider any arbitrary volume element\nv inside a conductor On the closed surface S bounding the volume\nelement v, electrostatic field is zero Thus the total electric flux through S\nis zero"}, {"Chapter": "1", "sentence_range": "1886-1889", "Text": "Consider any arbitrary volume element\nv inside a conductor On the closed surface S bounding the volume\nelement v, electrostatic field is zero Thus the total electric flux through S\nis zero Hence, by Gauss\u2019s law, there is no net charge enclosed by S"}, {"Chapter": "1", "sentence_range": "1887-1890", "Text": "On the closed surface S bounding the volume\nelement v, electrostatic field is zero Thus the total electric flux through S\nis zero Hence, by Gauss\u2019s law, there is no net charge enclosed by S But\nthe surface S can be made as small as you like, i"}, {"Chapter": "1", "sentence_range": "1888-1891", "Text": "Thus the total electric flux through S\nis zero Hence, by Gauss\u2019s law, there is no net charge enclosed by S But\nthe surface S can be made as small as you like, i e"}, {"Chapter": "1", "sentence_range": "1889-1892", "Text": "Hence, by Gauss\u2019s law, there is no net charge enclosed by S But\nthe surface S can be made as small as you like, i e , the volume v can be\nmade vanishingly small"}, {"Chapter": "1", "sentence_range": "1890-1893", "Text": "But\nthe surface S can be made as small as you like, i e , the volume v can be\nmade vanishingly small This means there is no net charge at any point\ninside the conductor, and any excess charge must reside at the surface"}, {"Chapter": "1", "sentence_range": "1891-1894", "Text": "e , the volume v can be\nmade vanishingly small This means there is no net charge at any point\ninside the conductor, and any excess charge must reside at the surface 4"}, {"Chapter": "1", "sentence_range": "1892-1895", "Text": ", the volume v can be\nmade vanishingly small This means there is no net charge at any point\ninside the conductor, and any excess charge must reside at the surface 4 Electrostatic potential is constant throughout the volume\nof the conductor and has the same value (as inside) on\nits surface\nThis follows from results 1 and 2 above"}, {"Chapter": "1", "sentence_range": "1893-1896", "Text": "This means there is no net charge at any point\ninside the conductor, and any excess charge must reside at the surface 4 Electrostatic potential is constant throughout the volume\nof the conductor and has the same value (as inside) on\nits surface\nThis follows from results 1 and 2 above Since E = 0 inside the conductor\nand has no tangential component on the surface, no work is done in\nmoving a small test charge within the conductor and on its surface"}, {"Chapter": "1", "sentence_range": "1894-1897", "Text": "4 Electrostatic potential is constant throughout the volume\nof the conductor and has the same value (as inside) on\nits surface\nThis follows from results 1 and 2 above Since E = 0 inside the conductor\nand has no tangential component on the surface, no work is done in\nmoving a small test charge within the conductor and on its surface That\nis, there is no potential difference between any two points inside or on\nthe surface of the conductor"}, {"Chapter": "1", "sentence_range": "1895-1898", "Text": "Electrostatic potential is constant throughout the volume\nof the conductor and has the same value (as inside) on\nits surface\nThis follows from results 1 and 2 above Since E = 0 inside the conductor\nand has no tangential component on the surface, no work is done in\nmoving a small test charge within the conductor and on its surface That\nis, there is no potential difference between any two points inside or on\nthe surface of the conductor Hence, the result"}, {"Chapter": "1", "sentence_range": "1896-1899", "Text": "Since E = 0 inside the conductor\nand has no tangential component on the surface, no work is done in\nmoving a small test charge within the conductor and on its surface That\nis, there is no potential difference between any two points inside or on\nthe surface of the conductor Hence, the result If the conductor is charged,\nRationalised 2023-24\nElectrostatic Potential\nand Capacitance\n63\nelectric field normal to the surface exists; this means potential will be\ndifferent for the surface and a point just outside the surface"}, {"Chapter": "1", "sentence_range": "1897-1900", "Text": "That\nis, there is no potential difference between any two points inside or on\nthe surface of the conductor Hence, the result If the conductor is charged,\nRationalised 2023-24\nElectrostatic Potential\nand Capacitance\n63\nelectric field normal to the surface exists; this means potential will be\ndifferent for the surface and a point just outside the surface In a system of conductors of arbitrary size, shape and\ncharge configuration, each conductor is characterised by a constant\nvalue of potential, but this constant may differ from one conductor to\nthe other"}, {"Chapter": "1", "sentence_range": "1898-1901", "Text": "Hence, the result If the conductor is charged,\nRationalised 2023-24\nElectrostatic Potential\nand Capacitance\n63\nelectric field normal to the surface exists; this means potential will be\ndifferent for the surface and a point just outside the surface In a system of conductors of arbitrary size, shape and\ncharge configuration, each conductor is characterised by a constant\nvalue of potential, but this constant may differ from one conductor to\nthe other 5"}, {"Chapter": "1", "sentence_range": "1899-1902", "Text": "If the conductor is charged,\nRationalised 2023-24\nElectrostatic Potential\nand Capacitance\n63\nelectric field normal to the surface exists; this means potential will be\ndifferent for the surface and a point just outside the surface In a system of conductors of arbitrary size, shape and\ncharge configuration, each conductor is characterised by a constant\nvalue of potential, but this constant may differ from one conductor to\nthe other 5 Electric field at the surface of a charged conductor\n0\nE=\u03b5\u03c3\u02c6\nn\n(2"}, {"Chapter": "1", "sentence_range": "1900-1903", "Text": "In a system of conductors of arbitrary size, shape and\ncharge configuration, each conductor is characterised by a constant\nvalue of potential, but this constant may differ from one conductor to\nthe other 5 Electric field at the surface of a charged conductor\n0\nE=\u03b5\u03c3\u02c6\nn\n(2 35)\nwhere s is the surface charge density and \u02c6n  is a unit vector normal\nto the surface in the outward direction"}, {"Chapter": "1", "sentence_range": "1901-1904", "Text": "5 Electric field at the surface of a charged conductor\n0\nE=\u03b5\u03c3\u02c6\nn\n(2 35)\nwhere s is the surface charge density and \u02c6n  is a unit vector normal\nto the surface in the outward direction To derive the result, choose a pill box (a short cylinder) as the Gaussian\nsurface about any point P on the surface, as shown in Fig"}, {"Chapter": "1", "sentence_range": "1902-1905", "Text": "Electric field at the surface of a charged conductor\n0\nE=\u03b5\u03c3\u02c6\nn\n(2 35)\nwhere s is the surface charge density and \u02c6n  is a unit vector normal\nto the surface in the outward direction To derive the result, choose a pill box (a short cylinder) as the Gaussian\nsurface about any point P on the surface, as shown in Fig 2"}, {"Chapter": "1", "sentence_range": "1903-1906", "Text": "35)\nwhere s is the surface charge density and \u02c6n  is a unit vector normal\nto the surface in the outward direction To derive the result, choose a pill box (a short cylinder) as the Gaussian\nsurface about any point P on the surface, as shown in Fig 2 17"}, {"Chapter": "1", "sentence_range": "1904-1907", "Text": "To derive the result, choose a pill box (a short cylinder) as the Gaussian\nsurface about any point P on the surface, as shown in Fig 2 17 The pill\nbox is partly inside and partly outside the surface of the conductor"}, {"Chapter": "1", "sentence_range": "1905-1908", "Text": "2 17 The pill\nbox is partly inside and partly outside the surface of the conductor It\nhas a small area of cross section d S and negligible height"}, {"Chapter": "1", "sentence_range": "1906-1909", "Text": "17 The pill\nbox is partly inside and partly outside the surface of the conductor It\nhas a small area of cross section d S and negligible height Just inside the surface, the electrostatic field is zero; just outside, the\nfield is normal to the surface with magnitude E"}, {"Chapter": "1", "sentence_range": "1907-1910", "Text": "The pill\nbox is partly inside and partly outside the surface of the conductor It\nhas a small area of cross section d S and negligible height Just inside the surface, the electrostatic field is zero; just outside, the\nfield is normal to the surface with magnitude E Thus,\nthe contribution to the total flux through the pill box\ncomes only from the outside (circular) cross-section\nof the pill box"}, {"Chapter": "1", "sentence_range": "1908-1911", "Text": "It\nhas a small area of cross section d S and negligible height Just inside the surface, the electrostatic field is zero; just outside, the\nfield is normal to the surface with magnitude E Thus,\nthe contribution to the total flux through the pill box\ncomes only from the outside (circular) cross-section\nof the pill box This equals  \u00b1 EdS (positive for s > 0,\nnegative for s < 0), since over the small area dS, E\nmay be considered constant and E and dS are parallel\nor antiparallel"}, {"Chapter": "1", "sentence_range": "1909-1912", "Text": "Just inside the surface, the electrostatic field is zero; just outside, the\nfield is normal to the surface with magnitude E Thus,\nthe contribution to the total flux through the pill box\ncomes only from the outside (circular) cross-section\nof the pill box This equals  \u00b1 EdS (positive for s > 0,\nnegative for s < 0), since over the small area dS, E\nmay be considered constant and E and dS are parallel\nor antiparallel The charge enclosed by the pill box\nis  sdS"}, {"Chapter": "1", "sentence_range": "1910-1913", "Text": "Thus,\nthe contribution to the total flux through the pill box\ncomes only from the outside (circular) cross-section\nof the pill box This equals  \u00b1 EdS (positive for s > 0,\nnegative for s < 0), since over the small area dS, E\nmay be considered constant and E and dS are parallel\nor antiparallel The charge enclosed by the pill box\nis  sdS By Gauss\u2019s law\nEdS = \n0\nS\n\u03c3 \u03b4\n\u03b5\nE = \n0\n\u03b5\u03c3\n(2"}, {"Chapter": "1", "sentence_range": "1911-1914", "Text": "This equals  \u00b1 EdS (positive for s > 0,\nnegative for s < 0), since over the small area dS, E\nmay be considered constant and E and dS are parallel\nor antiparallel The charge enclosed by the pill box\nis  sdS By Gauss\u2019s law\nEdS = \n0\nS\n\u03c3 \u03b4\n\u03b5\nE = \n0\n\u03b5\u03c3\n(2 36)\nIncluding the fact that electric field is normal to the\nsurface, we get the vector relation, Eq"}, {"Chapter": "1", "sentence_range": "1912-1915", "Text": "The charge enclosed by the pill box\nis  sdS By Gauss\u2019s law\nEdS = \n0\nS\n\u03c3 \u03b4\n\u03b5\nE = \n0\n\u03b5\u03c3\n(2 36)\nIncluding the fact that electric field is normal to the\nsurface, we get the vector relation, Eq (2"}, {"Chapter": "1", "sentence_range": "1913-1916", "Text": "By Gauss\u2019s law\nEdS = \n0\nS\n\u03c3 \u03b4\n\u03b5\nE = \n0\n\u03b5\u03c3\n(2 36)\nIncluding the fact that electric field is normal to the\nsurface, we get the vector relation, Eq (2 35), which\nis true for both signs of s"}, {"Chapter": "1", "sentence_range": "1914-1917", "Text": "36)\nIncluding the fact that electric field is normal to the\nsurface, we get the vector relation, Eq (2 35), which\nis true for both signs of s For s > 0, electric field is\nnormal to the surface outward; for s < 0, electric field\nis normal to the surface inward"}, {"Chapter": "1", "sentence_range": "1915-1918", "Text": "(2 35), which\nis true for both signs of s For s > 0, electric field is\nnormal to the surface outward; for s < 0, electric field\nis normal to the surface inward 6"}, {"Chapter": "1", "sentence_range": "1916-1919", "Text": "35), which\nis true for both signs of s For s > 0, electric field is\nnormal to the surface outward; for s < 0, electric field\nis normal to the surface inward 6 Electrostatic shielding\nConsider a conductor with a cavity, with no charges inside the cavity"}, {"Chapter": "1", "sentence_range": "1917-1920", "Text": "For s > 0, electric field is\nnormal to the surface outward; for s < 0, electric field\nis normal to the surface inward 6 Electrostatic shielding\nConsider a conductor with a cavity, with no charges inside the cavity A\nremarkable result is that the electric field inside the cavity is zero, whatever\nbe the size and shape of the cavity and whatever be the charge on the\nconductor and the external fields in which it might be placed"}, {"Chapter": "1", "sentence_range": "1918-1921", "Text": "6 Electrostatic shielding\nConsider a conductor with a cavity, with no charges inside the cavity A\nremarkable result is that the electric field inside the cavity is zero, whatever\nbe the size and shape of the cavity and whatever be the charge on the\nconductor and the external fields in which it might be placed We have\nproved a simple case of this result already: the electric field inside a charged\nspherical shell is zero"}, {"Chapter": "1", "sentence_range": "1919-1922", "Text": "Electrostatic shielding\nConsider a conductor with a cavity, with no charges inside the cavity A\nremarkable result is that the electric field inside the cavity is zero, whatever\nbe the size and shape of the cavity and whatever be the charge on the\nconductor and the external fields in which it might be placed We have\nproved a simple case of this result already: the electric field inside a charged\nspherical shell is zero The proof of the result for the shell makes use of\nthe spherical symmetry of the shell (see Chapter 1)"}, {"Chapter": "1", "sentence_range": "1920-1923", "Text": "A\nremarkable result is that the electric field inside the cavity is zero, whatever\nbe the size and shape of the cavity and whatever be the charge on the\nconductor and the external fields in which it might be placed We have\nproved a simple case of this result already: the electric field inside a charged\nspherical shell is zero The proof of the result for the shell makes use of\nthe spherical symmetry of the shell (see Chapter 1) But the vanishing of\nelectric field in the (charge-free) cavity of a conductor is, as mentioned\nabove, a very general result"}, {"Chapter": "1", "sentence_range": "1921-1924", "Text": "We have\nproved a simple case of this result already: the electric field inside a charged\nspherical shell is zero The proof of the result for the shell makes use of\nthe spherical symmetry of the shell (see Chapter 1) But the vanishing of\nelectric field in the (charge-free) cavity of a conductor is, as mentioned\nabove, a very general result A related result is that even if the conductor\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "1922-1925", "Text": "The proof of the result for the shell makes use of\nthe spherical symmetry of the shell (see Chapter 1) But the vanishing of\nelectric field in the (charge-free) cavity of a conductor is, as mentioned\nabove, a very general result A related result is that even if the conductor\nFIGURE 2 17 The Gaussian surface\n(a pill box) chosen to derive Eq"}, {"Chapter": "1", "sentence_range": "1923-1926", "Text": "But the vanishing of\nelectric field in the (charge-free) cavity of a conductor is, as mentioned\nabove, a very general result A related result is that even if the conductor\nFIGURE 2 17 The Gaussian surface\n(a pill box) chosen to derive Eq (2"}, {"Chapter": "1", "sentence_range": "1924-1927", "Text": "A related result is that even if the conductor\nFIGURE 2 17 The Gaussian surface\n(a pill box) chosen to derive Eq (2 35)\nfor electric field at the surface of a\ncharged conductor"}, {"Chapter": "1", "sentence_range": "1925-1928", "Text": "17 The Gaussian surface\n(a pill box) chosen to derive Eq (2 35)\nfor electric field at the surface of a\ncharged conductor Rationalised 2023-24\nPhysics\n64\n EXAMPLE 2"}, {"Chapter": "1", "sentence_range": "1926-1929", "Text": "(2 35)\nfor electric field at the surface of a\ncharged conductor Rationalised 2023-24\nPhysics\n64\n EXAMPLE 2 7\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "1927-1930", "Text": "35)\nfor electric field at the surface of a\ncharged conductor Rationalised 2023-24\nPhysics\n64\n EXAMPLE 2 7\nFIGURE 2 18 The electric field inside a\ncavity of any conductor is zero"}, {"Chapter": "1", "sentence_range": "1928-1931", "Text": "Rationalised 2023-24\nPhysics\n64\n EXAMPLE 2 7\nFIGURE 2 18 The electric field inside a\ncavity of any conductor is zero All\ncharges reside only on the outer surface\nof a conductor with cavity"}, {"Chapter": "1", "sentence_range": "1929-1932", "Text": "7\nFIGURE 2 18 The electric field inside a\ncavity of any conductor is zero All\ncharges reside only on the outer surface\nof a conductor with cavity (There are no\ncharges placed in the cavity"}, {"Chapter": "1", "sentence_range": "1930-1933", "Text": "18 The electric field inside a\ncavity of any conductor is zero All\ncharges reside only on the outer surface\nof a conductor with cavity (There are no\ncharges placed in the cavity )\nis charged or charges are induced on a neutral\nconductor by an external field, all charges reside\nonly on the outer surface of a conductor with cavity"}, {"Chapter": "1", "sentence_range": "1931-1934", "Text": "All\ncharges reside only on the outer surface\nof a conductor with cavity (There are no\ncharges placed in the cavity )\nis charged or charges are induced on a neutral\nconductor by an external field, all charges reside\nonly on the outer surface of a conductor with cavity The proofs of the results noted in Fig"}, {"Chapter": "1", "sentence_range": "1932-1935", "Text": "(There are no\ncharges placed in the cavity )\nis charged or charges are induced on a neutral\nconductor by an external field, all charges reside\nonly on the outer surface of a conductor with cavity The proofs of the results noted in Fig 2"}, {"Chapter": "1", "sentence_range": "1933-1936", "Text": ")\nis charged or charges are induced on a neutral\nconductor by an external field, all charges reside\nonly on the outer surface of a conductor with cavity The proofs of the results noted in Fig 2 18 are\nomitted here, but we note their important\nimplication"}, {"Chapter": "1", "sentence_range": "1934-1937", "Text": "The proofs of the results noted in Fig 2 18 are\nomitted here, but we note their important\nimplication Whatever be the charge and field\nconfiguration outside, any cavity in a conductor\nremains shielded from outside electric influence: the\nfield inside the cavity is always zero"}, {"Chapter": "1", "sentence_range": "1935-1938", "Text": "2 18 are\nomitted here, but we note their important\nimplication Whatever be the charge and field\nconfiguration outside, any cavity in a conductor\nremains shielded from outside electric influence: the\nfield inside the cavity is always zero This is known\nas electrostatic shielding"}, {"Chapter": "1", "sentence_range": "1936-1939", "Text": "18 are\nomitted here, but we note their important\nimplication Whatever be the charge and field\nconfiguration outside, any cavity in a conductor\nremains shielded from outside electric influence: the\nfield inside the cavity is always zero This is known\nas electrostatic shielding The effect can be made\nuse of in protecting sensitive instruments from\noutside electrical influence"}, {"Chapter": "1", "sentence_range": "1937-1940", "Text": "Whatever be the charge and field\nconfiguration outside, any cavity in a conductor\nremains shielded from outside electric influence: the\nfield inside the cavity is always zero This is known\nas electrostatic shielding The effect can be made\nuse of in protecting sensitive instruments from\noutside electrical influence Figure 2"}, {"Chapter": "1", "sentence_range": "1938-1941", "Text": "This is known\nas electrostatic shielding The effect can be made\nuse of in protecting sensitive instruments from\noutside electrical influence Figure 2 19 gives a\nsummary of the important electrostatic properties\nof a conductor"}, {"Chapter": "1", "sentence_range": "1939-1942", "Text": "The effect can be made\nuse of in protecting sensitive instruments from\noutside electrical influence Figure 2 19 gives a\nsummary of the important electrostatic properties\nof a conductor Example 2"}, {"Chapter": "1", "sentence_range": "1940-1943", "Text": "Figure 2 19 gives a\nsummary of the important electrostatic properties\nof a conductor Example 2 7\n(a) A comb run through one\u2019s dry hair attracts small bits of paper"}, {"Chapter": "1", "sentence_range": "1941-1944", "Text": "19 gives a\nsummary of the important electrostatic properties\nof a conductor Example 2 7\n(a) A comb run through one\u2019s dry hair attracts small bits of paper Why"}, {"Chapter": "1", "sentence_range": "1942-1945", "Text": "Example 2 7\n(a) A comb run through one\u2019s dry hair attracts small bits of paper Why What happens if the hair is wet or if it is a rainy day"}, {"Chapter": "1", "sentence_range": "1943-1946", "Text": "7\n(a) A comb run through one\u2019s dry hair attracts small bits of paper Why What happens if the hair is wet or if it is a rainy day (Remember,\na paper does not conduct electricity"}, {"Chapter": "1", "sentence_range": "1944-1947", "Text": "Why What happens if the hair is wet or if it is a rainy day (Remember,\na paper does not conduct electricity )\n(b) Ordinary rubber is an insulator"}, {"Chapter": "1", "sentence_range": "1945-1948", "Text": "What happens if the hair is wet or if it is a rainy day (Remember,\na paper does not conduct electricity )\n(b) Ordinary rubber is an insulator But special rubber tyres of\naircraft are made slightly conducting"}, {"Chapter": "1", "sentence_range": "1946-1949", "Text": "(Remember,\na paper does not conduct electricity )\n(b) Ordinary rubber is an insulator But special rubber tyres of\naircraft are made slightly conducting Why is this necessary"}, {"Chapter": "1", "sentence_range": "1947-1950", "Text": ")\n(b) Ordinary rubber is an insulator But special rubber tyres of\naircraft are made slightly conducting Why is this necessary (c) Vehicles carrying inflammable materials usually have metallic\nropes touching the ground during motion"}, {"Chapter": "1", "sentence_range": "1948-1951", "Text": "But special rubber tyres of\naircraft are made slightly conducting Why is this necessary (c) Vehicles carrying inflammable materials usually have metallic\nropes touching the ground during motion Why"}, {"Chapter": "1", "sentence_range": "1949-1952", "Text": "Why is this necessary (c) Vehicles carrying inflammable materials usually have metallic\nropes touching the ground during motion Why (d) A bird perches on a bare high power line, and nothing happens\nto the bird"}, {"Chapter": "1", "sentence_range": "1950-1953", "Text": "(c) Vehicles carrying inflammable materials usually have metallic\nropes touching the ground during motion Why (d) A bird perches on a bare high power line, and nothing happens\nto the bird A man standing on the ground touches the same line\nand gets a fatal shock"}, {"Chapter": "1", "sentence_range": "1951-1954", "Text": "Why (d) A bird perches on a bare high power line, and nothing happens\nto the bird A man standing on the ground touches the same line\nand gets a fatal shock Why"}, {"Chapter": "1", "sentence_range": "1952-1955", "Text": "(d) A bird perches on a bare high power line, and nothing happens\nto the bird A man standing on the ground touches the same line\nand gets a fatal shock Why Solution\n(a) This is because the comb gets charged by friction"}, {"Chapter": "1", "sentence_range": "1953-1956", "Text": "A man standing on the ground touches the same line\nand gets a fatal shock Why Solution\n(a) This is because the comb gets charged by friction The molecules\nin the paper gets polarised by the charged comb, resulting  in a\nnet force of attraction"}, {"Chapter": "1", "sentence_range": "1954-1957", "Text": "Why Solution\n(a) This is because the comb gets charged by friction The molecules\nin the paper gets polarised by the charged comb, resulting  in a\nnet force of attraction If the hair is wet, or if it is rainy day, friction\nbetween hair and the comb reduces"}, {"Chapter": "1", "sentence_range": "1955-1958", "Text": "Solution\n(a) This is because the comb gets charged by friction The molecules\nin the paper gets polarised by the charged comb, resulting  in a\nnet force of attraction If the hair is wet, or if it is rainy day, friction\nbetween hair and the comb reduces The comb does not get\ncharged and thus it will not attract small bits of paper"}, {"Chapter": "1", "sentence_range": "1956-1959", "Text": "The molecules\nin the paper gets polarised by the charged comb, resulting  in a\nnet force of attraction If the hair is wet, or if it is rainy day, friction\nbetween hair and the comb reduces The comb does not get\ncharged and thus it will not attract small bits of paper FIGURE 2"}, {"Chapter": "1", "sentence_range": "1957-1960", "Text": "If the hair is wet, or if it is rainy day, friction\nbetween hair and the comb reduces The comb does not get\ncharged and thus it will not attract small bits of paper FIGURE 2 19 Some important electrostatic properties of a conductor"}, {"Chapter": "1", "sentence_range": "1958-1961", "Text": "The comb does not get\ncharged and thus it will not attract small bits of paper FIGURE 2 19 Some important electrostatic properties of a conductor Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n65\n EXAMPLE 2"}, {"Chapter": "1", "sentence_range": "1959-1962", "Text": "FIGURE 2 19 Some important electrostatic properties of a conductor Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n65\n EXAMPLE 2 7\n(b) To enable them to conduct charge (produced by friction) to the\nground;  as too much of static electricity accumulated may result\nin spark and result in fire"}, {"Chapter": "1", "sentence_range": "1960-1963", "Text": "19 Some important electrostatic properties of a conductor Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n65\n EXAMPLE 2 7\n(b) To enable them to conduct charge (produced by friction) to the\nground;  as too much of static electricity accumulated may result\nin spark and result in fire (c) Reason similar to (b)"}, {"Chapter": "1", "sentence_range": "1961-1964", "Text": "Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n65\n EXAMPLE 2 7\n(b) To enable them to conduct charge (produced by friction) to the\nground;  as too much of static electricity accumulated may result\nin spark and result in fire (c) Reason similar to (b) (d) Current passes only when there is difference in potential"}, {"Chapter": "1", "sentence_range": "1962-1965", "Text": "7\n(b) To enable them to conduct charge (produced by friction) to the\nground;  as too much of static electricity accumulated may result\nin spark and result in fire (c) Reason similar to (b) (d) Current passes only when there is difference in potential 2"}, {"Chapter": "1", "sentence_range": "1963-1966", "Text": "(c) Reason similar to (b) (d) Current passes only when there is difference in potential 2 10  DIELECTRICS AND POLARISATION\nDielectrics are non-conducting substances"}, {"Chapter": "1", "sentence_range": "1964-1967", "Text": "(d) Current passes only when there is difference in potential 2 10  DIELECTRICS AND POLARISATION\nDielectrics are non-conducting substances In contrast to conductors,\nthey have no (or negligible number of ) charge carriers"}, {"Chapter": "1", "sentence_range": "1965-1968", "Text": "2 10  DIELECTRICS AND POLARISATION\nDielectrics are non-conducting substances In contrast to conductors,\nthey have no (or negligible number of ) charge carriers Recall from Section\n2"}, {"Chapter": "1", "sentence_range": "1966-1969", "Text": "10  DIELECTRICS AND POLARISATION\nDielectrics are non-conducting substances In contrast to conductors,\nthey have no (or negligible number of ) charge carriers Recall from Section\n2 9 what happens when a conductor is placed in an\nexternal electric field"}, {"Chapter": "1", "sentence_range": "1967-1970", "Text": "In contrast to conductors,\nthey have no (or negligible number of ) charge carriers Recall from Section\n2 9 what happens when a conductor is placed in an\nexternal electric field The free charge carriers move\nand charge distribution in the conductor adjusts\nitself in such a way that the electric field due to\ninduced charges opposes the external field within\nthe conductor"}, {"Chapter": "1", "sentence_range": "1968-1971", "Text": "Recall from Section\n2 9 what happens when a conductor is placed in an\nexternal electric field The free charge carriers move\nand charge distribution in the conductor adjusts\nitself in such a way that the electric field due to\ninduced charges opposes the external field within\nthe conductor This happens until, in the static\nsituation, the two fields cancel each other and the\nnet electrostatic field in the conductor is zero"}, {"Chapter": "1", "sentence_range": "1969-1972", "Text": "9 what happens when a conductor is placed in an\nexternal electric field The free charge carriers move\nand charge distribution in the conductor adjusts\nitself in such a way that the electric field due to\ninduced charges opposes the external field within\nthe conductor This happens until, in the static\nsituation, the two fields cancel each other and the\nnet electrostatic field in the conductor is zero In a\ndielectric, this free movement of charges is not\npossible"}, {"Chapter": "1", "sentence_range": "1970-1973", "Text": "The free charge carriers move\nand charge distribution in the conductor adjusts\nitself in such a way that the electric field due to\ninduced charges opposes the external field within\nthe conductor This happens until, in the static\nsituation, the two fields cancel each other and the\nnet electrostatic field in the conductor is zero In a\ndielectric, this free movement of charges is not\npossible It turns out that the external field induces\ndipole moment by stretching or re-orienting\nmolecules of the dielectric"}, {"Chapter": "1", "sentence_range": "1971-1974", "Text": "This happens until, in the static\nsituation, the two fields cancel each other and the\nnet electrostatic field in the conductor is zero In a\ndielectric, this free movement of charges is not\npossible It turns out that the external field induces\ndipole moment by stretching or re-orienting\nmolecules of the dielectric The collective effect of all\nthe molecular dipole moments is net charges on the\nsurface of the dielectric which produce a field that\nopposes the external field"}, {"Chapter": "1", "sentence_range": "1972-1975", "Text": "In a\ndielectric, this free movement of charges is not\npossible It turns out that the external field induces\ndipole moment by stretching or re-orienting\nmolecules of the dielectric The collective effect of all\nthe molecular dipole moments is net charges on the\nsurface of the dielectric which produce a field that\nopposes the external field Unlike in a conductor,\nhowever, the opposing field so induced does not\nexactly cancel the external field"}, {"Chapter": "1", "sentence_range": "1973-1976", "Text": "It turns out that the external field induces\ndipole moment by stretching or re-orienting\nmolecules of the dielectric The collective effect of all\nthe molecular dipole moments is net charges on the\nsurface of the dielectric which produce a field that\nopposes the external field Unlike in a conductor,\nhowever, the opposing field so induced does not\nexactly cancel the external field It only reduces it"}, {"Chapter": "1", "sentence_range": "1974-1977", "Text": "The collective effect of all\nthe molecular dipole moments is net charges on the\nsurface of the dielectric which produce a field that\nopposes the external field Unlike in a conductor,\nhowever, the opposing field so induced does not\nexactly cancel the external field It only reduces it The extent of the effect depends on the\nnature of the dielectric"}, {"Chapter": "1", "sentence_range": "1975-1978", "Text": "Unlike in a conductor,\nhowever, the opposing field so induced does not\nexactly cancel the external field It only reduces it The extent of the effect depends on the\nnature of the dielectric To understand the\neffect, we need to look at the charge\ndistribution of a dielectric at the\nmolecular level"}, {"Chapter": "1", "sentence_range": "1976-1979", "Text": "It only reduces it The extent of the effect depends on the\nnature of the dielectric To understand the\neffect, we need to look at the charge\ndistribution of a dielectric at the\nmolecular level The molecules of a substance may be\npolar or non-polar"}, {"Chapter": "1", "sentence_range": "1977-1980", "Text": "The extent of the effect depends on the\nnature of the dielectric To understand the\neffect, we need to look at the charge\ndistribution of a dielectric at the\nmolecular level The molecules of a substance may be\npolar or non-polar In a non-polar\nmolecule, the centres of positive and\nnegative charges coincide"}, {"Chapter": "1", "sentence_range": "1978-1981", "Text": "To understand the\neffect, we need to look at the charge\ndistribution of a dielectric at the\nmolecular level The molecules of a substance may be\npolar or non-polar In a non-polar\nmolecule, the centres of positive and\nnegative charges coincide The molecule\nthen has no permanent (or intrinsic) dipole\nmoment"}, {"Chapter": "1", "sentence_range": "1979-1982", "Text": "The molecules of a substance may be\npolar or non-polar In a non-polar\nmolecule, the centres of positive and\nnegative charges coincide The molecule\nthen has no permanent (or intrinsic) dipole\nmoment Examples of non-polar molecules\nare oxygen (O2) and hydrogen (H2)\nmolecules which, because of their\nsymmetry, have no dipole moment"}, {"Chapter": "1", "sentence_range": "1980-1983", "Text": "In a non-polar\nmolecule, the centres of positive and\nnegative charges coincide The molecule\nthen has no permanent (or intrinsic) dipole\nmoment Examples of non-polar molecules\nare oxygen (O2) and hydrogen (H2)\nmolecules which, because of their\nsymmetry, have no dipole moment On the\nother hand, a polar molecule is one in which\nthe centres of positive and negative charges\nare separated (even when there is no\nexternal field)"}, {"Chapter": "1", "sentence_range": "1981-1984", "Text": "The molecule\nthen has no permanent (or intrinsic) dipole\nmoment Examples of non-polar molecules\nare oxygen (O2) and hydrogen (H2)\nmolecules which, because of their\nsymmetry, have no dipole moment On the\nother hand, a polar molecule is one in which\nthe centres of positive and negative charges\nare separated (even when there is no\nexternal field) Such molecules have a\npermanent dipole moment"}, {"Chapter": "1", "sentence_range": "1982-1985", "Text": "Examples of non-polar molecules\nare oxygen (O2) and hydrogen (H2)\nmolecules which, because of their\nsymmetry, have no dipole moment On the\nother hand, a polar molecule is one in which\nthe centres of positive and negative charges\nare separated (even when there is no\nexternal field) Such molecules have a\npermanent dipole moment An ionic\nmolecule such as HCl or a molecule of water\n(H2O) are examples of polar molecules"}, {"Chapter": "1", "sentence_range": "1983-1986", "Text": "On the\nother hand, a polar molecule is one in which\nthe centres of positive and negative charges\nare separated (even when there is no\nexternal field) Such molecules have a\npermanent dipole moment An ionic\nmolecule such as HCl or a molecule of water\n(H2O) are examples of polar molecules FIGURE 2"}, {"Chapter": "1", "sentence_range": "1984-1987", "Text": "Such molecules have a\npermanent dipole moment An ionic\nmolecule such as HCl or a molecule of water\n(H2O) are examples of polar molecules FIGURE 2 20 Difference in behaviour\nof a conductor and a dielectric\nin an external electric field"}, {"Chapter": "1", "sentence_range": "1985-1988", "Text": "An ionic\nmolecule such as HCl or a molecule of water\n(H2O) are examples of polar molecules FIGURE 2 20 Difference in behaviour\nof a conductor and a dielectric\nin an external electric field FIGURE 2"}, {"Chapter": "1", "sentence_range": "1986-1989", "Text": "FIGURE 2 20 Difference in behaviour\nof a conductor and a dielectric\nin an external electric field FIGURE 2 21 Some examples of polar\nand non-polar molecules"}, {"Chapter": "1", "sentence_range": "1987-1990", "Text": "20 Difference in behaviour\nof a conductor and a dielectric\nin an external electric field FIGURE 2 21 Some examples of polar\nand non-polar molecules Rationalised 2023-24\nPhysics\n66\nIn an external electric field, the\npositive and negative charges of a non-\npolar molecule are displaced in opposite\ndirections"}, {"Chapter": "1", "sentence_range": "1988-1991", "Text": "FIGURE 2 21 Some examples of polar\nand non-polar molecules Rationalised 2023-24\nPhysics\n66\nIn an external electric field, the\npositive and negative charges of a non-\npolar molecule are displaced in opposite\ndirections The displacement stops when\nthe external force on the constituent\ncharges of the molecule is balanced by\nthe restoring force (due to internal fields\nin the molecule)"}, {"Chapter": "1", "sentence_range": "1989-1992", "Text": "21 Some examples of polar\nand non-polar molecules Rationalised 2023-24\nPhysics\n66\nIn an external electric field, the\npositive and negative charges of a non-\npolar molecule are displaced in opposite\ndirections The displacement stops when\nthe external force on the constituent\ncharges of the molecule is balanced by\nthe restoring force (due to internal fields\nin the molecule) The non-polar molecule\nthus develops an induced dipole moment"}, {"Chapter": "1", "sentence_range": "1990-1993", "Text": "Rationalised 2023-24\nPhysics\n66\nIn an external electric field, the\npositive and negative charges of a non-\npolar molecule are displaced in opposite\ndirections The displacement stops when\nthe external force on the constituent\ncharges of the molecule is balanced by\nthe restoring force (due to internal fields\nin the molecule) The non-polar molecule\nthus develops an induced dipole moment The dielectric is said to be polarised by\nthe external field"}, {"Chapter": "1", "sentence_range": "1991-1994", "Text": "The displacement stops when\nthe external force on the constituent\ncharges of the molecule is balanced by\nthe restoring force (due to internal fields\nin the molecule) The non-polar molecule\nthus develops an induced dipole moment The dielectric is said to be polarised by\nthe external field We consider only the\nsimple situation when the induced dipole\nmoment is in the direction of the field and\nis proportional to the field strength"}, {"Chapter": "1", "sentence_range": "1992-1995", "Text": "The non-polar molecule\nthus develops an induced dipole moment The dielectric is said to be polarised by\nthe external field We consider only the\nsimple situation when the induced dipole\nmoment is in the direction of the field and\nis proportional to the field strength (Substances for which this assumption\nis true are called linear isotropic\ndielectrics"}, {"Chapter": "1", "sentence_range": "1993-1996", "Text": "The dielectric is said to be polarised by\nthe external field We consider only the\nsimple situation when the induced dipole\nmoment is in the direction of the field and\nis proportional to the field strength (Substances for which this assumption\nis true are called linear isotropic\ndielectrics ) The induced dipole moments\nof different molecules add up giving a net\ndipole moment of the dielectric in the\npresence of the external field"}, {"Chapter": "1", "sentence_range": "1994-1997", "Text": "We consider only the\nsimple situation when the induced dipole\nmoment is in the direction of the field and\nis proportional to the field strength (Substances for which this assumption\nis true are called linear isotropic\ndielectrics ) The induced dipole moments\nof different molecules add up giving a net\ndipole moment of the dielectric in the\npresence of the external field A dielectric with polar molecules also\ndevelops a net dipole moment in an\nexternal field, but for a different reason"}, {"Chapter": "1", "sentence_range": "1995-1998", "Text": "(Substances for which this assumption\nis true are called linear isotropic\ndielectrics ) The induced dipole moments\nof different molecules add up giving a net\ndipole moment of the dielectric in the\npresence of the external field A dielectric with polar molecules also\ndevelops a net dipole moment in an\nexternal field, but for a different reason In the absence of any external field, the\ndifferent permanent dipoles are oriented\nrandomly due to thermal agitation; so\nthe total dipole moment is zero"}, {"Chapter": "1", "sentence_range": "1996-1999", "Text": ") The induced dipole moments\nof different molecules add up giving a net\ndipole moment of the dielectric in the\npresence of the external field A dielectric with polar molecules also\ndevelops a net dipole moment in an\nexternal field, but for a different reason In the absence of any external field, the\ndifferent permanent dipoles are oriented\nrandomly due to thermal agitation; so\nthe total dipole moment is zero When\nan external field is applied, the individual dipole moments tend  to align\nwith the field"}, {"Chapter": "1", "sentence_range": "1997-2000", "Text": "A dielectric with polar molecules also\ndevelops a net dipole moment in an\nexternal field, but for a different reason In the absence of any external field, the\ndifferent permanent dipoles are oriented\nrandomly due to thermal agitation; so\nthe total dipole moment is zero When\nan external field is applied, the individual dipole moments tend  to align\nwith the field When summed overall the molecules, there is then a net\ndipole moment in the direction of the external field, i"}, {"Chapter": "1", "sentence_range": "1998-2001", "Text": "In the absence of any external field, the\ndifferent permanent dipoles are oriented\nrandomly due to thermal agitation; so\nthe total dipole moment is zero When\nan external field is applied, the individual dipole moments tend  to align\nwith the field When summed overall the molecules, there is then a net\ndipole moment in the direction of the external field, i e"}, {"Chapter": "1", "sentence_range": "1999-2002", "Text": "When\nan external field is applied, the individual dipole moments tend  to align\nwith the field When summed overall the molecules, there is then a net\ndipole moment in the direction of the external field, i e , the dielectric is\npolarised"}, {"Chapter": "1", "sentence_range": "2000-2003", "Text": "When summed overall the molecules, there is then a net\ndipole moment in the direction of the external field, i e , the dielectric is\npolarised The extent of polarisation depends on the relative strength of\ntwo mutually opposite factors: the dipole potential energy in the external\nfield tending to align the dipoles with the field and thermal energy tending\nto disrupt the alignment"}, {"Chapter": "1", "sentence_range": "2001-2004", "Text": "e , the dielectric is\npolarised The extent of polarisation depends on the relative strength of\ntwo mutually opposite factors: the dipole potential energy in the external\nfield tending to align the dipoles with the field and thermal energy tending\nto disrupt the alignment There may be, in addition, the  \u2018induced dipole\nmoment\u2019 effect as for non-polar molecules, but generally the alignment\neffect is more important for polar molecules"}, {"Chapter": "1", "sentence_range": "2002-2005", "Text": ", the dielectric is\npolarised The extent of polarisation depends on the relative strength of\ntwo mutually opposite factors: the dipole potential energy in the external\nfield tending to align the dipoles with the field and thermal energy tending\nto disrupt the alignment There may be, in addition, the  \u2018induced dipole\nmoment\u2019 effect as for non-polar molecules, but generally the alignment\neffect is more important for polar molecules Thus in either case, whether polar or non-polar, a dielectric develops\na net dipole moment in the presence of an external field"}, {"Chapter": "1", "sentence_range": "2003-2006", "Text": "The extent of polarisation depends on the relative strength of\ntwo mutually opposite factors: the dipole potential energy in the external\nfield tending to align the dipoles with the field and thermal energy tending\nto disrupt the alignment There may be, in addition, the  \u2018induced dipole\nmoment\u2019 effect as for non-polar molecules, but generally the alignment\neffect is more important for polar molecules Thus in either case, whether polar or non-polar, a dielectric develops\na net dipole moment in the presence of an external field The dipole\nmoment per unit volume is called polarisation and is denoted by P"}, {"Chapter": "1", "sentence_range": "2004-2007", "Text": "There may be, in addition, the  \u2018induced dipole\nmoment\u2019 effect as for non-polar molecules, but generally the alignment\neffect is more important for polar molecules Thus in either case, whether polar or non-polar, a dielectric develops\na net dipole moment in the presence of an external field The dipole\nmoment per unit volume is called polarisation and is denoted by P For\nlinear isotropic dielectrics,\n0\nP=\u03b5 \u03c7\neE\n(2"}, {"Chapter": "1", "sentence_range": "2005-2008", "Text": "Thus in either case, whether polar or non-polar, a dielectric develops\na net dipole moment in the presence of an external field The dipole\nmoment per unit volume is called polarisation and is denoted by P For\nlinear isotropic dielectrics,\n0\nP=\u03b5 \u03c7\neE\n(2 37)\nwhere ce is a constant characteristic of the dielectric and is known as the\nelectric susceptibility of the dielectric medium"}, {"Chapter": "1", "sentence_range": "2006-2009", "Text": "The dipole\nmoment per unit volume is called polarisation and is denoted by P For\nlinear isotropic dielectrics,\n0\nP=\u03b5 \u03c7\neE\n(2 37)\nwhere ce is a constant characteristic of the dielectric and is known as the\nelectric susceptibility of the dielectric medium It is possible to relate ce to the molecular properties of the substance,\nbut we shall not pursue that here"}, {"Chapter": "1", "sentence_range": "2007-2010", "Text": "For\nlinear isotropic dielectrics,\n0\nP=\u03b5 \u03c7\neE\n(2 37)\nwhere ce is a constant characteristic of the dielectric and is known as the\nelectric susceptibility of the dielectric medium It is possible to relate ce to the molecular properties of the substance,\nbut we shall not pursue that here The question is: how does the polarised dielectric modify the original\nexternal field inside it"}, {"Chapter": "1", "sentence_range": "2008-2011", "Text": "37)\nwhere ce is a constant characteristic of the dielectric and is known as the\nelectric susceptibility of the dielectric medium It is possible to relate ce to the molecular properties of the substance,\nbut we shall not pursue that here The question is: how does the polarised dielectric modify the original\nexternal field inside it Let us consider, for simplicity, a rectangular\ndielectric slab placed in a uniform external field E0 parallel to two of its\nfaces"}, {"Chapter": "1", "sentence_range": "2009-2012", "Text": "It is possible to relate ce to the molecular properties of the substance,\nbut we shall not pursue that here The question is: how does the polarised dielectric modify the original\nexternal field inside it Let us consider, for simplicity, a rectangular\ndielectric slab placed in a uniform external field E0 parallel to two of its\nfaces The field causes a uniform polarisation P of the dielectric"}, {"Chapter": "1", "sentence_range": "2010-2013", "Text": "The question is: how does the polarised dielectric modify the original\nexternal field inside it Let us consider, for simplicity, a rectangular\ndielectric slab placed in a uniform external field E0 parallel to two of its\nfaces The field causes a uniform polarisation P of the dielectric Thus\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "2011-2014", "Text": "Let us consider, for simplicity, a rectangular\ndielectric slab placed in a uniform external field E0 parallel to two of its\nfaces The field causes a uniform polarisation P of the dielectric Thus\nFIGURE 2 22 A dielectric develops a net dipole\nmoment in an external electric field"}, {"Chapter": "1", "sentence_range": "2012-2015", "Text": "The field causes a uniform polarisation P of the dielectric Thus\nFIGURE 2 22 A dielectric develops a net dipole\nmoment in an external electric field (a) Non-polar\nmolecules, (b) Polar molecules"}, {"Chapter": "1", "sentence_range": "2013-2016", "Text": "Thus\nFIGURE 2 22 A dielectric develops a net dipole\nmoment in an external electric field (a) Non-polar\nmolecules, (b) Polar molecules Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n67\nevery volume element Dv of the slab has a dipole moment\nP Dv in the direction of the field"}, {"Chapter": "1", "sentence_range": "2014-2017", "Text": "22 A dielectric develops a net dipole\nmoment in an external electric field (a) Non-polar\nmolecules, (b) Polar molecules Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n67\nevery volume element Dv of the slab has a dipole moment\nP Dv in the direction of the field The volume element Dv  is\nmacroscopically small but contains a very large number of\nmolecular dipoles"}, {"Chapter": "1", "sentence_range": "2015-2018", "Text": "(a) Non-polar\nmolecules, (b) Polar molecules Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n67\nevery volume element Dv of the slab has a dipole moment\nP Dv in the direction of the field The volume element Dv  is\nmacroscopically small but contains a very large number of\nmolecular dipoles Anywhere inside the dielectric, the\nvolume element Dv has no net charge (though it has net\ndipole moment)"}, {"Chapter": "1", "sentence_range": "2016-2019", "Text": "Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n67\nevery volume element Dv of the slab has a dipole moment\nP Dv in the direction of the field The volume element Dv  is\nmacroscopically small but contains a very large number of\nmolecular dipoles Anywhere inside the dielectric, the\nvolume element Dv has no net charge (though it has net\ndipole moment) This is, because, the positive charge of one\ndipole sits close to the negative charge of the adjacent dipole"}, {"Chapter": "1", "sentence_range": "2017-2020", "Text": "The volume element Dv  is\nmacroscopically small but contains a very large number of\nmolecular dipoles Anywhere inside the dielectric, the\nvolume element Dv has no net charge (though it has net\ndipole moment) This is, because, the positive charge of one\ndipole sits close to the negative charge of the adjacent dipole However, at the surfaces of the dielectric normal to the\nelectric field, there is evidently a net charge density"}, {"Chapter": "1", "sentence_range": "2018-2021", "Text": "Anywhere inside the dielectric, the\nvolume element Dv has no net charge (though it has net\ndipole moment) This is, because, the positive charge of one\ndipole sits close to the negative charge of the adjacent dipole However, at the surfaces of the dielectric normal to the\nelectric field, there is evidently a net charge density As seen\nin Fig 2"}, {"Chapter": "1", "sentence_range": "2019-2022", "Text": "This is, because, the positive charge of one\ndipole sits close to the negative charge of the adjacent dipole However, at the surfaces of the dielectric normal to the\nelectric field, there is evidently a net charge density As seen\nin Fig 2 23, the positive ends of the dipoles remain\nunneutralised at the right surface and the negative ends at\nthe left surface"}, {"Chapter": "1", "sentence_range": "2020-2023", "Text": "However, at the surfaces of the dielectric normal to the\nelectric field, there is evidently a net charge density As seen\nin Fig 2 23, the positive ends of the dipoles remain\nunneutralised at the right surface and the negative ends at\nthe left surface The unbalanced charges are the induced\ncharges due to the external field"}, {"Chapter": "1", "sentence_range": "2021-2024", "Text": "As seen\nin Fig 2 23, the positive ends of the dipoles remain\nunneutralised at the right surface and the negative ends at\nthe left surface The unbalanced charges are the induced\ncharges due to the external field Thus, the polarised dielectric is equivalent to two charged\nsurfaces with induced surface charge densities, say sp\nand \u2013sp"}, {"Chapter": "1", "sentence_range": "2022-2025", "Text": "23, the positive ends of the dipoles remain\nunneutralised at the right surface and the negative ends at\nthe left surface The unbalanced charges are the induced\ncharges due to the external field Thus, the polarised dielectric is equivalent to two charged\nsurfaces with induced surface charge densities, say sp\nand \u2013sp Clearly, the field produced by these surface charges\nopposes the external field"}, {"Chapter": "1", "sentence_range": "2023-2026", "Text": "The unbalanced charges are the induced\ncharges due to the external field Thus, the polarised dielectric is equivalent to two charged\nsurfaces with induced surface charge densities, say sp\nand \u2013sp Clearly, the field produced by these surface charges\nopposes the external field The total field in the dielectric\nis, thereby, reduced from the case when no dielectric is\npresent"}, {"Chapter": "1", "sentence_range": "2024-2027", "Text": "Thus, the polarised dielectric is equivalent to two charged\nsurfaces with induced surface charge densities, say sp\nand \u2013sp Clearly, the field produced by these surface charges\nopposes the external field The total field in the dielectric\nis, thereby, reduced from the case when no dielectric is\npresent We should note that the surface charge density\n\u00b1sp arises from bound (not free charges) in the dielectric"}, {"Chapter": "1", "sentence_range": "2025-2028", "Text": "Clearly, the field produced by these surface charges\nopposes the external field The total field in the dielectric\nis, thereby, reduced from the case when no dielectric is\npresent We should note that the surface charge density\n\u00b1sp arises from bound (not free charges) in the dielectric 2"}, {"Chapter": "1", "sentence_range": "2026-2029", "Text": "The total field in the dielectric\nis, thereby, reduced from the case when no dielectric is\npresent We should note that the surface charge density\n\u00b1sp arises from bound (not free charges) in the dielectric 2 11  CAPACITORS AND CAPACITANCE\nA capacitor is a system of two conductors separated by an insulator\n(Fig"}, {"Chapter": "1", "sentence_range": "2027-2030", "Text": "We should note that the surface charge density\n\u00b1sp arises from bound (not free charges) in the dielectric 2 11  CAPACITORS AND CAPACITANCE\nA capacitor is a system of two conductors separated by an insulator\n(Fig 2"}, {"Chapter": "1", "sentence_range": "2028-2031", "Text": "2 11  CAPACITORS AND CAPACITANCE\nA capacitor is a system of two conductors separated by an insulator\n(Fig 2 24)"}, {"Chapter": "1", "sentence_range": "2029-2032", "Text": "11  CAPACITORS AND CAPACITANCE\nA capacitor is a system of two conductors separated by an insulator\n(Fig 2 24) The conductors have charges, say Q1 and Q2, and potentials\nV1 and V2"}, {"Chapter": "1", "sentence_range": "2030-2033", "Text": "2 24) The conductors have charges, say Q1 and Q2, and potentials\nV1 and V2 Usually, in practice, the two conductors have charges Q\nand \u2013 Q, with potential difference V = V1 \u2013 V2 between them"}, {"Chapter": "1", "sentence_range": "2031-2034", "Text": "24) The conductors have charges, say Q1 and Q2, and potentials\nV1 and V2 Usually, in practice, the two conductors have charges Q\nand \u2013 Q, with potential difference V = V1 \u2013 V2 between them We shall\nconsider only this kind of charge configuration of the capacitor"}, {"Chapter": "1", "sentence_range": "2032-2035", "Text": "The conductors have charges, say Q1 and Q2, and potentials\nV1 and V2 Usually, in practice, the two conductors have charges Q\nand \u2013 Q, with potential difference V = V1 \u2013 V2 between them We shall\nconsider only this kind of charge configuration of the capacitor (Even a\nsingle conductor can be used as a capacitor by assuming the other at\ninfinity"}, {"Chapter": "1", "sentence_range": "2033-2036", "Text": "Usually, in practice, the two conductors have charges Q\nand \u2013 Q, with potential difference V = V1 \u2013 V2 between them We shall\nconsider only this kind of charge configuration of the capacitor (Even a\nsingle conductor can be used as a capacitor by assuming the other at\ninfinity ) The conductors may be so charged by connecting them to the\ntwo terminals of a battery"}, {"Chapter": "1", "sentence_range": "2034-2037", "Text": "We shall\nconsider only this kind of charge configuration of the capacitor (Even a\nsingle conductor can be used as a capacitor by assuming the other at\ninfinity ) The conductors may be so charged by connecting them to the\ntwo terminals of a battery Q is called the charge of the capacitor, though\nthis, in fact, is the charge on one of the conductors \u2013 the total charge  of\nthe capacitor  is zero"}, {"Chapter": "1", "sentence_range": "2035-2038", "Text": "(Even a\nsingle conductor can be used as a capacitor by assuming the other at\ninfinity ) The conductors may be so charged by connecting them to the\ntwo terminals of a battery Q is called the charge of the capacitor, though\nthis, in fact, is the charge on one of the conductors \u2013 the total charge  of\nthe capacitor  is zero The electric field in the region between the\nconductors is proportional to the charge Q"}, {"Chapter": "1", "sentence_range": "2036-2039", "Text": ") The conductors may be so charged by connecting them to the\ntwo terminals of a battery Q is called the charge of the capacitor, though\nthis, in fact, is the charge on one of the conductors \u2013 the total charge  of\nthe capacitor  is zero The electric field in the region between the\nconductors is proportional to the charge Q That\nis, if the charge on the capacitor is, say  doubled,\nthe electric field will also be doubled at every point"}, {"Chapter": "1", "sentence_range": "2037-2040", "Text": "Q is called the charge of the capacitor, though\nthis, in fact, is the charge on one of the conductors \u2013 the total charge  of\nthe capacitor  is zero The electric field in the region between the\nconductors is proportional to the charge Q That\nis, if the charge on the capacitor is, say  doubled,\nthe electric field will also be doubled at every point (This follows from the direct proportionality\nbetween field and charge implied by Coulomb\u2019s\nlaw and the superposition principle"}, {"Chapter": "1", "sentence_range": "2038-2041", "Text": "The electric field in the region between the\nconductors is proportional to the charge Q That\nis, if the charge on the capacitor is, say  doubled,\nthe electric field will also be doubled at every point (This follows from the direct proportionality\nbetween field and charge implied by Coulomb\u2019s\nlaw and the superposition principle ) Now,\npotential difference V is the work done per unit\npositive charge in taking a small test charge from\nthe conductor 2 to 1 against the field"}, {"Chapter": "1", "sentence_range": "2039-2042", "Text": "That\nis, if the charge on the capacitor is, say  doubled,\nthe electric field will also be doubled at every point (This follows from the direct proportionality\nbetween field and charge implied by Coulomb\u2019s\nlaw and the superposition principle ) Now,\npotential difference V is the work done per unit\npositive charge in taking a small test charge from\nthe conductor 2 to 1 against the field Consequently, V is also proportional to Q, and the\nratio Q/V is a constant:\nQ\nC\n=V\n(2"}, {"Chapter": "1", "sentence_range": "2040-2043", "Text": "(This follows from the direct proportionality\nbetween field and charge implied by Coulomb\u2019s\nlaw and the superposition principle ) Now,\npotential difference V is the work done per unit\npositive charge in taking a small test charge from\nthe conductor 2 to 1 against the field Consequently, V is also proportional to Q, and the\nratio Q/V is a constant:\nQ\nC\n=V\n(2 38)\nThe constant C is called the capacitance of the capacitor"}, {"Chapter": "1", "sentence_range": "2041-2044", "Text": ") Now,\npotential difference V is the work done per unit\npositive charge in taking a small test charge from\nthe conductor 2 to 1 against the field Consequently, V is also proportional to Q, and the\nratio Q/V is a constant:\nQ\nC\n=V\n(2 38)\nThe constant C is called the capacitance of the capacitor C is independent\nof Q or V, as stated above"}, {"Chapter": "1", "sentence_range": "2042-2045", "Text": "Consequently, V is also proportional to Q, and the\nratio Q/V is a constant:\nQ\nC\n=V\n(2 38)\nThe constant C is called the capacitance of the capacitor C is independent\nof Q or V, as stated above The capacitance C depends only on the\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "2043-2046", "Text": "38)\nThe constant C is called the capacitance of the capacitor C is independent\nof Q or V, as stated above The capacitance C depends only on the\nFIGURE 2 23 A uniformly\npolarised dielectric amounts\nto induced surface charge\ndensity, but no volume\ncharge density"}, {"Chapter": "1", "sentence_range": "2044-2047", "Text": "C is independent\nof Q or V, as stated above The capacitance C depends only on the\nFIGURE 2 23 A uniformly\npolarised dielectric amounts\nto induced surface charge\ndensity, but no volume\ncharge density FIGURE 2"}, {"Chapter": "1", "sentence_range": "2045-2048", "Text": "The capacitance C depends only on the\nFIGURE 2 23 A uniformly\npolarised dielectric amounts\nto induced surface charge\ndensity, but no volume\ncharge density FIGURE 2 24 A system of two conductors\nseparated by an insulator forms a capacitor"}, {"Chapter": "1", "sentence_range": "2046-2049", "Text": "23 A uniformly\npolarised dielectric amounts\nto induced surface charge\ndensity, but no volume\ncharge density FIGURE 2 24 A system of two conductors\nseparated by an insulator forms a capacitor Rationalised 2023-24\nPhysics\n68\ngeometrical configuration (shape, size, separation) of the system of two\nconductors"}, {"Chapter": "1", "sentence_range": "2047-2050", "Text": "FIGURE 2 24 A system of two conductors\nseparated by an insulator forms a capacitor Rationalised 2023-24\nPhysics\n68\ngeometrical configuration (shape, size, separation) of the system of two\nconductors [As we shall see later, it also depends on the nature of the\ninsulator (dielectric) separating the two conductors"}, {"Chapter": "1", "sentence_range": "2048-2051", "Text": "24 A system of two conductors\nseparated by an insulator forms a capacitor Rationalised 2023-24\nPhysics\n68\ngeometrical configuration (shape, size, separation) of the system of two\nconductors [As we shall see later, it also depends on the nature of the\ninsulator (dielectric) separating the two conductors ]  The SI unit of\ncapacitance is 1 farad (=1 coulomb volt-1) or 1 F = 1 C V\u20131"}, {"Chapter": "1", "sentence_range": "2049-2052", "Text": "Rationalised 2023-24\nPhysics\n68\ngeometrical configuration (shape, size, separation) of the system of two\nconductors [As we shall see later, it also depends on the nature of the\ninsulator (dielectric) separating the two conductors ]  The SI unit of\ncapacitance is 1 farad (=1 coulomb volt-1) or 1 F = 1 C V\u20131 A capacitor\nwith fixed capacitance is symbolically shown as ---||---, while the one with\nvariable capacitance is shown as"}, {"Chapter": "1", "sentence_range": "2050-2053", "Text": "[As we shall see later, it also depends on the nature of the\ninsulator (dielectric) separating the two conductors ]  The SI unit of\ncapacitance is 1 farad (=1 coulomb volt-1) or 1 F = 1 C V\u20131 A capacitor\nwith fixed capacitance is symbolically shown as ---||---, while the one with\nvariable capacitance is shown as Equation (2"}, {"Chapter": "1", "sentence_range": "2051-2054", "Text": "]  The SI unit of\ncapacitance is 1 farad (=1 coulomb volt-1) or 1 F = 1 C V\u20131 A capacitor\nwith fixed capacitance is symbolically shown as ---||---, while the one with\nvariable capacitance is shown as Equation (2 38) shows that for large C, V is small for a given Q"}, {"Chapter": "1", "sentence_range": "2052-2055", "Text": "A capacitor\nwith fixed capacitance is symbolically shown as ---||---, while the one with\nvariable capacitance is shown as Equation (2 38) shows that for large C, V is small for a given Q This\nmeans a capacitor with large capacitance can hold large amount of charge\nQ at a relatively small V"}, {"Chapter": "1", "sentence_range": "2053-2056", "Text": "Equation (2 38) shows that for large C, V is small for a given Q This\nmeans a capacitor with large capacitance can hold large amount of charge\nQ at a relatively small V This is of practical importance"}, {"Chapter": "1", "sentence_range": "2054-2057", "Text": "38) shows that for large C, V is small for a given Q This\nmeans a capacitor with large capacitance can hold large amount of charge\nQ at a relatively small V This is of practical importance High potential\ndifference implies strong electric field around the conductors"}, {"Chapter": "1", "sentence_range": "2055-2058", "Text": "This\nmeans a capacitor with large capacitance can hold large amount of charge\nQ at a relatively small V This is of practical importance High potential\ndifference implies strong electric field around the conductors A strong\nelectric field can ionise the surrounding air and accelerate the charges so\nproduced to the oppositely charged plates, thereby neutralising the charge\non the capacitor plates, at least partly"}, {"Chapter": "1", "sentence_range": "2056-2059", "Text": "This is of practical importance High potential\ndifference implies strong electric field around the conductors A strong\nelectric field can ionise the surrounding air and accelerate the charges so\nproduced to the oppositely charged plates, thereby neutralising the charge\non the capacitor plates, at least partly In other words, the charge of the\ncapacitor leaks away due to the reduction in insulating power of the\nintervening medium"}, {"Chapter": "1", "sentence_range": "2057-2060", "Text": "High potential\ndifference implies strong electric field around the conductors A strong\nelectric field can ionise the surrounding air and accelerate the charges so\nproduced to the oppositely charged plates, thereby neutralising the charge\non the capacitor plates, at least partly In other words, the charge of the\ncapacitor leaks away due to the reduction in insulating power of the\nintervening medium The maximum electric field that a dielectric medium can withstand\nwithout break-down (of its insulating property) is called its dielectric\nstrength; for air it is about 3 \u00d7 106 Vm\u20131"}, {"Chapter": "1", "sentence_range": "2058-2061", "Text": "A strong\nelectric field can ionise the surrounding air and accelerate the charges so\nproduced to the oppositely charged plates, thereby neutralising the charge\non the capacitor plates, at least partly In other words, the charge of the\ncapacitor leaks away due to the reduction in insulating power of the\nintervening medium The maximum electric field that a dielectric medium can withstand\nwithout break-down (of its insulating property) is called its dielectric\nstrength; for air it is about 3 \u00d7 106 Vm\u20131 For a separation between\nconductors of the order of 1 cm or so, this field corresponds to a potential\ndifference of 3 \u00d7 104 V between the conductors"}, {"Chapter": "1", "sentence_range": "2059-2062", "Text": "In other words, the charge of the\ncapacitor leaks away due to the reduction in insulating power of the\nintervening medium The maximum electric field that a dielectric medium can withstand\nwithout break-down (of its insulating property) is called its dielectric\nstrength; for air it is about 3 \u00d7 106 Vm\u20131 For a separation between\nconductors of the order of 1 cm or so, this field corresponds to a potential\ndifference of 3 \u00d7 104 V between the conductors Thus, for a capacitor to\nstore a large amount of charge without  leaking, its capacitance should\nbe high enough so that the potential difference and hence the electric\nfield do not exceed the break-down limits"}, {"Chapter": "1", "sentence_range": "2060-2063", "Text": "The maximum electric field that a dielectric medium can withstand\nwithout break-down (of its insulating property) is called its dielectric\nstrength; for air it is about 3 \u00d7 106 Vm\u20131 For a separation between\nconductors of the order of 1 cm or so, this field corresponds to a potential\ndifference of 3 \u00d7 104 V between the conductors Thus, for a capacitor to\nstore a large amount of charge without  leaking, its capacitance should\nbe high enough so that the potential difference and hence the electric\nfield do not exceed the break-down limits Put differently, there is a limit\nto the amount of charge that can be stored on a given capacitor without\nsignificant leaking"}, {"Chapter": "1", "sentence_range": "2061-2064", "Text": "For a separation between\nconductors of the order of 1 cm or so, this field corresponds to a potential\ndifference of 3 \u00d7 104 V between the conductors Thus, for a capacitor to\nstore a large amount of charge without  leaking, its capacitance should\nbe high enough so that the potential difference and hence the electric\nfield do not exceed the break-down limits Put differently, there is a limit\nto the amount of charge that can be stored on a given capacitor without\nsignificant leaking In practice, a farad is a very big unit; the most common\nunits are its sub-multiples 1 mF = 10\u20136 F, 1 nF = 10\u20139 F, 1 pF = 10\u201312 F,\netc"}, {"Chapter": "1", "sentence_range": "2062-2065", "Text": "Thus, for a capacitor to\nstore a large amount of charge without  leaking, its capacitance should\nbe high enough so that the potential difference and hence the electric\nfield do not exceed the break-down limits Put differently, there is a limit\nto the amount of charge that can be stored on a given capacitor without\nsignificant leaking In practice, a farad is a very big unit; the most common\nunits are its sub-multiples 1 mF = 10\u20136 F, 1 nF = 10\u20139 F, 1 pF = 10\u201312 F,\netc Besides its use in storing charge, a capacitor is a key element of most\nac circuits with important functions, as described in Chapter 7"}, {"Chapter": "1", "sentence_range": "2063-2066", "Text": "Put differently, there is a limit\nto the amount of charge that can be stored on a given capacitor without\nsignificant leaking In practice, a farad is a very big unit; the most common\nunits are its sub-multiples 1 mF = 10\u20136 F, 1 nF = 10\u20139 F, 1 pF = 10\u201312 F,\netc Besides its use in storing charge, a capacitor is a key element of most\nac circuits with important functions, as described in Chapter 7 2"}, {"Chapter": "1", "sentence_range": "2064-2067", "Text": "In practice, a farad is a very big unit; the most common\nunits are its sub-multiples 1 mF = 10\u20136 F, 1 nF = 10\u20139 F, 1 pF = 10\u201312 F,\netc Besides its use in storing charge, a capacitor is a key element of most\nac circuits with important functions, as described in Chapter 7 2 12  THE PARALLEL PLATE CAPACITOR\nA parallel plate capacitor consists of two large plane parallel conducting\nplates separated by a small distance (Fig"}, {"Chapter": "1", "sentence_range": "2065-2068", "Text": "Besides its use in storing charge, a capacitor is a key element of most\nac circuits with important functions, as described in Chapter 7 2 12  THE PARALLEL PLATE CAPACITOR\nA parallel plate capacitor consists of two large plane parallel conducting\nplates separated by a small distance (Fig 2"}, {"Chapter": "1", "sentence_range": "2066-2069", "Text": "2 12  THE PARALLEL PLATE CAPACITOR\nA parallel plate capacitor consists of two large plane parallel conducting\nplates separated by a small distance (Fig 2 25)"}, {"Chapter": "1", "sentence_range": "2067-2070", "Text": "12  THE PARALLEL PLATE CAPACITOR\nA parallel plate capacitor consists of two large plane parallel conducting\nplates separated by a small distance (Fig 2 25) We first take the\nintervening medium between the plates to be\nvacuum"}, {"Chapter": "1", "sentence_range": "2068-2071", "Text": "2 25) We first take the\nintervening medium between the plates to be\nvacuum The effect of a dielectric medium between\nthe plates is discussed in the next section"}, {"Chapter": "1", "sentence_range": "2069-2072", "Text": "25) We first take the\nintervening medium between the plates to be\nvacuum The effect of a dielectric medium between\nthe plates is discussed in the next section Let A be\nthe area of each plate and d the separation between\nthem"}, {"Chapter": "1", "sentence_range": "2070-2073", "Text": "We first take the\nintervening medium between the plates to be\nvacuum The effect of a dielectric medium between\nthe plates is discussed in the next section Let A be\nthe area of each plate and d the separation between\nthem The two plates have charges Q and \u2013Q"}, {"Chapter": "1", "sentence_range": "2071-2074", "Text": "The effect of a dielectric medium between\nthe plates is discussed in the next section Let A be\nthe area of each plate and d the separation between\nthem The two plates have charges Q and \u2013Q Since\nd is much smaller than the linear dimension of the\nplates (d2 << A), we can use the result on electric\nfield by an infinite plane sheet of uniform surface\ncharge density (Section 1"}, {"Chapter": "1", "sentence_range": "2072-2075", "Text": "Let A be\nthe area of each plate and d the separation between\nthem The two plates have charges Q and \u2013Q Since\nd is much smaller than the linear dimension of the\nplates (d2 << A), we can use the result on electric\nfield by an infinite plane sheet of uniform surface\ncharge density (Section 1 15)"}, {"Chapter": "1", "sentence_range": "2073-2076", "Text": "The two plates have charges Q and \u2013Q Since\nd is much smaller than the linear dimension of the\nplates (d2 << A), we can use the result on electric\nfield by an infinite plane sheet of uniform surface\ncharge density (Section 1 15) Plate 1 has surface\ncharge density s = Q/A and plate 2 has a surface\ncharge density \u2013s"}, {"Chapter": "1", "sentence_range": "2074-2077", "Text": "Since\nd is much smaller than the linear dimension of the\nplates (d2 << A), we can use the result on electric\nfield by an infinite plane sheet of uniform surface\ncharge density (Section 1 15) Plate 1 has surface\ncharge density s = Q/A and plate 2 has a surface\ncharge density \u2013s Using Eq"}, {"Chapter": "1", "sentence_range": "2075-2078", "Text": "15) Plate 1 has surface\ncharge density s = Q/A and plate 2 has a surface\ncharge density \u2013s Using Eq (1"}, {"Chapter": "1", "sentence_range": "2076-2079", "Text": "Plate 1 has surface\ncharge density s = Q/A and plate 2 has a surface\ncharge density \u2013s Using Eq (1 33), the electric field\nin different regions is:\nOuter region I (region  above the plate 1),\n0\n0\n0\n2\n2\nE\n\u03c3\n\u03c3\n\u03b5\n\u03b5\n=\n\u2212\n=\n(2"}, {"Chapter": "1", "sentence_range": "2077-2080", "Text": "Using Eq (1 33), the electric field\nin different regions is:\nOuter region I (region  above the plate 1),\n0\n0\n0\n2\n2\nE\n\u03c3\n\u03c3\n\u03b5\n\u03b5\n=\n\u2212\n=\n(2 39)\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "2078-2081", "Text": "(1 33), the electric field\nin different regions is:\nOuter region I (region  above the plate 1),\n0\n0\n0\n2\n2\nE\n\u03c3\n\u03c3\n\u03b5\n\u03b5\n=\n\u2212\n=\n(2 39)\nFIGURE 2 25  The parallel plate capacitor"}, {"Chapter": "1", "sentence_range": "2079-2082", "Text": "33), the electric field\nin different regions is:\nOuter region I (region  above the plate 1),\n0\n0\n0\n2\n2\nE\n\u03c3\n\u03c3\n\u03b5\n\u03b5\n=\n\u2212\n=\n(2 39)\nFIGURE 2 25  The parallel plate capacitor Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n69\nOuter region II (region below the plate 2),\n0\n0\n0\n2\n2\nE\n\u03c3\n\u03c3\n\u03b5\n\u03b5\n=\n\u2212\n=\n(2"}, {"Chapter": "1", "sentence_range": "2080-2083", "Text": "39)\nFIGURE 2 25  The parallel plate capacitor Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n69\nOuter region II (region below the plate 2),\n0\n0\n0\n2\n2\nE\n\u03c3\n\u03c3\n\u03b5\n\u03b5\n=\n\u2212\n=\n(2 40)\nIn the inner region between the plates 1 and 2, the electric fields due\nto the two charged plates add up, giving\n0\n0\n0\n0\n2\n2\nQ\nE\nA\n\u03c3\n\u03c3\n\u03c3\n\u03b5\n\u03b5\n\u03b5\n\u03b5\n=\n+\n=\n=\n(2"}, {"Chapter": "1", "sentence_range": "2081-2084", "Text": "25  The parallel plate capacitor Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n69\nOuter region II (region below the plate 2),\n0\n0\n0\n2\n2\nE\n\u03c3\n\u03c3\n\u03b5\n\u03b5\n=\n\u2212\n=\n(2 40)\nIn the inner region between the plates 1 and 2, the electric fields due\nto the two charged plates add up, giving\n0\n0\n0\n0\n2\n2\nQ\nE\nA\n\u03c3\n\u03c3\n\u03c3\n\u03b5\n\u03b5\n\u03b5\n\u03b5\n=\n+\n=\n=\n(2 41)\nThe direction of electric field is from the positive to the negative plate"}, {"Chapter": "1", "sentence_range": "2082-2085", "Text": "Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n69\nOuter region II (region below the plate 2),\n0\n0\n0\n2\n2\nE\n\u03c3\n\u03c3\n\u03b5\n\u03b5\n=\n\u2212\n=\n(2 40)\nIn the inner region between the plates 1 and 2, the electric fields due\nto the two charged plates add up, giving\n0\n0\n0\n0\n2\n2\nQ\nE\nA\n\u03c3\n\u03c3\n\u03c3\n\u03b5\n\u03b5\n\u03b5\n\u03b5\n=\n+\n=\n=\n(2 41)\nThe direction of electric field is from the positive to the negative plate Thus, the electric field is localised between the two plates and is\nuniform throughout"}, {"Chapter": "1", "sentence_range": "2083-2086", "Text": "40)\nIn the inner region between the plates 1 and 2, the electric fields due\nto the two charged plates add up, giving\n0\n0\n0\n0\n2\n2\nQ\nE\nA\n\u03c3\n\u03c3\n\u03c3\n\u03b5\n\u03b5\n\u03b5\n\u03b5\n=\n+\n=\n=\n(2 41)\nThe direction of electric field is from the positive to the negative plate Thus, the electric field is localised between the two plates and is\nuniform throughout For plates with finite area, this will not be true near\nthe outer boundaries of the plates"}, {"Chapter": "1", "sentence_range": "2084-2087", "Text": "41)\nThe direction of electric field is from the positive to the negative plate Thus, the electric field is localised between the two plates and is\nuniform throughout For plates with finite area, this will not be true near\nthe outer boundaries of the plates The field lines bend outward at the\nedges \u2014 an effect called \u2018fringing of the field\u2019"}, {"Chapter": "1", "sentence_range": "2085-2088", "Text": "Thus, the electric field is localised between the two plates and is\nuniform throughout For plates with finite area, this will not be true near\nthe outer boundaries of the plates The field lines bend outward at the\nedges \u2014 an effect called \u2018fringing of the field\u2019 By the same token, s will\nnot be strictly uniform on the entire plate"}, {"Chapter": "1", "sentence_range": "2086-2089", "Text": "For plates with finite area, this will not be true near\nthe outer boundaries of the plates The field lines bend outward at the\nedges \u2014 an effect called \u2018fringing of the field\u2019 By the same token, s will\nnot be strictly uniform on the entire plate [E and s are related by Eq"}, {"Chapter": "1", "sentence_range": "2087-2090", "Text": "The field lines bend outward at the\nedges \u2014 an effect called \u2018fringing of the field\u2019 By the same token, s will\nnot be strictly uniform on the entire plate [E and s are related by Eq (2"}, {"Chapter": "1", "sentence_range": "2088-2091", "Text": "By the same token, s will\nnot be strictly uniform on the entire plate [E and s are related by Eq (2 35)"}, {"Chapter": "1", "sentence_range": "2089-2092", "Text": "[E and s are related by Eq (2 35) ] However, for d2 << A, these effects can be ignored in the regions\nsufficiently far from the edges, and the field there is given by Eq"}, {"Chapter": "1", "sentence_range": "2090-2093", "Text": "(2 35) ] However, for d2 << A, these effects can be ignored in the regions\nsufficiently far from the edges, and the field there is given by Eq (2"}, {"Chapter": "1", "sentence_range": "2091-2094", "Text": "35) ] However, for d2 << A, these effects can be ignored in the regions\nsufficiently far from the edges, and the field there is given by Eq (2 41)"}, {"Chapter": "1", "sentence_range": "2092-2095", "Text": "] However, for d2 << A, these effects can be ignored in the regions\nsufficiently far from the edges, and the field there is given by Eq (2 41) Now for uniform electric field, potential difference is simply the electric\nfield times the distance between the plates, that is,\n0\n1 Qd\nV\nE d\nA\n\u03b5\n=\n=\n(2"}, {"Chapter": "1", "sentence_range": "2093-2096", "Text": "(2 41) Now for uniform electric field, potential difference is simply the electric\nfield times the distance between the plates, that is,\n0\n1 Qd\nV\nE d\nA\n\u03b5\n=\n=\n(2 42)\nThe capacitance C of the parallel plate capacitor is then\nQ\nC\n=V\n = \n=\u03b5d0A\n(2"}, {"Chapter": "1", "sentence_range": "2094-2097", "Text": "41) Now for uniform electric field, potential difference is simply the electric\nfield times the distance between the plates, that is,\n0\n1 Qd\nV\nE d\nA\n\u03b5\n=\n=\n(2 42)\nThe capacitance C of the parallel plate capacitor is then\nQ\nC\n=V\n = \n=\u03b5d0A\n(2 43)\nwhich, as expected, depends only on the geometry of the system"}, {"Chapter": "1", "sentence_range": "2095-2098", "Text": "Now for uniform electric field, potential difference is simply the electric\nfield times the distance between the plates, that is,\n0\n1 Qd\nV\nE d\nA\n\u03b5\n=\n=\n(2 42)\nThe capacitance C of the parallel plate capacitor is then\nQ\nC\n=V\n = \n=\u03b5d0A\n(2 43)\nwhich, as expected, depends only on the geometry of the system For\ntypical values like A = 1 m2, d = 1 mm, we get\n12\n2\n\u20131\n\u20132\n2\n9\n3\n8"}, {"Chapter": "1", "sentence_range": "2096-2099", "Text": "42)\nThe capacitance C of the parallel plate capacitor is then\nQ\nC\n=V\n = \n=\u03b5d0A\n(2 43)\nwhich, as expected, depends only on the geometry of the system For\ntypical values like A = 1 m2, d = 1 mm, we get\n12\n2\n\u20131\n\u20132\n2\n9\n3\n8 85\n10\nC N m\n1m\n8"}, {"Chapter": "1", "sentence_range": "2097-2100", "Text": "43)\nwhich, as expected, depends only on the geometry of the system For\ntypical values like A = 1 m2, d = 1 mm, we get\n12\n2\n\u20131\n\u20132\n2\n9\n3\n8 85\n10\nC N m\n1m\n8 85\n10\nF\n10\nm\nC\n\u2212\n\u2212\n\u2212\n\u00d7\n\u00d7\n=\n=\n\u00d7\n(2"}, {"Chapter": "1", "sentence_range": "2098-2101", "Text": "For\ntypical values like A = 1 m2, d = 1 mm, we get\n12\n2\n\u20131\n\u20132\n2\n9\n3\n8 85\n10\nC N m\n1m\n8 85\n10\nF\n10\nm\nC\n\u2212\n\u2212\n\u2212\n\u00d7\n\u00d7\n=\n=\n\u00d7\n(2 44)\n(You can check that if 1F= 1C V\u20131 = 1C (NC\u20131m)\u20131 = 1 C2 N\u20131m\u20131"}, {"Chapter": "1", "sentence_range": "2099-2102", "Text": "85\n10\nC N m\n1m\n8 85\n10\nF\n10\nm\nC\n\u2212\n\u2212\n\u2212\n\u00d7\n\u00d7\n=\n=\n\u00d7\n(2 44)\n(You can check that if 1F= 1C V\u20131 = 1C (NC\u20131m)\u20131 = 1 C2 N\u20131m\u20131 )\nThis shows that 1F is too big a unit in practice, as remarked earlier"}, {"Chapter": "1", "sentence_range": "2100-2103", "Text": "85\n10\nF\n10\nm\nC\n\u2212\n\u2212\n\u2212\n\u00d7\n\u00d7\n=\n=\n\u00d7\n(2 44)\n(You can check that if 1F= 1C V\u20131 = 1C (NC\u20131m)\u20131 = 1 C2 N\u20131m\u20131 )\nThis shows that 1F is too big a unit in practice, as remarked earlier Another way of seeing the \u2018bigness\u2019 of 1F is to calculate the area of the\nplates needed to have C = 1F for a separation of, say 1 cm:\n0\nCd\nA\n=\u03b5\n=\n2\n9\n2\n12\n2\n\u20131\n\u20132\n1F\n10\nm\n10 m\n8"}, {"Chapter": "1", "sentence_range": "2101-2104", "Text": "44)\n(You can check that if 1F= 1C V\u20131 = 1C (NC\u20131m)\u20131 = 1 C2 N\u20131m\u20131 )\nThis shows that 1F is too big a unit in practice, as remarked earlier Another way of seeing the \u2018bigness\u2019 of 1F is to calculate the area of the\nplates needed to have C = 1F for a separation of, say 1 cm:\n0\nCd\nA\n=\u03b5\n=\n2\n9\n2\n12\n2\n\u20131\n\u20132\n1F\n10\nm\n10 m\n8 85\n10\nC N m\n\u2212\n\u00d7\u2212\n=\n\u00d7\n(2"}, {"Chapter": "1", "sentence_range": "2102-2105", "Text": ")\nThis shows that 1F is too big a unit in practice, as remarked earlier Another way of seeing the \u2018bigness\u2019 of 1F is to calculate the area of the\nplates needed to have C = 1F for a separation of, say 1 cm:\n0\nCd\nA\n=\u03b5\n=\n2\n9\n2\n12\n2\n\u20131\n\u20132\n1F\n10\nm\n10 m\n8 85\n10\nC N m\n\u2212\n\u00d7\u2212\n=\n\u00d7\n(2 45)\nwhich is a plate about 30 km in length and breadth"}, {"Chapter": "1", "sentence_range": "2103-2106", "Text": "Another way of seeing the \u2018bigness\u2019 of 1F is to calculate the area of the\nplates needed to have C = 1F for a separation of, say 1 cm:\n0\nCd\nA\n=\u03b5\n=\n2\n9\n2\n12\n2\n\u20131\n\u20132\n1F\n10\nm\n10 m\n8 85\n10\nC N m\n\u2212\n\u00d7\u2212\n=\n\u00d7\n(2 45)\nwhich is a plate about 30 km in length and breadth 2"}, {"Chapter": "1", "sentence_range": "2104-2107", "Text": "85\n10\nC N m\n\u2212\n\u00d7\u2212\n=\n\u00d7\n(2 45)\nwhich is a plate about 30 km in length and breadth 2 13  EFFECT OF DIELECTRIC ON CAPACITANCE\nWith the understanding of the behaviour of dielectrics in an external\nfield developed in Section 2"}, {"Chapter": "1", "sentence_range": "2105-2108", "Text": "45)\nwhich is a plate about 30 km in length and breadth 2 13  EFFECT OF DIELECTRIC ON CAPACITANCE\nWith the understanding of the behaviour of dielectrics in an external\nfield developed in Section 2 10, let us see how the capacitance of a parallel\nplate capacitor is modified when a dielectric is present"}, {"Chapter": "1", "sentence_range": "2106-2109", "Text": "2 13  EFFECT OF DIELECTRIC ON CAPACITANCE\nWith the understanding of the behaviour of dielectrics in an external\nfield developed in Section 2 10, let us see how the capacitance of a parallel\nplate capacitor is modified when a dielectric is present As before, we\nhave two large plates, each of area A, separated by a distance d"}, {"Chapter": "1", "sentence_range": "2107-2110", "Text": "13  EFFECT OF DIELECTRIC ON CAPACITANCE\nWith the understanding of the behaviour of dielectrics in an external\nfield developed in Section 2 10, let us see how the capacitance of a parallel\nplate capacitor is modified when a dielectric is present As before, we\nhave two large plates, each of area A, separated by a distance d The\ncharge on the plates is \u00b1Q, corresponding to the charge density \u00b1s (with\ns = Q/A)"}, {"Chapter": "1", "sentence_range": "2108-2111", "Text": "10, let us see how the capacitance of a parallel\nplate capacitor is modified when a dielectric is present As before, we\nhave two large plates, each of area A, separated by a distance d The\ncharge on the plates is \u00b1Q, corresponding to the charge density \u00b1s (with\ns = Q/A) When there is vacuum between the plates,\n0\n0\nE\n\u03b5\u03c3\n=\nFactors affecting capacitance, capacitors in action\nInteractive Java tutorial\nhttp://micro"}, {"Chapter": "1", "sentence_range": "2109-2112", "Text": "As before, we\nhave two large plates, each of area A, separated by a distance d The\ncharge on the plates is \u00b1Q, corresponding to the charge density \u00b1s (with\ns = Q/A) When there is vacuum between the plates,\n0\n0\nE\n\u03b5\u03c3\n=\nFactors affecting capacitance, capacitors in action\nInteractive Java tutorial\nhttp://micro magnet"}, {"Chapter": "1", "sentence_range": "2110-2113", "Text": "The\ncharge on the plates is \u00b1Q, corresponding to the charge density \u00b1s (with\ns = Q/A) When there is vacuum between the plates,\n0\n0\nE\n\u03b5\u03c3\n=\nFactors affecting capacitance, capacitors in action\nInteractive Java tutorial\nhttp://micro magnet fsu"}, {"Chapter": "1", "sentence_range": "2111-2114", "Text": "When there is vacuum between the plates,\n0\n0\nE\n\u03b5\u03c3\n=\nFactors affecting capacitance, capacitors in action\nInteractive Java tutorial\nhttp://micro magnet fsu edu/electromag/java/capacitance/\nRationalised 2023-24\nPhysics\n70\nand the potential difference  V0 is\nV0 = E0d\nThe capacitance C0 in this case is\n0\n0\n0\nQ\nA\nC\nV\n\u03b5d\n=\n=\n(2"}, {"Chapter": "1", "sentence_range": "2112-2115", "Text": "magnet fsu edu/electromag/java/capacitance/\nRationalised 2023-24\nPhysics\n70\nand the potential difference  V0 is\nV0 = E0d\nThe capacitance C0 in this case is\n0\n0\n0\nQ\nA\nC\nV\n\u03b5d\n=\n=\n(2 46)\nConsider next a dielectric inserted between the plates fully occupying\nthe intervening region"}, {"Chapter": "1", "sentence_range": "2113-2116", "Text": "fsu edu/electromag/java/capacitance/\nRationalised 2023-24\nPhysics\n70\nand the potential difference  V0 is\nV0 = E0d\nThe capacitance C0 in this case is\n0\n0\n0\nQ\nA\nC\nV\n\u03b5d\n=\n=\n(2 46)\nConsider next a dielectric inserted between the plates fully occupying\nthe intervening region The dielectric is polarised by the field and, as\nexplained in Section 2"}, {"Chapter": "1", "sentence_range": "2114-2117", "Text": "edu/electromag/java/capacitance/\nRationalised 2023-24\nPhysics\n70\nand the potential difference  V0 is\nV0 = E0d\nThe capacitance C0 in this case is\n0\n0\n0\nQ\nA\nC\nV\n\u03b5d\n=\n=\n(2 46)\nConsider next a dielectric inserted between the plates fully occupying\nthe intervening region The dielectric is polarised by the field and, as\nexplained in Section 2 10, the effect is equivalent to two charged sheets\n(at the surfaces of the dielectric normal to the field) with surface charge\ndensities sp and \u2013sp"}, {"Chapter": "1", "sentence_range": "2115-2118", "Text": "46)\nConsider next a dielectric inserted between the plates fully occupying\nthe intervening region The dielectric is polarised by the field and, as\nexplained in Section 2 10, the effect is equivalent to two charged sheets\n(at the surfaces of the dielectric normal to the field) with surface charge\ndensities sp and \u2013sp The electric field in the dielectric then corresponds\nto the case when the net surface charge density on the plates is \u00b1(s \u2013 sp)"}, {"Chapter": "1", "sentence_range": "2116-2119", "Text": "The dielectric is polarised by the field and, as\nexplained in Section 2 10, the effect is equivalent to two charged sheets\n(at the surfaces of the dielectric normal to the field) with surface charge\ndensities sp and \u2013sp The electric field in the dielectric then corresponds\nto the case when the net surface charge density on the plates is \u00b1(s \u2013 sp) That is,\n0\nP\nE\n\u03c3\n\u03c3\n\u2212\u03b5\n=\n(2"}, {"Chapter": "1", "sentence_range": "2117-2120", "Text": "10, the effect is equivalent to two charged sheets\n(at the surfaces of the dielectric normal to the field) with surface charge\ndensities sp and \u2013sp The electric field in the dielectric then corresponds\nto the case when the net surface charge density on the plates is \u00b1(s \u2013 sp) That is,\n0\nP\nE\n\u03c3\n\u03c3\n\u2212\u03b5\n=\n(2 47)\nso that the potential difference across the plates is\n0\nP\nV\nE d\nd\n\u03c3\n\u03c3\n\u2212\u03b5\n=\n=\n(2"}, {"Chapter": "1", "sentence_range": "2118-2121", "Text": "The electric field in the dielectric then corresponds\nto the case when the net surface charge density on the plates is \u00b1(s \u2013 sp) That is,\n0\nP\nE\n\u03c3\n\u03c3\n\u2212\u03b5\n=\n(2 47)\nso that the potential difference across the plates is\n0\nP\nV\nE d\nd\n\u03c3\n\u03c3\n\u2212\u03b5\n=\n=\n(2 48)\nFor linear dielectrics, we expect sp to be proportional to E0, i"}, {"Chapter": "1", "sentence_range": "2119-2122", "Text": "That is,\n0\nP\nE\n\u03c3\n\u03c3\n\u2212\u03b5\n=\n(2 47)\nso that the potential difference across the plates is\n0\nP\nV\nE d\nd\n\u03c3\n\u03c3\n\u2212\u03b5\n=\n=\n(2 48)\nFor linear dielectrics, we expect sp to be proportional to E0, i e"}, {"Chapter": "1", "sentence_range": "2120-2123", "Text": "47)\nso that the potential difference across the plates is\n0\nP\nV\nE d\nd\n\u03c3\n\u03c3\n\u2212\u03b5\n=\n=\n(2 48)\nFor linear dielectrics, we expect sp to be proportional to E0, i e , to s"}, {"Chapter": "1", "sentence_range": "2121-2124", "Text": "48)\nFor linear dielectrics, we expect sp to be proportional to E0, i e , to s Thus, (s \u2013 sp) is proportional to s  and we can write\nP\n\u03c3K\n\u03c3\n\u2212\u03c3\n=\n(2"}, {"Chapter": "1", "sentence_range": "2122-2125", "Text": "e , to s Thus, (s \u2013 sp) is proportional to s  and we can write\nP\n\u03c3K\n\u03c3\n\u2212\u03c3\n=\n(2 49)\nwhere K is a constant characteristic of the dielectric"}, {"Chapter": "1", "sentence_range": "2123-2126", "Text": ", to s Thus, (s \u2013 sp) is proportional to s  and we can write\nP\n\u03c3K\n\u03c3\n\u2212\u03c3\n=\n(2 49)\nwhere K is a constant characteristic of the dielectric Clearly,  K > 1"}, {"Chapter": "1", "sentence_range": "2124-2127", "Text": "Thus, (s \u2013 sp) is proportional to s  and we can write\nP\n\u03c3K\n\u03c3\n\u2212\u03c3\n=\n(2 49)\nwhere K is a constant characteristic of the dielectric Clearly,  K > 1 We\nthen have\n0\n0\nd\nQd\nV\nK\nA\nK\n\u03b5\u03c3\n\u03b5\n=\n=\n(2"}, {"Chapter": "1", "sentence_range": "2125-2128", "Text": "49)\nwhere K is a constant characteristic of the dielectric Clearly,  K > 1 We\nthen have\n0\n0\nd\nQd\nV\nK\nA\nK\n\u03b5\u03c3\n\u03b5\n=\n=\n(2 50)\nThe capacitance C, with dielectric between the plates, is then\n0KA\nQ\nC\nV\nd\n\u03b5\n=\n=\n(2"}, {"Chapter": "1", "sentence_range": "2126-2129", "Text": "Clearly,  K > 1 We\nthen have\n0\n0\nd\nQd\nV\nK\nA\nK\n\u03b5\u03c3\n\u03b5\n=\n=\n(2 50)\nThe capacitance C, with dielectric between the plates, is then\n0KA\nQ\nC\nV\nd\n\u03b5\n=\n=\n(2 51)\n The product e0K is called the permittivity of the medium and is\ndenoted by e\ne = e0 K\n(2"}, {"Chapter": "1", "sentence_range": "2127-2130", "Text": "We\nthen have\n0\n0\nd\nQd\nV\nK\nA\nK\n\u03b5\u03c3\n\u03b5\n=\n=\n(2 50)\nThe capacitance C, with dielectric between the plates, is then\n0KA\nQ\nC\nV\nd\n\u03b5\n=\n=\n(2 51)\n The product e0K is called the permittivity of the medium and is\ndenoted by e\ne = e0 K\n(2 52)\nFor vacuum K = 1 and e = e0; e0 is called the permittivity of the vacuum"}, {"Chapter": "1", "sentence_range": "2128-2131", "Text": "50)\nThe capacitance C, with dielectric between the plates, is then\n0KA\nQ\nC\nV\nd\n\u03b5\n=\n=\n(2 51)\n The product e0K is called the permittivity of the medium and is\ndenoted by e\ne = e0 K\n(2 52)\nFor vacuum K = 1 and e = e0; e0 is called the permittivity of the vacuum The dimensionless ratio\n0\nK\n=\u03b5\u03b5\n(2"}, {"Chapter": "1", "sentence_range": "2129-2132", "Text": "51)\n The product e0K is called the permittivity of the medium and is\ndenoted by e\ne = e0 K\n(2 52)\nFor vacuum K = 1 and e = e0; e0 is called the permittivity of the vacuum The dimensionless ratio\n0\nK\n=\u03b5\u03b5\n(2 53)\nis called the dielectric constant of the substance"}, {"Chapter": "1", "sentence_range": "2130-2133", "Text": "52)\nFor vacuum K = 1 and e = e0; e0 is called the permittivity of the vacuum The dimensionless ratio\n0\nK\n=\u03b5\u03b5\n(2 53)\nis called the dielectric constant of the substance As remarked before,\nfrom Eq"}, {"Chapter": "1", "sentence_range": "2131-2134", "Text": "The dimensionless ratio\n0\nK\n=\u03b5\u03b5\n(2 53)\nis called the dielectric constant of the substance As remarked before,\nfrom Eq (2"}, {"Chapter": "1", "sentence_range": "2132-2135", "Text": "53)\nis called the dielectric constant of the substance As remarked before,\nfrom Eq (2 49), it is clear that K is greater than 1"}, {"Chapter": "1", "sentence_range": "2133-2136", "Text": "As remarked before,\nfrom Eq (2 49), it is clear that K is greater than 1 From Eqs"}, {"Chapter": "1", "sentence_range": "2134-2137", "Text": "(2 49), it is clear that K is greater than 1 From Eqs (2"}, {"Chapter": "1", "sentence_range": "2135-2138", "Text": "49), it is clear that K is greater than 1 From Eqs (2 46) and\n(2"}, {"Chapter": "1", "sentence_range": "2136-2139", "Text": "From Eqs (2 46) and\n(2 51)\n0\nC\nK\n=C\n(2"}, {"Chapter": "1", "sentence_range": "2137-2140", "Text": "(2 46) and\n(2 51)\n0\nC\nK\n=C\n(2 54)\nThus, the dielectric constant of a substance is the factor (>1) by which\nthe capacitance increases from its vacuum value, when the dielectric is\ninserted fully between the plates of a capacitor"}, {"Chapter": "1", "sentence_range": "2138-2141", "Text": "46) and\n(2 51)\n0\nC\nK\n=C\n(2 54)\nThus, the dielectric constant of a substance is the factor (>1) by which\nthe capacitance increases from its vacuum value, when the dielectric is\ninserted fully between the plates of a capacitor Though we arrived at\nRationalised 2023-24\nElectrostatic Potential\nand Capacitance\n71\n EXAMPLE 2"}, {"Chapter": "1", "sentence_range": "2139-2142", "Text": "51)\n0\nC\nK\n=C\n(2 54)\nThus, the dielectric constant of a substance is the factor (>1) by which\nthe capacitance increases from its vacuum value, when the dielectric is\ninserted fully between the plates of a capacitor Though we arrived at\nRationalised 2023-24\nElectrostatic Potential\nand Capacitance\n71\n EXAMPLE 2 8\nEq"}, {"Chapter": "1", "sentence_range": "2140-2143", "Text": "54)\nThus, the dielectric constant of a substance is the factor (>1) by which\nthe capacitance increases from its vacuum value, when the dielectric is\ninserted fully between the plates of a capacitor Though we arrived at\nRationalised 2023-24\nElectrostatic Potential\nand Capacitance\n71\n EXAMPLE 2 8\nEq (2"}, {"Chapter": "1", "sentence_range": "2141-2144", "Text": "Though we arrived at\nRationalised 2023-24\nElectrostatic Potential\nand Capacitance\n71\n EXAMPLE 2 8\nEq (2 54) for the case of a parallel plate capacitor, it holds good for any\ntype of capacitor and can, in fact, be viewed in general as a definition of\nthe dielectric constant of a substance"}, {"Chapter": "1", "sentence_range": "2142-2145", "Text": "8\nEq (2 54) for the case of a parallel plate capacitor, it holds good for any\ntype of capacitor and can, in fact, be viewed in general as a definition of\nthe dielectric constant of a substance Example 2"}, {"Chapter": "1", "sentence_range": "2143-2146", "Text": "(2 54) for the case of a parallel plate capacitor, it holds good for any\ntype of capacitor and can, in fact, be viewed in general as a definition of\nthe dielectric constant of a substance Example 2 8 A slab of material of dielectric constant K has the same\narea as the plates of a parallel-plate capacitor but has a thickness\n(3/4)d, where d is the separation of the plates"}, {"Chapter": "1", "sentence_range": "2144-2147", "Text": "54) for the case of a parallel plate capacitor, it holds good for any\ntype of capacitor and can, in fact, be viewed in general as a definition of\nthe dielectric constant of a substance Example 2 8 A slab of material of dielectric constant K has the same\narea as the plates of a parallel-plate capacitor but has a thickness\n(3/4)d, where d is the separation of the plates How is the capacitance\nchanged when the slab is inserted between the plates"}, {"Chapter": "1", "sentence_range": "2145-2148", "Text": "Example 2 8 A slab of material of dielectric constant K has the same\narea as the plates of a parallel-plate capacitor but has a thickness\n(3/4)d, where d is the separation of the plates How is the capacitance\nchanged when the slab is inserted between the plates Solution Let  E0 = V0/d be the electric field between the plates when\nthere is no dielectric and the potential difference is V0"}, {"Chapter": "1", "sentence_range": "2146-2149", "Text": "8 A slab of material of dielectric constant K has the same\narea as the plates of a parallel-plate capacitor but has a thickness\n(3/4)d, where d is the separation of the plates How is the capacitance\nchanged when the slab is inserted between the plates Solution Let  E0 = V0/d be the electric field between the plates when\nthere is no dielectric and the potential difference is V0 If the dielectric\nis now inserted, the electric field in the dielectric will be E = E0/K"}, {"Chapter": "1", "sentence_range": "2147-2150", "Text": "How is the capacitance\nchanged when the slab is inserted between the plates Solution Let  E0 = V0/d be the electric field between the plates when\nthere is no dielectric and the potential difference is V0 If the dielectric\nis now inserted, the electric field in the dielectric will be E = E0/K The potential difference will then be\n0\n0\n1\n3\n(\n)\n(\n)\n4\n4\nE\nV\nE\nd\nd\nK\n=\n+\n0\n0\n1\n3\n3\n(\n)\n4\n4\nK4\nE d\nV\nK\n+K\n=\n+\n=\nThe potential difference decreases by the factor (K + 3)/4K while the\nfree charge Q0 on the plates remains unchanged"}, {"Chapter": "1", "sentence_range": "2148-2151", "Text": "Solution Let  E0 = V0/d be the electric field between the plates when\nthere is no dielectric and the potential difference is V0 If the dielectric\nis now inserted, the electric field in the dielectric will be E = E0/K The potential difference will then be\n0\n0\n1\n3\n(\n)\n(\n)\n4\n4\nE\nV\nE\nd\nd\nK\n=\n+\n0\n0\n1\n3\n3\n(\n)\n4\n4\nK4\nE d\nV\nK\n+K\n=\n+\n=\nThe potential difference decreases by the factor (K + 3)/4K while the\nfree charge Q0 on the plates remains unchanged The capacitance\nthus increases\n0\n0\n0\n0\n4\n4\n3\n3\nQ\nQ\nK\nK\nC\nC\nV\nK\nV\nK\n=\n=\n=\n+\n+\n2"}, {"Chapter": "1", "sentence_range": "2149-2152", "Text": "If the dielectric\nis now inserted, the electric field in the dielectric will be E = E0/K The potential difference will then be\n0\n0\n1\n3\n(\n)\n(\n)\n4\n4\nE\nV\nE\nd\nd\nK\n=\n+\n0\n0\n1\n3\n3\n(\n)\n4\n4\nK4\nE d\nV\nK\n+K\n=\n+\n=\nThe potential difference decreases by the factor (K + 3)/4K while the\nfree charge Q0 on the plates remains unchanged The capacitance\nthus increases\n0\n0\n0\n0\n4\n4\n3\n3\nQ\nQ\nK\nK\nC\nC\nV\nK\nV\nK\n=\n=\n=\n+\n+\n2 14  COMBINATION OF CAPACITORS\nWe can combine several capacitors of\ncapacitance C1, C2,\u2026, Cn to obtain a system with\nsome effective capacitance C"}, {"Chapter": "1", "sentence_range": "2150-2153", "Text": "The potential difference will then be\n0\n0\n1\n3\n(\n)\n(\n)\n4\n4\nE\nV\nE\nd\nd\nK\n=\n+\n0\n0\n1\n3\n3\n(\n)\n4\n4\nK4\nE d\nV\nK\n+K\n=\n+\n=\nThe potential difference decreases by the factor (K + 3)/4K while the\nfree charge Q0 on the plates remains unchanged The capacitance\nthus increases\n0\n0\n0\n0\n4\n4\n3\n3\nQ\nQ\nK\nK\nC\nC\nV\nK\nV\nK\n=\n=\n=\n+\n+\n2 14  COMBINATION OF CAPACITORS\nWe can combine several capacitors of\ncapacitance C1, C2,\u2026, Cn to obtain a system with\nsome effective capacitance C The effective\ncapacitance depends on the way the individual\ncapacitors are combined"}, {"Chapter": "1", "sentence_range": "2151-2154", "Text": "The capacitance\nthus increases\n0\n0\n0\n0\n4\n4\n3\n3\nQ\nQ\nK\nK\nC\nC\nV\nK\nV\nK\n=\n=\n=\n+\n+\n2 14  COMBINATION OF CAPACITORS\nWe can combine several capacitors of\ncapacitance C1, C2,\u2026, Cn to obtain a system with\nsome effective capacitance C The effective\ncapacitance depends on the way the individual\ncapacitors are combined Two simple\npossibilities are discussed below"}, {"Chapter": "1", "sentence_range": "2152-2155", "Text": "14  COMBINATION OF CAPACITORS\nWe can combine several capacitors of\ncapacitance C1, C2,\u2026, Cn to obtain a system with\nsome effective capacitance C The effective\ncapacitance depends on the way the individual\ncapacitors are combined Two simple\npossibilities are discussed below 2"}, {"Chapter": "1", "sentence_range": "2153-2156", "Text": "The effective\ncapacitance depends on the way the individual\ncapacitors are combined Two simple\npossibilities are discussed below 2 14"}, {"Chapter": "1", "sentence_range": "2154-2157", "Text": "Two simple\npossibilities are discussed below 2 14 1  Capacitors in series\nFigure 2"}, {"Chapter": "1", "sentence_range": "2155-2158", "Text": "2 14 1  Capacitors in series\nFigure 2 26 shows capacitors C1 and C2\ncombined in series"}, {"Chapter": "1", "sentence_range": "2156-2159", "Text": "14 1  Capacitors in series\nFigure 2 26 shows capacitors C1 and C2\ncombined in series The left plate of C1 and the right plate of C2\nare connected to two terminals of a battery and\nhave charges Q and \u2013Q , respectively"}, {"Chapter": "1", "sentence_range": "2157-2160", "Text": "1  Capacitors in series\nFigure 2 26 shows capacitors C1 and C2\ncombined in series The left plate of C1 and the right plate of C2\nare connected to two terminals of a battery and\nhave charges Q and \u2013Q , respectively It then\nfollows that the right plate of C1 has charge \u2013Q\nand the left plate of C2  has charge Q"}, {"Chapter": "1", "sentence_range": "2158-2161", "Text": "26 shows capacitors C1 and C2\ncombined in series The left plate of C1 and the right plate of C2\nare connected to two terminals of a battery and\nhave charges Q and \u2013Q , respectively It then\nfollows that the right plate of C1 has charge \u2013Q\nand the left plate of C2  has charge Q If this was\nnot so, the net charge on each capacitor would\nnot be zero"}, {"Chapter": "1", "sentence_range": "2159-2162", "Text": "The left plate of C1 and the right plate of C2\nare connected to two terminals of a battery and\nhave charges Q and \u2013Q , respectively It then\nfollows that the right plate of C1 has charge \u2013Q\nand the left plate of C2  has charge Q If this was\nnot so, the net charge on each capacitor would\nnot be zero This would result in an electric field\nin the conductor connecting C1and C2"}, {"Chapter": "1", "sentence_range": "2160-2163", "Text": "It then\nfollows that the right plate of C1 has charge \u2013Q\nand the left plate of C2  has charge Q If this was\nnot so, the net charge on each capacitor would\nnot be zero This would result in an electric field\nin the conductor connecting C1and C2 Charge\nwould flow until the net charge on both C1 and\nC2 is zero and there is no electric field in the\nconductor connecting C1 and C2"}, {"Chapter": "1", "sentence_range": "2161-2164", "Text": "If this was\nnot so, the net charge on each capacitor would\nnot be zero This would result in an electric field\nin the conductor connecting C1and C2 Charge\nwould flow until the net charge on both C1 and\nC2 is zero and there is no electric field in the\nconductor connecting C1 and C2 Thus, in the\nseries combination, charges on the two plates\n(\u00b1Q) are the same on each capacitor"}, {"Chapter": "1", "sentence_range": "2162-2165", "Text": "This would result in an electric field\nin the conductor connecting C1and C2 Charge\nwould flow until the net charge on both C1 and\nC2 is zero and there is no electric field in the\nconductor connecting C1 and C2 Thus, in the\nseries combination, charges on the two plates\n(\u00b1Q) are the same on each capacitor The total\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "2163-2166", "Text": "Charge\nwould flow until the net charge on both C1 and\nC2 is zero and there is no electric field in the\nconductor connecting C1 and C2 Thus, in the\nseries combination, charges on the two plates\n(\u00b1Q) are the same on each capacitor The total\nFIGURE 2 26  Combination of two\ncapacitors in series"}, {"Chapter": "1", "sentence_range": "2164-2167", "Text": "Thus, in the\nseries combination, charges on the two plates\n(\u00b1Q) are the same on each capacitor The total\nFIGURE 2 26  Combination of two\ncapacitors in series FIGURE 2"}, {"Chapter": "1", "sentence_range": "2165-2168", "Text": "The total\nFIGURE 2 26  Combination of two\ncapacitors in series FIGURE 2 27  Combination of n\ncapacitors in series"}, {"Chapter": "1", "sentence_range": "2166-2169", "Text": "26  Combination of two\ncapacitors in series FIGURE 2 27  Combination of n\ncapacitors in series Rationalised 2023-24\nPhysics\n72\npotential drop V across the combination is the sum of the potential drops\nV1 and V2 across C1 and C2, respectively"}, {"Chapter": "1", "sentence_range": "2167-2170", "Text": "FIGURE 2 27  Combination of n\ncapacitors in series Rationalised 2023-24\nPhysics\n72\npotential drop V across the combination is the sum of the potential drops\nV1 and V2 across C1 and C2, respectively V = V1 + V2 = \n1\n2\nQ\nQ\nC\n+C\n(2"}, {"Chapter": "1", "sentence_range": "2168-2171", "Text": "27  Combination of n\ncapacitors in series Rationalised 2023-24\nPhysics\n72\npotential drop V across the combination is the sum of the potential drops\nV1 and V2 across C1 and C2, respectively V = V1 + V2 = \n1\n2\nQ\nQ\nC\n+C\n(2 55)\ni"}, {"Chapter": "1", "sentence_range": "2169-2172", "Text": "Rationalised 2023-24\nPhysics\n72\npotential drop V across the combination is the sum of the potential drops\nV1 and V2 across C1 and C2, respectively V = V1 + V2 = \n1\n2\nQ\nQ\nC\n+C\n(2 55)\ni e"}, {"Chapter": "1", "sentence_range": "2170-2173", "Text": "V = V1 + V2 = \n1\n2\nQ\nQ\nC\n+C\n(2 55)\ni e , \n1\n2\n1\n1\nQV\nC\nC\n=\n+\n,\n(2"}, {"Chapter": "1", "sentence_range": "2171-2174", "Text": "55)\ni e , \n1\n2\n1\n1\nQV\nC\nC\n=\n+\n,\n(2 56)\nNow we can regard the combination as an effective capacitor with\ncharge Q and potential difference V"}, {"Chapter": "1", "sentence_range": "2172-2175", "Text": "e , \n1\n2\n1\n1\nQV\nC\nC\n=\n+\n,\n(2 56)\nNow we can regard the combination as an effective capacitor with\ncharge Q and potential difference V The effective capacitance of the\ncombination is\nQ\nC\n=V\n(2"}, {"Chapter": "1", "sentence_range": "2173-2176", "Text": ", \n1\n2\n1\n1\nQV\nC\nC\n=\n+\n,\n(2 56)\nNow we can regard the combination as an effective capacitor with\ncharge Q and potential difference V The effective capacitance of the\ncombination is\nQ\nC\n=V\n(2 57)\nWe compare Eq"}, {"Chapter": "1", "sentence_range": "2174-2177", "Text": "56)\nNow we can regard the combination as an effective capacitor with\ncharge Q and potential difference V The effective capacitance of the\ncombination is\nQ\nC\n=V\n(2 57)\nWe compare Eq (2"}, {"Chapter": "1", "sentence_range": "2175-2178", "Text": "The effective capacitance of the\ncombination is\nQ\nC\n=V\n(2 57)\nWe compare Eq (2 57) with Eq"}, {"Chapter": "1", "sentence_range": "2176-2179", "Text": "57)\nWe compare Eq (2 57) with Eq (2"}, {"Chapter": "1", "sentence_range": "2177-2180", "Text": "(2 57) with Eq (2 56), and obtain\n1\n2\n1\n1\n1\nC\nC\nC\n=\n+\n(2"}, {"Chapter": "1", "sentence_range": "2178-2181", "Text": "57) with Eq (2 56), and obtain\n1\n2\n1\n1\n1\nC\nC\nC\n=\n+\n(2 58)\nThe proof clearly goes through for any number of capacitors arranged\nin a similar way"}, {"Chapter": "1", "sentence_range": "2179-2182", "Text": "(2 56), and obtain\n1\n2\n1\n1\n1\nC\nC\nC\n=\n+\n(2 58)\nThe proof clearly goes through for any number of capacitors arranged\nin a similar way Equation (2"}, {"Chapter": "1", "sentence_range": "2180-2183", "Text": "56), and obtain\n1\n2\n1\n1\n1\nC\nC\nC\n=\n+\n(2 58)\nThe proof clearly goes through for any number of capacitors arranged\nin a similar way Equation (2 55), for n capacitors arranged in series,\ngeneralises to\n1\n2\nn\n1\n2\nn"}, {"Chapter": "1", "sentence_range": "2181-2184", "Text": "58)\nThe proof clearly goes through for any number of capacitors arranged\nin a similar way Equation (2 55), for n capacitors arranged in series,\ngeneralises to\n1\n2\nn\n1\n2\nn Q\nQ\nQ\nV\nV\nV\nV\nC\nC\nC\n=\n+\n+\n+\n=\n+\n+\n+\n(2"}, {"Chapter": "1", "sentence_range": "2182-2185", "Text": "Equation (2 55), for n capacitors arranged in series,\ngeneralises to\n1\n2\nn\n1\n2\nn Q\nQ\nQ\nV\nV\nV\nV\nC\nC\nC\n=\n+\n+\n+\n=\n+\n+\n+\n(2 59)\nFollowing the same steps as for the case of two\ncapacitors, we get the general formula for effective\ncapacitance of a series combination of n capacitors:\n1\n2\n3\nn\n1\n1\n1\n1\n1"}, {"Chapter": "1", "sentence_range": "2183-2186", "Text": "55), for n capacitors arranged in series,\ngeneralises to\n1\n2\nn\n1\n2\nn Q\nQ\nQ\nV\nV\nV\nV\nC\nC\nC\n=\n+\n+\n+\n=\n+\n+\n+\n(2 59)\nFollowing the same steps as for the case of two\ncapacitors, we get the general formula for effective\ncapacitance of a series combination of n capacitors:\n1\n2\n3\nn\n1\n1\n1\n1\n1 C\nC\nC\nC\nC\n=\n+\n+\n+\n+\n(2"}, {"Chapter": "1", "sentence_range": "2184-2187", "Text": "Q\nQ\nQ\nV\nV\nV\nV\nC\nC\nC\n=\n+\n+\n+\n=\n+\n+\n+\n(2 59)\nFollowing the same steps as for the case of two\ncapacitors, we get the general formula for effective\ncapacitance of a series combination of n capacitors:\n1\n2\n3\nn\n1\n1\n1\n1\n1 C\nC\nC\nC\nC\n=\n+\n+\n+\n+\n(2 60)\n2"}, {"Chapter": "1", "sentence_range": "2185-2188", "Text": "59)\nFollowing the same steps as for the case of two\ncapacitors, we get the general formula for effective\ncapacitance of a series combination of n capacitors:\n1\n2\n3\nn\n1\n1\n1\n1\n1 C\nC\nC\nC\nC\n=\n+\n+\n+\n+\n(2 60)\n2 14"}, {"Chapter": "1", "sentence_range": "2186-2189", "Text": "C\nC\nC\nC\nC\n=\n+\n+\n+\n+\n(2 60)\n2 14 2  Capacitors in parallel\nFigure 2"}, {"Chapter": "1", "sentence_range": "2187-2190", "Text": "60)\n2 14 2  Capacitors in parallel\nFigure 2 28 (a) shows two capacitors arranged in\nparallel"}, {"Chapter": "1", "sentence_range": "2188-2191", "Text": "14 2  Capacitors in parallel\nFigure 2 28 (a) shows two capacitors arranged in\nparallel In this case, the same potential difference is\napplied across both the capacitors"}, {"Chapter": "1", "sentence_range": "2189-2192", "Text": "2  Capacitors in parallel\nFigure 2 28 (a) shows two capacitors arranged in\nparallel In this case, the same potential difference is\napplied across both the capacitors But the plate charges\n(\u00b1Q1) on capacitor 1 and the plate charges (\u00b1Q2) on the\ncapacitor 2 are not necessarily the same:\nQ1 = C1V, Q2 = C2V\n(2"}, {"Chapter": "1", "sentence_range": "2190-2193", "Text": "28 (a) shows two capacitors arranged in\nparallel In this case, the same potential difference is\napplied across both the capacitors But the plate charges\n(\u00b1Q1) on capacitor 1 and the plate charges (\u00b1Q2) on the\ncapacitor 2 are not necessarily the same:\nQ1 = C1V, Q2 = C2V\n(2 61)\nThe equivalent capacitor is one with charge\nQ = Q1 + Q2\n(2"}, {"Chapter": "1", "sentence_range": "2191-2194", "Text": "In this case, the same potential difference is\napplied across both the capacitors But the plate charges\n(\u00b1Q1) on capacitor 1 and the plate charges (\u00b1Q2) on the\ncapacitor 2 are not necessarily the same:\nQ1 = C1V, Q2 = C2V\n(2 61)\nThe equivalent capacitor is one with charge\nQ = Q1 + Q2\n(2 62)\nand potential difference V"}, {"Chapter": "1", "sentence_range": "2192-2195", "Text": "But the plate charges\n(\u00b1Q1) on capacitor 1 and the plate charges (\u00b1Q2) on the\ncapacitor 2 are not necessarily the same:\nQ1 = C1V, Q2 = C2V\n(2 61)\nThe equivalent capacitor is one with charge\nQ = Q1 + Q2\n(2 62)\nand potential difference V Q = CV = C1V + C2V\n(2"}, {"Chapter": "1", "sentence_range": "2193-2196", "Text": "61)\nThe equivalent capacitor is one with charge\nQ = Q1 + Q2\n(2 62)\nand potential difference V Q = CV = C1V + C2V\n(2 63)\nThe effective capacitance C is, from Eq"}, {"Chapter": "1", "sentence_range": "2194-2197", "Text": "62)\nand potential difference V Q = CV = C1V + C2V\n(2 63)\nThe effective capacitance C is, from Eq (2"}, {"Chapter": "1", "sentence_range": "2195-2198", "Text": "Q = CV = C1V + C2V\n(2 63)\nThe effective capacitance C is, from Eq (2 63),\nC = C1 + C2\n(2"}, {"Chapter": "1", "sentence_range": "2196-2199", "Text": "63)\nThe effective capacitance C is, from Eq (2 63),\nC = C1 + C2\n(2 64)\nThe general formula for effective capacitance C for\nparallel combination of n capacitors [Fig"}, {"Chapter": "1", "sentence_range": "2197-2200", "Text": "(2 63),\nC = C1 + C2\n(2 64)\nThe general formula for effective capacitance C for\nparallel combination of n capacitors [Fig 2"}, {"Chapter": "1", "sentence_range": "2198-2201", "Text": "63),\nC = C1 + C2\n(2 64)\nThe general formula for effective capacitance C for\nparallel combination of n capacitors [Fig 2 28 (b)]\nfollows similarly,\nQ = Q1 + Q2 +"}, {"Chapter": "1", "sentence_range": "2199-2202", "Text": "64)\nThe general formula for effective capacitance C for\nparallel combination of n capacitors [Fig 2 28 (b)]\nfollows similarly,\nQ = Q1 + Q2 + + Qn\n(2"}, {"Chapter": "1", "sentence_range": "2200-2203", "Text": "2 28 (b)]\nfollows similarly,\nQ = Q1 + Q2 + + Qn\n(2 65)\ni"}, {"Chapter": "1", "sentence_range": "2201-2204", "Text": "28 (b)]\nfollows similarly,\nQ = Q1 + Q2 + + Qn\n(2 65)\ni e"}, {"Chapter": "1", "sentence_range": "2202-2205", "Text": "+ Qn\n(2 65)\ni e , CV = C1V + C2V +"}, {"Chapter": "1", "sentence_range": "2203-2206", "Text": "65)\ni e , CV = C1V + C2V + CnV(2"}, {"Chapter": "1", "sentence_range": "2204-2207", "Text": "e , CV = C1V + C2V + CnV(2 66)\nwhich gives\nC = C1 + C2 +"}, {"Chapter": "1", "sentence_range": "2205-2208", "Text": ", CV = C1V + C2V + CnV(2 66)\nwhich gives\nC = C1 + C2 + Cn\n(2"}, {"Chapter": "1", "sentence_range": "2206-2209", "Text": "CnV(2 66)\nwhich gives\nC = C1 + C2 + Cn\n(2 67)\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "2207-2210", "Text": "66)\nwhich gives\nC = C1 + C2 + Cn\n(2 67)\nFIGURE 2 28 Parallel combination of\n(a) two capacitors, (b) n capacitors"}, {"Chapter": "1", "sentence_range": "2208-2211", "Text": "Cn\n(2 67)\nFIGURE 2 28 Parallel combination of\n(a) two capacitors, (b) n capacitors Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n73\n EXAMPLE 2"}, {"Chapter": "1", "sentence_range": "2209-2212", "Text": "67)\nFIGURE 2 28 Parallel combination of\n(a) two capacitors, (b) n capacitors Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n73\n EXAMPLE 2 9\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "2210-2213", "Text": "28 Parallel combination of\n(a) two capacitors, (b) n capacitors Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n73\n EXAMPLE 2 9\nFIGURE 2 29\nExample 2"}, {"Chapter": "1", "sentence_range": "2211-2214", "Text": "Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n73\n EXAMPLE 2 9\nFIGURE 2 29\nExample 2 9 A network of four 10 mF capacitors is connected to a 500 V\nsupply, as shown in Fig"}, {"Chapter": "1", "sentence_range": "2212-2215", "Text": "9\nFIGURE 2 29\nExample 2 9 A network of four 10 mF capacitors is connected to a 500 V\nsupply, as shown in Fig 2"}, {"Chapter": "1", "sentence_range": "2213-2216", "Text": "29\nExample 2 9 A network of four 10 mF capacitors is connected to a 500 V\nsupply, as shown in Fig 2 29"}, {"Chapter": "1", "sentence_range": "2214-2217", "Text": "9 A network of four 10 mF capacitors is connected to a 500 V\nsupply, as shown in Fig 2 29 Determine (a) the equivalent capacitance\nof the network and (b) the charge on each capacitor"}, {"Chapter": "1", "sentence_range": "2215-2218", "Text": "2 29 Determine (a) the equivalent capacitance\nof the network and (b) the charge on each capacitor (Note, the charge on\na capacitor is the charge on the plate with higher potential, equal and\nopposite to the charge on the plate with lower potential"}, {"Chapter": "1", "sentence_range": "2216-2219", "Text": "29 Determine (a) the equivalent capacitance\nof the network and (b) the charge on each capacitor (Note, the charge on\na capacitor is the charge on the plate with higher potential, equal and\nopposite to the charge on the plate with lower potential )\nSolution\n(a) In the given network, C1, C2 and C3 are connected in series"}, {"Chapter": "1", "sentence_range": "2217-2220", "Text": "Determine (a) the equivalent capacitance\nof the network and (b) the charge on each capacitor (Note, the charge on\na capacitor is the charge on the plate with higher potential, equal and\nopposite to the charge on the plate with lower potential )\nSolution\n(a) In the given network, C1, C2 and C3 are connected in series The\neffective capacitance C\u00a2 of these three capacitors is given by\n1\n2\n3\n1\n1\n1\n1\nC\nC\nC\nC\n=\n+\n+\n\u2032\nFor C1 = C2 = C3 = 10 mF,  C\u00a2 = (10/3) mF"}, {"Chapter": "1", "sentence_range": "2218-2221", "Text": "(Note, the charge on\na capacitor is the charge on the plate with higher potential, equal and\nopposite to the charge on the plate with lower potential )\nSolution\n(a) In the given network, C1, C2 and C3 are connected in series The\neffective capacitance C\u00a2 of these three capacitors is given by\n1\n2\n3\n1\n1\n1\n1\nC\nC\nC\nC\n=\n+\n+\n\u2032\nFor C1 = C2 = C3 = 10 mF,  C\u00a2 = (10/3) mF The network has  C\u00a2 and C4\nconnected in parallel"}, {"Chapter": "1", "sentence_range": "2219-2222", "Text": ")\nSolution\n(a) In the given network, C1, C2 and C3 are connected in series The\neffective capacitance C\u00a2 of these three capacitors is given by\n1\n2\n3\n1\n1\n1\n1\nC\nC\nC\nC\n=\n+\n+\n\u2032\nFor C1 = C2 = C3 = 10 mF,  C\u00a2 = (10/3) mF The network has  C\u00a2 and C4\nconnected in parallel Thus, the equivalent capacitance C of the\nnetwork is\nC = C\u00a2 + C4 = 10\n3\n+10\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7  mF =13"}, {"Chapter": "1", "sentence_range": "2220-2223", "Text": "The\neffective capacitance C\u00a2 of these three capacitors is given by\n1\n2\n3\n1\n1\n1\n1\nC\nC\nC\nC\n=\n+\n+\n\u2032\nFor C1 = C2 = C3 = 10 mF,  C\u00a2 = (10/3) mF The network has  C\u00a2 and C4\nconnected in parallel Thus, the equivalent capacitance C of the\nnetwork is\nC = C\u00a2 + C4 = 10\n3\n+10\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7  mF =13 3mF\n(b) Clearly, from the figure, the charge on each of the capacitors, C1,\nC2 and C3 is the same, say Q"}, {"Chapter": "1", "sentence_range": "2221-2224", "Text": "The network has  C\u00a2 and C4\nconnected in parallel Thus, the equivalent capacitance C of the\nnetwork is\nC = C\u00a2 + C4 = 10\n3\n+10\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7  mF =13 3mF\n(b) Clearly, from the figure, the charge on each of the capacitors, C1,\nC2 and C3 is the same, say Q Let the charge on C4 be Q\u00a2"}, {"Chapter": "1", "sentence_range": "2222-2225", "Text": "Thus, the equivalent capacitance C of the\nnetwork is\nC = C\u00a2 + C4 = 10\n3\n+10\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7  mF =13 3mF\n(b) Clearly, from the figure, the charge on each of the capacitors, C1,\nC2 and C3 is the same, say Q Let the charge on C4 be Q\u00a2 Now, since\nthe potential difference across AB is  Q/C1, across BC is Q/C2, across\nCD is  Q/C3 , we have\n1\n2\n3\n500 V\nQ\nQ\nQ\nC\nC\nC\n+\n+\n="}, {"Chapter": "1", "sentence_range": "2223-2226", "Text": "3mF\n(b) Clearly, from the figure, the charge on each of the capacitors, C1,\nC2 and C3 is the same, say Q Let the charge on C4 be Q\u00a2 Now, since\nthe potential difference across AB is  Q/C1, across BC is Q/C2, across\nCD is  Q/C3 , we have\n1\n2\n3\n500 V\nQ\nQ\nQ\nC\nC\nC\n+\n+\n= Also, Q\u00a2/C4 = 500 V"}, {"Chapter": "1", "sentence_range": "2224-2227", "Text": "Let the charge on C4 be Q\u00a2 Now, since\nthe potential difference across AB is  Q/C1, across BC is Q/C2, across\nCD is  Q/C3 , we have\n1\n2\n3\n500 V\nQ\nQ\nQ\nC\nC\nC\n+\n+\n= Also, Q\u00a2/C4 = 500 V This gives for the given value of the capacitances,\n3\n10\n500\nF\n1"}, {"Chapter": "1", "sentence_range": "2225-2228", "Text": "Now, since\nthe potential difference across AB is  Q/C1, across BC is Q/C2, across\nCD is  Q/C3 , we have\n1\n2\n3\n500 V\nQ\nQ\nQ\nC\nC\nC\n+\n+\n= Also, Q\u00a2/C4 = 500 V This gives for the given value of the capacitances,\n3\n10\n500\nF\n1 7\n10\nC\n3\nQ\nV\n\u2212\n=\n\u00d7\n\u00b5\n=\n\u00d7\n and\n3\n500\n10 F\n5"}, {"Chapter": "1", "sentence_range": "2226-2229", "Text": "Also, Q\u00a2/C4 = 500 V This gives for the given value of the capacitances,\n3\n10\n500\nF\n1 7\n10\nC\n3\nQ\nV\n\u2212\n=\n\u00d7\n\u00b5\n=\n\u00d7\n and\n3\n500\n10 F\n5 0\n10\nC\nQ\nV\n\u2212\n=\n\u00d7\n\u00b5\n=\n\u00d7\n\u2032\n2"}, {"Chapter": "1", "sentence_range": "2227-2230", "Text": "This gives for the given value of the capacitances,\n3\n10\n500\nF\n1 7\n10\nC\n3\nQ\nV\n\u2212\n=\n\u00d7\n\u00b5\n=\n\u00d7\n and\n3\n500\n10 F\n5 0\n10\nC\nQ\nV\n\u2212\n=\n\u00d7\n\u00b5\n=\n\u00d7\n\u2032\n2 15  ENERGY STORED IN A CAPACITOR\nA capacitor, as we have seen above, is a system of two conductors with\ncharge Q and \u2013Q"}, {"Chapter": "1", "sentence_range": "2228-2231", "Text": "7\n10\nC\n3\nQ\nV\n\u2212\n=\n\u00d7\n\u00b5\n=\n\u00d7\n and\n3\n500\n10 F\n5 0\n10\nC\nQ\nV\n\u2212\n=\n\u00d7\n\u00b5\n=\n\u00d7\n\u2032\n2 15  ENERGY STORED IN A CAPACITOR\nA capacitor, as we have seen above, is a system of two conductors with\ncharge Q and \u2013Q To determine the energy stored in this configuration,\nconsider initially two uncharged conductors 1 and 2"}, {"Chapter": "1", "sentence_range": "2229-2232", "Text": "0\n10\nC\nQ\nV\n\u2212\n=\n\u00d7\n\u00b5\n=\n\u00d7\n\u2032\n2 15  ENERGY STORED IN A CAPACITOR\nA capacitor, as we have seen above, is a system of two conductors with\ncharge Q and \u2013Q To determine the energy stored in this configuration,\nconsider initially two uncharged conductors 1 and 2 Imagine next a\nprocess of transferring charge from conductor 2 to conductor 1 bit by\nRationalised 2023-24\nPhysics\n74\nbit, so that at the end, conductor 1 gets charge Q"}, {"Chapter": "1", "sentence_range": "2230-2233", "Text": "15  ENERGY STORED IN A CAPACITOR\nA capacitor, as we have seen above, is a system of two conductors with\ncharge Q and \u2013Q To determine the energy stored in this configuration,\nconsider initially two uncharged conductors 1 and 2 Imagine next a\nprocess of transferring charge from conductor 2 to conductor 1 bit by\nRationalised 2023-24\nPhysics\n74\nbit, so that at the end, conductor 1 gets charge Q By\ncharge conservation, conductor 2 has charge \u2013Q at\nthe end (Fig 2"}, {"Chapter": "1", "sentence_range": "2231-2234", "Text": "To determine the energy stored in this configuration,\nconsider initially two uncharged conductors 1 and 2 Imagine next a\nprocess of transferring charge from conductor 2 to conductor 1 bit by\nRationalised 2023-24\nPhysics\n74\nbit, so that at the end, conductor 1 gets charge Q By\ncharge conservation, conductor 2 has charge \u2013Q at\nthe end (Fig 2 30 )"}, {"Chapter": "1", "sentence_range": "2232-2235", "Text": "Imagine next a\nprocess of transferring charge from conductor 2 to conductor 1 bit by\nRationalised 2023-24\nPhysics\n74\nbit, so that at the end, conductor 1 gets charge Q By\ncharge conservation, conductor 2 has charge \u2013Q at\nthe end (Fig 2 30 ) In transferring positive charge from conductor 2\nto conductor 1, work will be done externally, since at\nany stage conductor 1 is at a higher potential than\nconductor 2"}, {"Chapter": "1", "sentence_range": "2233-2236", "Text": "By\ncharge conservation, conductor 2 has charge \u2013Q at\nthe end (Fig 2 30 ) In transferring positive charge from conductor 2\nto conductor 1, work will be done externally, since at\nany stage conductor 1 is at a higher potential than\nconductor 2 To calculate the total work done, we first\ncalculate the work done in a small step involving\ntransfer of an infinitesimal (i"}, {"Chapter": "1", "sentence_range": "2234-2237", "Text": "30 ) In transferring positive charge from conductor 2\nto conductor 1, work will be done externally, since at\nany stage conductor 1 is at a higher potential than\nconductor 2 To calculate the total work done, we first\ncalculate the work done in a small step involving\ntransfer of an infinitesimal (i e"}, {"Chapter": "1", "sentence_range": "2235-2238", "Text": "In transferring positive charge from conductor 2\nto conductor 1, work will be done externally, since at\nany stage conductor 1 is at a higher potential than\nconductor 2 To calculate the total work done, we first\ncalculate the work done in a small step involving\ntransfer of an infinitesimal (i e , vanishingly small)\namount of charge"}, {"Chapter": "1", "sentence_range": "2236-2239", "Text": "To calculate the total work done, we first\ncalculate the work done in a small step involving\ntransfer of an infinitesimal (i e , vanishingly small)\namount of charge Consider the intermediate situation\nwhen the conductors 1 and 2 have charges Q\u00a2 and\n\u2013Q\u00a2 respectively"}, {"Chapter": "1", "sentence_range": "2237-2240", "Text": "e , vanishingly small)\namount of charge Consider the intermediate situation\nwhen the conductors 1 and 2 have charges Q\u00a2 and\n\u2013Q\u00a2 respectively At this stage, the potential difference\nV\u00a2 between conductors 1 to 2 is Q\u00a2/C, where C is the\ncapacitance of the system"}, {"Chapter": "1", "sentence_range": "2238-2241", "Text": ", vanishingly small)\namount of charge Consider the intermediate situation\nwhen the conductors 1 and 2 have charges Q\u00a2 and\n\u2013Q\u00a2 respectively At this stage, the potential difference\nV\u00a2 between conductors 1 to 2 is Q\u00a2/C, where C is the\ncapacitance of the system Next imagine that a small\ncharge d Q\u00a2 is transferred from conductor 2 to 1"}, {"Chapter": "1", "sentence_range": "2239-2242", "Text": "Consider the intermediate situation\nwhen the conductors 1 and 2 have charges Q\u00a2 and\n\u2013Q\u00a2 respectively At this stage, the potential difference\nV\u00a2 between conductors 1 to 2 is Q\u00a2/C, where C is the\ncapacitance of the system Next imagine that a small\ncharge d Q\u00a2 is transferred from conductor 2 to 1 Work\ndone in this step (d W), resulting in charge Q\u00a2 on\nconductor 1 increasing to Q\u00a2+ d Q\u00a2, is given by\nQ\nW\nV\nQ\nQ\nC\n\u03b4\n\u03b4\n\u2032\u03b4\n=\n=\n\u2032\n\u2032\n\u2032\n(2"}, {"Chapter": "1", "sentence_range": "2240-2243", "Text": "At this stage, the potential difference\nV\u00a2 between conductors 1 to 2 is Q\u00a2/C, where C is the\ncapacitance of the system Next imagine that a small\ncharge d Q\u00a2 is transferred from conductor 2 to 1 Work\ndone in this step (d W), resulting in charge Q\u00a2 on\nconductor 1 increasing to Q\u00a2+ d Q\u00a2, is given by\nQ\nW\nV\nQ\nQ\nC\n\u03b4\n\u03b4\n\u2032\u03b4\n=\n=\n\u2032\n\u2032\n\u2032\n(2 68)\nIntegrating eq"}, {"Chapter": "1", "sentence_range": "2241-2244", "Text": "Next imagine that a small\ncharge d Q\u00a2 is transferred from conductor 2 to 1 Work\ndone in this step (d W), resulting in charge Q\u00a2 on\nconductor 1 increasing to Q\u00a2+ d Q\u00a2, is given by\nQ\nW\nV\nQ\nQ\nC\n\u03b4\n\u03b4\n\u2032\u03b4\n=\n=\n\u2032\n\u2032\n\u2032\n(2 68)\nIntegrating eq (2"}, {"Chapter": "1", "sentence_range": "2242-2245", "Text": "Work\ndone in this step (d W), resulting in charge Q\u00a2 on\nconductor 1 increasing to Q\u00a2+ d Q\u00a2, is given by\nQ\nW\nV\nQ\nQ\nC\n\u03b4\n\u03b4\n\u2032\u03b4\n=\n=\n\u2032\n\u2032\n\u2032\n(2 68)\nIntegrating eq (2 68)\nW\nQ\nC\nQ\nC\nQ\nQ\nC\nQ\nQ\n=\n\u2032\n=\n\u2032\n=\n\u222b\n0\n2\n0\n2\n1\n2\n2\n\u03b4\n\u2019\nWe can write the final result, in different ways\n2\n2\n1\n1\n2\n2\n2\nQ\nW\nCV\nQV\nC\n=\n=\n=\n(2"}, {"Chapter": "1", "sentence_range": "2243-2246", "Text": "68)\nIntegrating eq (2 68)\nW\nQ\nC\nQ\nC\nQ\nQ\nC\nQ\nQ\n=\n\u2032\n=\n\u2032\n=\n\u222b\n0\n2\n0\n2\n1\n2\n2\n\u03b4\n\u2019\nWe can write the final result, in different ways\n2\n2\n1\n1\n2\n2\n2\nQ\nW\nCV\nQV\nC\n=\n=\n=\n(2 69)\nSince electrostatic force is conservative, this work is stored in the form\nof potential energy of the system"}, {"Chapter": "1", "sentence_range": "2244-2247", "Text": "(2 68)\nW\nQ\nC\nQ\nC\nQ\nQ\nC\nQ\nQ\n=\n\u2032\n=\n\u2032\n=\n\u222b\n0\n2\n0\n2\n1\n2\n2\n\u03b4\n\u2019\nWe can write the final result, in different ways\n2\n2\n1\n1\n2\n2\n2\nQ\nW\nCV\nQV\nC\n=\n=\n=\n(2 69)\nSince electrostatic force is conservative, this work is stored in the form\nof potential energy of the system For the same reason, the final result for\npotential energy [Eq"}, {"Chapter": "1", "sentence_range": "2245-2248", "Text": "68)\nW\nQ\nC\nQ\nC\nQ\nQ\nC\nQ\nQ\n=\n\u2032\n=\n\u2032\n=\n\u222b\n0\n2\n0\n2\n1\n2\n2\n\u03b4\n\u2019\nWe can write the final result, in different ways\n2\n2\n1\n1\n2\n2\n2\nQ\nW\nCV\nQV\nC\n=\n=\n=\n(2 69)\nSince electrostatic force is conservative, this work is stored in the form\nof potential energy of the system For the same reason, the final result for\npotential energy [Eq (2"}, {"Chapter": "1", "sentence_range": "2246-2249", "Text": "69)\nSince electrostatic force is conservative, this work is stored in the form\nof potential energy of the system For the same reason, the final result for\npotential energy [Eq (2 69)] is independent of the manner in which the\ncharge configuration of the capacitor is built up"}, {"Chapter": "1", "sentence_range": "2247-2250", "Text": "For the same reason, the final result for\npotential energy [Eq (2 69)] is independent of the manner in which the\ncharge configuration of the capacitor is built up When the capacitor\ndischarges, this stored-up energy is released"}, {"Chapter": "1", "sentence_range": "2248-2251", "Text": "(2 69)] is independent of the manner in which the\ncharge configuration of the capacitor is built up When the capacitor\ndischarges, this stored-up energy is released It is possible to view the\npotential energy of the capacitor as \u2018stored\u2019 in the electric field between\nthe plates"}, {"Chapter": "1", "sentence_range": "2249-2252", "Text": "69)] is independent of the manner in which the\ncharge configuration of the capacitor is built up When the capacitor\ndischarges, this stored-up energy is released It is possible to view the\npotential energy of the capacitor as \u2018stored\u2019 in the electric field between\nthe plates To see this, consider for simplicity, a parallel plate capacitor\n[of area A (of each plate) and separation d between the plates]"}, {"Chapter": "1", "sentence_range": "2250-2253", "Text": "When the capacitor\ndischarges, this stored-up energy is released It is possible to view the\npotential energy of the capacitor as \u2018stored\u2019 in the electric field between\nthe plates To see this, consider for simplicity, a parallel plate capacitor\n[of area A (of each plate) and separation d between the plates] Energy stored in the capacitor\n= \n2\n2\n0\n1\n(\n)\n2\n2\nQ\nA\nd\nC\nA\n\u03c3\n\u03b5\n=\n\u00d7\n(2"}, {"Chapter": "1", "sentence_range": "2251-2254", "Text": "It is possible to view the\npotential energy of the capacitor as \u2018stored\u2019 in the electric field between\nthe plates To see this, consider for simplicity, a parallel plate capacitor\n[of area A (of each plate) and separation d between the plates] Energy stored in the capacitor\n= \n2\n2\n0\n1\n(\n)\n2\n2\nQ\nA\nd\nC\nA\n\u03c3\n\u03b5\n=\n\u00d7\n(2 70)\nThe surface charge density s is related to the electric field E between\nthe plates,\n0\nE\n=\u03b5\u03c3\n(2"}, {"Chapter": "1", "sentence_range": "2252-2255", "Text": "To see this, consider for simplicity, a parallel plate capacitor\n[of area A (of each plate) and separation d between the plates] Energy stored in the capacitor\n= \n2\n2\n0\n1\n(\n)\n2\n2\nQ\nA\nd\nC\nA\n\u03c3\n\u03b5\n=\n\u00d7\n(2 70)\nThe surface charge density s is related to the electric field E between\nthe plates,\n0\nE\n=\u03b5\u03c3\n(2 71)\nFrom Eqs"}, {"Chapter": "1", "sentence_range": "2253-2256", "Text": "Energy stored in the capacitor\n= \n2\n2\n0\n1\n(\n)\n2\n2\nQ\nA\nd\nC\nA\n\u03c3\n\u03b5\n=\n\u00d7\n(2 70)\nThe surface charge density s is related to the electric field E between\nthe plates,\n0\nE\n=\u03b5\u03c3\n(2 71)\nFrom Eqs (2"}, {"Chapter": "1", "sentence_range": "2254-2257", "Text": "70)\nThe surface charge density s is related to the electric field E between\nthe plates,\n0\nE\n=\u03b5\u03c3\n(2 71)\nFrom Eqs (2 70) and (2"}, {"Chapter": "1", "sentence_range": "2255-2258", "Text": "71)\nFrom Eqs (2 70) and (2 71) , we get\nEnergy stored in the capacitor\nU = (\n)\n2\n0\n1/2\nE\nA d\n\u03b5\n\u00d7\n(2"}, {"Chapter": "1", "sentence_range": "2256-2259", "Text": "(2 70) and (2 71) , we get\nEnergy stored in the capacitor\nU = (\n)\n2\n0\n1/2\nE\nA d\n\u03b5\n\u00d7\n(2 72)\nFIGURE 2"}, {"Chapter": "1", "sentence_range": "2257-2260", "Text": "70) and (2 71) , we get\nEnergy stored in the capacitor\nU = (\n)\n2\n0\n1/2\nE\nA d\n\u03b5\n\u00d7\n(2 72)\nFIGURE 2 30 (a) Work done in a small\nstep of building charge on conductor 1\nfrom Q\u00a2 to Q\u00a2 + d Q\u00a2"}, {"Chapter": "1", "sentence_range": "2258-2261", "Text": "71) , we get\nEnergy stored in the capacitor\nU = (\n)\n2\n0\n1/2\nE\nA d\n\u03b5\n\u00d7\n(2 72)\nFIGURE 2 30 (a) Work done in a small\nstep of building charge on conductor 1\nfrom Q\u00a2 to Q\u00a2 + d Q\u00a2 (b)  Total work done\nin charging the capacitor may be\nviewed as stored in the energy of\nelectric field between the plates"}, {"Chapter": "1", "sentence_range": "2259-2262", "Text": "72)\nFIGURE 2 30 (a) Work done in a small\nstep of building charge on conductor 1\nfrom Q\u00a2 to Q\u00a2 + d Q\u00a2 (b)  Total work done\nin charging the capacitor may be\nviewed as stored in the energy of\nelectric field between the plates Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n75\n EXAMPLE 2"}, {"Chapter": "1", "sentence_range": "2260-2263", "Text": "30 (a) Work done in a small\nstep of building charge on conductor 1\nfrom Q\u00a2 to Q\u00a2 + d Q\u00a2 (b)  Total work done\nin charging the capacitor may be\nviewed as stored in the energy of\nelectric field between the plates Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n75\n EXAMPLE 2 10\nNote that Ad is the volume of the region between the plates (where\nelectric field alone exists)"}, {"Chapter": "1", "sentence_range": "2261-2264", "Text": "(b)  Total work done\nin charging the capacitor may be\nviewed as stored in the energy of\nelectric field between the plates Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n75\n EXAMPLE 2 10\nNote that Ad is the volume of the region between the plates (where\nelectric field alone exists) If we define energy density as energy stored\nper unit volume of space, Eq (2"}, {"Chapter": "1", "sentence_range": "2262-2265", "Text": "Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n75\n EXAMPLE 2 10\nNote that Ad is the volume of the region between the plates (where\nelectric field alone exists) If we define energy density as energy stored\nper unit volume of space, Eq (2 72) shows that\nEnergy density of electric field,\nu =(1/2)e0E 2\n(2"}, {"Chapter": "1", "sentence_range": "2263-2266", "Text": "10\nNote that Ad is the volume of the region between the plates (where\nelectric field alone exists) If we define energy density as energy stored\nper unit volume of space, Eq (2 72) shows that\nEnergy density of electric field,\nu =(1/2)e0E 2\n(2 73)\nThough we derived Eq"}, {"Chapter": "1", "sentence_range": "2264-2267", "Text": "If we define energy density as energy stored\nper unit volume of space, Eq (2 72) shows that\nEnergy density of electric field,\nu =(1/2)e0E 2\n(2 73)\nThough we derived Eq (2"}, {"Chapter": "1", "sentence_range": "2265-2268", "Text": "72) shows that\nEnergy density of electric field,\nu =(1/2)e0E 2\n(2 73)\nThough we derived Eq (2 73) for the case of a parallel plate\ncapacitor, the result on energy density of an electric field is, in fact,\nvery general and holds true for electric field due to any configuration\nof charges"}, {"Chapter": "1", "sentence_range": "2266-2269", "Text": "73)\nThough we derived Eq (2 73) for the case of a parallel plate\ncapacitor, the result on energy density of an electric field is, in fact,\nvery general and holds true for electric field due to any configuration\nof charges Example 2"}, {"Chapter": "1", "sentence_range": "2267-2270", "Text": "(2 73) for the case of a parallel plate\ncapacitor, the result on energy density of an electric field is, in fact,\nvery general and holds true for electric field due to any configuration\nof charges Example 2 10 (a) A 900 pF capacitor is charged by 100 V battery\n[Fig"}, {"Chapter": "1", "sentence_range": "2268-2271", "Text": "73) for the case of a parallel plate\ncapacitor, the result on energy density of an electric field is, in fact,\nvery general and holds true for electric field due to any configuration\nof charges Example 2 10 (a) A 900 pF capacitor is charged by 100 V battery\n[Fig 2"}, {"Chapter": "1", "sentence_range": "2269-2272", "Text": "Example 2 10 (a) A 900 pF capacitor is charged by 100 V battery\n[Fig 2 31(a)]"}, {"Chapter": "1", "sentence_range": "2270-2273", "Text": "10 (a) A 900 pF capacitor is charged by 100 V battery\n[Fig 2 31(a)] How much electrostatic energy is stored by the capacitor"}, {"Chapter": "1", "sentence_range": "2271-2274", "Text": "2 31(a)] How much electrostatic energy is stored by the capacitor (b) The capacitor is disconnected from the battery and connected to\nanother 900 pF capacitor [Fig"}, {"Chapter": "1", "sentence_range": "2272-2275", "Text": "31(a)] How much electrostatic energy is stored by the capacitor (b) The capacitor is disconnected from the battery and connected to\nanother 900 pF capacitor [Fig 2"}, {"Chapter": "1", "sentence_range": "2273-2276", "Text": "How much electrostatic energy is stored by the capacitor (b) The capacitor is disconnected from the battery and connected to\nanother 900 pF capacitor [Fig 2 31(b)]"}, {"Chapter": "1", "sentence_range": "2274-2277", "Text": "(b) The capacitor is disconnected from the battery and connected to\nanother 900 pF capacitor [Fig 2 31(b)] What is the electrostatic\nenergy stored by the system"}, {"Chapter": "1", "sentence_range": "2275-2278", "Text": "2 31(b)] What is the electrostatic\nenergy stored by the system FIGURE 2"}, {"Chapter": "1", "sentence_range": "2276-2279", "Text": "31(b)] What is the electrostatic\nenergy stored by the system FIGURE 2 31\nSolution\n(a) The charge on the capacitor is\n      Q = CV = 900 \u00d7 10\u201312 F \u00d7 100 V = 9 \u00d7 10\u20138 C\nThe energy stored by the capacitor is\n      = (1/2) CV 2 = (1/2) QV\n= (1/2) \u00d7 9 \u00d7 10\u20138C \u00d7 100 V = 4"}, {"Chapter": "1", "sentence_range": "2277-2280", "Text": "What is the electrostatic\nenergy stored by the system FIGURE 2 31\nSolution\n(a) The charge on the capacitor is\n      Q = CV = 900 \u00d7 10\u201312 F \u00d7 100 V = 9 \u00d7 10\u20138 C\nThe energy stored by the capacitor is\n      = (1/2) CV 2 = (1/2) QV\n= (1/2) \u00d7 9 \u00d7 10\u20138C \u00d7 100 V = 4 5 \u00d7 10\u20136 J\n(b) In the steady situation, the two capacitors have their positive\nplates at the same potential, and their negative plates at the\nsame potential"}, {"Chapter": "1", "sentence_range": "2278-2281", "Text": "FIGURE 2 31\nSolution\n(a) The charge on the capacitor is\n      Q = CV = 900 \u00d7 10\u201312 F \u00d7 100 V = 9 \u00d7 10\u20138 C\nThe energy stored by the capacitor is\n      = (1/2) CV 2 = (1/2) QV\n= (1/2) \u00d7 9 \u00d7 10\u20138C \u00d7 100 V = 4 5 \u00d7 10\u20136 J\n(b) In the steady situation, the two capacitors have their positive\nplates at the same potential, and their negative plates at the\nsame potential Let the common potential difference be V\u00a2"}, {"Chapter": "1", "sentence_range": "2279-2282", "Text": "31\nSolution\n(a) The charge on the capacitor is\n      Q = CV = 900 \u00d7 10\u201312 F \u00d7 100 V = 9 \u00d7 10\u20138 C\nThe energy stored by the capacitor is\n      = (1/2) CV 2 = (1/2) QV\n= (1/2) \u00d7 9 \u00d7 10\u20138C \u00d7 100 V = 4 5 \u00d7 10\u20136 J\n(b) In the steady situation, the two capacitors have their positive\nplates at the same potential, and their negative plates at the\nsame potential Let the common potential difference be V\u00a2 The\nRationalised 2023-24\nPhysics\n76\ncharge on each capacitor is then Q\u00a2 = CV\u00a2"}, {"Chapter": "1", "sentence_range": "2280-2283", "Text": "5 \u00d7 10\u20136 J\n(b) In the steady situation, the two capacitors have their positive\nplates at the same potential, and their negative plates at the\nsame potential Let the common potential difference be V\u00a2 The\nRationalised 2023-24\nPhysics\n76\ncharge on each capacitor is then Q\u00a2 = CV\u00a2 By charge conservation,\nQ\u00a2  = Q/2"}, {"Chapter": "1", "sentence_range": "2281-2284", "Text": "Let the common potential difference be V\u00a2 The\nRationalised 2023-24\nPhysics\n76\ncharge on each capacitor is then Q\u00a2 = CV\u00a2 By charge conservation,\nQ\u00a2  = Q/2 This implies V\u00a2  = V/2"}, {"Chapter": "1", "sentence_range": "2282-2285", "Text": "The\nRationalised 2023-24\nPhysics\n76\ncharge on each capacitor is then Q\u00a2 = CV\u00a2 By charge conservation,\nQ\u00a2  = Q/2 This implies V\u00a2  = V/2 The total energy of the system is\n6\n1\n1\n2\n'\n'\n2"}, {"Chapter": "1", "sentence_range": "2283-2286", "Text": "By charge conservation,\nQ\u00a2  = Q/2 This implies V\u00a2  = V/2 The total energy of the system is\n6\n1\n1\n2\n'\n'\n2 25\n10\nJ\n2\n4\nQ V\nQV\n\u2212\n=\n\u00d7\n=\n=\n\u00d7\nThus in going from (a) to (b), though no charge is lost; the final\nenergy is only half the initial energy"}, {"Chapter": "1", "sentence_range": "2284-2287", "Text": "This implies V\u00a2  = V/2 The total energy of the system is\n6\n1\n1\n2\n'\n'\n2 25\n10\nJ\n2\n4\nQ V\nQV\n\u2212\n=\n\u00d7\n=\n=\n\u00d7\nThus in going from (a) to (b), though no charge is lost; the final\nenergy is only half the initial energy Where has the remaining energy\ngone"}, {"Chapter": "1", "sentence_range": "2285-2288", "Text": "The total energy of the system is\n6\n1\n1\n2\n'\n'\n2 25\n10\nJ\n2\n4\nQ V\nQV\n\u2212\n=\n\u00d7\n=\n=\n\u00d7\nThus in going from (a) to (b), though no charge is lost; the final\nenergy is only half the initial energy Where has the remaining energy\ngone There is a transient period before the system settles to the\nsituation (b)"}, {"Chapter": "1", "sentence_range": "2286-2289", "Text": "25\n10\nJ\n2\n4\nQ V\nQV\n\u2212\n=\n\u00d7\n=\n=\n\u00d7\nThus in going from (a) to (b), though no charge is lost; the final\nenergy is only half the initial energy Where has the remaining energy\ngone There is a transient period before the system settles to the\nsituation (b) During this period, a transient current flows from\nthe first capacitor to the second"}, {"Chapter": "1", "sentence_range": "2287-2290", "Text": "Where has the remaining energy\ngone There is a transient period before the system settles to the\nsituation (b) During this period, a transient current flows from\nthe first capacitor to the second Energy is lost during this time in\nthe form of heat and electromagnetic radiation"}, {"Chapter": "1", "sentence_range": "2288-2291", "Text": "There is a transient period before the system settles to the\nsituation (b) During this period, a transient current flows from\nthe first capacitor to the second Energy is lost during this time in\nthe form of heat and electromagnetic radiation EXAMPLE 2"}, {"Chapter": "1", "sentence_range": "2289-2292", "Text": "During this period, a transient current flows from\nthe first capacitor to the second Energy is lost during this time in\nthe form of heat and electromagnetic radiation EXAMPLE 2 10\nSUMMARY\n1"}, {"Chapter": "1", "sentence_range": "2290-2293", "Text": "Energy is lost during this time in\nthe form of heat and electromagnetic radiation EXAMPLE 2 10\nSUMMARY\n1 Electrostatic force is a conservative force"}, {"Chapter": "1", "sentence_range": "2291-2294", "Text": "EXAMPLE 2 10\nSUMMARY\n1 Electrostatic force is a conservative force Work done by an external\nforce (equal and opposite to the electrostatic force) in bringing a charge\nq from a point R to a point P is q(VP\u2013VR), which is the difference in\npotential energy of charge q between the final and initial points"}, {"Chapter": "1", "sentence_range": "2292-2295", "Text": "10\nSUMMARY\n1 Electrostatic force is a conservative force Work done by an external\nforce (equal and opposite to the electrostatic force) in bringing a charge\nq from a point R to a point P is q(VP\u2013VR), which is the difference in\npotential energy of charge q between the final and initial points 2"}, {"Chapter": "1", "sentence_range": "2293-2296", "Text": "Electrostatic force is a conservative force Work done by an external\nforce (equal and opposite to the electrostatic force) in bringing a charge\nq from a point R to a point P is q(VP\u2013VR), which is the difference in\npotential energy of charge q between the final and initial points 2 Potential at a point is the work done per unit charge (by an external\nagency) in bringing a charge from infinity to that point"}, {"Chapter": "1", "sentence_range": "2294-2297", "Text": "Work done by an external\nforce (equal and opposite to the electrostatic force) in bringing a charge\nq from a point R to a point P is q(VP\u2013VR), which is the difference in\npotential energy of charge q between the final and initial points 2 Potential at a point is the work done per unit charge (by an external\nagency) in bringing a charge from infinity to that point Potential at a\npoint is arbitrary to within an additive constant, since it is the potential\ndifference between two points which is physically significant"}, {"Chapter": "1", "sentence_range": "2295-2298", "Text": "2 Potential at a point is the work done per unit charge (by an external\nagency) in bringing a charge from infinity to that point Potential at a\npoint is arbitrary to within an additive constant, since it is the potential\ndifference between two points which is physically significant If potential\nat infinity is chosen to be zero; potential at a point with position vector\nr due to a point charge Q placed at the origin is given is given by\n1\n( )\n4\no\nQ\nV\nr\n\u03b5\n=\n\u03c0\nr\n3"}, {"Chapter": "1", "sentence_range": "2296-2299", "Text": "Potential at a point is the work done per unit charge (by an external\nagency) in bringing a charge from infinity to that point Potential at a\npoint is arbitrary to within an additive constant, since it is the potential\ndifference between two points which is physically significant If potential\nat infinity is chosen to be zero; potential at a point with position vector\nr due to a point charge Q placed at the origin is given is given by\n1\n( )\n4\no\nQ\nV\nr\n\u03b5\n=\n\u03c0\nr\n3 The electrostatic potential at a point with position vector r due to a\npoint dipole of dipole moment p placed at the origin is\n\u02c62\n1\n( )\n=4 \u03b5\n\u03c0\np"}, {"Chapter": "1", "sentence_range": "2297-2300", "Text": "Potential at a\npoint is arbitrary to within an additive constant, since it is the potential\ndifference between two points which is physically significant If potential\nat infinity is chosen to be zero; potential at a point with position vector\nr due to a point charge Q placed at the origin is given is given by\n1\n( )\n4\no\nQ\nV\nr\n\u03b5\n=\n\u03c0\nr\n3 The electrostatic potential at a point with position vector r due to a\npoint dipole of dipole moment p placed at the origin is\n\u02c62\n1\n( )\n=4 \u03b5\n\u03c0\np r\nr\no\nV\nr\nThe result is true also for a dipole (with charges \u2013q and q separated by\n2a)  for r >> a"}, {"Chapter": "1", "sentence_range": "2298-2301", "Text": "If potential\nat infinity is chosen to be zero; potential at a point with position vector\nr due to a point charge Q placed at the origin is given is given by\n1\n( )\n4\no\nQ\nV\nr\n\u03b5\n=\n\u03c0\nr\n3 The electrostatic potential at a point with position vector r due to a\npoint dipole of dipole moment p placed at the origin is\n\u02c62\n1\n( )\n=4 \u03b5\n\u03c0\np r\nr\no\nV\nr\nThe result is true also for a dipole (with charges \u2013q and q separated by\n2a)  for r >> a 4"}, {"Chapter": "1", "sentence_range": "2299-2302", "Text": "The electrostatic potential at a point with position vector r due to a\npoint dipole of dipole moment p placed at the origin is\n\u02c62\n1\n( )\n=4 \u03b5\n\u03c0\np r\nr\no\nV\nr\nThe result is true also for a dipole (with charges \u2013q and q separated by\n2a)  for r >> a 4 For a charge configuration q1, q2,"}, {"Chapter": "1", "sentence_range": "2300-2303", "Text": "r\nr\no\nV\nr\nThe result is true also for a dipole (with charges \u2013q and q separated by\n2a)  for r >> a 4 For a charge configuration q1, q2, , qn with position vectors r1,\nr2,"}, {"Chapter": "1", "sentence_range": "2301-2304", "Text": "4 For a charge configuration q1, q2, , qn with position vectors r1,\nr2, rn, the potential at a point P is given by the superposition principle\n1\n2\n0\n1P\n2P\nP\n1\n("}, {"Chapter": "1", "sentence_range": "2302-2305", "Text": "For a charge configuration q1, q2, , qn with position vectors r1,\nr2, rn, the potential at a point P is given by the superposition principle\n1\n2\n0\n1P\n2P\nP\n1\n( )\n4\nn\nn\nq\nq\nq\nV\nr\nr\nr\n\u03b5\n=\n+\n+\n+\n\u03c0\nwhere r1P is the distance between q1 and P, as and so on"}, {"Chapter": "1", "sentence_range": "2303-2306", "Text": ", qn with position vectors r1,\nr2, rn, the potential at a point P is given by the superposition principle\n1\n2\n0\n1P\n2P\nP\n1\n( )\n4\nn\nn\nq\nq\nq\nV\nr\nr\nr\n\u03b5\n=\n+\n+\n+\n\u03c0\nwhere r1P is the distance between q1 and P, as and so on 5"}, {"Chapter": "1", "sentence_range": "2304-2307", "Text": "rn, the potential at a point P is given by the superposition principle\n1\n2\n0\n1P\n2P\nP\n1\n( )\n4\nn\nn\nq\nq\nq\nV\nr\nr\nr\n\u03b5\n=\n+\n+\n+\n\u03c0\nwhere r1P is the distance between q1 and P, as and so on 5 An equipotential surface is a surface over which potential has a constant\nvalue"}, {"Chapter": "1", "sentence_range": "2305-2308", "Text": ")\n4\nn\nn\nq\nq\nq\nV\nr\nr\nr\n\u03b5\n=\n+\n+\n+\n\u03c0\nwhere r1P is the distance between q1 and P, as and so on 5 An equipotential surface is a surface over which potential has a constant\nvalue For a point charge, concentric spheres centred at a location of the\ncharge are equipotential surfaces"}, {"Chapter": "1", "sentence_range": "2306-2309", "Text": "5 An equipotential surface is a surface over which potential has a constant\nvalue For a point charge, concentric spheres centred at a location of the\ncharge are equipotential surfaces The electric field E at a point is\nperpendicular to the equipotential surface through the point"}, {"Chapter": "1", "sentence_range": "2307-2310", "Text": "An equipotential surface is a surface over which potential has a constant\nvalue For a point charge, concentric spheres centred at a location of the\ncharge are equipotential surfaces The electric field E at a point is\nperpendicular to the equipotential surface through the point E is in the\ndirection of the steepest decrease of potential"}, {"Chapter": "1", "sentence_range": "2308-2311", "Text": "For a point charge, concentric spheres centred at a location of the\ncharge are equipotential surfaces The electric field E at a point is\nperpendicular to the equipotential surface through the point E is in the\ndirection of the steepest decrease of potential Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n77\n6"}, {"Chapter": "1", "sentence_range": "2309-2312", "Text": "The electric field E at a point is\nperpendicular to the equipotential surface through the point E is in the\ndirection of the steepest decrease of potential Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n77\n6 Potential energy stored in a system of charges is the work done (by an\nexternal agency) in assembling the charges at their locations"}, {"Chapter": "1", "sentence_range": "2310-2313", "Text": "E is in the\ndirection of the steepest decrease of potential Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n77\n6 Potential energy stored in a system of charges is the work done (by an\nexternal agency) in assembling the charges at their locations Potential\nenergy of two charges q1, q2 at r1, r2 is given by\n1\n2\n0\n12\n1\n4\nq q\nU\nr\n\u03b5\n=\n\u03c0\nwhere r12 is distance between q1 and q2"}, {"Chapter": "1", "sentence_range": "2311-2314", "Text": "Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n77\n6 Potential energy stored in a system of charges is the work done (by an\nexternal agency) in assembling the charges at their locations Potential\nenergy of two charges q1, q2 at r1, r2 is given by\n1\n2\n0\n12\n1\n4\nq q\nU\nr\n\u03b5\n=\n\u03c0\nwhere r12 is distance between q1 and q2 7"}, {"Chapter": "1", "sentence_range": "2312-2315", "Text": "Potential energy stored in a system of charges is the work done (by an\nexternal agency) in assembling the charges at their locations Potential\nenergy of two charges q1, q2 at r1, r2 is given by\n1\n2\n0\n12\n1\n4\nq q\nU\nr\n\u03b5\n=\n\u03c0\nwhere r12 is distance between q1 and q2 7 The potential energy of a charge q in an external potential V(r) is qV(r)"}, {"Chapter": "1", "sentence_range": "2313-2316", "Text": "Potential\nenergy of two charges q1, q2 at r1, r2 is given by\n1\n2\n0\n12\n1\n4\nq q\nU\nr\n\u03b5\n=\n\u03c0\nwhere r12 is distance between q1 and q2 7 The potential energy of a charge q in an external potential V(r) is qV(r) The potential energy of a dipole moment p in a uniform electric field E\nis  \u2013p"}, {"Chapter": "1", "sentence_range": "2314-2317", "Text": "7 The potential energy of a charge q in an external potential V(r) is qV(r) The potential energy of a dipole moment p in a uniform electric field E\nis  \u2013p E"}, {"Chapter": "1", "sentence_range": "2315-2318", "Text": "The potential energy of a charge q in an external potential V(r) is qV(r) The potential energy of a dipole moment p in a uniform electric field E\nis  \u2013p E 8"}, {"Chapter": "1", "sentence_range": "2316-2319", "Text": "The potential energy of a dipole moment p in a uniform electric field E\nis  \u2013p E 8 Electrostatics field E is zero in the interior of a conductor; just outside\nthe surface of a charged conductor, E is normal to the surface given by\n0\nE=\u03b5\u03c3\u02c6\nn  where \u02c6n  is the unit vector along the outward normal to the\nsurface and s is the surface charge density"}, {"Chapter": "1", "sentence_range": "2317-2320", "Text": "E 8 Electrostatics field E is zero in the interior of a conductor; just outside\nthe surface of a charged conductor, E is normal to the surface given by\n0\nE=\u03b5\u03c3\u02c6\nn  where \u02c6n  is the unit vector along the outward normal to the\nsurface and s is the surface charge density Charges in a conductor can\nreside only at its surface"}, {"Chapter": "1", "sentence_range": "2318-2321", "Text": "8 Electrostatics field E is zero in the interior of a conductor; just outside\nthe surface of a charged conductor, E is normal to the surface given by\n0\nE=\u03b5\u03c3\u02c6\nn  where \u02c6n  is the unit vector along the outward normal to the\nsurface and s is the surface charge density Charges in a conductor can\nreside only at its surface Potential is constant within and on the surface\nof a conductor"}, {"Chapter": "1", "sentence_range": "2319-2322", "Text": "Electrostatics field E is zero in the interior of a conductor; just outside\nthe surface of a charged conductor, E is normal to the surface given by\n0\nE=\u03b5\u03c3\u02c6\nn  where \u02c6n  is the unit vector along the outward normal to the\nsurface and s is the surface charge density Charges in a conductor can\nreside only at its surface Potential is constant within and on the surface\nof a conductor In a cavity within a conductor (with no charges), the\nelectric field is zero"}, {"Chapter": "1", "sentence_range": "2320-2323", "Text": "Charges in a conductor can\nreside only at its surface Potential is constant within and on the surface\nof a conductor In a cavity within a conductor (with no charges), the\nelectric field is zero 9"}, {"Chapter": "1", "sentence_range": "2321-2324", "Text": "Potential is constant within and on the surface\nof a conductor In a cavity within a conductor (with no charges), the\nelectric field is zero 9 A capacitor is a system of two conductors separated by an insulator"}, {"Chapter": "1", "sentence_range": "2322-2325", "Text": "In a cavity within a conductor (with no charges), the\nelectric field is zero 9 A capacitor is a system of two conductors separated by an insulator Its\ncapacitance is defined by C = Q/V, where Q and \u2013Q are the charges on the\ntwo conductors and V is the potential difference between them"}, {"Chapter": "1", "sentence_range": "2323-2326", "Text": "9 A capacitor is a system of two conductors separated by an insulator Its\ncapacitance is defined by C = Q/V, where Q and \u2013Q are the charges on the\ntwo conductors and V is the potential difference between them C is\ndetermined purely geometrically, by the shapes, sizes and relative\npositions of the two conductors"}, {"Chapter": "1", "sentence_range": "2324-2327", "Text": "A capacitor is a system of two conductors separated by an insulator Its\ncapacitance is defined by C = Q/V, where Q and \u2013Q are the charges on the\ntwo conductors and V is the potential difference between them C is\ndetermined purely geometrically, by the shapes, sizes and relative\npositions of the two conductors The unit of capacitance is farad:,\n1 F = 1 C V \u20131"}, {"Chapter": "1", "sentence_range": "2325-2328", "Text": "Its\ncapacitance is defined by C = Q/V, where Q and \u2013Q are the charges on the\ntwo conductors and V is the potential difference between them C is\ndetermined purely geometrically, by the shapes, sizes and relative\npositions of the two conductors The unit of capacitance is farad:,\n1 F = 1 C V \u20131 For a parallel plate capacitor (with vacuum between the\nplates),\nC = \n0\ndA\n\u03b5\nwhere A is the area of each plate and d the separation between them"}, {"Chapter": "1", "sentence_range": "2326-2329", "Text": "C is\ndetermined purely geometrically, by the shapes, sizes and relative\npositions of the two conductors The unit of capacitance is farad:,\n1 F = 1 C V \u20131 For a parallel plate capacitor (with vacuum between the\nplates),\nC = \n0\ndA\n\u03b5\nwhere A is the area of each plate and d the separation between them 10"}, {"Chapter": "1", "sentence_range": "2327-2330", "Text": "The unit of capacitance is farad:,\n1 F = 1 C V \u20131 For a parallel plate capacitor (with vacuum between the\nplates),\nC = \n0\ndA\n\u03b5\nwhere A is the area of each plate and d the separation between them 10 If the medium between the plates of a capacitor is filled with an insulating\nsubstance (dielectric), the electric field due to the charged plates induces\na net dipole moment in the dielectric"}, {"Chapter": "1", "sentence_range": "2328-2331", "Text": "For a parallel plate capacitor (with vacuum between the\nplates),\nC = \n0\ndA\n\u03b5\nwhere A is the area of each plate and d the separation between them 10 If the medium between the plates of a capacitor is filled with an insulating\nsubstance (dielectric), the electric field due to the charged plates induces\na net dipole moment in the dielectric This effect, called polarisation,\ngives rise to a field in the opposite direction"}, {"Chapter": "1", "sentence_range": "2329-2332", "Text": "10 If the medium between the plates of a capacitor is filled with an insulating\nsubstance (dielectric), the electric field due to the charged plates induces\na net dipole moment in the dielectric This effect, called polarisation,\ngives rise to a field in the opposite direction The net electric field inside\nthe dielectric and hence the potential difference between the plates is\nthus reduced"}, {"Chapter": "1", "sentence_range": "2330-2333", "Text": "If the medium between the plates of a capacitor is filled with an insulating\nsubstance (dielectric), the electric field due to the charged plates induces\na net dipole moment in the dielectric This effect, called polarisation,\ngives rise to a field in the opposite direction The net electric field inside\nthe dielectric and hence the potential difference between the plates is\nthus reduced Consequently, the capacitance C increases from its value\nC0 when there is no medium (vacuum),\nC = KC0\nwhere K is the dielectric constant of the insulating substance"}, {"Chapter": "1", "sentence_range": "2331-2334", "Text": "This effect, called polarisation,\ngives rise to a field in the opposite direction The net electric field inside\nthe dielectric and hence the potential difference between the plates is\nthus reduced Consequently, the capacitance C increases from its value\nC0 when there is no medium (vacuum),\nC = KC0\nwhere K is the dielectric constant of the insulating substance 11"}, {"Chapter": "1", "sentence_range": "2332-2335", "Text": "The net electric field inside\nthe dielectric and hence the potential difference between the plates is\nthus reduced Consequently, the capacitance C increases from its value\nC0 when there is no medium (vacuum),\nC = KC0\nwhere K is the dielectric constant of the insulating substance 11 For capacitors in the series combination, the total capacitance C is given by\n1\n2\n3\n1\n1\n1\n1"}, {"Chapter": "1", "sentence_range": "2333-2336", "Text": "Consequently, the capacitance C increases from its value\nC0 when there is no medium (vacuum),\nC = KC0\nwhere K is the dielectric constant of the insulating substance 11 For capacitors in the series combination, the total capacitance C is given by\n1\n2\n3\n1\n1\n1\n1 C\nC\nC\nC\n=\n+\n+\n+\nIn the parallel combination, the total capacitance C is:\nC =  C1 + C2 + C3 +"}, {"Chapter": "1", "sentence_range": "2334-2337", "Text": "11 For capacitors in the series combination, the total capacitance C is given by\n1\n2\n3\n1\n1\n1\n1 C\nC\nC\nC\n=\n+\n+\n+\nIn the parallel combination, the total capacitance C is:\nC =  C1 + C2 + C3 + where C1, C2, C3"}, {"Chapter": "1", "sentence_range": "2335-2338", "Text": "For capacitors in the series combination, the total capacitance C is given by\n1\n2\n3\n1\n1\n1\n1 C\nC\nC\nC\n=\n+\n+\n+\nIn the parallel combination, the total capacitance C is:\nC =  C1 + C2 + C3 + where C1, C2, C3 are individual capacitances"}, {"Chapter": "1", "sentence_range": "2336-2339", "Text": "C\nC\nC\nC\n=\n+\n+\n+\nIn the parallel combination, the total capacitance C is:\nC =  C1 + C2 + C3 + where C1, C2, C3 are individual capacitances Rationalised 2023-24\nPhysics\n78\n12"}, {"Chapter": "1", "sentence_range": "2337-2340", "Text": "where C1, C2, C3 are individual capacitances Rationalised 2023-24\nPhysics\n78\n12 The energy U stored in a capacitor of capacitance C, with charge Q and\nvoltage V is\nU\nQV\nCV\nQ\nC\n=\n=\n=\n21\n21\n1\n2\n2\n2\nThe electric energy density (energy per unit volume) in a region with\nelectric field is (1/2)e0E2"}, {"Chapter": "1", "sentence_range": "2338-2341", "Text": "are individual capacitances Rationalised 2023-24\nPhysics\n78\n12 The energy U stored in a capacitor of capacitance C, with charge Q and\nvoltage V is\nU\nQV\nCV\nQ\nC\n=\n=\n=\n21\n21\n1\n2\n2\n2\nThe electric energy density (energy per unit volume) in a region with\nelectric field is (1/2)e0E2 Physical quantity\nSymbol\nDimensions\nUnit\n Remark\nPotential\n or V\n[M1 L2 T\u20133 A\u20131]\nV\nPotential difference is\nphysically significant\nCapacitance\nC\n[M\u20131 L\u20132 T\u20134 A2]\nF\nPolarisation\nP\n[L\u20132 AT]\nC m-2\nDipole moment per unit\nvolume\nDielectric constant\nK\n[Dimensionless]\nPOINTS TO PONDER\n1"}, {"Chapter": "1", "sentence_range": "2339-2342", "Text": "Rationalised 2023-24\nPhysics\n78\n12 The energy U stored in a capacitor of capacitance C, with charge Q and\nvoltage V is\nU\nQV\nCV\nQ\nC\n=\n=\n=\n21\n21\n1\n2\n2\n2\nThe electric energy density (energy per unit volume) in a region with\nelectric field is (1/2)e0E2 Physical quantity\nSymbol\nDimensions\nUnit\n Remark\nPotential\n or V\n[M1 L2 T\u20133 A\u20131]\nV\nPotential difference is\nphysically significant\nCapacitance\nC\n[M\u20131 L\u20132 T\u20134 A2]\nF\nPolarisation\nP\n[L\u20132 AT]\nC m-2\nDipole moment per unit\nvolume\nDielectric constant\nK\n[Dimensionless]\nPOINTS TO PONDER\n1 Electrostatics deals with forces between charges at rest"}, {"Chapter": "1", "sentence_range": "2340-2343", "Text": "The energy U stored in a capacitor of capacitance C, with charge Q and\nvoltage V is\nU\nQV\nCV\nQ\nC\n=\n=\n=\n21\n21\n1\n2\n2\n2\nThe electric energy density (energy per unit volume) in a region with\nelectric field is (1/2)e0E2 Physical quantity\nSymbol\nDimensions\nUnit\n Remark\nPotential\n or V\n[M1 L2 T\u20133 A\u20131]\nV\nPotential difference is\nphysically significant\nCapacitance\nC\n[M\u20131 L\u20132 T\u20134 A2]\nF\nPolarisation\nP\n[L\u20132 AT]\nC m-2\nDipole moment per unit\nvolume\nDielectric constant\nK\n[Dimensionless]\nPOINTS TO PONDER\n1 Electrostatics deals with forces between charges at rest But if there is a\nforce on a charge, how can it be at rest"}, {"Chapter": "1", "sentence_range": "2341-2344", "Text": "Physical quantity\nSymbol\nDimensions\nUnit\n Remark\nPotential\n or V\n[M1 L2 T\u20133 A\u20131]\nV\nPotential difference is\nphysically significant\nCapacitance\nC\n[M\u20131 L\u20132 T\u20134 A2]\nF\nPolarisation\nP\n[L\u20132 AT]\nC m-2\nDipole moment per unit\nvolume\nDielectric constant\nK\n[Dimensionless]\nPOINTS TO PONDER\n1 Electrostatics deals with forces between charges at rest But if there is a\nforce on a charge, how can it be at rest Thus, when we are talking of\nelectrostatic force between charges, it should be understood that each\ncharge is being kept at rest by some unspecified force that opposes the\nnet Coulomb force on the charge"}, {"Chapter": "1", "sentence_range": "2342-2345", "Text": "Electrostatics deals with forces between charges at rest But if there is a\nforce on a charge, how can it be at rest Thus, when we are talking of\nelectrostatic force between charges, it should be understood that each\ncharge is being kept at rest by some unspecified force that opposes the\nnet Coulomb force on the charge 2"}, {"Chapter": "1", "sentence_range": "2343-2346", "Text": "But if there is a\nforce on a charge, how can it be at rest Thus, when we are talking of\nelectrostatic force between charges, it should be understood that each\ncharge is being kept at rest by some unspecified force that opposes the\nnet Coulomb force on the charge 2 A capacitor is so configured that it confines the electric field lines within\na small region of space"}, {"Chapter": "1", "sentence_range": "2344-2347", "Text": "Thus, when we are talking of\nelectrostatic force between charges, it should be understood that each\ncharge is being kept at rest by some unspecified force that opposes the\nnet Coulomb force on the charge 2 A capacitor is so configured that it confines the electric field lines within\na small region of space Thus, even though field may have considerable\nstrength, the potential difference between the two conductors of a\ncapacitor is small"}, {"Chapter": "1", "sentence_range": "2345-2348", "Text": "2 A capacitor is so configured that it confines the electric field lines within\na small region of space Thus, even though field may have considerable\nstrength, the potential difference between the two conductors of a\ncapacitor is small 3"}, {"Chapter": "1", "sentence_range": "2346-2349", "Text": "A capacitor is so configured that it confines the electric field lines within\na small region of space Thus, even though field may have considerable\nstrength, the potential difference between the two conductors of a\ncapacitor is small 3 Electric field is discontinuous across the surface of a spherical charged\nshell"}, {"Chapter": "1", "sentence_range": "2347-2350", "Text": "Thus, even though field may have considerable\nstrength, the potential difference between the two conductors of a\ncapacitor is small 3 Electric field is discontinuous across the surface of a spherical charged\nshell It is zero inside and \u03c3\u03b50 \u02c6n outside"}, {"Chapter": "1", "sentence_range": "2348-2351", "Text": "3 Electric field is discontinuous across the surface of a spherical charged\nshell It is zero inside and \u03c3\u03b50 \u02c6n outside Electric potential is, however\ncontinuous across the surface, equal to q/4pe0R  at the surface"}, {"Chapter": "1", "sentence_range": "2349-2352", "Text": "Electric field is discontinuous across the surface of a spherical charged\nshell It is zero inside and \u03c3\u03b50 \u02c6n outside Electric potential is, however\ncontinuous across the surface, equal to q/4pe0R  at the surface 4"}, {"Chapter": "1", "sentence_range": "2350-2353", "Text": "It is zero inside and \u03c3\u03b50 \u02c6n outside Electric potential is, however\ncontinuous across the surface, equal to q/4pe0R  at the surface 4 The torque p \u00d7 E on a dipole causes it to oscillate about E"}, {"Chapter": "1", "sentence_range": "2351-2354", "Text": "Electric potential is, however\ncontinuous across the surface, equal to q/4pe0R  at the surface 4 The torque p \u00d7 E on a dipole causes it to oscillate about E Only if there\nis a dissipative mechanism, the oscillations are damped and the dipole\neventually aligns with E"}, {"Chapter": "1", "sentence_range": "2352-2355", "Text": "4 The torque p \u00d7 E on a dipole causes it to oscillate about E Only if there\nis a dissipative mechanism, the oscillations are damped and the dipole\neventually aligns with E 5"}, {"Chapter": "1", "sentence_range": "2353-2356", "Text": "The torque p \u00d7 E on a dipole causes it to oscillate about E Only if there\nis a dissipative mechanism, the oscillations are damped and the dipole\neventually aligns with E 5 Potential due to a charge q at its own location is not defined \u2013 it is\ninfinite"}, {"Chapter": "1", "sentence_range": "2354-2357", "Text": "Only if there\nis a dissipative mechanism, the oscillations are damped and the dipole\neventually aligns with E 5 Potential due to a charge q at its own location is not defined \u2013 it is\ninfinite 6"}, {"Chapter": "1", "sentence_range": "2355-2358", "Text": "5 Potential due to a charge q at its own location is not defined \u2013 it is\ninfinite 6 In the expression qV (r) for potential energy of a charge q, V (r) is the\npotential due to external charges and not the potential due to q"}, {"Chapter": "1", "sentence_range": "2356-2359", "Text": "Potential due to a charge q at its own location is not defined \u2013 it is\ninfinite 6 In the expression qV (r) for potential energy of a charge q, V (r) is the\npotential due to external charges and not the potential due to q As seen\nin point 5, this expression will be ill-defined if V (r) includes potential\ndue to a charge q itself"}, {"Chapter": "1", "sentence_range": "2357-2360", "Text": "6 In the expression qV (r) for potential energy of a charge q, V (r) is the\npotential due to external charges and not the potential due to q As seen\nin point 5, this expression will be ill-defined if V (r) includes potential\ndue to a charge q itself Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n79\n7"}, {"Chapter": "1", "sentence_range": "2358-2361", "Text": "In the expression qV (r) for potential energy of a charge q, V (r) is the\npotential due to external charges and not the potential due to q As seen\nin point 5, this expression will be ill-defined if V (r) includes potential\ndue to a charge q itself Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n79\n7 A cavity inside a conductor is shielded from outside electrical influences"}, {"Chapter": "1", "sentence_range": "2359-2362", "Text": "As seen\nin point 5, this expression will be ill-defined if V (r) includes potential\ndue to a charge q itself Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n79\n7 A cavity inside a conductor is shielded from outside electrical influences It is worth noting that electrostatic shielding does not work the other\nway round; that is, if you put charges inside the cavity, the exterior of\nthe conductor is not shielded from the fields by the inside charges"}, {"Chapter": "1", "sentence_range": "2360-2363", "Text": "Rationalised 2023-24\nElectrostatic Potential\nand Capacitance\n79\n7 A cavity inside a conductor is shielded from outside electrical influences It is worth noting that electrostatic shielding does not work the other\nway round; that is, if you put charges inside the cavity, the exterior of\nthe conductor is not shielded from the fields by the inside charges EXERCISES\n2"}, {"Chapter": "1", "sentence_range": "2361-2364", "Text": "A cavity inside a conductor is shielded from outside electrical influences It is worth noting that electrostatic shielding does not work the other\nway round; that is, if you put charges inside the cavity, the exterior of\nthe conductor is not shielded from the fields by the inside charges EXERCISES\n2 1\nTwo charges 5 \u00d7 10\u20138 C and \u20133 \u00d7 10\u20138 C are located 16 cm apart"}, {"Chapter": "1", "sentence_range": "2362-2365", "Text": "It is worth noting that electrostatic shielding does not work the other\nway round; that is, if you put charges inside the cavity, the exterior of\nthe conductor is not shielded from the fields by the inside charges EXERCISES\n2 1\nTwo charges 5 \u00d7 10\u20138 C and \u20133 \u00d7 10\u20138 C are located 16 cm apart At\nwhat point(s) on the line joining the two charges is the electric\npotential zero"}, {"Chapter": "1", "sentence_range": "2363-2366", "Text": "EXERCISES\n2 1\nTwo charges 5 \u00d7 10\u20138 C and \u20133 \u00d7 10\u20138 C are located 16 cm apart At\nwhat point(s) on the line joining the two charges is the electric\npotential zero Take the potential at infinity to be zero"}, {"Chapter": "1", "sentence_range": "2364-2367", "Text": "1\nTwo charges 5 \u00d7 10\u20138 C and \u20133 \u00d7 10\u20138 C are located 16 cm apart At\nwhat point(s) on the line joining the two charges is the electric\npotential zero Take the potential at infinity to be zero 2"}, {"Chapter": "1", "sentence_range": "2365-2368", "Text": "At\nwhat point(s) on the line joining the two charges is the electric\npotential zero Take the potential at infinity to be zero 2 2\nA regular hexagon of side 10 cm has a charge 5 mC at each of its\nvertices"}, {"Chapter": "1", "sentence_range": "2366-2369", "Text": "Take the potential at infinity to be zero 2 2\nA regular hexagon of side 10 cm has a charge 5 mC at each of its\nvertices Calculate the potential at the centre of the hexagon"}, {"Chapter": "1", "sentence_range": "2367-2370", "Text": "2 2\nA regular hexagon of side 10 cm has a charge 5 mC at each of its\nvertices Calculate the potential at the centre of the hexagon 2"}, {"Chapter": "1", "sentence_range": "2368-2371", "Text": "2\nA regular hexagon of side 10 cm has a charge 5 mC at each of its\nvertices Calculate the potential at the centre of the hexagon 2 3\nTwo charges 2 mC and \u20132 mC are placed at points A and B 6 cm\napart"}, {"Chapter": "1", "sentence_range": "2369-2372", "Text": "Calculate the potential at the centre of the hexagon 2 3\nTwo charges 2 mC and \u20132 mC are placed at points A and B 6 cm\napart (a)\nIdentify an equipotential surface of the system"}, {"Chapter": "1", "sentence_range": "2370-2373", "Text": "2 3\nTwo charges 2 mC and \u20132 mC are placed at points A and B 6 cm\napart (a)\nIdentify an equipotential surface of the system (b)\nWhat is the direction of the electric field at every point on this\nsurface"}, {"Chapter": "1", "sentence_range": "2371-2374", "Text": "3\nTwo charges 2 mC and \u20132 mC are placed at points A and B 6 cm\napart (a)\nIdentify an equipotential surface of the system (b)\nWhat is the direction of the electric field at every point on this\nsurface 2"}, {"Chapter": "1", "sentence_range": "2372-2375", "Text": "(a)\nIdentify an equipotential surface of the system (b)\nWhat is the direction of the electric field at every point on this\nsurface 2 4\nA spherical conductor of radius 12 cm has a charge of 1"}, {"Chapter": "1", "sentence_range": "2373-2376", "Text": "(b)\nWhat is the direction of the electric field at every point on this\nsurface 2 4\nA spherical conductor of radius 12 cm has a charge of 1 6 \u00d7 10\u20137C\ndistributed uniformly on its surface"}, {"Chapter": "1", "sentence_range": "2374-2377", "Text": "2 4\nA spherical conductor of radius 12 cm has a charge of 1 6 \u00d7 10\u20137C\ndistributed uniformly on its surface What is the electric field\n(a)\ninside the sphere\n(b)\njust outside the sphere\n(c)\nat a point 18 cm from the centre of the sphere"}, {"Chapter": "1", "sentence_range": "2375-2378", "Text": "4\nA spherical conductor of radius 12 cm has a charge of 1 6 \u00d7 10\u20137C\ndistributed uniformly on its surface What is the electric field\n(a)\ninside the sphere\n(b)\njust outside the sphere\n(c)\nat a point 18 cm from the centre of the sphere 2"}, {"Chapter": "1", "sentence_range": "2376-2379", "Text": "6 \u00d7 10\u20137C\ndistributed uniformly on its surface What is the electric field\n(a)\ninside the sphere\n(b)\njust outside the sphere\n(c)\nat a point 18 cm from the centre of the sphere 2 5\nA parallel plate capacitor with air between the plates has a\ncapacitance of 8 pF (1pF = 10\u201312 F)"}, {"Chapter": "1", "sentence_range": "2377-2380", "Text": "What is the electric field\n(a)\ninside the sphere\n(b)\njust outside the sphere\n(c)\nat a point 18 cm from the centre of the sphere 2 5\nA parallel plate capacitor with air between the plates has a\ncapacitance of 8 pF (1pF = 10\u201312 F) What will be the capacitance if\nthe distance between the plates is reduced by half, and the space\nbetween them is filled with a substance of dielectric constant 6"}, {"Chapter": "1", "sentence_range": "2378-2381", "Text": "2 5\nA parallel plate capacitor with air between the plates has a\ncapacitance of 8 pF (1pF = 10\u201312 F) What will be the capacitance if\nthe distance between the plates is reduced by half, and the space\nbetween them is filled with a substance of dielectric constant 6 2"}, {"Chapter": "1", "sentence_range": "2379-2382", "Text": "5\nA parallel plate capacitor with air between the plates has a\ncapacitance of 8 pF (1pF = 10\u201312 F) What will be the capacitance if\nthe distance between the plates is reduced by half, and the space\nbetween them is filled with a substance of dielectric constant 6 2 6\nThree capacitors each of capacitance 9 pF are connected in series"}, {"Chapter": "1", "sentence_range": "2380-2383", "Text": "What will be the capacitance if\nthe distance between the plates is reduced by half, and the space\nbetween them is filled with a substance of dielectric constant 6 2 6\nThree capacitors each of capacitance 9 pF are connected in series (a)\nWhat is the total capacitance of the combination"}, {"Chapter": "1", "sentence_range": "2381-2384", "Text": "2 6\nThree capacitors each of capacitance 9 pF are connected in series (a)\nWhat is the total capacitance of the combination (b)\nWhat is the potential difference across each capacitor if the\ncombination is connected to a 120 V supply"}, {"Chapter": "1", "sentence_range": "2382-2385", "Text": "6\nThree capacitors each of capacitance 9 pF are connected in series (a)\nWhat is the total capacitance of the combination (b)\nWhat is the potential difference across each capacitor if the\ncombination is connected to a 120 V supply 2"}, {"Chapter": "1", "sentence_range": "2383-2386", "Text": "(a)\nWhat is the total capacitance of the combination (b)\nWhat is the potential difference across each capacitor if the\ncombination is connected to a 120 V supply 2 7\nThree capacitors of capacitances 2 pF, 3 pF and 4 pF are connected\nin parallel"}, {"Chapter": "1", "sentence_range": "2384-2387", "Text": "(b)\nWhat is the potential difference across each capacitor if the\ncombination is connected to a 120 V supply 2 7\nThree capacitors of capacitances 2 pF, 3 pF and 4 pF are connected\nin parallel (a)\nWhat is the total capacitance of the combination"}, {"Chapter": "1", "sentence_range": "2385-2388", "Text": "2 7\nThree capacitors of capacitances 2 pF, 3 pF and 4 pF are connected\nin parallel (a)\nWhat is the total capacitance of the combination (b)\nDetermine the charge on each capacitor if the combination is\nconnected to a 100 V supply"}, {"Chapter": "1", "sentence_range": "2386-2389", "Text": "7\nThree capacitors of capacitances 2 pF, 3 pF and 4 pF are connected\nin parallel (a)\nWhat is the total capacitance of the combination (b)\nDetermine the charge on each capacitor if the combination is\nconnected to a 100 V supply 2"}, {"Chapter": "1", "sentence_range": "2387-2390", "Text": "(a)\nWhat is the total capacitance of the combination (b)\nDetermine the charge on each capacitor if the combination is\nconnected to a 100 V supply 2 8\nIn a parallel plate capacitor with air between the plates, each plate\nhas an area of 6 \u00d7 10\u20133 m2 and the distance between the plates is 3 mm"}, {"Chapter": "1", "sentence_range": "2388-2391", "Text": "(b)\nDetermine the charge on each capacitor if the combination is\nconnected to a 100 V supply 2 8\nIn a parallel plate capacitor with air between the plates, each plate\nhas an area of 6 \u00d7 10\u20133 m2 and the distance between the plates is 3 mm Calculate the capacitance of the capacitor"}, {"Chapter": "1", "sentence_range": "2389-2392", "Text": "2 8\nIn a parallel plate capacitor with air between the plates, each plate\nhas an area of 6 \u00d7 10\u20133 m2 and the distance between the plates is 3 mm Calculate the capacitance of the capacitor If this capacitor is\nconnected to a 100 V supply, what is the charge on each plate of the\ncapacitor"}, {"Chapter": "1", "sentence_range": "2390-2393", "Text": "8\nIn a parallel plate capacitor with air between the plates, each plate\nhas an area of 6 \u00d7 10\u20133 m2 and the distance between the plates is 3 mm Calculate the capacitance of the capacitor If this capacitor is\nconnected to a 100 V supply, what is the charge on each plate of the\ncapacitor Rationalised 2023-24\nPhysics\n80\n2"}, {"Chapter": "1", "sentence_range": "2391-2394", "Text": "Calculate the capacitance of the capacitor If this capacitor is\nconnected to a 100 V supply, what is the charge on each plate of the\ncapacitor Rationalised 2023-24\nPhysics\n80\n2 9\nExplain what would happen if in the capacitor given in Exercise\n2"}, {"Chapter": "1", "sentence_range": "2392-2395", "Text": "If this capacitor is\nconnected to a 100 V supply, what is the charge on each plate of the\ncapacitor Rationalised 2023-24\nPhysics\n80\n2 9\nExplain what would happen if in the capacitor given in Exercise\n2 8, a 3 mm thick mica sheet (of dielectric constant = 6) were inserted\nbetween the plates,\n(a)\nwhile the voltage supply remained connected"}, {"Chapter": "1", "sentence_range": "2393-2396", "Text": "Rationalised 2023-24\nPhysics\n80\n2 9\nExplain what would happen if in the capacitor given in Exercise\n2 8, a 3 mm thick mica sheet (of dielectric constant = 6) were inserted\nbetween the plates,\n(a)\nwhile the voltage supply remained connected (b)\nafter the supply was disconnected"}, {"Chapter": "1", "sentence_range": "2394-2397", "Text": "9\nExplain what would happen if in the capacitor given in Exercise\n2 8, a 3 mm thick mica sheet (of dielectric constant = 6) were inserted\nbetween the plates,\n(a)\nwhile the voltage supply remained connected (b)\nafter the supply was disconnected 2"}, {"Chapter": "1", "sentence_range": "2395-2398", "Text": "8, a 3 mm thick mica sheet (of dielectric constant = 6) were inserted\nbetween the plates,\n(a)\nwhile the voltage supply remained connected (b)\nafter the supply was disconnected 2 10\nA 12pF capacitor is connected to a 50V battery"}, {"Chapter": "1", "sentence_range": "2396-2399", "Text": "(b)\nafter the supply was disconnected 2 10\nA 12pF capacitor is connected to a 50V battery How much\nelectrostatic energy is stored in the capacitor"}, {"Chapter": "1", "sentence_range": "2397-2400", "Text": "2 10\nA 12pF capacitor is connected to a 50V battery How much\nelectrostatic energy is stored in the capacitor 2"}, {"Chapter": "1", "sentence_range": "2398-2401", "Text": "10\nA 12pF capacitor is connected to a 50V battery How much\nelectrostatic energy is stored in the capacitor 2 11\nA 600pF capacitor is charged by a 200V supply"}, {"Chapter": "1", "sentence_range": "2399-2402", "Text": "How much\nelectrostatic energy is stored in the capacitor 2 11\nA 600pF capacitor is charged by a 200V supply It is then\ndisconnected from the supply and is connected to another\nuncharged 600 pF capacitor"}, {"Chapter": "1", "sentence_range": "2400-2403", "Text": "2 11\nA 600pF capacitor is charged by a 200V supply It is then\ndisconnected from the supply and is connected to another\nuncharged 600 pF capacitor How much electrostatic energy is lost\nin the process"}, {"Chapter": "1", "sentence_range": "2401-2404", "Text": "11\nA 600pF capacitor is charged by a 200V supply It is then\ndisconnected from the supply and is connected to another\nuncharged 600 pF capacitor How much electrostatic energy is lost\nin the process Rationalised 2023-24\n3"}, {"Chapter": "1", "sentence_range": "2402-2405", "Text": "It is then\ndisconnected from the supply and is connected to another\nuncharged 600 pF capacitor How much electrostatic energy is lost\nin the process Rationalised 2023-24\n3 1  INTRODUCTION\nIn Chapter 1, all charges whether free or bound, were considered to be at\nrest"}, {"Chapter": "1", "sentence_range": "2403-2406", "Text": "How much electrostatic energy is lost\nin the process Rationalised 2023-24\n3 1  INTRODUCTION\nIn Chapter 1, all charges whether free or bound, were considered to be at\nrest Charges in motion constitute an electric current"}, {"Chapter": "1", "sentence_range": "2404-2407", "Text": "Rationalised 2023-24\n3 1  INTRODUCTION\nIn Chapter 1, all charges whether free or bound, were considered to be at\nrest Charges in motion constitute an electric current Such currents occur\nnaturally in many situations"}, {"Chapter": "1", "sentence_range": "2405-2408", "Text": "1  INTRODUCTION\nIn Chapter 1, all charges whether free or bound, were considered to be at\nrest Charges in motion constitute an electric current Such currents occur\nnaturally in many situations Lightning is one such phenomenon in\nwhich charges flow from the clouds to the earth through the atmosphere,\nsometimes with disastrous results"}, {"Chapter": "1", "sentence_range": "2406-2409", "Text": "Charges in motion constitute an electric current Such currents occur\nnaturally in many situations Lightning is one such phenomenon in\nwhich charges flow from the clouds to the earth through the atmosphere,\nsometimes with disastrous results The flow of charges in lightning is not\nsteady, but in our everyday life we see many devices where charges flow\nin a steady manner, like water flowing smoothly in a river"}, {"Chapter": "1", "sentence_range": "2407-2410", "Text": "Such currents occur\nnaturally in many situations Lightning is one such phenomenon in\nwhich charges flow from the clouds to the earth through the atmosphere,\nsometimes with disastrous results The flow of charges in lightning is not\nsteady, but in our everyday life we see many devices where charges flow\nin a steady manner, like water flowing smoothly in a river A torch and a\ncell-driven clock are examples of such devices"}, {"Chapter": "1", "sentence_range": "2408-2411", "Text": "Lightning is one such phenomenon in\nwhich charges flow from the clouds to the earth through the atmosphere,\nsometimes with disastrous results The flow of charges in lightning is not\nsteady, but in our everyday life we see many devices where charges flow\nin a steady manner, like water flowing smoothly in a river A torch and a\ncell-driven clock are examples of such devices In the present chapter, we\nshall study some of the basic laws concerning steady electric currents"}, {"Chapter": "1", "sentence_range": "2409-2412", "Text": "The flow of charges in lightning is not\nsteady, but in our everyday life we see many devices where charges flow\nin a steady manner, like water flowing smoothly in a river A torch and a\ncell-driven clock are examples of such devices In the present chapter, we\nshall study some of the basic laws concerning steady electric currents 3"}, {"Chapter": "1", "sentence_range": "2410-2413", "Text": "A torch and a\ncell-driven clock are examples of such devices In the present chapter, we\nshall study some of the basic laws concerning steady electric currents 3 2  ELECTRIC CURRENT\nImagine a small area held normal to the direction of flow of charges"}, {"Chapter": "1", "sentence_range": "2411-2414", "Text": "In the present chapter, we\nshall study some of the basic laws concerning steady electric currents 3 2  ELECTRIC CURRENT\nImagine a small area held normal to the direction of flow of charges Both\nthe positive and the negative charges may flow forward and backward\nacross the area"}, {"Chapter": "1", "sentence_range": "2412-2415", "Text": "3 2  ELECTRIC CURRENT\nImagine a small area held normal to the direction of flow of charges Both\nthe positive and the negative charges may flow forward and backward\nacross the area In a given time interval t, let q+ be the net amount (i"}, {"Chapter": "1", "sentence_range": "2413-2416", "Text": "2  ELECTRIC CURRENT\nImagine a small area held normal to the direction of flow of charges Both\nthe positive and the negative charges may flow forward and backward\nacross the area In a given time interval t, let q+ be the net amount (i e"}, {"Chapter": "1", "sentence_range": "2414-2417", "Text": "Both\nthe positive and the negative charges may flow forward and backward\nacross the area In a given time interval t, let q+ be the net amount (i e ,\nforward minus backward) of positive charge that flows in the forward\ndirection across the area"}, {"Chapter": "1", "sentence_range": "2415-2418", "Text": "In a given time interval t, let q+ be the net amount (i e ,\nforward minus backward) of positive charge that flows in the forward\ndirection across the area Similarly, let q\u2013 be the net amount of negative\ncharge flowing across the area in the forward direction"}, {"Chapter": "1", "sentence_range": "2416-2419", "Text": "e ,\nforward minus backward) of positive charge that flows in the forward\ndirection across the area Similarly, let q\u2013 be the net amount of negative\ncharge flowing across the area in the forward direction The net amount\nof charge flowing across the area in the forward direction in the time\ninterval t, then, is q = q+\u2013 q\u2013"}, {"Chapter": "1", "sentence_range": "2417-2420", "Text": ",\nforward minus backward) of positive charge that flows in the forward\ndirection across the area Similarly, let q\u2013 be the net amount of negative\ncharge flowing across the area in the forward direction The net amount\nof charge flowing across the area in the forward direction in the time\ninterval t, then, is q = q+\u2013 q\u2013 This is proportional to t for steady current\nChapter Three\nCURRENT\nELECTRICITY\nRationalised 2023-24\nPhysics\n82\nand the quotient\nq\nI\n=t\n(3"}, {"Chapter": "1", "sentence_range": "2418-2421", "Text": "Similarly, let q\u2013 be the net amount of negative\ncharge flowing across the area in the forward direction The net amount\nof charge flowing across the area in the forward direction in the time\ninterval t, then, is q = q+\u2013 q\u2013 This is proportional to t for steady current\nChapter Three\nCURRENT\nELECTRICITY\nRationalised 2023-24\nPhysics\n82\nand the quotient\nq\nI\n=t\n(3 1)\nis defined to be the current across the area in the forward direction"}, {"Chapter": "1", "sentence_range": "2419-2422", "Text": "The net amount\nof charge flowing across the area in the forward direction in the time\ninterval t, then, is q = q+\u2013 q\u2013 This is proportional to t for steady current\nChapter Three\nCURRENT\nELECTRICITY\nRationalised 2023-24\nPhysics\n82\nand the quotient\nq\nI\n=t\n(3 1)\nis defined to be the current across the area in the forward direction (If it\nturn out to be a negative number, it implies a current in the backward\ndirection"}, {"Chapter": "1", "sentence_range": "2420-2423", "Text": "This is proportional to t for steady current\nChapter Three\nCURRENT\nELECTRICITY\nRationalised 2023-24\nPhysics\n82\nand the quotient\nq\nI\n=t\n(3 1)\nis defined to be the current across the area in the forward direction (If it\nturn out to be a negative number, it implies a current in the backward\ndirection )\nCurrents are not always steady and hence more generally, we define\nthe current as follows"}, {"Chapter": "1", "sentence_range": "2421-2424", "Text": "1)\nis defined to be the current across the area in the forward direction (If it\nturn out to be a negative number, it implies a current in the backward\ndirection )\nCurrents are not always steady and hence more generally, we define\nthe current as follows Let DQ be the net charge flowing across a cross-\nsection of a conductor during the time interval Dt [i"}, {"Chapter": "1", "sentence_range": "2422-2425", "Text": "(If it\nturn out to be a negative number, it implies a current in the backward\ndirection )\nCurrents are not always steady and hence more generally, we define\nthe current as follows Let DQ be the net charge flowing across a cross-\nsection of a conductor during the time interval Dt [i e"}, {"Chapter": "1", "sentence_range": "2423-2426", "Text": ")\nCurrents are not always steady and hence more generally, we define\nthe current as follows Let DQ be the net charge flowing across a cross-\nsection of a conductor during the time interval Dt [i e , between times t\nand (t + Dt)]"}, {"Chapter": "1", "sentence_range": "2424-2427", "Text": "Let DQ be the net charge flowing across a cross-\nsection of a conductor during the time interval Dt [i e , between times t\nand (t + Dt)] Then, the current at time t across the cross-section of the\nconductor is defined as the value of the ratio of DQ to Dt in the limit of Dt\ntending to zero,\n( )\nlim0\nt\nQ\nI t\nt\n\u2206 \u2192\n\u2206\n\u2261\n\u2206\n(3"}, {"Chapter": "1", "sentence_range": "2425-2428", "Text": "e , between times t\nand (t + Dt)] Then, the current at time t across the cross-section of the\nconductor is defined as the value of the ratio of DQ to Dt in the limit of Dt\ntending to zero,\n( )\nlim0\nt\nQ\nI t\nt\n\u2206 \u2192\n\u2206\n\u2261\n\u2206\n(3 2)\nIn SI units, the unit of current is ampere"}, {"Chapter": "1", "sentence_range": "2426-2429", "Text": ", between times t\nand (t + Dt)] Then, the current at time t across the cross-section of the\nconductor is defined as the value of the ratio of DQ to Dt in the limit of Dt\ntending to zero,\n( )\nlim0\nt\nQ\nI t\nt\n\u2206 \u2192\n\u2206\n\u2261\n\u2206\n(3 2)\nIn SI units, the unit of current is ampere An ampere is defined\nthrough magnetic effects of currents that we will study in the following\nchapter"}, {"Chapter": "1", "sentence_range": "2427-2430", "Text": "Then, the current at time t across the cross-section of the\nconductor is defined as the value of the ratio of DQ to Dt in the limit of Dt\ntending to zero,\n( )\nlim0\nt\nQ\nI t\nt\n\u2206 \u2192\n\u2206\n\u2261\n\u2206\n(3 2)\nIn SI units, the unit of current is ampere An ampere is defined\nthrough magnetic effects of currents that we will study in the following\nchapter An ampere is typically the order of magnitude of currents in\ndomestic appliances"}, {"Chapter": "1", "sentence_range": "2428-2431", "Text": "2)\nIn SI units, the unit of current is ampere An ampere is defined\nthrough magnetic effects of currents that we will study in the following\nchapter An ampere is typically the order of magnitude of currents in\ndomestic appliances An average lightning carries currents of the order\nof tens of thousands of amperes and at the other extreme, currents in\nour nerves are in microamperes"}, {"Chapter": "1", "sentence_range": "2429-2432", "Text": "An ampere is defined\nthrough magnetic effects of currents that we will study in the following\nchapter An ampere is typically the order of magnitude of currents in\ndomestic appliances An average lightning carries currents of the order\nof tens of thousands of amperes and at the other extreme, currents in\nour nerves are in microamperes 3"}, {"Chapter": "1", "sentence_range": "2430-2433", "Text": "An ampere is typically the order of magnitude of currents in\ndomestic appliances An average lightning carries currents of the order\nof tens of thousands of amperes and at the other extreme, currents in\nour nerves are in microamperes 3 3 ELECTRIC CURRENTS IN CONDUCTORS\nAn electric charge will experience a force if an electric field is applied"}, {"Chapter": "1", "sentence_range": "2431-2434", "Text": "An average lightning carries currents of the order\nof tens of thousands of amperes and at the other extreme, currents in\nour nerves are in microamperes 3 3 ELECTRIC CURRENTS IN CONDUCTORS\nAn electric charge will experience a force if an electric field is applied If it is\nfree to move, it will thus move contributing to a current"}, {"Chapter": "1", "sentence_range": "2432-2435", "Text": "3 3 ELECTRIC CURRENTS IN CONDUCTORS\nAn electric charge will experience a force if an electric field is applied If it is\nfree to move, it will thus move contributing to a current In nature, free\ncharged particles do exist like in upper strata of atmosphere called the\nionosphere"}, {"Chapter": "1", "sentence_range": "2433-2436", "Text": "3 ELECTRIC CURRENTS IN CONDUCTORS\nAn electric charge will experience a force if an electric field is applied If it is\nfree to move, it will thus move contributing to a current In nature, free\ncharged particles do exist like in upper strata of atmosphere called the\nionosphere However, in atoms and molecules, the negatively charged\nelectrons and the positively charged nuclei are bound to each other and\nare thus not free to move"}, {"Chapter": "1", "sentence_range": "2434-2437", "Text": "If it is\nfree to move, it will thus move contributing to a current In nature, free\ncharged particles do exist like in upper strata of atmosphere called the\nionosphere However, in atoms and molecules, the negatively charged\nelectrons and the positively charged nuclei are bound to each other and\nare thus not free to move Bulk matter is made up of many molecules, a\ngram of water, for example, contains approximately 1022 molecules"}, {"Chapter": "1", "sentence_range": "2435-2438", "Text": "In nature, free\ncharged particles do exist like in upper strata of atmosphere called the\nionosphere However, in atoms and molecules, the negatively charged\nelectrons and the positively charged nuclei are bound to each other and\nare thus not free to move Bulk matter is made up of many molecules, a\ngram of water, for example, contains approximately 1022 molecules These\nmolecules are so closely packed that the electrons are no longer attached\nto individual nuclei"}, {"Chapter": "1", "sentence_range": "2436-2439", "Text": "However, in atoms and molecules, the negatively charged\nelectrons and the positively charged nuclei are bound to each other and\nare thus not free to move Bulk matter is made up of many molecules, a\ngram of water, for example, contains approximately 1022 molecules These\nmolecules are so closely packed that the electrons are no longer attached\nto individual nuclei In some materials, the electrons will still be bound,\ni"}, {"Chapter": "1", "sentence_range": "2437-2440", "Text": "Bulk matter is made up of many molecules, a\ngram of water, for example, contains approximately 1022 molecules These\nmolecules are so closely packed that the electrons are no longer attached\nto individual nuclei In some materials, the electrons will still be bound,\ni e"}, {"Chapter": "1", "sentence_range": "2438-2441", "Text": "These\nmolecules are so closely packed that the electrons are no longer attached\nto individual nuclei In some materials, the electrons will still be bound,\ni e , they will not accelerate even if an electric field is applied"}, {"Chapter": "1", "sentence_range": "2439-2442", "Text": "In some materials, the electrons will still be bound,\ni e , they will not accelerate even if an electric field is applied In other\nmaterials, notably metals, some of the electrons are practically free to move\nwithin the bulk material"}, {"Chapter": "1", "sentence_range": "2440-2443", "Text": "e , they will not accelerate even if an electric field is applied In other\nmaterials, notably metals, some of the electrons are practically free to move\nwithin the bulk material These materials, generally called conductors,\ndevelop electric currents in them when an electric field is applied"}, {"Chapter": "1", "sentence_range": "2441-2444", "Text": ", they will not accelerate even if an electric field is applied In other\nmaterials, notably metals, some of the electrons are practically free to move\nwithin the bulk material These materials, generally called conductors,\ndevelop electric currents in them when an electric field is applied If we consider solid conductors, then of course the atoms are tightly\nbound to each other so that the current is carried by the negatively\ncharged electrons"}, {"Chapter": "1", "sentence_range": "2442-2445", "Text": "In other\nmaterials, notably metals, some of the electrons are practically free to move\nwithin the bulk material These materials, generally called conductors,\ndevelop electric currents in them when an electric field is applied If we consider solid conductors, then of course the atoms are tightly\nbound to each other so that the current is carried by the negatively\ncharged electrons There are, however, other types of conductors like\nelectrolytic solutions where positive and negative charges both can move"}, {"Chapter": "1", "sentence_range": "2443-2446", "Text": "These materials, generally called conductors,\ndevelop electric currents in them when an electric field is applied If we consider solid conductors, then of course the atoms are tightly\nbound to each other so that the current is carried by the negatively\ncharged electrons There are, however, other types of conductors like\nelectrolytic solutions where positive and negative charges both can move In our discussions, we will focus only on solid conductors so that the\ncurrent is carried by the negatively charged electrons in the background\nof fixed positive ions"}, {"Chapter": "1", "sentence_range": "2444-2447", "Text": "If we consider solid conductors, then of course the atoms are tightly\nbound to each other so that the current is carried by the negatively\ncharged electrons There are, however, other types of conductors like\nelectrolytic solutions where positive and negative charges both can move In our discussions, we will focus only on solid conductors so that the\ncurrent is carried by the negatively charged electrons in the background\nof fixed positive ions Consider first the case when no electric field is present"}, {"Chapter": "1", "sentence_range": "2445-2448", "Text": "There are, however, other types of conductors like\nelectrolytic solutions where positive and negative charges both can move In our discussions, we will focus only on solid conductors so that the\ncurrent is carried by the negatively charged electrons in the background\nof fixed positive ions Consider first the case when no electric field is present The electrons\nwill be moving due to thermal motion during which they collide with the\nfixed ions"}, {"Chapter": "1", "sentence_range": "2446-2449", "Text": "In our discussions, we will focus only on solid conductors so that the\ncurrent is carried by the negatively charged electrons in the background\nof fixed positive ions Consider first the case when no electric field is present The electrons\nwill be moving due to thermal motion during which they collide with the\nfixed ions An electron colliding with an ion emerges with the same speed\nas before the collision"}, {"Chapter": "1", "sentence_range": "2447-2450", "Text": "Consider first the case when no electric field is present The electrons\nwill be moving due to thermal motion during which they collide with the\nfixed ions An electron colliding with an ion emerges with the same speed\nas before the collision However, the direction of its velocity after the\ncollision is completely random"}, {"Chapter": "1", "sentence_range": "2448-2451", "Text": "The electrons\nwill be moving due to thermal motion during which they collide with the\nfixed ions An electron colliding with an ion emerges with the same speed\nas before the collision However, the direction of its velocity after the\ncollision is completely random At a given time, there is no preferential\ndirection for the velocities of the electrons"}, {"Chapter": "1", "sentence_range": "2449-2452", "Text": "An electron colliding with an ion emerges with the same speed\nas before the collision However, the direction of its velocity after the\ncollision is completely random At a given time, there is no preferential\ndirection for the velocities of the electrons Thus on the average, the\nRationalised 2023-24\nCurrent\nElectricity\n83\nnumber of electrons travelling in any direction will be equal to the number\nof electrons travelling in the opposite direction"}, {"Chapter": "1", "sentence_range": "2450-2453", "Text": "However, the direction of its velocity after the\ncollision is completely random At a given time, there is no preferential\ndirection for the velocities of the electrons Thus on the average, the\nRationalised 2023-24\nCurrent\nElectricity\n83\nnumber of electrons travelling in any direction will be equal to the number\nof electrons travelling in the opposite direction So, there will be no net\nelectric current"}, {"Chapter": "1", "sentence_range": "2451-2454", "Text": "At a given time, there is no preferential\ndirection for the velocities of the electrons Thus on the average, the\nRationalised 2023-24\nCurrent\nElectricity\n83\nnumber of electrons travelling in any direction will be equal to the number\nof electrons travelling in the opposite direction So, there will be no net\nelectric current Let us now see what happens to such a\npiece of conductor if an electric field is applied"}, {"Chapter": "1", "sentence_range": "2452-2455", "Text": "Thus on the average, the\nRationalised 2023-24\nCurrent\nElectricity\n83\nnumber of electrons travelling in any direction will be equal to the number\nof electrons travelling in the opposite direction So, there will be no net\nelectric current Let us now see what happens to such a\npiece of conductor if an electric field is applied To focus our thoughts, imagine the conductor\nin the shape of a cylinder of radius R (Fig"}, {"Chapter": "1", "sentence_range": "2453-2456", "Text": "So, there will be no net\nelectric current Let us now see what happens to such a\npiece of conductor if an electric field is applied To focus our thoughts, imagine the conductor\nin the shape of a cylinder of radius R (Fig 3"}, {"Chapter": "1", "sentence_range": "2454-2457", "Text": "Let us now see what happens to such a\npiece of conductor if an electric field is applied To focus our thoughts, imagine the conductor\nin the shape of a cylinder of radius R (Fig 3 1)"}, {"Chapter": "1", "sentence_range": "2455-2458", "Text": "To focus our thoughts, imagine the conductor\nin the shape of a cylinder of radius R (Fig 3 1) Suppose we now take two thin circular discs\nof a dielectric of the same radius and put\npositive charge +Q distributed over one disc\nand similarly \u2013Q at the other disc"}, {"Chapter": "1", "sentence_range": "2456-2459", "Text": "3 1) Suppose we now take two thin circular discs\nof a dielectric of the same radius and put\npositive charge +Q distributed over one disc\nand similarly \u2013Q at the other disc We attach\nthe two discs on the two flat surfaces of the\ncylinder"}, {"Chapter": "1", "sentence_range": "2457-2460", "Text": "1) Suppose we now take two thin circular discs\nof a dielectric of the same radius and put\npositive charge +Q distributed over one disc\nand similarly \u2013Q at the other disc We attach\nthe two discs on the two flat surfaces of the\ncylinder An electric field will be created and\nis directed from the positive towards the\nnegative charge"}, {"Chapter": "1", "sentence_range": "2458-2461", "Text": "Suppose we now take two thin circular discs\nof a dielectric of the same radius and put\npositive charge +Q distributed over one disc\nand similarly \u2013Q at the other disc We attach\nthe two discs on the two flat surfaces of the\ncylinder An electric field will be created and\nis directed from the positive towards the\nnegative charge The electrons will be accelerated due to this field towards\n+Q"}, {"Chapter": "1", "sentence_range": "2459-2462", "Text": "We attach\nthe two discs on the two flat surfaces of the\ncylinder An electric field will be created and\nis directed from the positive towards the\nnegative charge The electrons will be accelerated due to this field towards\n+Q They will thus move to neutralise the charges"}, {"Chapter": "1", "sentence_range": "2460-2463", "Text": "An electric field will be created and\nis directed from the positive towards the\nnegative charge The electrons will be accelerated due to this field towards\n+Q They will thus move to neutralise the charges The electrons, as long\nas they are moving, will constitute an electric current"}, {"Chapter": "1", "sentence_range": "2461-2464", "Text": "The electrons will be accelerated due to this field towards\n+Q They will thus move to neutralise the charges The electrons, as long\nas they are moving, will constitute an electric current Hence in the\nsituation considered, there will be a current for a very short while and no\ncurrent thereafter"}, {"Chapter": "1", "sentence_range": "2462-2465", "Text": "They will thus move to neutralise the charges The electrons, as long\nas they are moving, will constitute an electric current Hence in the\nsituation considered, there will be a current for a very short while and no\ncurrent thereafter We can also imagine a mechanism where the ends of the cylinder are\nsupplied with fresh charges to make up for any charges neutralised by\nelectrons moving inside the conductor"}, {"Chapter": "1", "sentence_range": "2463-2466", "Text": "The electrons, as long\nas they are moving, will constitute an electric current Hence in the\nsituation considered, there will be a current for a very short while and no\ncurrent thereafter We can also imagine a mechanism where the ends of the cylinder are\nsupplied with fresh charges to make up for any charges neutralised by\nelectrons moving inside the conductor In that case, there will be a steady\nelectric field in the body of the conductor"}, {"Chapter": "1", "sentence_range": "2464-2467", "Text": "Hence in the\nsituation considered, there will be a current for a very short while and no\ncurrent thereafter We can also imagine a mechanism where the ends of the cylinder are\nsupplied with fresh charges to make up for any charges neutralised by\nelectrons moving inside the conductor In that case, there will be a steady\nelectric field in the body of the conductor This will result in a continuous\ncurrent rather than a current for a short period of time"}, {"Chapter": "1", "sentence_range": "2465-2468", "Text": "We can also imagine a mechanism where the ends of the cylinder are\nsupplied with fresh charges to make up for any charges neutralised by\nelectrons moving inside the conductor In that case, there will be a steady\nelectric field in the body of the conductor This will result in a continuous\ncurrent rather than a current for a short period of time Mechanisms,\nwhich maintain a steady electric field are cells or batteries that we shall\nstudy  later in this chapter"}, {"Chapter": "1", "sentence_range": "2466-2469", "Text": "In that case, there will be a steady\nelectric field in the body of the conductor This will result in a continuous\ncurrent rather than a current for a short period of time Mechanisms,\nwhich maintain a steady electric field are cells or batteries that we shall\nstudy  later in this chapter In the next sections, we shall study the steady\ncurrent that results from a steady electric field in conductors"}, {"Chapter": "1", "sentence_range": "2467-2470", "Text": "This will result in a continuous\ncurrent rather than a current for a short period of time Mechanisms,\nwhich maintain a steady electric field are cells or batteries that we shall\nstudy  later in this chapter In the next sections, we shall study the steady\ncurrent that results from a steady electric field in conductors 3"}, {"Chapter": "1", "sentence_range": "2468-2471", "Text": "Mechanisms,\nwhich maintain a steady electric field are cells or batteries that we shall\nstudy  later in this chapter In the next sections, we shall study the steady\ncurrent that results from a steady electric field in conductors 3 4  OHM\u2019S LAW\nA basic law regarding flow of currents was discovered by G"}, {"Chapter": "1", "sentence_range": "2469-2472", "Text": "In the next sections, we shall study the steady\ncurrent that results from a steady electric field in conductors 3 4  OHM\u2019S LAW\nA basic law regarding flow of currents was discovered by G S"}, {"Chapter": "1", "sentence_range": "2470-2473", "Text": "3 4  OHM\u2019S LAW\nA basic law regarding flow of currents was discovered by G S Ohm in\n1828, long before the physical mechanism responsible for flow of currents\nwas discovered"}, {"Chapter": "1", "sentence_range": "2471-2474", "Text": "4  OHM\u2019S LAW\nA basic law regarding flow of currents was discovered by G S Ohm in\n1828, long before the physical mechanism responsible for flow of currents\nwas discovered Imagine a conductor through which a current I is flowing\nand let V be the potential difference between the ends of the conductor"}, {"Chapter": "1", "sentence_range": "2472-2475", "Text": "S Ohm in\n1828, long before the physical mechanism responsible for flow of currents\nwas discovered Imagine a conductor through which a current I is flowing\nand let V be the potential difference between the ends of the conductor Then Ohm\u2019s law states that\n     V \u00b5 I\nor, V = R I\n(3"}, {"Chapter": "1", "sentence_range": "2473-2476", "Text": "Ohm in\n1828, long before the physical mechanism responsible for flow of currents\nwas discovered Imagine a conductor through which a current I is flowing\nand let V be the potential difference between the ends of the conductor Then Ohm\u2019s law states that\n     V \u00b5 I\nor, V = R I\n(3 3)\nwhere the constant of proportionality R is called the resistance of the\nconductor"}, {"Chapter": "1", "sentence_range": "2474-2477", "Text": "Imagine a conductor through which a current I is flowing\nand let V be the potential difference between the ends of the conductor Then Ohm\u2019s law states that\n     V \u00b5 I\nor, V = R I\n(3 3)\nwhere the constant of proportionality R is called the resistance of the\nconductor The SI units of resistance is ohm, and is denoted by the symbol\nW"}, {"Chapter": "1", "sentence_range": "2475-2478", "Text": "Then Ohm\u2019s law states that\n     V \u00b5 I\nor, V = R I\n(3 3)\nwhere the constant of proportionality R is called the resistance of the\nconductor The SI units of resistance is ohm, and is denoted by the symbol\nW The resistance R not only depends on the material of the conductor\nbut also on the dimensions of the conductor"}, {"Chapter": "1", "sentence_range": "2476-2479", "Text": "3)\nwhere the constant of proportionality R is called the resistance of the\nconductor The SI units of resistance is ohm, and is denoted by the symbol\nW The resistance R not only depends on the material of the conductor\nbut also on the dimensions of the conductor The dependence of R on the\ndimensions of the conductor can easily be determined as follows"}, {"Chapter": "1", "sentence_range": "2477-2480", "Text": "The SI units of resistance is ohm, and is denoted by the symbol\nW The resistance R not only depends on the material of the conductor\nbut also on the dimensions of the conductor The dependence of R on the\ndimensions of the conductor can easily be determined as follows Consider a conductor satisfying Eq"}, {"Chapter": "1", "sentence_range": "2478-2481", "Text": "The resistance R not only depends on the material of the conductor\nbut also on the dimensions of the conductor The dependence of R on the\ndimensions of the conductor can easily be determined as follows Consider a conductor satisfying Eq (3"}, {"Chapter": "1", "sentence_range": "2479-2482", "Text": "The dependence of R on the\ndimensions of the conductor can easily be determined as follows Consider a conductor satisfying Eq (3 3) to be in the form of a slab of\nlength l and cross sectional area A [Fig"}, {"Chapter": "1", "sentence_range": "2480-2483", "Text": "Consider a conductor satisfying Eq (3 3) to be in the form of a slab of\nlength l and cross sectional area A [Fig 3"}, {"Chapter": "1", "sentence_range": "2481-2484", "Text": "(3 3) to be in the form of a slab of\nlength l and cross sectional area A [Fig 3 2(a)]"}, {"Chapter": "1", "sentence_range": "2482-2485", "Text": "3) to be in the form of a slab of\nlength l and cross sectional area A [Fig 3 2(a)] Imagine placing two such\nidentical slabs side by side [Fig"}, {"Chapter": "1", "sentence_range": "2483-2486", "Text": "3 2(a)] Imagine placing two such\nidentical slabs side by side [Fig 3"}, {"Chapter": "1", "sentence_range": "2484-2487", "Text": "2(a)] Imagine placing two such\nidentical slabs side by side [Fig 3 2(b)], so that the length of the\ncombination is 2l"}, {"Chapter": "1", "sentence_range": "2485-2488", "Text": "Imagine placing two such\nidentical slabs side by side [Fig 3 2(b)], so that the length of the\ncombination is 2l The current flowing through the  combination is the\nsame as that flowing through either of the slabs"}, {"Chapter": "1", "sentence_range": "2486-2489", "Text": "3 2(b)], so that the length of the\ncombination is 2l The current flowing through the  combination is the\nsame as that flowing through either of the slabs If V is the potential\ndifference across the ends of the first slab, then V is also the potential\ndifference across the ends of the second slab since the second slab is\nFIGURE 3"}, {"Chapter": "1", "sentence_range": "2487-2490", "Text": "2(b)], so that the length of the\ncombination is 2l The current flowing through the  combination is the\nsame as that flowing through either of the slabs If V is the potential\ndifference across the ends of the first slab, then V is also the potential\ndifference across the ends of the second slab since the second slab is\nFIGURE 3 1 Charges +Q and \u2013Q put at the ends\nof a metallic cylinder"}, {"Chapter": "1", "sentence_range": "2488-2491", "Text": "The current flowing through the  combination is the\nsame as that flowing through either of the slabs If V is the potential\ndifference across the ends of the first slab, then V is also the potential\ndifference across the ends of the second slab since the second slab is\nFIGURE 3 1 Charges +Q and \u2013Q put at the ends\nof a metallic cylinder The electrons will drift\nbecause of the electric field created to\nneutralise the charges"}, {"Chapter": "1", "sentence_range": "2489-2492", "Text": "If V is the potential\ndifference across the ends of the first slab, then V is also the potential\ndifference across the ends of the second slab since the second slab is\nFIGURE 3 1 Charges +Q and \u2013Q put at the ends\nof a metallic cylinder The electrons will drift\nbecause of the electric field created to\nneutralise the charges The current thus\nwill stop after a while unless the charges +Q\nand \u2013Q are continuously replenished"}, {"Chapter": "1", "sentence_range": "2490-2493", "Text": "1 Charges +Q and \u2013Q put at the ends\nof a metallic cylinder The electrons will drift\nbecause of the electric field created to\nneutralise the charges The current thus\nwill stop after a while unless the charges +Q\nand \u2013Q are continuously replenished FIGURE 3"}, {"Chapter": "1", "sentence_range": "2491-2494", "Text": "The electrons will drift\nbecause of the electric field created to\nneutralise the charges The current thus\nwill stop after a while unless the charges +Q\nand \u2013Q are continuously replenished FIGURE 3 2\nIllustrating the\nrelation R = rl/A for\na rectangular slab\nof length l and area\nof cross-section A"}, {"Chapter": "1", "sentence_range": "2492-2495", "Text": "The current thus\nwill stop after a while unless the charges +Q\nand \u2013Q are continuously replenished FIGURE 3 2\nIllustrating the\nrelation R = rl/A for\na rectangular slab\nof length l and area\nof cross-section A Rationalised 2023-24\nPhysics\n84\nidentical to the first and the same current I flows through\nboth"}, {"Chapter": "1", "sentence_range": "2493-2496", "Text": "FIGURE 3 2\nIllustrating the\nrelation R = rl/A for\na rectangular slab\nof length l and area\nof cross-section A Rationalised 2023-24\nPhysics\n84\nidentical to the first and the same current I flows through\nboth The potential difference across the ends of the\ncombination is clearly sum of the potential difference\nacross the two individual slabs and hence equals 2V"}, {"Chapter": "1", "sentence_range": "2494-2497", "Text": "2\nIllustrating the\nrelation R = rl/A for\na rectangular slab\nof length l and area\nof cross-section A Rationalised 2023-24\nPhysics\n84\nidentical to the first and the same current I flows through\nboth The potential difference across the ends of the\ncombination is clearly sum of the potential difference\nacross the two individual slabs and hence equals 2V The\ncurrent through the combination is I and the resistance\nof the combination RC is [from Eq"}, {"Chapter": "1", "sentence_range": "2495-2498", "Text": "Rationalised 2023-24\nPhysics\n84\nidentical to the first and the same current I flows through\nboth The potential difference across the ends of the\ncombination is clearly sum of the potential difference\nacross the two individual slabs and hence equals 2V The\ncurrent through the combination is I and the resistance\nof the combination RC is [from Eq (3"}, {"Chapter": "1", "sentence_range": "2496-2499", "Text": "The potential difference across the ends of the\ncombination is clearly sum of the potential difference\nacross the two individual slabs and hence equals 2V The\ncurrent through the combination is I and the resistance\nof the combination RC is [from Eq (3 3)],\n2\n2\nC\nV\nR\nR\n=I\n=\n(3"}, {"Chapter": "1", "sentence_range": "2497-2500", "Text": "The\ncurrent through the combination is I and the resistance\nof the combination RC is [from Eq (3 3)],\n2\n2\nC\nV\nR\nR\n=I\n=\n(3 4)\nsince V/I = R, the resistance of either of the slabs"}, {"Chapter": "1", "sentence_range": "2498-2501", "Text": "(3 3)],\n2\n2\nC\nV\nR\nR\n=I\n=\n(3 4)\nsince V/I = R, the resistance of either of the slabs Thus,\ndoubling the length of a conductor doubles the\nresistance"}, {"Chapter": "1", "sentence_range": "2499-2502", "Text": "3)],\n2\n2\nC\nV\nR\nR\n=I\n=\n(3 4)\nsince V/I = R, the resistance of either of the slabs Thus,\ndoubling the length of a conductor doubles the\nresistance In general, then resistance is proportional to\nlength,\nR\n\u221dl\n(3"}, {"Chapter": "1", "sentence_range": "2500-2503", "Text": "4)\nsince V/I = R, the resistance of either of the slabs Thus,\ndoubling the length of a conductor doubles the\nresistance In general, then resistance is proportional to\nlength,\nR\n\u221dl\n(3 5)\nNext, imagine dividing the slab into two by cutting it\nlengthwise so that the slab can be considered as a\ncombination of two identical slabs of length l, but each\nhaving a cross sectional  area of A/2 [Fig"}, {"Chapter": "1", "sentence_range": "2501-2504", "Text": "Thus,\ndoubling the length of a conductor doubles the\nresistance In general, then resistance is proportional to\nlength,\nR\n\u221dl\n(3 5)\nNext, imagine dividing the slab into two by cutting it\nlengthwise so that the slab can be considered as a\ncombination of two identical slabs of length l, but each\nhaving a cross sectional  area of A/2 [Fig 3"}, {"Chapter": "1", "sentence_range": "2502-2505", "Text": "In general, then resistance is proportional to\nlength,\nR\n\u221dl\n(3 5)\nNext, imagine dividing the slab into two by cutting it\nlengthwise so that the slab can be considered as a\ncombination of two identical slabs of length l, but each\nhaving a cross sectional  area of A/2 [Fig 3 2(c)]"}, {"Chapter": "1", "sentence_range": "2503-2506", "Text": "5)\nNext, imagine dividing the slab into two by cutting it\nlengthwise so that the slab can be considered as a\ncombination of two identical slabs of length l, but each\nhaving a cross sectional  area of A/2 [Fig 3 2(c)] For a given voltage V across the slab, if I is the current\nthrough the entire slab, then clearly the current flowing\nthrough each of the two half-slabs is I/2"}, {"Chapter": "1", "sentence_range": "2504-2507", "Text": "3 2(c)] For a given voltage V across the slab, if I is the current\nthrough the entire slab, then clearly the current flowing\nthrough each of the two half-slabs is I/2 Since the\npotential difference across the ends of the half-slabs is V,\ni"}, {"Chapter": "1", "sentence_range": "2505-2508", "Text": "2(c)] For a given voltage V across the slab, if I is the current\nthrough the entire slab, then clearly the current flowing\nthrough each of the two half-slabs is I/2 Since the\npotential difference across the ends of the half-slabs is V,\ni e"}, {"Chapter": "1", "sentence_range": "2506-2509", "Text": "For a given voltage V across the slab, if I is the current\nthrough the entire slab, then clearly the current flowing\nthrough each of the two half-slabs is I/2 Since the\npotential difference across the ends of the half-slabs is V,\ni e , the same as across the full slab, the resistance of each\nof the half-slabs R1 is\n1\n2\n2"}, {"Chapter": "1", "sentence_range": "2507-2510", "Text": "Since the\npotential difference across the ends of the half-slabs is V,\ni e , the same as across the full slab, the resistance of each\nof the half-slabs R1 is\n1\n2\n2 ( /2)\nV\nV\nR\nR\nI\nI\n=\n=\n=\n(3"}, {"Chapter": "1", "sentence_range": "2508-2511", "Text": "e , the same as across the full slab, the resistance of each\nof the half-slabs R1 is\n1\n2\n2 ( /2)\nV\nV\nR\nR\nI\nI\n=\n=\n=\n(3 6)\nThus, halving the area of the cross-section of a conductor doubles\nthe resistance"}, {"Chapter": "1", "sentence_range": "2509-2512", "Text": ", the same as across the full slab, the resistance of each\nof the half-slabs R1 is\n1\n2\n2 ( /2)\nV\nV\nR\nR\nI\nI\n=\n=\n=\n(3 6)\nThus, halving the area of the cross-section of a conductor doubles\nthe resistance In general, then the resistance R is inversely proportional\nto the cross-sectional area,\n1\nR\n\u221dA\n(3"}, {"Chapter": "1", "sentence_range": "2510-2513", "Text": "( /2)\nV\nV\nR\nR\nI\nI\n=\n=\n=\n(3 6)\nThus, halving the area of the cross-section of a conductor doubles\nthe resistance In general, then the resistance R is inversely proportional\nto the cross-sectional area,\n1\nR\n\u221dA\n(3 7)\nCombining Eqs"}, {"Chapter": "1", "sentence_range": "2511-2514", "Text": "6)\nThus, halving the area of the cross-section of a conductor doubles\nthe resistance In general, then the resistance R is inversely proportional\nto the cross-sectional area,\n1\nR\n\u221dA\n(3 7)\nCombining Eqs (3"}, {"Chapter": "1", "sentence_range": "2512-2515", "Text": "In general, then the resistance R is inversely proportional\nto the cross-sectional area,\n1\nR\n\u221dA\n(3 7)\nCombining Eqs (3 5) and (3"}, {"Chapter": "1", "sentence_range": "2513-2516", "Text": "7)\nCombining Eqs (3 5) and (3 7), we have\nl\nR\n\u221dA\n(3"}, {"Chapter": "1", "sentence_range": "2514-2517", "Text": "(3 5) and (3 7), we have\nl\nR\n\u221dA\n(3 8)\nand hence for a given conductor\nl\nR\n=\u03c1A\n(3"}, {"Chapter": "1", "sentence_range": "2515-2518", "Text": "5) and (3 7), we have\nl\nR\n\u221dA\n(3 8)\nand hence for a given conductor\nl\nR\n=\u03c1A\n(3 9)\nwhere the constant of proportionality r depends on the material of the\nconductor but not on its dimensions"}, {"Chapter": "1", "sentence_range": "2516-2519", "Text": "7), we have\nl\nR\n\u221dA\n(3 8)\nand hence for a given conductor\nl\nR\n=\u03c1A\n(3 9)\nwhere the constant of proportionality r depends on the material of the\nconductor but not on its dimensions r is called resistivity"}, {"Chapter": "1", "sentence_range": "2517-2520", "Text": "8)\nand hence for a given conductor\nl\nR\n=\u03c1A\n(3 9)\nwhere the constant of proportionality r depends on the material of the\nconductor but not on its dimensions r is called resistivity Using the last equation, Ohm\u2019s law reads\nI l\nV\nI\nR\n\u03c1A\n=\n\u00d7\n=\n(3"}, {"Chapter": "1", "sentence_range": "2518-2521", "Text": "9)\nwhere the constant of proportionality r depends on the material of the\nconductor but not on its dimensions r is called resistivity Using the last equation, Ohm\u2019s law reads\nI l\nV\nI\nR\n\u03c1A\n=\n\u00d7\n=\n(3 10)\nCurrent per unit area (taken normal to the current), I/A, is called\ncurrent density and is denoted by j"}, {"Chapter": "1", "sentence_range": "2519-2522", "Text": "r is called resistivity Using the last equation, Ohm\u2019s law reads\nI l\nV\nI\nR\n\u03c1A\n=\n\u00d7\n=\n(3 10)\nCurrent per unit area (taken normal to the current), I/A, is called\ncurrent density and is denoted by j The SI units of the current density\nare A/m2"}, {"Chapter": "1", "sentence_range": "2520-2523", "Text": "Using the last equation, Ohm\u2019s law reads\nI l\nV\nI\nR\n\u03c1A\n=\n\u00d7\n=\n(3 10)\nCurrent per unit area (taken normal to the current), I/A, is called\ncurrent density and is denoted by j The SI units of the current density\nare A/m2 Further, if E is the magnitude of uniform electric field in the\nconductor whose length is l, then the potential difference V across its\nends is El"}, {"Chapter": "1", "sentence_range": "2521-2524", "Text": "10)\nCurrent per unit area (taken normal to the current), I/A, is called\ncurrent density and is denoted by j The SI units of the current density\nare A/m2 Further, if E is the magnitude of uniform electric field in the\nconductor whose length is l, then the potential difference V across its\nends is El Using these, the last equation reads\nGEORG SIMON OHM (1787\u20131854)\nGeorg Simon Ohm  (1787\u2013\n1854) German physicist,\nprofessor at Munich"}, {"Chapter": "1", "sentence_range": "2522-2525", "Text": "The SI units of the current density\nare A/m2 Further, if E is the magnitude of uniform electric field in the\nconductor whose length is l, then the potential difference V across its\nends is El Using these, the last equation reads\nGEORG SIMON OHM (1787\u20131854)\nGeorg Simon Ohm  (1787\u2013\n1854) German physicist,\nprofessor at Munich Ohm\nwas led to his law by an\nanalogy \nbetween \nthe\nconduction of heat: the\nelectric field is analogous to\nthe temperature gradient,\nand the electric current is\nanalogous to the heat flow"}, {"Chapter": "1", "sentence_range": "2523-2526", "Text": "Further, if E is the magnitude of uniform electric field in the\nconductor whose length is l, then the potential difference V across its\nends is El Using these, the last equation reads\nGEORG SIMON OHM (1787\u20131854)\nGeorg Simon Ohm  (1787\u2013\n1854) German physicist,\nprofessor at Munich Ohm\nwas led to his law by an\nanalogy \nbetween \nthe\nconduction of heat: the\nelectric field is analogous to\nthe temperature gradient,\nand the electric current is\nanalogous to the heat flow Rationalised 2023-24\nCurrent\nElectricity\n85\n        E l = j r l\nor,    E = j r\n(3"}, {"Chapter": "1", "sentence_range": "2524-2527", "Text": "Using these, the last equation reads\nGEORG SIMON OHM (1787\u20131854)\nGeorg Simon Ohm  (1787\u2013\n1854) German physicist,\nprofessor at Munich Ohm\nwas led to his law by an\nanalogy \nbetween \nthe\nconduction of heat: the\nelectric field is analogous to\nthe temperature gradient,\nand the electric current is\nanalogous to the heat flow Rationalised 2023-24\nCurrent\nElectricity\n85\n        E l = j r l\nor,    E = j r\n(3 11)\nThe above relation for magnitudes E and j can indeed be cast in a\nvector  form"}, {"Chapter": "1", "sentence_range": "2525-2528", "Text": "Ohm\nwas led to his law by an\nanalogy \nbetween \nthe\nconduction of heat: the\nelectric field is analogous to\nthe temperature gradient,\nand the electric current is\nanalogous to the heat flow Rationalised 2023-24\nCurrent\nElectricity\n85\n        E l = j r l\nor,    E = j r\n(3 11)\nThe above relation for magnitudes E and j can indeed be cast in a\nvector  form The current density, (which we have defined as the current\nthrough unit area normal to the current) is also directed along E, and is\nalso a vector j (\u00ba\u00ba\u00ba\u00ba\u00ba j E/E)"}, {"Chapter": "1", "sentence_range": "2526-2529", "Text": "Rationalised 2023-24\nCurrent\nElectricity\n85\n        E l = j r l\nor,    E = j r\n(3 11)\nThe above relation for magnitudes E and j can indeed be cast in a\nvector  form The current density, (which we have defined as the current\nthrough unit area normal to the current) is also directed along E, and is\nalso a vector j (\u00ba\u00ba\u00ba\u00ba\u00ba j E/E) Thus, the last equation can be written as,\n        E = jr\n(3"}, {"Chapter": "1", "sentence_range": "2527-2530", "Text": "11)\nThe above relation for magnitudes E and j can indeed be cast in a\nvector  form The current density, (which we have defined as the current\nthrough unit area normal to the current) is also directed along E, and is\nalso a vector j (\u00ba\u00ba\u00ba\u00ba\u00ba j E/E) Thus, the last equation can be written as,\n        E = jr\n(3 12)\nor,   j = s E\n(3"}, {"Chapter": "1", "sentence_range": "2528-2531", "Text": "The current density, (which we have defined as the current\nthrough unit area normal to the current) is also directed along E, and is\nalso a vector j (\u00ba\u00ba\u00ba\u00ba\u00ba j E/E) Thus, the last equation can be written as,\n        E = jr\n(3 12)\nor,   j = s E\n(3 13)\nwhere s \u00ba1/r is called the conductivity"}, {"Chapter": "1", "sentence_range": "2529-2532", "Text": "Thus, the last equation can be written as,\n        E = jr\n(3 12)\nor,   j = s E\n(3 13)\nwhere s \u00ba1/r is called the conductivity Ohm\u2019s law is often stated in an\nequivalent form, Eq"}, {"Chapter": "1", "sentence_range": "2530-2533", "Text": "12)\nor,   j = s E\n(3 13)\nwhere s \u00ba1/r is called the conductivity Ohm\u2019s law is often stated in an\nequivalent form, Eq (3"}, {"Chapter": "1", "sentence_range": "2531-2534", "Text": "13)\nwhere s \u00ba1/r is called the conductivity Ohm\u2019s law is often stated in an\nequivalent form, Eq (3 13) in addition to Eq"}, {"Chapter": "1", "sentence_range": "2532-2535", "Text": "Ohm\u2019s law is often stated in an\nequivalent form, Eq (3 13) in addition to Eq (3"}, {"Chapter": "1", "sentence_range": "2533-2536", "Text": "(3 13) in addition to Eq (3 3)"}, {"Chapter": "1", "sentence_range": "2534-2537", "Text": "13) in addition to Eq (3 3) In the next section, we\nwill try to understand the origin of the Ohm\u2019s law as arising from the\ncharacteristics of the drift of electrons"}, {"Chapter": "1", "sentence_range": "2535-2538", "Text": "(3 3) In the next section, we\nwill try to understand the origin of the Ohm\u2019s law as arising from the\ncharacteristics of the drift of electrons 3"}, {"Chapter": "1", "sentence_range": "2536-2539", "Text": "3) In the next section, we\nwill try to understand the origin of the Ohm\u2019s law as arising from the\ncharacteristics of the drift of electrons 3 5 DRIFT OF ELECTRONS AND THE ORIGIN\nOF RESISTIVITY\nAs remarked before, an electron will suffer collisions with the heavy fixed\nions, but after collision, it will emerge with the same speed but in random\ndirections"}, {"Chapter": "1", "sentence_range": "2537-2540", "Text": "In the next section, we\nwill try to understand the origin of the Ohm\u2019s law as arising from the\ncharacteristics of the drift of electrons 3 5 DRIFT OF ELECTRONS AND THE ORIGIN\nOF RESISTIVITY\nAs remarked before, an electron will suffer collisions with the heavy fixed\nions, but after collision, it will emerge with the same speed but in random\ndirections If we consider all the electrons, their average velocity will be\nzero since their directions are random"}, {"Chapter": "1", "sentence_range": "2538-2541", "Text": "3 5 DRIFT OF ELECTRONS AND THE ORIGIN\nOF RESISTIVITY\nAs remarked before, an electron will suffer collisions with the heavy fixed\nions, but after collision, it will emerge with the same speed but in random\ndirections If we consider all the electrons, their average velocity will be\nzero since their directions are random Thus, if there are N electrons and\nthe velocity of the ith electron (i = 1, 2, 3,"}, {"Chapter": "1", "sentence_range": "2539-2542", "Text": "5 DRIFT OF ELECTRONS AND THE ORIGIN\nOF RESISTIVITY\nAs remarked before, an electron will suffer collisions with the heavy fixed\nions, but after collision, it will emerge with the same speed but in random\ndirections If we consider all the electrons, their average velocity will be\nzero since their directions are random Thus, if there are N electrons and\nthe velocity of the ith electron (i = 1, 2, 3, N ) at a given time is vi, then\n1\n0\n1\nN\ni\ni\nv =\n=\u2211\nN\n(3"}, {"Chapter": "1", "sentence_range": "2540-2543", "Text": "If we consider all the electrons, their average velocity will be\nzero since their directions are random Thus, if there are N electrons and\nthe velocity of the ith electron (i = 1, 2, 3, N ) at a given time is vi, then\n1\n0\n1\nN\ni\ni\nv =\n=\u2211\nN\n(3 14)\nConsider now the situation when an electric field is\npresent"}, {"Chapter": "1", "sentence_range": "2541-2544", "Text": "Thus, if there are N electrons and\nthe velocity of the ith electron (i = 1, 2, 3, N ) at a given time is vi, then\n1\n0\n1\nN\ni\ni\nv =\n=\u2211\nN\n(3 14)\nConsider now the situation when an electric field is\npresent Electrons will be accelerated due to this\nfield by\na= \u2013 E\ne\nm\n(3"}, {"Chapter": "1", "sentence_range": "2542-2545", "Text": "N ) at a given time is vi, then\n1\n0\n1\nN\ni\ni\nv =\n=\u2211\nN\n(3 14)\nConsider now the situation when an electric field is\npresent Electrons will be accelerated due to this\nfield by\na= \u2013 E\ne\nm\n(3 15)\nwhere \u2013e is the charge and m is the mass of an electron"}, {"Chapter": "1", "sentence_range": "2543-2546", "Text": "14)\nConsider now the situation when an electric field is\npresent Electrons will be accelerated due to this\nfield by\na= \u2013 E\ne\nm\n(3 15)\nwhere \u2013e is the charge and m is the mass of an electron Consider again the ith electron at a given time t"}, {"Chapter": "1", "sentence_range": "2544-2547", "Text": "Electrons will be accelerated due to this\nfield by\na= \u2013 E\ne\nm\n(3 15)\nwhere \u2013e is the charge and m is the mass of an electron Consider again the ith electron at a given time t This\nelectron would have had its last collision some time\nbefore t, and let ti be the time elapsed after its last\ncollision"}, {"Chapter": "1", "sentence_range": "2545-2548", "Text": "15)\nwhere \u2013e is the charge and m is the mass of an electron Consider again the ith electron at a given time t This\nelectron would have had its last collision some time\nbefore t, and let ti be the time elapsed after its last\ncollision If vi was its velocity immediately after the last\ncollision, then its velocity Vi at time t is\n \u2212\n \n=\n+  \n \n \n \nE\nV\nv\ni\ni\ni\ne\nt\nm\n(3"}, {"Chapter": "1", "sentence_range": "2546-2549", "Text": "Consider again the ith electron at a given time t This\nelectron would have had its last collision some time\nbefore t, and let ti be the time elapsed after its last\ncollision If vi was its velocity immediately after the last\ncollision, then its velocity Vi at time t is\n \u2212\n \n=\n+  \n \n \n \nE\nV\nv\ni\ni\ni\ne\nt\nm\n(3 16)\nsince starting with its last collision it was accelerated\n(Fig"}, {"Chapter": "1", "sentence_range": "2547-2550", "Text": "This\nelectron would have had its last collision some time\nbefore t, and let ti be the time elapsed after its last\ncollision If vi was its velocity immediately after the last\ncollision, then its velocity Vi at time t is\n \u2212\n \n=\n+  \n \n \n \nE\nV\nv\ni\ni\ni\ne\nt\nm\n(3 16)\nsince starting with its last collision it was accelerated\n(Fig 3"}, {"Chapter": "1", "sentence_range": "2548-2551", "Text": "If vi was its velocity immediately after the last\ncollision, then its velocity Vi at time t is\n \u2212\n \n=\n+  \n \n \n \nE\nV\nv\ni\ni\ni\ne\nt\nm\n(3 16)\nsince starting with its last collision it was accelerated\n(Fig 3 3) with an acceleration given by Eq"}, {"Chapter": "1", "sentence_range": "2549-2552", "Text": "16)\nsince starting with its last collision it was accelerated\n(Fig 3 3) with an acceleration given by Eq (3"}, {"Chapter": "1", "sentence_range": "2550-2553", "Text": "3 3) with an acceleration given by Eq (3 15) for a\ntime interval ti"}, {"Chapter": "1", "sentence_range": "2551-2554", "Text": "3) with an acceleration given by Eq (3 15) for a\ntime interval ti The average velocity of the electrons at\ntime t is the average of all the Vi\u2019s"}, {"Chapter": "1", "sentence_range": "2552-2555", "Text": "(3 15) for a\ntime interval ti The average velocity of the electrons at\ntime t is the average of all the Vi\u2019s The average of vi\u2019s is\nzero [Eq"}, {"Chapter": "1", "sentence_range": "2553-2556", "Text": "15) for a\ntime interval ti The average velocity of the electrons at\ntime t is the average of all the Vi\u2019s The average of vi\u2019s is\nzero [Eq (3"}, {"Chapter": "1", "sentence_range": "2554-2557", "Text": "The average velocity of the electrons at\ntime t is the average of all the Vi\u2019s The average of vi\u2019s is\nzero [Eq (3 14)] since immediately after any collision,\nthe direction of the velocity of an electron is completely\nrandom"}, {"Chapter": "1", "sentence_range": "2555-2558", "Text": "The average of vi\u2019s is\nzero [Eq (3 14)] since immediately after any collision,\nthe direction of the velocity of an electron is completely\nrandom The collisions of the electrons do not occur at\nregular intervals but at random times"}, {"Chapter": "1", "sentence_range": "2556-2559", "Text": "(3 14)] since immediately after any collision,\nthe direction of the velocity of an electron is completely\nrandom The collisions of the electrons do not occur at\nregular intervals but at random times Let us denote by\nt, the average time between successive collisions"}, {"Chapter": "1", "sentence_range": "2557-2560", "Text": "14)] since immediately after any collision,\nthe direction of the velocity of an electron is completely\nrandom The collisions of the electrons do not occur at\nregular intervals but at random times Let us denote by\nt, the average time between successive collisions Then\nat a given time, some of the electrons would have spent\nFIGURE 3"}, {"Chapter": "1", "sentence_range": "2558-2561", "Text": "The collisions of the electrons do not occur at\nregular intervals but at random times Let us denote by\nt, the average time between successive collisions Then\nat a given time, some of the electrons would have spent\nFIGURE 3 3 A schematic picture of\nan electron moving from a point A to\nanother point B through repeated\ncollisions, and straight line travel\nbetween collisions (full lines)"}, {"Chapter": "1", "sentence_range": "2559-2562", "Text": "Let us denote by\nt, the average time between successive collisions Then\nat a given time, some of the electrons would have spent\nFIGURE 3 3 A schematic picture of\nan electron moving from a point A to\nanother point B through repeated\ncollisions, and straight line travel\nbetween collisions (full lines) If an\nelectric field is applied as shown, the\nelectron ends up at point B\u00a2 (dotted\nlines)"}, {"Chapter": "1", "sentence_range": "2560-2563", "Text": "Then\nat a given time, some of the electrons would have spent\nFIGURE 3 3 A schematic picture of\nan electron moving from a point A to\nanother point B through repeated\ncollisions, and straight line travel\nbetween collisions (full lines) If an\nelectric field is applied as shown, the\nelectron ends up at point B\u00a2 (dotted\nlines) A slight drift in a direction\nopposite the electric field is visible"}, {"Chapter": "1", "sentence_range": "2561-2564", "Text": "3 A schematic picture of\nan electron moving from a point A to\nanother point B through repeated\ncollisions, and straight line travel\nbetween collisions (full lines) If an\nelectric field is applied as shown, the\nelectron ends up at point B\u00a2 (dotted\nlines) A slight drift in a direction\nopposite the electric field is visible Rationalised 2023-24\nPhysics\n86\ntime more than t and some less than t"}, {"Chapter": "1", "sentence_range": "2562-2565", "Text": "If an\nelectric field is applied as shown, the\nelectron ends up at point B\u00a2 (dotted\nlines) A slight drift in a direction\nopposite the electric field is visible Rationalised 2023-24\nPhysics\n86\ntime more than t and some less than t In other words, the time ti in\nEq"}, {"Chapter": "1", "sentence_range": "2563-2566", "Text": "A slight drift in a direction\nopposite the electric field is visible Rationalised 2023-24\nPhysics\n86\ntime more than t and some less than t In other words, the time ti in\nEq (3"}, {"Chapter": "1", "sentence_range": "2564-2567", "Text": "Rationalised 2023-24\nPhysics\n86\ntime more than t and some less than t In other words, the time ti in\nEq (3 16) will be less than t for some and more than t for others as we go\nthrough the values of i = 1, 2"}, {"Chapter": "1", "sentence_range": "2565-2568", "Text": "In other words, the time ti in\nEq (3 16) will be less than t for some and more than t for others as we go\nthrough the values of i = 1, 2 N"}, {"Chapter": "1", "sentence_range": "2566-2569", "Text": "(3 16) will be less than t for some and more than t for others as we go\nthrough the values of i = 1, 2 N The average value of ti then is t\n(known as relaxation time)"}, {"Chapter": "1", "sentence_range": "2567-2570", "Text": "16) will be less than t for some and more than t for others as we go\nthrough the values of i = 1, 2 N The average value of ti then is t\n(known as relaxation time) Thus, averaging Eq"}, {"Chapter": "1", "sentence_range": "2568-2571", "Text": "N The average value of ti then is t\n(known as relaxation time) Thus, averaging Eq (3"}, {"Chapter": "1", "sentence_range": "2569-2572", "Text": "The average value of ti then is t\n(known as relaxation time) Thus, averaging Eq (3 16) over the\nN-electrons at any given time t gives us for the average velocity vd\n(\n)\n(\n)\n( )\n\u2261\n=\n\u2212\nE\nv\nV\nv\nd\ni\ni\ni\naverage\naverage\naverage\ne\nt\nm\n0 \u2013\n\u03c4\n\u03c4\n=\n= \u2212\nE\nE\ne\ne\nm\nm\n(3"}, {"Chapter": "1", "sentence_range": "2570-2573", "Text": "Thus, averaging Eq (3 16) over the\nN-electrons at any given time t gives us for the average velocity vd\n(\n)\n(\n)\n( )\n\u2261\n=\n\u2212\nE\nv\nV\nv\nd\ni\ni\ni\naverage\naverage\naverage\ne\nt\nm\n0 \u2013\n\u03c4\n\u03c4\n=\n= \u2212\nE\nE\ne\ne\nm\nm\n(3 17)\nThis last result is surprising"}, {"Chapter": "1", "sentence_range": "2571-2574", "Text": "(3 16) over the\nN-electrons at any given time t gives us for the average velocity vd\n(\n)\n(\n)\n( )\n\u2261\n=\n\u2212\nE\nv\nV\nv\nd\ni\ni\ni\naverage\naverage\naverage\ne\nt\nm\n0 \u2013\n\u03c4\n\u03c4\n=\n= \u2212\nE\nE\ne\ne\nm\nm\n(3 17)\nThis last result is surprising It tells us that the\nelectrons move with an average velocity which is\nindependent of time, although electrons are\naccelerated"}, {"Chapter": "1", "sentence_range": "2572-2575", "Text": "16) over the\nN-electrons at any given time t gives us for the average velocity vd\n(\n)\n(\n)\n( )\n\u2261\n=\n\u2212\nE\nv\nV\nv\nd\ni\ni\ni\naverage\naverage\naverage\ne\nt\nm\n0 \u2013\n\u03c4\n\u03c4\n=\n= \u2212\nE\nE\ne\ne\nm\nm\n(3 17)\nThis last result is surprising It tells us that the\nelectrons move with an average velocity which is\nindependent of time, although electrons are\naccelerated This is the phenomenon of drift and the\nvelocity vd in Eq"}, {"Chapter": "1", "sentence_range": "2573-2576", "Text": "17)\nThis last result is surprising It tells us that the\nelectrons move with an average velocity which is\nindependent of time, although electrons are\naccelerated This is the phenomenon of drift and the\nvelocity vd in Eq (3"}, {"Chapter": "1", "sentence_range": "2574-2577", "Text": "It tells us that the\nelectrons move with an average velocity which is\nindependent of time, although electrons are\naccelerated This is the phenomenon of drift and the\nvelocity vd in Eq (3 17) is called the drift velocity"}, {"Chapter": "1", "sentence_range": "2575-2578", "Text": "This is the phenomenon of drift and the\nvelocity vd in Eq (3 17) is called the drift velocity Because of the drift, there will be net transport of\ncharges across any area perpendicular to E"}, {"Chapter": "1", "sentence_range": "2576-2579", "Text": "(3 17) is called the drift velocity Because of the drift, there will be net transport of\ncharges across any area perpendicular to E Consider\na planar area A, located inside the conductor such that\nthe normal to the area is parallel to E (Fig"}, {"Chapter": "1", "sentence_range": "2577-2580", "Text": "17) is called the drift velocity Because of the drift, there will be net transport of\ncharges across any area perpendicular to E Consider\na planar area A, located inside the conductor such that\nthe normal to the area is parallel to E (Fig 3"}, {"Chapter": "1", "sentence_range": "2578-2581", "Text": "Because of the drift, there will be net transport of\ncharges across any area perpendicular to E Consider\na planar area A, located inside the conductor such that\nthe normal to the area is parallel to E (Fig 3 4)"}, {"Chapter": "1", "sentence_range": "2579-2582", "Text": "Consider\na planar area A, located inside the conductor such that\nthe normal to the area is parallel to E (Fig 3 4) Then\nbecause of the drift, in an infinitesimal amount of time\nDt, all electrons to the left of the area at distances upto\n|vd|Dt would have crossed the area"}, {"Chapter": "1", "sentence_range": "2580-2583", "Text": "3 4) Then\nbecause of the drift, in an infinitesimal amount of time\nDt, all electrons to the left of the area at distances upto\n|vd|Dt would have crossed the area If n is the number\nof free electrons per unit volume in the metal, then\nthere are n Dt |vd|A such electrons"}, {"Chapter": "1", "sentence_range": "2581-2584", "Text": "4) Then\nbecause of the drift, in an infinitesimal amount of time\nDt, all electrons to the left of the area at distances upto\n|vd|Dt would have crossed the area If n is the number\nof free electrons per unit volume in the metal, then\nthere are n Dt |vd|A such electrons Since each\nelectron carries a charge \u2013e, the total charge transported across this area\nA to the right in time Dt is \u2013ne A|vd|Dt"}, {"Chapter": "1", "sentence_range": "2582-2585", "Text": "Then\nbecause of the drift, in an infinitesimal amount of time\nDt, all electrons to the left of the area at distances upto\n|vd|Dt would have crossed the area If n is the number\nof free electrons per unit volume in the metal, then\nthere are n Dt |vd|A such electrons Since each\nelectron carries a charge \u2013e, the total charge transported across this area\nA to the right in time Dt is \u2013ne A|vd|Dt E is directed towards the left and\nhence the total charge transported along E across the area is negative of\nthis"}, {"Chapter": "1", "sentence_range": "2583-2586", "Text": "If n is the number\nof free electrons per unit volume in the metal, then\nthere are n Dt |vd|A such electrons Since each\nelectron carries a charge \u2013e, the total charge transported across this area\nA to the right in time Dt is \u2013ne A|vd|Dt E is directed towards the left and\nhence the total charge transported along E across the area is negative of\nthis The amount of charge crossing the area A in time Dt is by definition\n[Eq"}, {"Chapter": "1", "sentence_range": "2584-2587", "Text": "Since each\nelectron carries a charge \u2013e, the total charge transported across this area\nA to the right in time Dt is \u2013ne A|vd|Dt E is directed towards the left and\nhence the total charge transported along E across the area is negative of\nthis The amount of charge crossing the area A in time Dt is by definition\n[Eq (3"}, {"Chapter": "1", "sentence_range": "2585-2588", "Text": "E is directed towards the left and\nhence the total charge transported along E across the area is negative of\nthis The amount of charge crossing the area A in time Dt is by definition\n[Eq (3 2)] I Dt, where I is the magnitude of the current"}, {"Chapter": "1", "sentence_range": "2586-2589", "Text": "The amount of charge crossing the area A in time Dt is by definition\n[Eq (3 2)] I Dt, where I is the magnitude of the current Hence,\nv\n\u2206 = +\n\u2206\nd\nI\nt\nn e A\nt\n(3"}, {"Chapter": "1", "sentence_range": "2587-2590", "Text": "(3 2)] I Dt, where I is the magnitude of the current Hence,\nv\n\u2206 = +\n\u2206\nd\nI\nt\nn e A\nt\n(3 18)\nSubstituting the value of |vd| from Eq"}, {"Chapter": "1", "sentence_range": "2588-2591", "Text": "2)] I Dt, where I is the magnitude of the current Hence,\nv\n\u2206 = +\n\u2206\nd\nI\nt\nn e A\nt\n(3 18)\nSubstituting the value of |vd| from Eq (3"}, {"Chapter": "1", "sentence_range": "2589-2592", "Text": "Hence,\nv\n\u2206 = +\n\u2206\nd\nI\nt\nn e A\nt\n(3 18)\nSubstituting the value of |vd| from Eq (3 17)\n2\nE\n\u03c4\n\u2206 =\n\u2206\ne A\nI t\nn\nt\nm\n(3"}, {"Chapter": "1", "sentence_range": "2590-2593", "Text": "18)\nSubstituting the value of |vd| from Eq (3 17)\n2\nE\n\u03c4\n\u2206 =\n\u2206\ne A\nI t\nn\nt\nm\n(3 19)\nBy definition I is related to the magnitude |j| of the current density by\nI = |j|A\n(3"}, {"Chapter": "1", "sentence_range": "2591-2594", "Text": "(3 17)\n2\nE\n\u03c4\n\u2206 =\n\u2206\ne A\nI t\nn\nt\nm\n(3 19)\nBy definition I is related to the magnitude |j| of the current density by\nI = |j|A\n(3 20)\nHence, from Eqs"}, {"Chapter": "1", "sentence_range": "2592-2595", "Text": "17)\n2\nE\n\u03c4\n\u2206 =\n\u2206\ne A\nI t\nn\nt\nm\n(3 19)\nBy definition I is related to the magnitude |j| of the current density by\nI = |j|A\n(3 20)\nHence, from Eqs (3"}, {"Chapter": "1", "sentence_range": "2593-2596", "Text": "19)\nBy definition I is related to the magnitude |j| of the current density by\nI = |j|A\n(3 20)\nHence, from Eqs (3 19) and (3"}, {"Chapter": "1", "sentence_range": "2594-2597", "Text": "20)\nHence, from Eqs (3 19) and (3 20),\n2\nj\n= ne\u03c4E\nm\n(3"}, {"Chapter": "1", "sentence_range": "2595-2598", "Text": "(3 19) and (3 20),\n2\nj\n= ne\u03c4E\nm\n(3 21)\nThe vector j is parallel to E and hence we can write Eq"}, {"Chapter": "1", "sentence_range": "2596-2599", "Text": "19) and (3 20),\n2\nj\n= ne\u03c4E\nm\n(3 21)\nThe vector j is parallel to E and hence we can write Eq (3"}, {"Chapter": "1", "sentence_range": "2597-2600", "Text": "20),\n2\nj\n= ne\u03c4E\nm\n(3 21)\nThe vector j is parallel to E and hence we can write Eq (3 21) in the\nvector form\n2\n\u03c4\nj=\nE\nne\nm\n(3"}, {"Chapter": "1", "sentence_range": "2598-2601", "Text": "21)\nThe vector j is parallel to E and hence we can write Eq (3 21) in the\nvector form\n2\n\u03c4\nj=\nE\nne\nm\n(3 22)\nComparison with Eq"}, {"Chapter": "1", "sentence_range": "2599-2602", "Text": "(3 21) in the\nvector form\n2\n\u03c4\nj=\nE\nne\nm\n(3 22)\nComparison with Eq (3"}, {"Chapter": "1", "sentence_range": "2600-2603", "Text": "21) in the\nvector form\n2\n\u03c4\nj=\nE\nne\nm\n(3 22)\nComparison with Eq (3 13) shows that Eq"}, {"Chapter": "1", "sentence_range": "2601-2604", "Text": "22)\nComparison with Eq (3 13) shows that Eq (3"}, {"Chapter": "1", "sentence_range": "2602-2605", "Text": "(3 13) shows that Eq (3 22) is exactly the Ohm\u2019s\nlaw, if we identify the conductivity s  as\nFIGURE 3"}, {"Chapter": "1", "sentence_range": "2603-2606", "Text": "13) shows that Eq (3 22) is exactly the Ohm\u2019s\nlaw, if we identify the conductivity s  as\nFIGURE 3 4 Current in a metallic\nconductor"}, {"Chapter": "1", "sentence_range": "2604-2607", "Text": "(3 22) is exactly the Ohm\u2019s\nlaw, if we identify the conductivity s  as\nFIGURE 3 4 Current in a metallic\nconductor The magnitude of current\ndensity in a metal is the magnitude of\ncharge contained in a cylinder of unit\narea and length vd"}, {"Chapter": "1", "sentence_range": "2605-2608", "Text": "22) is exactly the Ohm\u2019s\nlaw, if we identify the conductivity s  as\nFIGURE 3 4 Current in a metallic\nconductor The magnitude of current\ndensity in a metal is the magnitude of\ncharge contained in a cylinder of unit\narea and length vd Rationalised 2023-24\nCurrent\nElectricity\n87\n EXAMPLE 3"}, {"Chapter": "1", "sentence_range": "2606-2609", "Text": "4 Current in a metallic\nconductor The magnitude of current\ndensity in a metal is the magnitude of\ncharge contained in a cylinder of unit\narea and length vd Rationalised 2023-24\nCurrent\nElectricity\n87\n EXAMPLE 3 1\nne2\nm\n\u03c3\n\u03c4\n=\n(3"}, {"Chapter": "1", "sentence_range": "2607-2610", "Text": "The magnitude of current\ndensity in a metal is the magnitude of\ncharge contained in a cylinder of unit\narea and length vd Rationalised 2023-24\nCurrent\nElectricity\n87\n EXAMPLE 3 1\nne2\nm\n\u03c3\n\u03c4\n=\n(3 23)\nWe thus see that a very simple picture of electrical conduction\nreproduces Ohm\u2019s law"}, {"Chapter": "1", "sentence_range": "2608-2611", "Text": "Rationalised 2023-24\nCurrent\nElectricity\n87\n EXAMPLE 3 1\nne2\nm\n\u03c3\n\u03c4\n=\n(3 23)\nWe thus see that a very simple picture of electrical conduction\nreproduces Ohm\u2019s law We have, of course, made assumptions that t\nand n are constants, independent of E"}, {"Chapter": "1", "sentence_range": "2609-2612", "Text": "1\nne2\nm\n\u03c3\n\u03c4\n=\n(3 23)\nWe thus see that a very simple picture of electrical conduction\nreproduces Ohm\u2019s law We have, of course, made assumptions that t\nand n are constants, independent of E We shall, in the next section,\ndiscuss the limitations of Ohm\u2019s law"}, {"Chapter": "1", "sentence_range": "2610-2613", "Text": "23)\nWe thus see that a very simple picture of electrical conduction\nreproduces Ohm\u2019s law We have, of course, made assumptions that t\nand n are constants, independent of E We shall, in the next section,\ndiscuss the limitations of Ohm\u2019s law Example 3"}, {"Chapter": "1", "sentence_range": "2611-2614", "Text": "We have, of course, made assumptions that t\nand n are constants, independent of E We shall, in the next section,\ndiscuss the limitations of Ohm\u2019s law Example 3 1 (a) Estimate the average drift speed of conduction\nelectrons in a copper wire of cross-sectional area 1"}, {"Chapter": "1", "sentence_range": "2612-2615", "Text": "We shall, in the next section,\ndiscuss the limitations of Ohm\u2019s law Example 3 1 (a) Estimate the average drift speed of conduction\nelectrons in a copper wire of cross-sectional area 1 0 \u00d7 10\u20137 m2 carrying\na current of 1"}, {"Chapter": "1", "sentence_range": "2613-2616", "Text": "Example 3 1 (a) Estimate the average drift speed of conduction\nelectrons in a copper wire of cross-sectional area 1 0 \u00d7 10\u20137 m2 carrying\na current of 1 5 A"}, {"Chapter": "1", "sentence_range": "2614-2617", "Text": "1 (a) Estimate the average drift speed of conduction\nelectrons in a copper wire of cross-sectional area 1 0 \u00d7 10\u20137 m2 carrying\na current of 1 5 A Assume that each copper atom contributes roughly\none conduction electron"}, {"Chapter": "1", "sentence_range": "2615-2618", "Text": "0 \u00d7 10\u20137 m2 carrying\na current of 1 5 A Assume that each copper atom contributes roughly\none conduction electron The density of copper is 9"}, {"Chapter": "1", "sentence_range": "2616-2619", "Text": "5 A Assume that each copper atom contributes roughly\none conduction electron The density of copper is 9 0 \u00d7 103 kg/m3,\nand its atomic mass is 63"}, {"Chapter": "1", "sentence_range": "2617-2620", "Text": "Assume that each copper atom contributes roughly\none conduction electron The density of copper is 9 0 \u00d7 103 kg/m3,\nand its atomic mass is 63 5 u"}, {"Chapter": "1", "sentence_range": "2618-2621", "Text": "The density of copper is 9 0 \u00d7 103 kg/m3,\nand its atomic mass is 63 5 u (b) Compare the drift speed obtained\nabove with, (i) thermal speeds of copper atoms at ordinary\ntemperatures, (ii) speed of propagation of electric field along the\nconductor which causes the drift motion"}, {"Chapter": "1", "sentence_range": "2619-2622", "Text": "0 \u00d7 103 kg/m3,\nand its atomic mass is 63 5 u (b) Compare the drift speed obtained\nabove with, (i) thermal speeds of copper atoms at ordinary\ntemperatures, (ii) speed of propagation of electric field along the\nconductor which causes the drift motion Solution\n(a) The direction of drift velocity of conduction electrons is opposite\nto the electric field direction, i"}, {"Chapter": "1", "sentence_range": "2620-2623", "Text": "5 u (b) Compare the drift speed obtained\nabove with, (i) thermal speeds of copper atoms at ordinary\ntemperatures, (ii) speed of propagation of electric field along the\nconductor which causes the drift motion Solution\n(a) The direction of drift velocity of conduction electrons is opposite\nto the electric field direction, i e"}, {"Chapter": "1", "sentence_range": "2621-2624", "Text": "(b) Compare the drift speed obtained\nabove with, (i) thermal speeds of copper atoms at ordinary\ntemperatures, (ii) speed of propagation of electric field along the\nconductor which causes the drift motion Solution\n(a) The direction of drift velocity of conduction electrons is opposite\nto the electric field direction, i e , electrons drift in the direction\nof increasing potential"}, {"Chapter": "1", "sentence_range": "2622-2625", "Text": "Solution\n(a) The direction of drift velocity of conduction electrons is opposite\nto the electric field direction, i e , electrons drift in the direction\nof increasing potential The drift speed vd is given by Eq"}, {"Chapter": "1", "sentence_range": "2623-2626", "Text": "e , electrons drift in the direction\nof increasing potential The drift speed vd is given by Eq (3"}, {"Chapter": "1", "sentence_range": "2624-2627", "Text": ", electrons drift in the direction\nof increasing potential The drift speed vd is given by Eq (3 18)\nvd  = (I/neA)\nNow, e = 1"}, {"Chapter": "1", "sentence_range": "2625-2628", "Text": "The drift speed vd is given by Eq (3 18)\nvd  = (I/neA)\nNow, e = 1 6 \u00d7 10\u201319 C, A = 1"}, {"Chapter": "1", "sentence_range": "2626-2629", "Text": "(3 18)\nvd  = (I/neA)\nNow, e = 1 6 \u00d7 10\u201319 C, A = 1 0 \u00d7 10\u20137m2, I = 1"}, {"Chapter": "1", "sentence_range": "2627-2630", "Text": "18)\nvd  = (I/neA)\nNow, e = 1 6 \u00d7 10\u201319 C, A = 1 0 \u00d7 10\u20137m2, I = 1 5 A"}, {"Chapter": "1", "sentence_range": "2628-2631", "Text": "6 \u00d7 10\u201319 C, A = 1 0 \u00d7 10\u20137m2, I = 1 5 A The density of\nconduction electrons, n is equal to the number of atoms per cubic\nmetre (assuming one conduction electron per Cu atom as is\nreasonable from its valence electron count of one)"}, {"Chapter": "1", "sentence_range": "2629-2632", "Text": "0 \u00d7 10\u20137m2, I = 1 5 A The density of\nconduction electrons, n is equal to the number of atoms per cubic\nmetre (assuming one conduction electron per Cu atom as is\nreasonable from its valence electron count of one) A cubic metre\nof copper has a mass of 9"}, {"Chapter": "1", "sentence_range": "2630-2633", "Text": "5 A The density of\nconduction electrons, n is equal to the number of atoms per cubic\nmetre (assuming one conduction electron per Cu atom as is\nreasonable from its valence electron count of one) A cubic metre\nof copper has a mass of 9 0 \u00d7 103 kg"}, {"Chapter": "1", "sentence_range": "2631-2634", "Text": "The density of\nconduction electrons, n is equal to the number of atoms per cubic\nmetre (assuming one conduction electron per Cu atom as is\nreasonable from its valence electron count of one) A cubic metre\nof copper has a mass of 9 0 \u00d7 103 kg Since 6"}, {"Chapter": "1", "sentence_range": "2632-2635", "Text": "A cubic metre\nof copper has a mass of 9 0 \u00d7 103 kg Since 6 0 \u00d7 1023 copper\natoms have a mass of 63"}, {"Chapter": "1", "sentence_range": "2633-2636", "Text": "0 \u00d7 103 kg Since 6 0 \u00d7 1023 copper\natoms have a mass of 63 5 g,\n23\n6\n6"}, {"Chapter": "1", "sentence_range": "2634-2637", "Text": "Since 6 0 \u00d7 1023 copper\natoms have a mass of 63 5 g,\n23\n6\n6 0\n10\n9"}, {"Chapter": "1", "sentence_range": "2635-2638", "Text": "0 \u00d7 1023 copper\natoms have a mass of 63 5 g,\n23\n6\n6 0\n10\n9 0\n10\n63"}, {"Chapter": "1", "sentence_range": "2636-2639", "Text": "5 g,\n23\n6\n6 0\n10\n9 0\n10\n63 5\nn\n\u00d7\n=\n\u00d7\n\u00d7\n   = 8"}, {"Chapter": "1", "sentence_range": "2637-2640", "Text": "0\n10\n9 0\n10\n63 5\nn\n\u00d7\n=\n\u00d7\n\u00d7\n   = 8 5 \u00d7 1028 m\u20133\nwhich gives,\n28\n\u201319\n\u20137\n1"}, {"Chapter": "1", "sentence_range": "2638-2641", "Text": "0\n10\n63 5\nn\n\u00d7\n=\n\u00d7\n\u00d7\n   = 8 5 \u00d7 1028 m\u20133\nwhich gives,\n28\n\u201319\n\u20137\n1 5\n8"}, {"Chapter": "1", "sentence_range": "2639-2642", "Text": "5\nn\n\u00d7\n=\n\u00d7\n\u00d7\n   = 8 5 \u00d7 1028 m\u20133\nwhich gives,\n28\n\u201319\n\u20137\n1 5\n8 5\n10\n1"}, {"Chapter": "1", "sentence_range": "2640-2643", "Text": "5 \u00d7 1028 m\u20133\nwhich gives,\n28\n\u201319\n\u20137\n1 5\n8 5\n10\n1 6\n10\n1"}, {"Chapter": "1", "sentence_range": "2641-2644", "Text": "5\n8 5\n10\n1 6\n10\n1 0\n10\n=\n\u00d7\n\u00d7\n\u00d7\n\u00d7\n\u00d7\nd\nv\n    = 1"}, {"Chapter": "1", "sentence_range": "2642-2645", "Text": "5\n10\n1 6\n10\n1 0\n10\n=\n\u00d7\n\u00d7\n\u00d7\n\u00d7\n\u00d7\nd\nv\n    = 1 1 \u00d7 10\u20133 m s\u20131  = 1"}, {"Chapter": "1", "sentence_range": "2643-2646", "Text": "6\n10\n1 0\n10\n=\n\u00d7\n\u00d7\n\u00d7\n\u00d7\n\u00d7\nd\nv\n    = 1 1 \u00d7 10\u20133 m s\u20131  = 1 1 mm s\u20131\n(b) (i) At a temperature T, the thermal speed* of a copper atom of\nmass M is obtained from [<(1/2) Mv2 > = (3/2) kBT ] and is thus\ntypically of the order of \n/\nB\nk T M , where kB is the Boltzmann\nconstant"}, {"Chapter": "1", "sentence_range": "2644-2647", "Text": "0\n10\n=\n\u00d7\n\u00d7\n\u00d7\n\u00d7\n\u00d7\nd\nv\n    = 1 1 \u00d7 10\u20133 m s\u20131  = 1 1 mm s\u20131\n(b) (i) At a temperature T, the thermal speed* of a copper atom of\nmass M is obtained from [<(1/2) Mv2 > = (3/2) kBT ] and is thus\ntypically of the order of \n/\nB\nk T M , where kB is the Boltzmann\nconstant For copper at 300 K, this is about 2 \u00d7 102 m/s"}, {"Chapter": "1", "sentence_range": "2645-2648", "Text": "1 \u00d7 10\u20133 m s\u20131  = 1 1 mm s\u20131\n(b) (i) At a temperature T, the thermal speed* of a copper atom of\nmass M is obtained from [<(1/2) Mv2 > = (3/2) kBT ] and is thus\ntypically of the order of \n/\nB\nk T M , where kB is the Boltzmann\nconstant For copper at 300 K, this is about 2 \u00d7 102 m/s This\nfigure indicates the random vibrational speeds of copper atoms\nin a conductor"}, {"Chapter": "1", "sentence_range": "2646-2649", "Text": "1 mm s\u20131\n(b) (i) At a temperature T, the thermal speed* of a copper atom of\nmass M is obtained from [<(1/2) Mv2 > = (3/2) kBT ] and is thus\ntypically of the order of \n/\nB\nk T M , where kB is the Boltzmann\nconstant For copper at 300 K, this is about 2 \u00d7 102 m/s This\nfigure indicates the random vibrational speeds of copper atoms\nin a conductor Note that the drift speed of electrons is much\nsmaller, about 10\u20135 times the typical thermal speed at ordinary\ntemperatures"}, {"Chapter": "1", "sentence_range": "2647-2650", "Text": "For copper at 300 K, this is about 2 \u00d7 102 m/s This\nfigure indicates the random vibrational speeds of copper atoms\nin a conductor Note that the drift speed of electrons is much\nsmaller, about 10\u20135 times the typical thermal speed at ordinary\ntemperatures (ii) An electric field travelling along the conductor has a speed of\nan electromagnetic wave, namely equal to 3"}, {"Chapter": "1", "sentence_range": "2648-2651", "Text": "This\nfigure indicates the random vibrational speeds of copper atoms\nin a conductor Note that the drift speed of electrons is much\nsmaller, about 10\u20135 times the typical thermal speed at ordinary\ntemperatures (ii) An electric field travelling along the conductor has a speed of\nan electromagnetic wave, namely equal to 3 0 \u00d7 108 m s\u20131\n(You will learn about this in Chapter 8)"}, {"Chapter": "1", "sentence_range": "2649-2652", "Text": "Note that the drift speed of electrons is much\nsmaller, about 10\u20135 times the typical thermal speed at ordinary\ntemperatures (ii) An electric field travelling along the conductor has a speed of\nan electromagnetic wave, namely equal to 3 0 \u00d7 108 m s\u20131\n(You will learn about this in Chapter 8) The drift speed is, in\ncomparison, extremely small; smaller by a factor of 10\u201311"}, {"Chapter": "1", "sentence_range": "2650-2653", "Text": "(ii) An electric field travelling along the conductor has a speed of\nan electromagnetic wave, namely equal to 3 0 \u00d7 108 m s\u20131\n(You will learn about this in Chapter 8) The drift speed is, in\ncomparison, extremely small; smaller by a factor of 10\u201311 *\nSee Eq"}, {"Chapter": "1", "sentence_range": "2651-2654", "Text": "0 \u00d7 108 m s\u20131\n(You will learn about this in Chapter 8) The drift speed is, in\ncomparison, extremely small; smaller by a factor of 10\u201311 *\nSee Eq (12"}, {"Chapter": "1", "sentence_range": "2652-2655", "Text": "The drift speed is, in\ncomparison, extremely small; smaller by a factor of 10\u201311 *\nSee Eq (12 23) of Chapter 12 from Class XI book"}, {"Chapter": "1", "sentence_range": "2653-2656", "Text": "*\nSee Eq (12 23) of Chapter 12 from Class XI book Rationalised 2023-24\nPhysics\n88\n EXAMPLE 3"}, {"Chapter": "1", "sentence_range": "2654-2657", "Text": "(12 23) of Chapter 12 from Class XI book Rationalised 2023-24\nPhysics\n88\n EXAMPLE 3 2\nExample 3"}, {"Chapter": "1", "sentence_range": "2655-2658", "Text": "23) of Chapter 12 from Class XI book Rationalised 2023-24\nPhysics\n88\n EXAMPLE 3 2\nExample 3 2\n(a) In Example 3"}, {"Chapter": "1", "sentence_range": "2656-2659", "Text": "Rationalised 2023-24\nPhysics\n88\n EXAMPLE 3 2\nExample 3 2\n(a) In Example 3 1, the electron drift speed is estimated to be only a\nfew mm s\u20131 for currents in the range of a few amperes"}, {"Chapter": "1", "sentence_range": "2657-2660", "Text": "2\nExample 3 2\n(a) In Example 3 1, the electron drift speed is estimated to be only a\nfew mm s\u20131 for currents in the range of a few amperes How then\nis current established almost the instant a circuit is closed"}, {"Chapter": "1", "sentence_range": "2658-2661", "Text": "2\n(a) In Example 3 1, the electron drift speed is estimated to be only a\nfew mm s\u20131 for currents in the range of a few amperes How then\nis current established almost the instant a circuit is closed (b) The electron drift arises due to the force experienced by electrons\nin the electric field inside the conductor"}, {"Chapter": "1", "sentence_range": "2659-2662", "Text": "1, the electron drift speed is estimated to be only a\nfew mm s\u20131 for currents in the range of a few amperes How then\nis current established almost the instant a circuit is closed (b) The electron drift arises due to the force experienced by electrons\nin the electric field inside the conductor But force should cause\nacceleration"}, {"Chapter": "1", "sentence_range": "2660-2663", "Text": "How then\nis current established almost the instant a circuit is closed (b) The electron drift arises due to the force experienced by electrons\nin the electric field inside the conductor But force should cause\nacceleration Why then do the electrons acquire a steady average\ndrift speed"}, {"Chapter": "1", "sentence_range": "2661-2664", "Text": "(b) The electron drift arises due to the force experienced by electrons\nin the electric field inside the conductor But force should cause\nacceleration Why then do the electrons acquire a steady average\ndrift speed (c) If the electron drift speed is so small, and the electron\u2019s charge is\nsmall, how can we still obtain large amounts of current in a\nconductor"}, {"Chapter": "1", "sentence_range": "2662-2665", "Text": "But force should cause\nacceleration Why then do the electrons acquire a steady average\ndrift speed (c) If the electron drift speed is so small, and the electron\u2019s charge is\nsmall, how can we still obtain large amounts of current in a\nconductor (d) When electrons drift in a metal from lower to higher potential,\ndoes it mean that all the \u2018free\u2019 electrons of the metal are moving\nin the same direction"}, {"Chapter": "1", "sentence_range": "2663-2666", "Text": "Why then do the electrons acquire a steady average\ndrift speed (c) If the electron drift speed is so small, and the electron\u2019s charge is\nsmall, how can we still obtain large amounts of current in a\nconductor (d) When electrons drift in a metal from lower to higher potential,\ndoes it mean that all the \u2018free\u2019 electrons of the metal are moving\nin the same direction (e) Are the paths of electrons straight lines between successive\ncollisions (with the positive ions of the metal) in the (i) absence of\nelectric field, (ii) presence of electric field"}, {"Chapter": "1", "sentence_range": "2664-2667", "Text": "(c) If the electron drift speed is so small, and the electron\u2019s charge is\nsmall, how can we still obtain large amounts of current in a\nconductor (d) When electrons drift in a metal from lower to higher potential,\ndoes it mean that all the \u2018free\u2019 electrons of the metal are moving\nin the same direction (e) Are the paths of electrons straight lines between successive\ncollisions (with the positive ions of the metal) in the (i) absence of\nelectric field, (ii) presence of electric field Solution\n(a) Electric field is established throughout the circuit, almost instantly\n(with the speed of light) causing at every point a local electron\ndrift"}, {"Chapter": "1", "sentence_range": "2665-2668", "Text": "(d) When electrons drift in a metal from lower to higher potential,\ndoes it mean that all the \u2018free\u2019 electrons of the metal are moving\nin the same direction (e) Are the paths of electrons straight lines between successive\ncollisions (with the positive ions of the metal) in the (i) absence of\nelectric field, (ii) presence of electric field Solution\n(a) Electric field is established throughout the circuit, almost instantly\n(with the speed of light) causing at every point a local electron\ndrift Establishment of a current does not have to wait for electrons\nfrom one end of the conductor travelling to the other end"}, {"Chapter": "1", "sentence_range": "2666-2669", "Text": "(e) Are the paths of electrons straight lines between successive\ncollisions (with the positive ions of the metal) in the (i) absence of\nelectric field, (ii) presence of electric field Solution\n(a) Electric field is established throughout the circuit, almost instantly\n(with the speed of light) causing at every point a local electron\ndrift Establishment of a current does not have to wait for electrons\nfrom one end of the conductor travelling to the other end However,\nit does take a little while for the current to reach its steady value"}, {"Chapter": "1", "sentence_range": "2667-2670", "Text": "Solution\n(a) Electric field is established throughout the circuit, almost instantly\n(with the speed of light) causing at every point a local electron\ndrift Establishment of a current does not have to wait for electrons\nfrom one end of the conductor travelling to the other end However,\nit does take a little while for the current to reach its steady value (b) Each \u2018free\u2019 electron does accelerate, increasing its drift speed until\nit collides with a positive ion of the metal"}, {"Chapter": "1", "sentence_range": "2668-2671", "Text": "Establishment of a current does not have to wait for electrons\nfrom one end of the conductor travelling to the other end However,\nit does take a little while for the current to reach its steady value (b) Each \u2018free\u2019 electron does accelerate, increasing its drift speed until\nit collides with a positive ion of the metal It loses its drift speed\nafter collision but starts to accelerate and increases its drift speed\nagain only to suffer a collision again and so on"}, {"Chapter": "1", "sentence_range": "2669-2672", "Text": "However,\nit does take a little while for the current to reach its steady value (b) Each \u2018free\u2019 electron does accelerate, increasing its drift speed until\nit collides with a positive ion of the metal It loses its drift speed\nafter collision but starts to accelerate and increases its drift speed\nagain only to suffer a collision again and so on On the average,\ntherefore, electrons acquire only a drift speed"}, {"Chapter": "1", "sentence_range": "2670-2673", "Text": "(b) Each \u2018free\u2019 electron does accelerate, increasing its drift speed until\nit collides with a positive ion of the metal It loses its drift speed\nafter collision but starts to accelerate and increases its drift speed\nagain only to suffer a collision again and so on On the average,\ntherefore, electrons acquire only a drift speed (c) Simple, because the electron number density is enormous,\n~1029 m\u20133"}, {"Chapter": "1", "sentence_range": "2671-2674", "Text": "It loses its drift speed\nafter collision but starts to accelerate and increases its drift speed\nagain only to suffer a collision again and so on On the average,\ntherefore, electrons acquire only a drift speed (c) Simple, because the electron number density is enormous,\n~1029 m\u20133 (d) By no means"}, {"Chapter": "1", "sentence_range": "2672-2675", "Text": "On the average,\ntherefore, electrons acquire only a drift speed (c) Simple, because the electron number density is enormous,\n~1029 m\u20133 (d) By no means The drift velocity is superposed over the large\nrandom velocities of electrons"}, {"Chapter": "1", "sentence_range": "2673-2676", "Text": "(c) Simple, because the electron number density is enormous,\n~1029 m\u20133 (d) By no means The drift velocity is superposed over the large\nrandom velocities of electrons (e) In the absence of electric field, the paths are straight lines; in the\npresence of electric field, the paths are, in general, curved"}, {"Chapter": "1", "sentence_range": "2674-2677", "Text": "(d) By no means The drift velocity is superposed over the large\nrandom velocities of electrons (e) In the absence of electric field, the paths are straight lines; in the\npresence of electric field, the paths are, in general, curved 3"}, {"Chapter": "1", "sentence_range": "2675-2678", "Text": "The drift velocity is superposed over the large\nrandom velocities of electrons (e) In the absence of electric field, the paths are straight lines; in the\npresence of electric field, the paths are, in general, curved 3 5"}, {"Chapter": "1", "sentence_range": "2676-2679", "Text": "(e) In the absence of electric field, the paths are straight lines; in the\npresence of electric field, the paths are, in general, curved 3 5 1  Mobility\nAs we have seen, conductivity arises from mobile charge carriers"}, {"Chapter": "1", "sentence_range": "2677-2680", "Text": "3 5 1  Mobility\nAs we have seen, conductivity arises from mobile charge carriers In\nmetals, these mobile charge carriers are electrons; in an ionised gas, they\nare electrons and positive charged ions; in an electrolyte, these can be\nboth positive and negative ions"}, {"Chapter": "1", "sentence_range": "2678-2681", "Text": "5 1  Mobility\nAs we have seen, conductivity arises from mobile charge carriers In\nmetals, these mobile charge carriers are electrons; in an ionised gas, they\nare electrons and positive charged ions; in an electrolyte, these can be\nboth positive and negative ions An important quantity is the mobility m defined as the magnitude of\nthe drift velocity per unit electric field:\n|\nEd|\n\u00b5 = v\n(3"}, {"Chapter": "1", "sentence_range": "2679-2682", "Text": "1  Mobility\nAs we have seen, conductivity arises from mobile charge carriers In\nmetals, these mobile charge carriers are electrons; in an ionised gas, they\nare electrons and positive charged ions; in an electrolyte, these can be\nboth positive and negative ions An important quantity is the mobility m defined as the magnitude of\nthe drift velocity per unit electric field:\n|\nEd|\n\u00b5 = v\n(3 24)\nThe SI unit of mobility is m2/Vs and is 104 of the mobility in practical\nunits (cm2/Vs)"}, {"Chapter": "1", "sentence_range": "2680-2683", "Text": "In\nmetals, these mobile charge carriers are electrons; in an ionised gas, they\nare electrons and positive charged ions; in an electrolyte, these can be\nboth positive and negative ions An important quantity is the mobility m defined as the magnitude of\nthe drift velocity per unit electric field:\n|\nEd|\n\u00b5 = v\n(3 24)\nThe SI unit of mobility is m2/Vs and is 104 of the mobility in practical\nunits (cm2/Vs) Mobility is positive"}, {"Chapter": "1", "sentence_range": "2681-2684", "Text": "An important quantity is the mobility m defined as the magnitude of\nthe drift velocity per unit electric field:\n|\nEd|\n\u00b5 = v\n(3 24)\nThe SI unit of mobility is m2/Vs and is 104 of the mobility in practical\nunits (cm2/Vs) Mobility is positive From Eq"}, {"Chapter": "1", "sentence_range": "2682-2685", "Text": "24)\nThe SI unit of mobility is m2/Vs and is 104 of the mobility in practical\nunits (cm2/Vs) Mobility is positive From Eq (3"}, {"Chapter": "1", "sentence_range": "2683-2686", "Text": "Mobility is positive From Eq (3 17), we have\nvd = \n\u03c4\ne E\nm\nRationalised 2023-24\nCurrent\nElectricity\n89\nHence,\n\u03c4\n\u00b5 =\nvd=\ne\nE\nm\n(3"}, {"Chapter": "1", "sentence_range": "2684-2687", "Text": "From Eq (3 17), we have\nvd = \n\u03c4\ne E\nm\nRationalised 2023-24\nCurrent\nElectricity\n89\nHence,\n\u03c4\n\u00b5 =\nvd=\ne\nE\nm\n(3 25)\nwhere t is the average collision time for electrons"}, {"Chapter": "1", "sentence_range": "2685-2688", "Text": "(3 17), we have\nvd = \n\u03c4\ne E\nm\nRationalised 2023-24\nCurrent\nElectricity\n89\nHence,\n\u03c4\n\u00b5 =\nvd=\ne\nE\nm\n(3 25)\nwhere t is the average collision time for electrons 3"}, {"Chapter": "1", "sentence_range": "2686-2689", "Text": "17), we have\nvd = \n\u03c4\ne E\nm\nRationalised 2023-24\nCurrent\nElectricity\n89\nHence,\n\u03c4\n\u00b5 =\nvd=\ne\nE\nm\n(3 25)\nwhere t is the average collision time for electrons 3 6  LIMITATIONS OF OHM\u2019S LAW\nAlthough Ohm\u2019s law has been found valid over a large class\nof materials, there do exist materials and devices used in\nelectric circuits where the proportionality of V and I does not\nhold"}, {"Chapter": "1", "sentence_range": "2687-2690", "Text": "25)\nwhere t is the average collision time for electrons 3 6  LIMITATIONS OF OHM\u2019S LAW\nAlthough Ohm\u2019s law has been found valid over a large class\nof materials, there do exist materials and devices used in\nelectric circuits where the proportionality of V and I does not\nhold The deviations broadly are one or more of the following\ntypes:\n(a) V ceases to be proportional to I (Fig"}, {"Chapter": "1", "sentence_range": "2688-2691", "Text": "3 6  LIMITATIONS OF OHM\u2019S LAW\nAlthough Ohm\u2019s law has been found valid over a large class\nof materials, there do exist materials and devices used in\nelectric circuits where the proportionality of V and I does not\nhold The deviations broadly are one or more of the following\ntypes:\n(a) V ceases to be proportional to I (Fig 3"}, {"Chapter": "1", "sentence_range": "2689-2692", "Text": "6  LIMITATIONS OF OHM\u2019S LAW\nAlthough Ohm\u2019s law has been found valid over a large class\nof materials, there do exist materials and devices used in\nelectric circuits where the proportionality of V and I does not\nhold The deviations broadly are one or more of the following\ntypes:\n(a) V ceases to be proportional to I (Fig 3 5)"}, {"Chapter": "1", "sentence_range": "2690-2693", "Text": "The deviations broadly are one or more of the following\ntypes:\n(a) V ceases to be proportional to I (Fig 3 5) (b) The relation between V and I depends on the sign of V"}, {"Chapter": "1", "sentence_range": "2691-2694", "Text": "3 5) (b) The relation between V and I depends on the sign of V In\nother words, if I is the current for a certain V, then reversing\nthe direction of V keeping its magnitude fixed, does not\nproduce a current of the same magnitude as I in the opposite direction\n(Fig"}, {"Chapter": "1", "sentence_range": "2692-2695", "Text": "5) (b) The relation between V and I depends on the sign of V In\nother words, if I is the current for a certain V, then reversing\nthe direction of V keeping its magnitude fixed, does not\nproduce a current of the same magnitude as I in the opposite direction\n(Fig 3"}, {"Chapter": "1", "sentence_range": "2693-2696", "Text": "(b) The relation between V and I depends on the sign of V In\nother words, if I is the current for a certain V, then reversing\nthe direction of V keeping its magnitude fixed, does not\nproduce a current of the same magnitude as I in the opposite direction\n(Fig 3 6)"}, {"Chapter": "1", "sentence_range": "2694-2697", "Text": "In\nother words, if I is the current for a certain V, then reversing\nthe direction of V keeping its magnitude fixed, does not\nproduce a current of the same magnitude as I in the opposite direction\n(Fig 3 6) This happens, for example, in a diode which we will study\nin Chapter 14"}, {"Chapter": "1", "sentence_range": "2695-2698", "Text": "3 6) This happens, for example, in a diode which we will study\nin Chapter 14 (c) The relation between V and I is not unique, i"}, {"Chapter": "1", "sentence_range": "2696-2699", "Text": "6) This happens, for example, in a diode which we will study\nin Chapter 14 (c) The relation between V and I is not unique, i e"}, {"Chapter": "1", "sentence_range": "2697-2700", "Text": "This happens, for example, in a diode which we will study\nin Chapter 14 (c) The relation between V and I is not unique, i e , there is more than\none value of V for the same current I (Fig"}, {"Chapter": "1", "sentence_range": "2698-2701", "Text": "(c) The relation between V and I is not unique, i e , there is more than\none value of V for the same current I (Fig 3"}, {"Chapter": "1", "sentence_range": "2699-2702", "Text": "e , there is more than\none value of V for the same current I (Fig 3 7)"}, {"Chapter": "1", "sentence_range": "2700-2703", "Text": ", there is more than\none value of V for the same current I (Fig 3 7) A material exhibiting\nsuch behaviour is GaAs"}, {"Chapter": "1", "sentence_range": "2701-2704", "Text": "3 7) A material exhibiting\nsuch behaviour is GaAs Materials and devices not obeying Ohm\u2019s law in the form of Eq"}, {"Chapter": "1", "sentence_range": "2702-2705", "Text": "7) A material exhibiting\nsuch behaviour is GaAs Materials and devices not obeying Ohm\u2019s law in the form of Eq (3"}, {"Chapter": "1", "sentence_range": "2703-2706", "Text": "A material exhibiting\nsuch behaviour is GaAs Materials and devices not obeying Ohm\u2019s law in the form of Eq (3 3)\nare actually widely used in electronic circuits"}, {"Chapter": "1", "sentence_range": "2704-2707", "Text": "Materials and devices not obeying Ohm\u2019s law in the form of Eq (3 3)\nare actually widely used in electronic circuits In this and a few\nsubsequent chapters, however, we will study the electrical currents in\nmaterials that obey Ohm\u2019s law"}, {"Chapter": "1", "sentence_range": "2705-2708", "Text": "(3 3)\nare actually widely used in electronic circuits In this and a few\nsubsequent chapters, however, we will study the electrical currents in\nmaterials that obey Ohm\u2019s law 3"}, {"Chapter": "1", "sentence_range": "2706-2709", "Text": "3)\nare actually widely used in electronic circuits In this and a few\nsubsequent chapters, however, we will study the electrical currents in\nmaterials that obey Ohm\u2019s law 3 7  RESISTIVITY OF VARIOUS MATERIALS\nThe materials are classified as conductors, semiconductors and insulators\ndepending on their resistivities, in an increasing order of their values"}, {"Chapter": "1", "sentence_range": "2707-2710", "Text": "In this and a few\nsubsequent chapters, however, we will study the electrical currents in\nmaterials that obey Ohm\u2019s law 3 7  RESISTIVITY OF VARIOUS MATERIALS\nThe materials are classified as conductors, semiconductors and insulators\ndepending on their resistivities, in an increasing order of their values FIGURE 3"}, {"Chapter": "1", "sentence_range": "2708-2711", "Text": "3 7  RESISTIVITY OF VARIOUS MATERIALS\nThe materials are classified as conductors, semiconductors and insulators\ndepending on their resistivities, in an increasing order of their values FIGURE 3 5 The dashed line\nrepresents the linear Ohm\u2019s\nlaw"}, {"Chapter": "1", "sentence_range": "2709-2712", "Text": "7  RESISTIVITY OF VARIOUS MATERIALS\nThe materials are classified as conductors, semiconductors and insulators\ndepending on their resistivities, in an increasing order of their values FIGURE 3 5 The dashed line\nrepresents the linear Ohm\u2019s\nlaw The solid line is the voltage\nV versus current I for a good\nconductor"}, {"Chapter": "1", "sentence_range": "2710-2713", "Text": "FIGURE 3 5 The dashed line\nrepresents the linear Ohm\u2019s\nlaw The solid line is the voltage\nV versus current I for a good\nconductor FIGURE 3"}, {"Chapter": "1", "sentence_range": "2711-2714", "Text": "5 The dashed line\nrepresents the linear Ohm\u2019s\nlaw The solid line is the voltage\nV versus current I for a good\nconductor FIGURE 3 6 Characteristic curve\nof a diode"}, {"Chapter": "1", "sentence_range": "2712-2715", "Text": "The solid line is the voltage\nV versus current I for a good\nconductor FIGURE 3 6 Characteristic curve\nof a diode Note the different\nscales for negative and positive\nvalues of the voltage and current"}, {"Chapter": "1", "sentence_range": "2713-2716", "Text": "FIGURE 3 6 Characteristic curve\nof a diode Note the different\nscales for negative and positive\nvalues of the voltage and current FIGURE 3"}, {"Chapter": "1", "sentence_range": "2714-2717", "Text": "6 Characteristic curve\nof a diode Note the different\nscales for negative and positive\nvalues of the voltage and current FIGURE 3 7 Variation of current\nversus voltage for GaAs"}, {"Chapter": "1", "sentence_range": "2715-2718", "Text": "Note the different\nscales for negative and positive\nvalues of the voltage and current FIGURE 3 7 Variation of current\nversus voltage for GaAs Rationalised 2023-24\nPhysics\n90\nMetals have low resistivities in the range of 10\u20138 Wm to 10\u20136 Wm"}, {"Chapter": "1", "sentence_range": "2716-2719", "Text": "FIGURE 3 7 Variation of current\nversus voltage for GaAs Rationalised 2023-24\nPhysics\n90\nMetals have low resistivities in the range of 10\u20138 Wm to 10\u20136 Wm At the\nother end are insulators like ceramic, rubber and plastics having\nresistivities 1018 times greater than metals or more"}, {"Chapter": "1", "sentence_range": "2717-2720", "Text": "7 Variation of current\nversus voltage for GaAs Rationalised 2023-24\nPhysics\n90\nMetals have low resistivities in the range of 10\u20138 Wm to 10\u20136 Wm At the\nother end are insulators like ceramic, rubber and plastics having\nresistivities 1018 times greater than metals or more In between the two\nare the semiconductors"}, {"Chapter": "1", "sentence_range": "2718-2721", "Text": "Rationalised 2023-24\nPhysics\n90\nMetals have low resistivities in the range of 10\u20138 Wm to 10\u20136 Wm At the\nother end are insulators like ceramic, rubber and plastics having\nresistivities 1018 times greater than metals or more In between the two\nare the semiconductors These, however, have resistivities\ncharacteristically decreasing with a rise in temperature"}, {"Chapter": "1", "sentence_range": "2719-2722", "Text": "At the\nother end are insulators like ceramic, rubber and plastics having\nresistivities 1018 times greater than metals or more In between the two\nare the semiconductors These, however, have resistivities\ncharacteristically decreasing with a rise in temperature The resistivities\nof semiconductors can be decreased by adding small amount of suitable\nimpurities"}, {"Chapter": "1", "sentence_range": "2720-2723", "Text": "In between the two\nare the semiconductors These, however, have resistivities\ncharacteristically decreasing with a rise in temperature The resistivities\nof semiconductors can be decreased by adding small amount of suitable\nimpurities This last feature is exploited in use of semiconductors for\nelectronic devices"}, {"Chapter": "1", "sentence_range": "2721-2724", "Text": "These, however, have resistivities\ncharacteristically decreasing with a rise in temperature The resistivities\nof semiconductors can be decreased by adding small amount of suitable\nimpurities This last feature is exploited in use of semiconductors for\nelectronic devices 3"}, {"Chapter": "1", "sentence_range": "2722-2725", "Text": "The resistivities\nof semiconductors can be decreased by adding small amount of suitable\nimpurities This last feature is exploited in use of semiconductors for\nelectronic devices 3 8\nTEMPERATURE DEPENDENCE OF RESISTIVITY\nThe resistivity of a material is found to be dependent on the temperature"}, {"Chapter": "1", "sentence_range": "2723-2726", "Text": "This last feature is exploited in use of semiconductors for\nelectronic devices 3 8\nTEMPERATURE DEPENDENCE OF RESISTIVITY\nThe resistivity of a material is found to be dependent on the temperature Different materials do not exhibit the same dependence on temperatures"}, {"Chapter": "1", "sentence_range": "2724-2727", "Text": "3 8\nTEMPERATURE DEPENDENCE OF RESISTIVITY\nThe resistivity of a material is found to be dependent on the temperature Different materials do not exhibit the same dependence on temperatures Over a limited range of temperatures, that is not too large, the resistivity\nof a metallic conductor is approximately given by,\nrT = r0 [1 + a (T\u2013T0)]\n(3"}, {"Chapter": "1", "sentence_range": "2725-2728", "Text": "8\nTEMPERATURE DEPENDENCE OF RESISTIVITY\nThe resistivity of a material is found to be dependent on the temperature Different materials do not exhibit the same dependence on temperatures Over a limited range of temperatures, that is not too large, the resistivity\nof a metallic conductor is approximately given by,\nrT = r0 [1 + a (T\u2013T0)]\n(3 26)\nwhere rT is the resistivity at a temperature T and r0 is the same at a\nreference temperature T0"}, {"Chapter": "1", "sentence_range": "2726-2729", "Text": "Different materials do not exhibit the same dependence on temperatures Over a limited range of temperatures, that is not too large, the resistivity\nof a metallic conductor is approximately given by,\nrT = r0 [1 + a (T\u2013T0)]\n(3 26)\nwhere rT is the resistivity at a temperature T and r0 is the same at a\nreference temperature T0 a is called the temperature co-efficient of\nresistivity, and from Eq"}, {"Chapter": "1", "sentence_range": "2727-2730", "Text": "Over a limited range of temperatures, that is not too large, the resistivity\nof a metallic conductor is approximately given by,\nrT = r0 [1 + a (T\u2013T0)]\n(3 26)\nwhere rT is the resistivity at a temperature T and r0 is the same at a\nreference temperature T0 a is called the temperature co-efficient of\nresistivity, and from Eq (3"}, {"Chapter": "1", "sentence_range": "2728-2731", "Text": "26)\nwhere rT is the resistivity at a temperature T and r0 is the same at a\nreference temperature T0 a is called the temperature co-efficient of\nresistivity, and from Eq (3 26), the dimension of a is (Temperature)\u20131"}, {"Chapter": "1", "sentence_range": "2729-2732", "Text": "a is called the temperature co-efficient of\nresistivity, and from Eq (3 26), the dimension of a is (Temperature)\u20131 For  metals, a is positive"}, {"Chapter": "1", "sentence_range": "2730-2733", "Text": "(3 26), the dimension of a is (Temperature)\u20131 For  metals, a is positive The relation of Eq"}, {"Chapter": "1", "sentence_range": "2731-2734", "Text": "26), the dimension of a is (Temperature)\u20131 For  metals, a is positive The relation of Eq (3"}, {"Chapter": "1", "sentence_range": "2732-2735", "Text": "For  metals, a is positive The relation of Eq (3 26) implies that a graph of rT plotted against T\nwould be a straight line"}, {"Chapter": "1", "sentence_range": "2733-2736", "Text": "The relation of Eq (3 26) implies that a graph of rT plotted against T\nwould be a straight line At temperatures much lower than 0\u00b0C, the graph,\nhowever, deviates considerably from a straight line (Fig"}, {"Chapter": "1", "sentence_range": "2734-2737", "Text": "(3 26) implies that a graph of rT plotted against T\nwould be a straight line At temperatures much lower than 0\u00b0C, the graph,\nhowever, deviates considerably from a straight line (Fig 3"}, {"Chapter": "1", "sentence_range": "2735-2738", "Text": "26) implies that a graph of rT plotted against T\nwould be a straight line At temperatures much lower than 0\u00b0C, the graph,\nhowever, deviates considerably from a straight line (Fig 3 8)"}, {"Chapter": "1", "sentence_range": "2736-2739", "Text": "At temperatures much lower than 0\u00b0C, the graph,\nhowever, deviates considerably from a straight line (Fig 3 8) Equation (3"}, {"Chapter": "1", "sentence_range": "2737-2740", "Text": "3 8) Equation (3 26) thus, can be used approximately over a limited range\nof T around any reference temperature T0, where the graph can be\napproximated as a straight line"}, {"Chapter": "1", "sentence_range": "2738-2741", "Text": "8) Equation (3 26) thus, can be used approximately over a limited range\nof T around any reference temperature T0, where the graph can be\napproximated as a straight line FIGURE 3"}, {"Chapter": "1", "sentence_range": "2739-2742", "Text": "Equation (3 26) thus, can be used approximately over a limited range\nof T around any reference temperature T0, where the graph can be\napproximated as a straight line FIGURE 3 8\nResistivity rT of\ncopper as a function\nof temperature T"}, {"Chapter": "1", "sentence_range": "2740-2743", "Text": "26) thus, can be used approximately over a limited range\nof T around any reference temperature T0, where the graph can be\napproximated as a straight line FIGURE 3 8\nResistivity rT of\ncopper as a function\nof temperature T FIGURE 3"}, {"Chapter": "1", "sentence_range": "2741-2744", "Text": "FIGURE 3 8\nResistivity rT of\ncopper as a function\nof temperature T FIGURE 3 9 Resistivity\nrT of nichrome as a\nfunction of absolute\ntemperature T"}, {"Chapter": "1", "sentence_range": "2742-2745", "Text": "8\nResistivity rT of\ncopper as a function\nof temperature T FIGURE 3 9 Resistivity\nrT of nichrome as a\nfunction of absolute\ntemperature T FIGURE 3"}, {"Chapter": "1", "sentence_range": "2743-2746", "Text": "FIGURE 3 9 Resistivity\nrT of nichrome as a\nfunction of absolute\ntemperature T FIGURE 3 10\nTemperature dependence\nof resistivity for a typical\nsemiconductor"}, {"Chapter": "1", "sentence_range": "2744-2747", "Text": "9 Resistivity\nrT of nichrome as a\nfunction of absolute\ntemperature T FIGURE 3 10\nTemperature dependence\nof resistivity for a typical\nsemiconductor \uf072\nSome materials like Nichrome (which is an alloy of nickel, iron and\nchromium) exhibit a very weak dependence of resistivity with temperature\n(Fig"}, {"Chapter": "1", "sentence_range": "2745-2748", "Text": "FIGURE 3 10\nTemperature dependence\nof resistivity for a typical\nsemiconductor \uf072\nSome materials like Nichrome (which is an alloy of nickel, iron and\nchromium) exhibit a very weak dependence of resistivity with temperature\n(Fig 3"}, {"Chapter": "1", "sentence_range": "2746-2749", "Text": "10\nTemperature dependence\nof resistivity for a typical\nsemiconductor \uf072\nSome materials like Nichrome (which is an alloy of nickel, iron and\nchromium) exhibit a very weak dependence of resistivity with temperature\n(Fig 3 9)"}, {"Chapter": "1", "sentence_range": "2747-2750", "Text": "\uf072\nSome materials like Nichrome (which is an alloy of nickel, iron and\nchromium) exhibit a very weak dependence of resistivity with temperature\n(Fig 3 9) Manganin and constantan have similar properties"}, {"Chapter": "1", "sentence_range": "2748-2751", "Text": "3 9) Manganin and constantan have similar properties These\nmaterials are thus widely used in wire bound standard resistors since\ntheir resistance values would change very little with temperatures"}, {"Chapter": "1", "sentence_range": "2749-2752", "Text": "9) Manganin and constantan have similar properties These\nmaterials are thus widely used in wire bound standard resistors since\ntheir resistance values would change very little with temperatures Rationalised 2023-24\nCurrent\nElectricity\n91\n EXAMPLE 3"}, {"Chapter": "1", "sentence_range": "2750-2753", "Text": "Manganin and constantan have similar properties These\nmaterials are thus widely used in wire bound standard resistors since\ntheir resistance values would change very little with temperatures Rationalised 2023-24\nCurrent\nElectricity\n91\n EXAMPLE 3 3\nUnlike metals, the resistivities of semiconductors decrease with\nincreasing temperatures"}, {"Chapter": "1", "sentence_range": "2751-2754", "Text": "These\nmaterials are thus widely used in wire bound standard resistors since\ntheir resistance values would change very little with temperatures Rationalised 2023-24\nCurrent\nElectricity\n91\n EXAMPLE 3 3\nUnlike metals, the resistivities of semiconductors decrease with\nincreasing temperatures A typical dependence is shown in Fig"}, {"Chapter": "1", "sentence_range": "2752-2755", "Text": "Rationalised 2023-24\nCurrent\nElectricity\n91\n EXAMPLE 3 3\nUnlike metals, the resistivities of semiconductors decrease with\nincreasing temperatures A typical dependence is shown in Fig 3"}, {"Chapter": "1", "sentence_range": "2753-2756", "Text": "3\nUnlike metals, the resistivities of semiconductors decrease with\nincreasing temperatures A typical dependence is shown in Fig 3 10"}, {"Chapter": "1", "sentence_range": "2754-2757", "Text": "A typical dependence is shown in Fig 3 10 We can qualitatively understand the temperature dependence of\nresistivity, in the light of our derivation of Eq"}, {"Chapter": "1", "sentence_range": "2755-2758", "Text": "3 10 We can qualitatively understand the temperature dependence of\nresistivity, in the light of our derivation of Eq (3"}, {"Chapter": "1", "sentence_range": "2756-2759", "Text": "10 We can qualitatively understand the temperature dependence of\nresistivity, in the light of our derivation of Eq (3 23)"}, {"Chapter": "1", "sentence_range": "2757-2760", "Text": "We can qualitatively understand the temperature dependence of\nresistivity, in the light of our derivation of Eq (3 23) From this equation,\nresistivity of a material is given by\n2\n1\nm\nn e\n\u03c1\n\u03c3\n\u03c4\n=\n=\n(3"}, {"Chapter": "1", "sentence_range": "2758-2761", "Text": "(3 23) From this equation,\nresistivity of a material is given by\n2\n1\nm\nn e\n\u03c1\n\u03c3\n\u03c4\n=\n=\n(3 27)\nr thus depends inversely both on the number n of free electrons per unit\nvolume and on the average time t between collisions"}, {"Chapter": "1", "sentence_range": "2759-2762", "Text": "23) From this equation,\nresistivity of a material is given by\n2\n1\nm\nn e\n\u03c1\n\u03c3\n\u03c4\n=\n=\n(3 27)\nr thus depends inversely both on the number n of free electrons per unit\nvolume and on the average time t between collisions As we increase\ntemperature, average speed of the electrons, which act as the carriers of\ncurrent, increases resulting in more frequent collisions"}, {"Chapter": "1", "sentence_range": "2760-2763", "Text": "From this equation,\nresistivity of a material is given by\n2\n1\nm\nn e\n\u03c1\n\u03c3\n\u03c4\n=\n=\n(3 27)\nr thus depends inversely both on the number n of free electrons per unit\nvolume and on the average time t between collisions As we increase\ntemperature, average speed of the electrons, which act as the carriers of\ncurrent, increases resulting in more frequent collisions The average time\nof collisions t, thus decreases with temperature"}, {"Chapter": "1", "sentence_range": "2761-2764", "Text": "27)\nr thus depends inversely both on the number n of free electrons per unit\nvolume and on the average time t between collisions As we increase\ntemperature, average speed of the electrons, which act as the carriers of\ncurrent, increases resulting in more frequent collisions The average time\nof collisions t, thus decreases with temperature In a metal, n is not dependent on temperature to any appreciable\nextent and thus the decrease in the value of t with rise in temperature\ncauses r to increase as we have observed"}, {"Chapter": "1", "sentence_range": "2762-2765", "Text": "As we increase\ntemperature, average speed of the electrons, which act as the carriers of\ncurrent, increases resulting in more frequent collisions The average time\nof collisions t, thus decreases with temperature In a metal, n is not dependent on temperature to any appreciable\nextent and thus the decrease in the value of t with rise in temperature\ncauses r to increase as we have observed For insulators and semiconductors, however, n increases with\ntemperature"}, {"Chapter": "1", "sentence_range": "2763-2766", "Text": "The average time\nof collisions t, thus decreases with temperature In a metal, n is not dependent on temperature to any appreciable\nextent and thus the decrease in the value of t with rise in temperature\ncauses r to increase as we have observed For insulators and semiconductors, however, n increases with\ntemperature This increase more than compensates any decrease in t in\nEq"}, {"Chapter": "1", "sentence_range": "2764-2767", "Text": "In a metal, n is not dependent on temperature to any appreciable\nextent and thus the decrease in the value of t with rise in temperature\ncauses r to increase as we have observed For insulators and semiconductors, however, n increases with\ntemperature This increase more than compensates any decrease in t in\nEq (3"}, {"Chapter": "1", "sentence_range": "2765-2768", "Text": "For insulators and semiconductors, however, n increases with\ntemperature This increase more than compensates any decrease in t in\nEq (3 23) so that for such materials, r decreases with temperature"}, {"Chapter": "1", "sentence_range": "2766-2769", "Text": "This increase more than compensates any decrease in t in\nEq (3 23) so that for such materials, r decreases with temperature Example 3"}, {"Chapter": "1", "sentence_range": "2767-2770", "Text": "(3 23) so that for such materials, r decreases with temperature Example 3 3 An electric toaster uses nichrome for its heating\nelement"}, {"Chapter": "1", "sentence_range": "2768-2771", "Text": "23) so that for such materials, r decreases with temperature Example 3 3 An electric toaster uses nichrome for its heating\nelement When a negligibly small current passes through it, its\nresistance at room temperature (27"}, {"Chapter": "1", "sentence_range": "2769-2772", "Text": "Example 3 3 An electric toaster uses nichrome for its heating\nelement When a negligibly small current passes through it, its\nresistance at room temperature (27 0 \u00b0C) is found to be 75"}, {"Chapter": "1", "sentence_range": "2770-2773", "Text": "3 An electric toaster uses nichrome for its heating\nelement When a negligibly small current passes through it, its\nresistance at room temperature (27 0 \u00b0C) is found to be 75 3 W"}, {"Chapter": "1", "sentence_range": "2771-2774", "Text": "When a negligibly small current passes through it, its\nresistance at room temperature (27 0 \u00b0C) is found to be 75 3 W When\nthe toaster is connected to a 230 V supply, the current settles, after\na few seconds, to a steady value of 2"}, {"Chapter": "1", "sentence_range": "2772-2775", "Text": "0 \u00b0C) is found to be 75 3 W When\nthe toaster is connected to a 230 V supply, the current settles, after\na few seconds, to a steady value of 2 68 A"}, {"Chapter": "1", "sentence_range": "2773-2776", "Text": "3 W When\nthe toaster is connected to a 230 V supply, the current settles, after\na few seconds, to a steady value of 2 68 A What is the steady\ntemperature of the nichrome element"}, {"Chapter": "1", "sentence_range": "2774-2777", "Text": "When\nthe toaster is connected to a 230 V supply, the current settles, after\na few seconds, to a steady value of 2 68 A What is the steady\ntemperature of the nichrome element The temperature coefficient\nof resistance of nichrome averaged over the temperature range\ninvolved, is 1"}, {"Chapter": "1", "sentence_range": "2775-2778", "Text": "68 A What is the steady\ntemperature of the nichrome element The temperature coefficient\nof resistance of nichrome averaged over the temperature range\ninvolved, is 1 70 \u00d7 10\u20134 \u00b0C\u20131"}, {"Chapter": "1", "sentence_range": "2776-2779", "Text": "What is the steady\ntemperature of the nichrome element The temperature coefficient\nof resistance of nichrome averaged over the temperature range\ninvolved, is 1 70 \u00d7 10\u20134 \u00b0C\u20131 Solution  When the current through the element is very small, heating\neffects can be ignored and the temperature T1 of the element is the\nsame as room temperature"}, {"Chapter": "1", "sentence_range": "2777-2780", "Text": "The temperature coefficient\nof resistance of nichrome averaged over the temperature range\ninvolved, is 1 70 \u00d7 10\u20134 \u00b0C\u20131 Solution  When the current through the element is very small, heating\neffects can be ignored and the temperature T1 of the element is the\nsame as room temperature When the toaster is connected to the\nsupply, its initial current will be slightly higher than its steady value\nof 2"}, {"Chapter": "1", "sentence_range": "2778-2781", "Text": "70 \u00d7 10\u20134 \u00b0C\u20131 Solution  When the current through the element is very small, heating\neffects can be ignored and the temperature T1 of the element is the\nsame as room temperature When the toaster is connected to the\nsupply, its initial current will be slightly higher than its steady value\nof 2 68 A"}, {"Chapter": "1", "sentence_range": "2779-2782", "Text": "Solution  When the current through the element is very small, heating\neffects can be ignored and the temperature T1 of the element is the\nsame as room temperature When the toaster is connected to the\nsupply, its initial current will be slightly higher than its steady value\nof 2 68 A But due to heating effect of the current, the temperature\nwill rise"}, {"Chapter": "1", "sentence_range": "2780-2783", "Text": "When the toaster is connected to the\nsupply, its initial current will be slightly higher than its steady value\nof 2 68 A But due to heating effect of the current, the temperature\nwill rise This will cause an increase in resistance and a slight\ndecrease in current"}, {"Chapter": "1", "sentence_range": "2781-2784", "Text": "68 A But due to heating effect of the current, the temperature\nwill rise This will cause an increase in resistance and a slight\ndecrease in current In a few seconds, a steady state will be reached\nwhen temperature will rise no further, and both the resistance of the\nelement and the current drawn will achieve steady values"}, {"Chapter": "1", "sentence_range": "2782-2785", "Text": "But due to heating effect of the current, the temperature\nwill rise This will cause an increase in resistance and a slight\ndecrease in current In a few seconds, a steady state will be reached\nwhen temperature will rise no further, and both the resistance of the\nelement and the current drawn will achieve steady values The\nresistance R2 at the steady temperature T2 is\nR2 \n230 V\n85"}, {"Chapter": "1", "sentence_range": "2783-2786", "Text": "This will cause an increase in resistance and a slight\ndecrease in current In a few seconds, a steady state will be reached\nwhen temperature will rise no further, and both the resistance of the\nelement and the current drawn will achieve steady values The\nresistance R2 at the steady temperature T2 is\nR2 \n230 V\n85 8\n=2"}, {"Chapter": "1", "sentence_range": "2784-2787", "Text": "In a few seconds, a steady state will be reached\nwhen temperature will rise no further, and both the resistance of the\nelement and the current drawn will achieve steady values The\nresistance R2 at the steady temperature T2 is\nR2 \n230 V\n85 8\n=2 68 A\n=\n\u2126\nUsing the relation\nR2 = R1 [1 + a (T2 \u2013 T1)]\nwith a = 1"}, {"Chapter": "1", "sentence_range": "2785-2788", "Text": "The\nresistance R2 at the steady temperature T2 is\nR2 \n230 V\n85 8\n=2 68 A\n=\n\u2126\nUsing the relation\nR2 = R1 [1 + a (T2 \u2013 T1)]\nwith a = 1 70 \u00d7 10\u20134 \u00b0C\u20131, we get\nT2 \u2013 T1\n\u20134\n(85"}, {"Chapter": "1", "sentence_range": "2786-2789", "Text": "8\n=2 68 A\n=\n\u2126\nUsing the relation\nR2 = R1 [1 + a (T2 \u2013 T1)]\nwith a = 1 70 \u00d7 10\u20134 \u00b0C\u20131, we get\nT2 \u2013 T1\n\u20134\n(85 8 \u2013 75"}, {"Chapter": "1", "sentence_range": "2787-2790", "Text": "68 A\n=\n\u2126\nUsing the relation\nR2 = R1 [1 + a (T2 \u2013 T1)]\nwith a = 1 70 \u00d7 10\u20134 \u00b0C\u20131, we get\nT2 \u2013 T1\n\u20134\n(85 8 \u2013 75 3)\n=(75"}, {"Chapter": "1", "sentence_range": "2788-2791", "Text": "70 \u00d7 10\u20134 \u00b0C\u20131, we get\nT2 \u2013 T1\n\u20134\n(85 8 \u2013 75 3)\n=(75 3) 1"}, {"Chapter": "1", "sentence_range": "2789-2792", "Text": "8 \u2013 75 3)\n=(75 3) 1 70 10\n\u00d7\n\u00d7\n  = 820 \u00b0C\nthat is,  T2 = (820 + 27"}, {"Chapter": "1", "sentence_range": "2790-2793", "Text": "3)\n=(75 3) 1 70 10\n\u00d7\n\u00d7\n  = 820 \u00b0C\nthat is,  T2 = (820 + 27 0) \u00b0C = 847 \u00b0C\nThus, the steady temperature of the heating element (when heating\neffect due to the current equals heat loss to the surroundings) is\n847 \u00b0C"}, {"Chapter": "1", "sentence_range": "2791-2794", "Text": "3) 1 70 10\n\u00d7\n\u00d7\n  = 820 \u00b0C\nthat is,  T2 = (820 + 27 0) \u00b0C = 847 \u00b0C\nThus, the steady temperature of the heating element (when heating\neffect due to the current equals heat loss to the surroundings) is\n847 \u00b0C Rationalised 2023-24\nPhysics\n92\n EXAMPLE 3"}, {"Chapter": "1", "sentence_range": "2792-2795", "Text": "70 10\n\u00d7\n\u00d7\n  = 820 \u00b0C\nthat is,  T2 = (820 + 27 0) \u00b0C = 847 \u00b0C\nThus, the steady temperature of the heating element (when heating\neffect due to the current equals heat loss to the surroundings) is\n847 \u00b0C Rationalised 2023-24\nPhysics\n92\n EXAMPLE 3 4\nExample 3"}, {"Chapter": "1", "sentence_range": "2793-2796", "Text": "0) \u00b0C = 847 \u00b0C\nThus, the steady temperature of the heating element (when heating\neffect due to the current equals heat loss to the surroundings) is\n847 \u00b0C Rationalised 2023-24\nPhysics\n92\n EXAMPLE 3 4\nExample 3 4 The resistance of the platinum wire of a platinum\nresistance thermometer at the ice point is 5 W and at steam point is\n5"}, {"Chapter": "1", "sentence_range": "2794-2797", "Text": "Rationalised 2023-24\nPhysics\n92\n EXAMPLE 3 4\nExample 3 4 The resistance of the platinum wire of a platinum\nresistance thermometer at the ice point is 5 W and at steam point is\n5 23 W"}, {"Chapter": "1", "sentence_range": "2795-2798", "Text": "4\nExample 3 4 The resistance of the platinum wire of a platinum\nresistance thermometer at the ice point is 5 W and at steam point is\n5 23 W When the thermometer is inserted in a hot bath, the resistance\nof the platinum wire is 5"}, {"Chapter": "1", "sentence_range": "2796-2799", "Text": "4 The resistance of the platinum wire of a platinum\nresistance thermometer at the ice point is 5 W and at steam point is\n5 23 W When the thermometer is inserted in a hot bath, the resistance\nof the platinum wire is 5 795 W"}, {"Chapter": "1", "sentence_range": "2797-2800", "Text": "23 W When the thermometer is inserted in a hot bath, the resistance\nof the platinum wire is 5 795 W Calculate the temperature of the\nbath"}, {"Chapter": "1", "sentence_range": "2798-2801", "Text": "When the thermometer is inserted in a hot bath, the resistance\nof the platinum wire is 5 795 W Calculate the temperature of the\nbath Solution R0 = 5 W, R100 = 5"}, {"Chapter": "1", "sentence_range": "2799-2802", "Text": "795 W Calculate the temperature of the\nbath Solution R0 = 5 W, R100 = 5 23 W and Rt = 5"}, {"Chapter": "1", "sentence_range": "2800-2803", "Text": "Calculate the temperature of the\nbath Solution R0 = 5 W, R100 = 5 23 W and Rt = 5 795 W\nNow,\n0\n0\n100\n0\n100,\n(1\n)\nt\nt\nR\nR\nt\nR\nR\nt\nR\nR\n\u03b1\n\u2212\n=\n\u00d7\n=\n+\n\u2212\n5"}, {"Chapter": "1", "sentence_range": "2801-2804", "Text": "Solution R0 = 5 W, R100 = 5 23 W and Rt = 5 795 W\nNow,\n0\n0\n100\n0\n100,\n(1\n)\nt\nt\nR\nR\nt\nR\nR\nt\nR\nR\n\u03b1\n\u2212\n=\n\u00d7\n=\n+\n\u2212\n5 795\n5\n100\n5"}, {"Chapter": "1", "sentence_range": "2802-2805", "Text": "23 W and Rt = 5 795 W\nNow,\n0\n0\n100\n0\n100,\n(1\n)\nt\nt\nR\nR\nt\nR\nR\nt\nR\nR\n\u03b1\n\u2212\n=\n\u00d7\n=\n+\n\u2212\n5 795\n5\n100\n5 23\n\u22125\n=\n\u00d7\n\u2212\n= 0"}, {"Chapter": "1", "sentence_range": "2803-2806", "Text": "795 W\nNow,\n0\n0\n100\n0\n100,\n(1\n)\nt\nt\nR\nR\nt\nR\nR\nt\nR\nR\n\u03b1\n\u2212\n=\n\u00d7\n=\n+\n\u2212\n5 795\n5\n100\n5 23\n\u22125\n=\n\u00d7\n\u2212\n= 0 795\n0"}, {"Chapter": "1", "sentence_range": "2804-2807", "Text": "795\n5\n100\n5 23\n\u22125\n=\n\u00d7\n\u2212\n= 0 795\n0 23 \u00d7100\n = 345"}, {"Chapter": "1", "sentence_range": "2805-2808", "Text": "23\n\u22125\n=\n\u00d7\n\u2212\n= 0 795\n0 23 \u00d7100\n = 345 65 \u00b0C\n3"}, {"Chapter": "1", "sentence_range": "2806-2809", "Text": "795\n0 23 \u00d7100\n = 345 65 \u00b0C\n3 9  ELECTRICAL ENERGY, POWER\nConsider a conductor with end points A and B, in which a current I is\nflowing from A to B"}, {"Chapter": "1", "sentence_range": "2807-2810", "Text": "23 \u00d7100\n = 345 65 \u00b0C\n3 9  ELECTRICAL ENERGY, POWER\nConsider a conductor with end points A and B, in which a current I is\nflowing from A to B The electric potential at A and B are denoted by V(A)\nand V(B) respectively"}, {"Chapter": "1", "sentence_range": "2808-2811", "Text": "65 \u00b0C\n3 9  ELECTRICAL ENERGY, POWER\nConsider a conductor with end points A and B, in which a current I is\nflowing from A to B The electric potential at A and B are denoted by V(A)\nand V(B) respectively Since current is flowing from A to B, V(A) > V(B)\nand the potential difference across  AB is V = V(A) \u2013 V(B) > 0"}, {"Chapter": "1", "sentence_range": "2809-2812", "Text": "9  ELECTRICAL ENERGY, POWER\nConsider a conductor with end points A and B, in which a current I is\nflowing from A to B The electric potential at A and B are denoted by V(A)\nand V(B) respectively Since current is flowing from A to B, V(A) > V(B)\nand the potential difference across  AB is V = V(A) \u2013 V(B) > 0 In a time interval Dt, an amount of charge DQ = I Dt travels from A to\nB"}, {"Chapter": "1", "sentence_range": "2810-2813", "Text": "The electric potential at A and B are denoted by V(A)\nand V(B) respectively Since current is flowing from A to B, V(A) > V(B)\nand the potential difference across  AB is V = V(A) \u2013 V(B) > 0 In a time interval Dt, an amount of charge DQ = I Dt travels from A to\nB The potential energy of the charge at A, by definition, was Q V(A) and\nsimilarly at B, it is Q V(B)"}, {"Chapter": "1", "sentence_range": "2811-2814", "Text": "Since current is flowing from A to B, V(A) > V(B)\nand the potential difference across  AB is V = V(A) \u2013 V(B) > 0 In a time interval Dt, an amount of charge DQ = I Dt travels from A to\nB The potential energy of the charge at A, by definition, was Q V(A) and\nsimilarly at B, it is Q V(B) Thus, change in its potential energy DUpot is\nDUpot = Final potential energy \u2013 Initial potential energy\n         = DQ[(V (B) \u2013 V (A)] = \u2013DQ V\n         = \u2013I VDt < 0\n(3"}, {"Chapter": "1", "sentence_range": "2812-2815", "Text": "In a time interval Dt, an amount of charge DQ = I Dt travels from A to\nB The potential energy of the charge at A, by definition, was Q V(A) and\nsimilarly at B, it is Q V(B) Thus, change in its potential energy DUpot is\nDUpot = Final potential energy \u2013 Initial potential energy\n         = DQ[(V (B) \u2013 V (A)] = \u2013DQ V\n         = \u2013I VDt < 0\n(3 28)\nIf charges moved without collisions through the conductor, their\nkinetic energy would also change so that the total energy is unchanged"}, {"Chapter": "1", "sentence_range": "2813-2816", "Text": "The potential energy of the charge at A, by definition, was Q V(A) and\nsimilarly at B, it is Q V(B) Thus, change in its potential energy DUpot is\nDUpot = Final potential energy \u2013 Initial potential energy\n         = DQ[(V (B) \u2013 V (A)] = \u2013DQ V\n         = \u2013I VDt < 0\n(3 28)\nIf charges moved without collisions through the conductor, their\nkinetic energy would also change so that the total energy is unchanged Conservation of total energy would then imply that,\nDK =  \u2013DUpot\n(3"}, {"Chapter": "1", "sentence_range": "2814-2817", "Text": "Thus, change in its potential energy DUpot is\nDUpot = Final potential energy \u2013 Initial potential energy\n         = DQ[(V (B) \u2013 V (A)] = \u2013DQ V\n         = \u2013I VDt < 0\n(3 28)\nIf charges moved without collisions through the conductor, their\nkinetic energy would also change so that the total energy is unchanged Conservation of total energy would then imply that,\nDK =  \u2013DUpot\n(3 29)\nthat is,\nDK = I VDt > 0\n(3"}, {"Chapter": "1", "sentence_range": "2815-2818", "Text": "28)\nIf charges moved without collisions through the conductor, their\nkinetic energy would also change so that the total energy is unchanged Conservation of total energy would then imply that,\nDK =  \u2013DUpot\n(3 29)\nthat is,\nDK = I VDt > 0\n(3 30)\nThus, in case charges were moving freely through the conductor under\nthe action of electric field, their kinetic energy would increase as they\nmove"}, {"Chapter": "1", "sentence_range": "2816-2819", "Text": "Conservation of total energy would then imply that,\nDK =  \u2013DUpot\n(3 29)\nthat is,\nDK = I VDt > 0\n(3 30)\nThus, in case charges were moving freely through the conductor under\nthe action of electric field, their kinetic energy would increase as they\nmove We have, however, seen earlier that on the average, charge carriers\ndo not move with acceleration but with a steady drift velocity"}, {"Chapter": "1", "sentence_range": "2817-2820", "Text": "29)\nthat is,\nDK = I VDt > 0\n(3 30)\nThus, in case charges were moving freely through the conductor under\nthe action of electric field, their kinetic energy would increase as they\nmove We have, however, seen earlier that on the average, charge carriers\ndo not move with acceleration but with a steady drift velocity This is\nbecause of the collisions with ions and atoms during transit"}, {"Chapter": "1", "sentence_range": "2818-2821", "Text": "30)\nThus, in case charges were moving freely through the conductor under\nthe action of electric field, their kinetic energy would increase as they\nmove We have, however, seen earlier that on the average, charge carriers\ndo not move with acceleration but with a steady drift velocity This is\nbecause of the collisions with ions and atoms during transit During\ncollisions, the energy gained by the charges thus is shared with the atoms"}, {"Chapter": "1", "sentence_range": "2819-2822", "Text": "We have, however, seen earlier that on the average, charge carriers\ndo not move with acceleration but with a steady drift velocity This is\nbecause of the collisions with ions and atoms during transit During\ncollisions, the energy gained by the charges thus is shared with the atoms The atoms vibrate more vigorously, i"}, {"Chapter": "1", "sentence_range": "2820-2823", "Text": "This is\nbecause of the collisions with ions and atoms during transit During\ncollisions, the energy gained by the charges thus is shared with the atoms The atoms vibrate more vigorously, i e"}, {"Chapter": "1", "sentence_range": "2821-2824", "Text": "During\ncollisions, the energy gained by the charges thus is shared with the atoms The atoms vibrate more vigorously, i e , the conductor heats up"}, {"Chapter": "1", "sentence_range": "2822-2825", "Text": "The atoms vibrate more vigorously, i e , the conductor heats up Thus,\nin an actual conductor, an amount of energy dissipated as heat in the\nconductor during the time interval Dt is,\nDW = I VDt\n(3"}, {"Chapter": "1", "sentence_range": "2823-2826", "Text": "e , the conductor heats up Thus,\nin an actual conductor, an amount of energy dissipated as heat in the\nconductor during the time interval Dt is,\nDW = I VDt\n(3 31)\nThe energy dissipated per unit time is the power dissipated\nP = DW/Dt  and we have,\nP = I V\n(3"}, {"Chapter": "1", "sentence_range": "2824-2827", "Text": ", the conductor heats up Thus,\nin an actual conductor, an amount of energy dissipated as heat in the\nconductor during the time interval Dt is,\nDW = I VDt\n(3 31)\nThe energy dissipated per unit time is the power dissipated\nP = DW/Dt  and we have,\nP = I V\n(3 32)\nRationalised 2023-24\nCurrent\nElectricity\n93\nUsing Ohm\u2019s law V = IR, we get\nP = I 2 R = V 2/R\n(3"}, {"Chapter": "1", "sentence_range": "2825-2828", "Text": "Thus,\nin an actual conductor, an amount of energy dissipated as heat in the\nconductor during the time interval Dt is,\nDW = I VDt\n(3 31)\nThe energy dissipated per unit time is the power dissipated\nP = DW/Dt  and we have,\nP = I V\n(3 32)\nRationalised 2023-24\nCurrent\nElectricity\n93\nUsing Ohm\u2019s law V = IR, we get\nP = I 2 R = V 2/R\n(3 33)\nas the power loss (\u201cohmic loss\u201d) in a conductor of resistance R carrying a\ncurrent I"}, {"Chapter": "1", "sentence_range": "2826-2829", "Text": "31)\nThe energy dissipated per unit time is the power dissipated\nP = DW/Dt  and we have,\nP = I V\n(3 32)\nRationalised 2023-24\nCurrent\nElectricity\n93\nUsing Ohm\u2019s law V = IR, we get\nP = I 2 R = V 2/R\n(3 33)\nas the power loss (\u201cohmic loss\u201d) in a conductor of resistance R carrying a\ncurrent I It is this power which heats up, for example, the coil of an\nelectric bulb to incandescence, radiating out heat and light"}, {"Chapter": "1", "sentence_range": "2827-2830", "Text": "32)\nRationalised 2023-24\nCurrent\nElectricity\n93\nUsing Ohm\u2019s law V = IR, we get\nP = I 2 R = V 2/R\n(3 33)\nas the power loss (\u201cohmic loss\u201d) in a conductor of resistance R carrying a\ncurrent I It is this power which heats up, for example, the coil of an\nelectric bulb to incandescence, radiating out heat and light Where does the power come from"}, {"Chapter": "1", "sentence_range": "2828-2831", "Text": "33)\nas the power loss (\u201cohmic loss\u201d) in a conductor of resistance R carrying a\ncurrent I It is this power which heats up, for example, the coil of an\nelectric bulb to incandescence, radiating out heat and light Where does the power come from As we have\nreasoned before, we need an external source to keep\na steady current through the conductor"}, {"Chapter": "1", "sentence_range": "2829-2832", "Text": "It is this power which heats up, for example, the coil of an\nelectric bulb to incandescence, radiating out heat and light Where does the power come from As we have\nreasoned before, we need an external source to keep\na steady current through the conductor It is clearly\nthis source which must supply this power"}, {"Chapter": "1", "sentence_range": "2830-2833", "Text": "Where does the power come from As we have\nreasoned before, we need an external source to keep\na steady current through the conductor It is clearly\nthis source which must supply this power In the\nsimple circuit shown with a cell (Fig"}, {"Chapter": "1", "sentence_range": "2831-2834", "Text": "As we have\nreasoned before, we need an external source to keep\na steady current through the conductor It is clearly\nthis source which must supply this power In the\nsimple circuit shown with a cell (Fig 3"}, {"Chapter": "1", "sentence_range": "2832-2835", "Text": "It is clearly\nthis source which must supply this power In the\nsimple circuit shown with a cell (Fig 3 11), it is the\nchemical energy of the cell which supplies this power\nfor as long as it can"}, {"Chapter": "1", "sentence_range": "2833-2836", "Text": "In the\nsimple circuit shown with a cell (Fig 3 11), it is the\nchemical energy of the cell which supplies this power\nfor as long as it can The expressions for power, Eqs"}, {"Chapter": "1", "sentence_range": "2834-2837", "Text": "3 11), it is the\nchemical energy of the cell which supplies this power\nfor as long as it can The expressions for power, Eqs (3"}, {"Chapter": "1", "sentence_range": "2835-2838", "Text": "11), it is the\nchemical energy of the cell which supplies this power\nfor as long as it can The expressions for power, Eqs (3 32) and (3"}, {"Chapter": "1", "sentence_range": "2836-2839", "Text": "The expressions for power, Eqs (3 32) and (3 33),\nshow the dependence of the power dissipated in a\nresistor R on the current through it and the voltage\nacross it"}, {"Chapter": "1", "sentence_range": "2837-2840", "Text": "(3 32) and (3 33),\nshow the dependence of the power dissipated in a\nresistor R on the current through it and the voltage\nacross it Equation (3"}, {"Chapter": "1", "sentence_range": "2838-2841", "Text": "32) and (3 33),\nshow the dependence of the power dissipated in a\nresistor R on the current through it and the voltage\nacross it Equation (3 33) has an important application to\npower transmission"}, {"Chapter": "1", "sentence_range": "2839-2842", "Text": "33),\nshow the dependence of the power dissipated in a\nresistor R on the current through it and the voltage\nacross it Equation (3 33) has an important application to\npower transmission Electrical power is transmitted\nfrom power stations to homes and factories, which\nmay be hundreds of miles away, via transmission\ncables"}, {"Chapter": "1", "sentence_range": "2840-2843", "Text": "Equation (3 33) has an important application to\npower transmission Electrical power is transmitted\nfrom power stations to homes and factories, which\nmay be hundreds of miles away, via transmission\ncables One obviously wants to minimise the power\nloss in the transmission cables connecting the power stations to homes\nand factories"}, {"Chapter": "1", "sentence_range": "2841-2844", "Text": "33) has an important application to\npower transmission Electrical power is transmitted\nfrom power stations to homes and factories, which\nmay be hundreds of miles away, via transmission\ncables One obviously wants to minimise the power\nloss in the transmission cables connecting the power stations to homes\nand factories We shall  see now how this can be achieved"}, {"Chapter": "1", "sentence_range": "2842-2845", "Text": "Electrical power is transmitted\nfrom power stations to homes and factories, which\nmay be hundreds of miles away, via transmission\ncables One obviously wants to minimise the power\nloss in the transmission cables connecting the power stations to homes\nand factories We shall  see now how this can be achieved Consider a\ndevice R, to which a power P is to be delivered via transmission cables\nhaving a resistance Rc to be dissipated by it finally"}, {"Chapter": "1", "sentence_range": "2843-2846", "Text": "One obviously wants to minimise the power\nloss in the transmission cables connecting the power stations to homes\nand factories We shall  see now how this can be achieved Consider a\ndevice R, to which a power P is to be delivered via transmission cables\nhaving a resistance Rc to be dissipated by it finally If V is the voltage\nacross R and I the current through it, then\nP = V I\n(3"}, {"Chapter": "1", "sentence_range": "2844-2847", "Text": "We shall  see now how this can be achieved Consider a\ndevice R, to which a power P is to be delivered via transmission cables\nhaving a resistance Rc to be dissipated by it finally If V is the voltage\nacross R and I the current through it, then\nP = V I\n(3 34)\nThe connecting wires from the power station to the device has a finite\nresistance Rc"}, {"Chapter": "1", "sentence_range": "2845-2848", "Text": "Consider a\ndevice R, to which a power P is to be delivered via transmission cables\nhaving a resistance Rc to be dissipated by it finally If V is the voltage\nacross R and I the current through it, then\nP = V I\n(3 34)\nThe connecting wires from the power station to the device has a finite\nresistance Rc The power dissipated in the connecting wires, which is\nwasted is Pc  with\nPc = I 2 Rc\n   \n2\n2\nc\nP R\nV\n=\n(3"}, {"Chapter": "1", "sentence_range": "2846-2849", "Text": "If V is the voltage\nacross R and I the current through it, then\nP = V I\n(3 34)\nThe connecting wires from the power station to the device has a finite\nresistance Rc The power dissipated in the connecting wires, which is\nwasted is Pc  with\nPc = I 2 Rc\n   \n2\n2\nc\nP R\nV\n=\n(3 35)\nfrom Eq"}, {"Chapter": "1", "sentence_range": "2847-2850", "Text": "34)\nThe connecting wires from the power station to the device has a finite\nresistance Rc The power dissipated in the connecting wires, which is\nwasted is Pc  with\nPc = I 2 Rc\n   \n2\n2\nc\nP R\nV\n=\n(3 35)\nfrom Eq (3"}, {"Chapter": "1", "sentence_range": "2848-2851", "Text": "The power dissipated in the connecting wires, which is\nwasted is Pc  with\nPc = I 2 Rc\n   \n2\n2\nc\nP R\nV\n=\n(3 35)\nfrom Eq (3 32)"}, {"Chapter": "1", "sentence_range": "2849-2852", "Text": "35)\nfrom Eq (3 32) Thus, to drive a device of power P, the power wasted in the\nconnecting wires is inversely proportional to V 2"}, {"Chapter": "1", "sentence_range": "2850-2853", "Text": "(3 32) Thus, to drive a device of power P, the power wasted in the\nconnecting wires is inversely proportional to V 2 The  transmission cables\nfrom power stations are hundreds of miles long and their resistance Rc is\nconsiderable"}, {"Chapter": "1", "sentence_range": "2851-2854", "Text": "32) Thus, to drive a device of power P, the power wasted in the\nconnecting wires is inversely proportional to V 2 The  transmission cables\nfrom power stations are hundreds of miles long and their resistance Rc is\nconsiderable To reduce Pc, these wires carry current at enormous values\nof V and this is the reason for the high voltage danger signs on transmission\nlines \u2014 a common sight as we move away from populated areas"}, {"Chapter": "1", "sentence_range": "2852-2855", "Text": "Thus, to drive a device of power P, the power wasted in the\nconnecting wires is inversely proportional to V 2 The  transmission cables\nfrom power stations are hundreds of miles long and their resistance Rc is\nconsiderable To reduce Pc, these wires carry current at enormous values\nof V and this is the reason for the high voltage danger signs on transmission\nlines \u2014 a common sight as we move away from populated areas Using\nelectricity at such voltages is not safe and hence at the other end, a device\ncalled a transformer lowers the voltage to a value suitable for use"}, {"Chapter": "1", "sentence_range": "2853-2856", "Text": "The  transmission cables\nfrom power stations are hundreds of miles long and their resistance Rc is\nconsiderable To reduce Pc, these wires carry current at enormous values\nof V and this is the reason for the high voltage danger signs on transmission\nlines \u2014 a common sight as we move away from populated areas Using\nelectricity at such voltages is not safe and hence at the other end, a device\ncalled a transformer lowers the voltage to a value suitable for use 3"}, {"Chapter": "1", "sentence_range": "2854-2857", "Text": "To reduce Pc, these wires carry current at enormous values\nof V and this is the reason for the high voltage danger signs on transmission\nlines \u2014 a common sight as we move away from populated areas Using\nelectricity at such voltages is not safe and hence at the other end, a device\ncalled a transformer lowers the voltage to a value suitable for use 3 10 CELLS, EMF, INTERNAL RESISTANCE\nWe have already mentioned that a simple device to maintain a steady\ncurrent in an electric circuit is the electrolytic cell"}, {"Chapter": "1", "sentence_range": "2855-2858", "Text": "Using\nelectricity at such voltages is not safe and hence at the other end, a device\ncalled a transformer lowers the voltage to a value suitable for use 3 10 CELLS, EMF, INTERNAL RESISTANCE\nWe have already mentioned that a simple device to maintain a steady\ncurrent in an electric circuit is the electrolytic cell Basically a cell has\ntwo electrodes, called the positive (P) and the negative (N), as shown in\nFIGURE 3"}, {"Chapter": "1", "sentence_range": "2856-2859", "Text": "3 10 CELLS, EMF, INTERNAL RESISTANCE\nWe have already mentioned that a simple device to maintain a steady\ncurrent in an electric circuit is the electrolytic cell Basically a cell has\ntwo electrodes, called the positive (P) and the negative (N), as shown in\nFIGURE 3 11 Heat is produced in the\nresistor R which is connected across\nthe terminals of a cell"}, {"Chapter": "1", "sentence_range": "2857-2860", "Text": "10 CELLS, EMF, INTERNAL RESISTANCE\nWe have already mentioned that a simple device to maintain a steady\ncurrent in an electric circuit is the electrolytic cell Basically a cell has\ntwo electrodes, called the positive (P) and the negative (N), as shown in\nFIGURE 3 11 Heat is produced in the\nresistor R which is connected across\nthe terminals of a cell The energy\ndissipated in the resistor R comes from\nthe chemical energy of the electrolyte"}, {"Chapter": "1", "sentence_range": "2858-2861", "Text": "Basically a cell has\ntwo electrodes, called the positive (P) and the negative (N), as shown in\nFIGURE 3 11 Heat is produced in the\nresistor R which is connected across\nthe terminals of a cell The energy\ndissipated in the resistor R comes from\nthe chemical energy of the electrolyte Rationalised 2023-24\nPhysics\n94\nFig"}, {"Chapter": "1", "sentence_range": "2859-2862", "Text": "11 Heat is produced in the\nresistor R which is connected across\nthe terminals of a cell The energy\ndissipated in the resistor R comes from\nthe chemical energy of the electrolyte Rationalised 2023-24\nPhysics\n94\nFig 3"}, {"Chapter": "1", "sentence_range": "2860-2863", "Text": "The energy\ndissipated in the resistor R comes from\nthe chemical energy of the electrolyte Rationalised 2023-24\nPhysics\n94\nFig 3 12"}, {"Chapter": "1", "sentence_range": "2861-2864", "Text": "Rationalised 2023-24\nPhysics\n94\nFig 3 12 They are immersed in an electrolytic solution"}, {"Chapter": "1", "sentence_range": "2862-2865", "Text": "3 12 They are immersed in an electrolytic solution Dipped in\nthe solution, the electrodes exchange charges with the electrolyte"}, {"Chapter": "1", "sentence_range": "2863-2866", "Text": "12 They are immersed in an electrolytic solution Dipped in\nthe solution, the electrodes exchange charges with the electrolyte The positive electrode has a potential difference V+ (V+ > 0) between\nitself and the electrolyte solution immediately adjacent to it marked\nA in the figure"}, {"Chapter": "1", "sentence_range": "2864-2867", "Text": "They are immersed in an electrolytic solution Dipped in\nthe solution, the electrodes exchange charges with the electrolyte The positive electrode has a potential difference V+ (V+ > 0) between\nitself and the electrolyte solution immediately adjacent to it marked\nA in the figure Similarly, the negative electrode develops a negative\npotential  \u2013 (V\u2013 ) (V\u2013 \u2265  0) relative to the electrolyte adjacent to it,\nmarked as B in the figure"}, {"Chapter": "1", "sentence_range": "2865-2868", "Text": "Dipped in\nthe solution, the electrodes exchange charges with the electrolyte The positive electrode has a potential difference V+ (V+ > 0) between\nitself and the electrolyte solution immediately adjacent to it marked\nA in the figure Similarly, the negative electrode develops a negative\npotential  \u2013 (V\u2013 ) (V\u2013 \u2265  0) relative to the electrolyte adjacent to it,\nmarked as B in the figure When there is no current, the electrolyte\nhas the same potential throughout, so that the potential difference\nbetween P and N is V+ \u2013 (\u2013V\u2013) = V+ + V\u2013"}, {"Chapter": "1", "sentence_range": "2866-2869", "Text": "The positive electrode has a potential difference V+ (V+ > 0) between\nitself and the electrolyte solution immediately adjacent to it marked\nA in the figure Similarly, the negative electrode develops a negative\npotential  \u2013 (V\u2013 ) (V\u2013 \u2265  0) relative to the electrolyte adjacent to it,\nmarked as B in the figure When there is no current, the electrolyte\nhas the same potential throughout, so that the potential difference\nbetween P and N is V+ \u2013 (\u2013V\u2013) = V+ + V\u2013 This difference is called the\nelectromotive force (emf) of the cell and is denoted by e"}, {"Chapter": "1", "sentence_range": "2867-2870", "Text": "Similarly, the negative electrode develops a negative\npotential  \u2013 (V\u2013 ) (V\u2013 \u2265  0) relative to the electrolyte adjacent to it,\nmarked as B in the figure When there is no current, the electrolyte\nhas the same potential throughout, so that the potential difference\nbetween P and N is V+ \u2013 (\u2013V\u2013) = V+ + V\u2013 This difference is called the\nelectromotive force (emf) of the cell and is denoted by e Thus\ne = V++V\u2013 >  0\n(3"}, {"Chapter": "1", "sentence_range": "2868-2871", "Text": "When there is no current, the electrolyte\nhas the same potential throughout, so that the potential difference\nbetween P and N is V+ \u2013 (\u2013V\u2013) = V+ + V\u2013 This difference is called the\nelectromotive force (emf) of the cell and is denoted by e Thus\ne = V++V\u2013 >  0\n(3 36)\nNote that e is, actually, a potential difference and not a force"}, {"Chapter": "1", "sentence_range": "2869-2872", "Text": "This difference is called the\nelectromotive force (emf) of the cell and is denoted by e Thus\ne = V++V\u2013 >  0\n(3 36)\nNote that e is, actually, a potential difference and not a force The\nname emf, however, is used because of historical reasons, and was\ngiven at a time when the phenomenon was not understood properly"}, {"Chapter": "1", "sentence_range": "2870-2873", "Text": "Thus\ne = V++V\u2013 >  0\n(3 36)\nNote that e is, actually, a potential difference and not a force The\nname emf, however, is used because of historical reasons, and was\ngiven at a time when the phenomenon was not understood properly To understand the significance of e, consider a resistor R\nconnected across the cell (Fig"}, {"Chapter": "1", "sentence_range": "2871-2874", "Text": "36)\nNote that e is, actually, a potential difference and not a force The\nname emf, however, is used because of historical reasons, and was\ngiven at a time when the phenomenon was not understood properly To understand the significance of e, consider a resistor R\nconnected across the cell (Fig 3"}, {"Chapter": "1", "sentence_range": "2872-2875", "Text": "The\nname emf, however, is used because of historical reasons, and was\ngiven at a time when the phenomenon was not understood properly To understand the significance of e, consider a resistor R\nconnected across the cell (Fig 3 12)"}, {"Chapter": "1", "sentence_range": "2873-2876", "Text": "To understand the significance of e, consider a resistor R\nconnected across the cell (Fig 3 12) A current I flows across R\nfrom C to D"}, {"Chapter": "1", "sentence_range": "2874-2877", "Text": "3 12) A current I flows across R\nfrom C to D As explained before, a steady current is maintained\nbecause current flows from N to P through the electrolyte"}, {"Chapter": "1", "sentence_range": "2875-2878", "Text": "12) A current I flows across R\nfrom C to D As explained before, a steady current is maintained\nbecause current flows from N to P through the electrolyte Clearly,\nacross the electrolyte the same current flows through the electrolyte\nbut from N to P, whereas through R, it flows from P to N"}, {"Chapter": "1", "sentence_range": "2876-2879", "Text": "A current I flows across R\nfrom C to D As explained before, a steady current is maintained\nbecause current flows from N to P through the electrolyte Clearly,\nacross the electrolyte the same current flows through the electrolyte\nbut from N to P, whereas through R, it flows from P to N The electrolyte through which a current flows has a finite\nresistance r, called the internal resistance"}, {"Chapter": "1", "sentence_range": "2877-2880", "Text": "As explained before, a steady current is maintained\nbecause current flows from N to P through the electrolyte Clearly,\nacross the electrolyte the same current flows through the electrolyte\nbut from N to P, whereas through R, it flows from P to N The electrolyte through which a current flows has a finite\nresistance r, called the internal resistance Consider first the\nsituation when R is infinite so that I = V/R = 0, where V is the\npotential difference between P and N"}, {"Chapter": "1", "sentence_range": "2878-2881", "Text": "Clearly,\nacross the electrolyte the same current flows through the electrolyte\nbut from N to P, whereas through R, it flows from P to N The electrolyte through which a current flows has a finite\nresistance r, called the internal resistance Consider first the\nsituation when R is infinite so that I = V/R = 0, where V is the\npotential difference between P and N Now,\nV = Potential difference between P and A\n       + Potential difference between A and B\n       + Potential  difference between B and N\n   = e\n(3"}, {"Chapter": "1", "sentence_range": "2879-2882", "Text": "The electrolyte through which a current flows has a finite\nresistance r, called the internal resistance Consider first the\nsituation when R is infinite so that I = V/R = 0, where V is the\npotential difference between P and N Now,\nV = Potential difference between P and A\n       + Potential difference between A and B\n       + Potential  difference between B and N\n   = e\n(3 37)\nThus, emf e is the potential difference between the positive and\nnegative electrodes in an open circuit, i"}, {"Chapter": "1", "sentence_range": "2880-2883", "Text": "Consider first the\nsituation when R is infinite so that I = V/R = 0, where V is the\npotential difference between P and N Now,\nV = Potential difference between P and A\n       + Potential difference between A and B\n       + Potential  difference between B and N\n   = e\n(3 37)\nThus, emf e is the potential difference between the positive and\nnegative electrodes in an open circuit, i e"}, {"Chapter": "1", "sentence_range": "2881-2884", "Text": "Now,\nV = Potential difference between P and A\n       + Potential difference between A and B\n       + Potential  difference between B and N\n   = e\n(3 37)\nThus, emf e is the potential difference between the positive and\nnegative electrodes in an open circuit, i e , when no current is\nflowing through the cell"}, {"Chapter": "1", "sentence_range": "2882-2885", "Text": "37)\nThus, emf e is the potential difference between the positive and\nnegative electrodes in an open circuit, i e , when no current is\nflowing through the cell If however R is finite, I is not zero"}, {"Chapter": "1", "sentence_range": "2883-2886", "Text": "e , when no current is\nflowing through the cell If however R is finite, I is not zero In that case the potential difference\nbetween P and N is\nV = V++ V\u2013 \u2013 I r\n   = e \u2013 I r\n(3"}, {"Chapter": "1", "sentence_range": "2884-2887", "Text": ", when no current is\nflowing through the cell If however R is finite, I is not zero In that case the potential difference\nbetween P and N is\nV = V++ V\u2013 \u2013 I r\n   = e \u2013 I r\n(3 38)\nNote the negative sign in the expression (I r) for the potential difference\nbetween A and B"}, {"Chapter": "1", "sentence_range": "2885-2888", "Text": "If however R is finite, I is not zero In that case the potential difference\nbetween P and N is\nV = V++ V\u2013 \u2013 I r\n   = e \u2013 I r\n(3 38)\nNote the negative sign in the expression (I r) for the potential difference\nbetween A and B This is because the current I flows from B to A in the\nelectrolyte"}, {"Chapter": "1", "sentence_range": "2886-2889", "Text": "In that case the potential difference\nbetween P and N is\nV = V++ V\u2013 \u2013 I r\n   = e \u2013 I r\n(3 38)\nNote the negative sign in the expression (I r) for the potential difference\nbetween A and B This is because the current I flows from B to A in the\nelectrolyte In  practical calculations, internal resistances of cells in the circuit\nmay be neglected when the  current I is such that e >> I r"}, {"Chapter": "1", "sentence_range": "2887-2890", "Text": "38)\nNote the negative sign in the expression (I r) for the potential difference\nbetween A and B This is because the current I flows from B to A in the\nelectrolyte In  practical calculations, internal resistances of cells in the circuit\nmay be neglected when the  current I is such that e >> I r The actual\nvalues of the internal resistances of cells vary from cell to cell"}, {"Chapter": "1", "sentence_range": "2888-2891", "Text": "This is because the current I flows from B to A in the\nelectrolyte In  practical calculations, internal resistances of cells in the circuit\nmay be neglected when the  current I is such that e >> I r The actual\nvalues of the internal resistances of cells vary from cell to cell The internal\nresistance of dry cells, however, is much higher than the common\nelectrolytic cells"}, {"Chapter": "1", "sentence_range": "2889-2892", "Text": "In  practical calculations, internal resistances of cells in the circuit\nmay be neglected when the  current I is such that e >> I r The actual\nvalues of the internal resistances of cells vary from cell to cell The internal\nresistance of dry cells, however, is much higher than the common\nelectrolytic cells We also observe that since V is the potential difference across R, we\nhave from Ohm\u2019s law\nV = I  R\n(3"}, {"Chapter": "1", "sentence_range": "2890-2893", "Text": "The actual\nvalues of the internal resistances of cells vary from cell to cell The internal\nresistance of dry cells, however, is much higher than the common\nelectrolytic cells We also observe that since V is the potential difference across R, we\nhave from Ohm\u2019s law\nV = I  R\n(3 39)\nCombining Eqs"}, {"Chapter": "1", "sentence_range": "2891-2894", "Text": "The internal\nresistance of dry cells, however, is much higher than the common\nelectrolytic cells We also observe that since V is the potential difference across R, we\nhave from Ohm\u2019s law\nV = I  R\n(3 39)\nCombining Eqs (3"}, {"Chapter": "1", "sentence_range": "2892-2895", "Text": "We also observe that since V is the potential difference across R, we\nhave from Ohm\u2019s law\nV = I  R\n(3 39)\nCombining Eqs (3 38) and (3"}, {"Chapter": "1", "sentence_range": "2893-2896", "Text": "39)\nCombining Eqs (3 38) and (3 39), we get\nFIGURE 3"}, {"Chapter": "1", "sentence_range": "2894-2897", "Text": "(3 38) and (3 39), we get\nFIGURE 3 12 (a) Sketch of\nan electrolyte cell with\npositive terminal P and\nnegative terminal N"}, {"Chapter": "1", "sentence_range": "2895-2898", "Text": "38) and (3 39), we get\nFIGURE 3 12 (a) Sketch of\nan electrolyte cell with\npositive terminal P and\nnegative terminal N The\ngap between the electrodes\nis exaggerated for clarity"}, {"Chapter": "1", "sentence_range": "2896-2899", "Text": "39), we get\nFIGURE 3 12 (a) Sketch of\nan electrolyte cell with\npositive terminal P and\nnegative terminal N The\ngap between the electrodes\nis exaggerated for clarity A\nand B are points in the\nelectrolyte typically close to\nP and N"}, {"Chapter": "1", "sentence_range": "2897-2900", "Text": "12 (a) Sketch of\nan electrolyte cell with\npositive terminal P and\nnegative terminal N The\ngap between the electrodes\nis exaggerated for clarity A\nand B are points in the\nelectrolyte typically close to\nP and N (b) the symbol for\na cell, + referring to P and\n\u2013 referring to the N\nelectrode"}, {"Chapter": "1", "sentence_range": "2898-2901", "Text": "The\ngap between the electrodes\nis exaggerated for clarity A\nand B are points in the\nelectrolyte typically close to\nP and N (b) the symbol for\na cell, + referring to P and\n\u2013 referring to the N\nelectrode Electrical\nconnections to the cell are\nmade at P and N"}, {"Chapter": "1", "sentence_range": "2899-2902", "Text": "A\nand B are points in the\nelectrolyte typically close to\nP and N (b) the symbol for\na cell, + referring to P and\n\u2013 referring to the N\nelectrode Electrical\nconnections to the cell are\nmade at P and N Rationalised 2023-24\nCurrent\nElectricity\n95\nI  R   =  e \u2013 I  r\nOr, I\nR\nr\n=\n\u03b5+\n(3"}, {"Chapter": "1", "sentence_range": "2900-2903", "Text": "(b) the symbol for\na cell, + referring to P and\n\u2013 referring to the N\nelectrode Electrical\nconnections to the cell are\nmade at P and N Rationalised 2023-24\nCurrent\nElectricity\n95\nI  R   =  e \u2013 I  r\nOr, I\nR\nr\n=\n\u03b5+\n(3 40)\nThe maximum current that can be drawn from a cell is for R = 0 and\nit is Imax = e/r"}, {"Chapter": "1", "sentence_range": "2901-2904", "Text": "Electrical\nconnections to the cell are\nmade at P and N Rationalised 2023-24\nCurrent\nElectricity\n95\nI  R   =  e \u2013 I  r\nOr, I\nR\nr\n=\n\u03b5+\n(3 40)\nThe maximum current that can be drawn from a cell is for R = 0 and\nit is Imax = e/r However, in most cells the maximum allowed current is\nmuch lower than this to prevent permanent damage to the cell"}, {"Chapter": "1", "sentence_range": "2902-2905", "Text": "Rationalised 2023-24\nCurrent\nElectricity\n95\nI  R   =  e \u2013 I  r\nOr, I\nR\nr\n=\n\u03b5+\n(3 40)\nThe maximum current that can be drawn from a cell is for R = 0 and\nit is Imax = e/r However, in most cells the maximum allowed current is\nmuch lower than this to prevent permanent damage to the cell 3"}, {"Chapter": "1", "sentence_range": "2903-2906", "Text": "40)\nThe maximum current that can be drawn from a cell is for R = 0 and\nit is Imax = e/r However, in most cells the maximum allowed current is\nmuch lower than this to prevent permanent damage to the cell 3 11  CELLS IN SERIES AND IN PARALLEL\nLike resistors, cells can be combined together in an electric circuit"}, {"Chapter": "1", "sentence_range": "2904-2907", "Text": "However, in most cells the maximum allowed current is\nmuch lower than this to prevent permanent damage to the cell 3 11  CELLS IN SERIES AND IN PARALLEL\nLike resistors, cells can be combined together in an electric circuit And\nlike resistors, one can, for calculating currents and voltages in a circuit,\nreplace a combination of cells by an equivalent cell"}, {"Chapter": "1", "sentence_range": "2905-2908", "Text": "3 11  CELLS IN SERIES AND IN PARALLEL\nLike resistors, cells can be combined together in an electric circuit And\nlike resistors, one can, for calculating currents and voltages in a circuit,\nreplace a combination of cells by an equivalent cell FIGURE 3"}, {"Chapter": "1", "sentence_range": "2906-2909", "Text": "11  CELLS IN SERIES AND IN PARALLEL\nLike resistors, cells can be combined together in an electric circuit And\nlike resistors, one can, for calculating currents and voltages in a circuit,\nreplace a combination of cells by an equivalent cell FIGURE 3 13 Two cells of emf\u2019s e1 and e2 in the series"}, {"Chapter": "1", "sentence_range": "2907-2910", "Text": "And\nlike resistors, one can, for calculating currents and voltages in a circuit,\nreplace a combination of cells by an equivalent cell FIGURE 3 13 Two cells of emf\u2019s e1 and e2 in the series r1, r2 are their\ninternal resistances"}, {"Chapter": "1", "sentence_range": "2908-2911", "Text": "FIGURE 3 13 Two cells of emf\u2019s e1 and e2 in the series r1, r2 are their\ninternal resistances For connections across A and C, the combination\ncan be considered as one cell of emf eeq and an internal resistance req"}, {"Chapter": "1", "sentence_range": "2909-2912", "Text": "13 Two cells of emf\u2019s e1 and e2 in the series r1, r2 are their\ninternal resistances For connections across A and C, the combination\ncan be considered as one cell of emf eeq and an internal resistance req Consider first  two cells in series (Fig"}, {"Chapter": "1", "sentence_range": "2910-2913", "Text": "r1, r2 are their\ninternal resistances For connections across A and C, the combination\ncan be considered as one cell of emf eeq and an internal resistance req Consider first  two cells in series (Fig 3"}, {"Chapter": "1", "sentence_range": "2911-2914", "Text": "For connections across A and C, the combination\ncan be considered as one cell of emf eeq and an internal resistance req Consider first  two cells in series (Fig 3 13), where one terminal of the\ntwo cells is joined together leaving the other terminal in either cell free"}, {"Chapter": "1", "sentence_range": "2912-2915", "Text": "Consider first  two cells in series (Fig 3 13), where one terminal of the\ntwo cells is joined together leaving the other terminal in either cell free e1, e2 are the emf\u2019s of the two cells and r1, r2 their internal resistances,\nrespectively"}, {"Chapter": "1", "sentence_range": "2913-2916", "Text": "3 13), where one terminal of the\ntwo cells is joined together leaving the other terminal in either cell free e1, e2 are the emf\u2019s of the two cells and r1, r2 their internal resistances,\nrespectively Let V (A), V (B), V (C) be the potentials at points A, B and C shown in\nFig"}, {"Chapter": "1", "sentence_range": "2914-2917", "Text": "13), where one terminal of the\ntwo cells is joined together leaving the other terminal in either cell free e1, e2 are the emf\u2019s of the two cells and r1, r2 their internal resistances,\nrespectively Let V (A), V (B), V (C) be the potentials at points A, B and C shown in\nFig 3"}, {"Chapter": "1", "sentence_range": "2915-2918", "Text": "e1, e2 are the emf\u2019s of the two cells and r1, r2 their internal resistances,\nrespectively Let V (A), V (B), V (C) be the potentials at points A, B and C shown in\nFig 3 13"}, {"Chapter": "1", "sentence_range": "2916-2919", "Text": "Let V (A), V (B), V (C) be the potentials at points A, B and C shown in\nFig 3 13 Then V (A) \u2013 V (B) is the potential difference between the positive\nand negative terminals of the first cell"}, {"Chapter": "1", "sentence_range": "2917-2920", "Text": "3 13 Then V (A) \u2013 V (B) is the potential difference between the positive\nand negative terminals of the first cell We have already calculated it in\nEq"}, {"Chapter": "1", "sentence_range": "2918-2921", "Text": "13 Then V (A) \u2013 V (B) is the potential difference between the positive\nand negative terminals of the first cell We have already calculated it in\nEq (3"}, {"Chapter": "1", "sentence_range": "2919-2922", "Text": "Then V (A) \u2013 V (B) is the potential difference between the positive\nand negative terminals of the first cell We have already calculated it in\nEq (3 38) and hence,\nV\nV\nV\nI r\nAB\nA\nB\n\u2261\n=\n(\n) \ufffd\n( )\n\u03b51\ufffd\n1\n(3"}, {"Chapter": "1", "sentence_range": "2920-2923", "Text": "We have already calculated it in\nEq (3 38) and hence,\nV\nV\nV\nI r\nAB\nA\nB\n\u2261\n=\n(\n) \ufffd\n( )\n\u03b51\ufffd\n1\n(3 41)\nSimilarly,\nV\nV\nV\nI r\nBC\nB\nC\n\u2261\n=\n( ) \ufffd\n( )\n\u03b52\ufffd\n2\n(3"}, {"Chapter": "1", "sentence_range": "2921-2924", "Text": "(3 38) and hence,\nV\nV\nV\nI r\nAB\nA\nB\n\u2261\n=\n(\n) \ufffd\n( )\n\u03b51\ufffd\n1\n(3 41)\nSimilarly,\nV\nV\nV\nI r\nBC\nB\nC\n\u2261\n=\n( ) \ufffd\n( )\n\u03b52\ufffd\n2\n(3 42)\nHence, the potential difference between the terminals A and C of the\ncombination is\n( )\n( )\n( )\n( )\nAC\n(A)\u2013\n(C)\nA \u2013\nB\nB \u2013\nC\nV\nV\nV\nV\nV\nV\nV\n\u2261\n=\n+\n \n \n \n \n \n \n \n \n       \n(\n)\n(\n)\n1\n2\n1\n2\n\u2013 I r\nr\n\u03b5\n\u03b5\n=\n+\n+\n(3"}, {"Chapter": "1", "sentence_range": "2922-2925", "Text": "38) and hence,\nV\nV\nV\nI r\nAB\nA\nB\n\u2261\n=\n(\n) \ufffd\n( )\n\u03b51\ufffd\n1\n(3 41)\nSimilarly,\nV\nV\nV\nI r\nBC\nB\nC\n\u2261\n=\n( ) \ufffd\n( )\n\u03b52\ufffd\n2\n(3 42)\nHence, the potential difference between the terminals A and C of the\ncombination is\n( )\n( )\n( )\n( )\nAC\n(A)\u2013\n(C)\nA \u2013\nB\nB \u2013\nC\nV\nV\nV\nV\nV\nV\nV\n\u2261\n=\n+\n \n \n \n \n \n \n \n \n       \n(\n)\n(\n)\n1\n2\n1\n2\n\u2013 I r\nr\n\u03b5\n\u03b5\n=\n+\n+\n(3 43)\nIf we wish to replace the combination by a single cell between A and\nC of emf eeq and internal resistance req, we would have\nVAC = eeq\u2013 I req\n(3"}, {"Chapter": "1", "sentence_range": "2923-2926", "Text": "41)\nSimilarly,\nV\nV\nV\nI r\nBC\nB\nC\n\u2261\n=\n( ) \ufffd\n( )\n\u03b52\ufffd\n2\n(3 42)\nHence, the potential difference between the terminals A and C of the\ncombination is\n( )\n( )\n( )\n( )\nAC\n(A)\u2013\n(C)\nA \u2013\nB\nB \u2013\nC\nV\nV\nV\nV\nV\nV\nV\n\u2261\n=\n+\n \n \n \n \n \n \n \n \n       \n(\n)\n(\n)\n1\n2\n1\n2\n\u2013 I r\nr\n\u03b5\n\u03b5\n=\n+\n+\n(3 43)\nIf we wish to replace the combination by a single cell between A and\nC of emf eeq and internal resistance req, we would have\nVAC = eeq\u2013 I req\n(3 44)\nComparing the last two equations, we get\n  eeq = e1 + e2\n(3"}, {"Chapter": "1", "sentence_range": "2924-2927", "Text": "42)\nHence, the potential difference between the terminals A and C of the\ncombination is\n( )\n( )\n( )\n( )\nAC\n(A)\u2013\n(C)\nA \u2013\nB\nB \u2013\nC\nV\nV\nV\nV\nV\nV\nV\n\u2261\n=\n+\n \n \n \n \n \n \n \n \n       \n(\n)\n(\n)\n1\n2\n1\n2\n\u2013 I r\nr\n\u03b5\n\u03b5\n=\n+\n+\n(3 43)\nIf we wish to replace the combination by a single cell between A and\nC of emf eeq and internal resistance req, we would have\nVAC = eeq\u2013 I req\n(3 44)\nComparing the last two equations, we get\n  eeq = e1 + e2\n(3 45)\nand  req = r1 + r2\n(3"}, {"Chapter": "1", "sentence_range": "2925-2928", "Text": "43)\nIf we wish to replace the combination by a single cell between A and\nC of emf eeq and internal resistance req, we would have\nVAC = eeq\u2013 I req\n(3 44)\nComparing the last two equations, we get\n  eeq = e1 + e2\n(3 45)\nand  req = r1 + r2\n(3 46)\nIn Fig"}, {"Chapter": "1", "sentence_range": "2926-2929", "Text": "44)\nComparing the last two equations, we get\n  eeq = e1 + e2\n(3 45)\nand  req = r1 + r2\n(3 46)\nIn Fig 3"}, {"Chapter": "1", "sentence_range": "2927-2930", "Text": "45)\nand  req = r1 + r2\n(3 46)\nIn Fig 3 13, we had connected the negative electrode of the first to the\npositive electrode of the second"}, {"Chapter": "1", "sentence_range": "2928-2931", "Text": "46)\nIn Fig 3 13, we had connected the negative electrode of the first to the\npositive electrode of the second If instead we connect the two negatives,\nRationalised 2023-24\nPhysics\n96\nEq"}, {"Chapter": "1", "sentence_range": "2929-2932", "Text": "3 13, we had connected the negative electrode of the first to the\npositive electrode of the second If instead we connect the two negatives,\nRationalised 2023-24\nPhysics\n96\nEq (3"}, {"Chapter": "1", "sentence_range": "2930-2933", "Text": "13, we had connected the negative electrode of the first to the\npositive electrode of the second If instead we connect the two negatives,\nRationalised 2023-24\nPhysics\n96\nEq (3 42) would change to VBC = \u2013e2\u2013Ir2 and we will get\neeq = e1 \u2013 e2         (e1 > e2)\n(3"}, {"Chapter": "1", "sentence_range": "2931-2934", "Text": "If instead we connect the two negatives,\nRationalised 2023-24\nPhysics\n96\nEq (3 42) would change to VBC = \u2013e2\u2013Ir2 and we will get\neeq = e1 \u2013 e2         (e1 > e2)\n(3 47)\nThe rule for series combination clearly can be extended to any number\nof cells:\n(i)\nThe equivalent emf of a series combination of n cells is just the sum of\ntheir individual emf\u2019s, and\n(ii) The equivalent internal resistance of a series combination of n cells is\njust the sum of their internal resistances"}, {"Chapter": "1", "sentence_range": "2932-2935", "Text": "(3 42) would change to VBC = \u2013e2\u2013Ir2 and we will get\neeq = e1 \u2013 e2         (e1 > e2)\n(3 47)\nThe rule for series combination clearly can be extended to any number\nof cells:\n(i)\nThe equivalent emf of a series combination of n cells is just the sum of\ntheir individual emf\u2019s, and\n(ii) The equivalent internal resistance of a series combination of n cells is\njust the sum of their internal resistances This is so, when the current leaves each cell from the positive\nelectrode"}, {"Chapter": "1", "sentence_range": "2933-2936", "Text": "42) would change to VBC = \u2013e2\u2013Ir2 and we will get\neeq = e1 \u2013 e2         (e1 > e2)\n(3 47)\nThe rule for series combination clearly can be extended to any number\nof cells:\n(i)\nThe equivalent emf of a series combination of n cells is just the sum of\ntheir individual emf\u2019s, and\n(ii) The equivalent internal resistance of a series combination of n cells is\njust the sum of their internal resistances This is so, when the current leaves each cell from the positive\nelectrode If in the combination, the current leaves any cell from\nthe negative electrode, the emf of the cell enters the expression\nfor eeq with a negative sign, as in Eq"}, {"Chapter": "1", "sentence_range": "2934-2937", "Text": "47)\nThe rule for series combination clearly can be extended to any number\nof cells:\n(i)\nThe equivalent emf of a series combination of n cells is just the sum of\ntheir individual emf\u2019s, and\n(ii) The equivalent internal resistance of a series combination of n cells is\njust the sum of their internal resistances This is so, when the current leaves each cell from the positive\nelectrode If in the combination, the current leaves any cell from\nthe negative electrode, the emf of the cell enters the expression\nfor eeq with a negative sign, as in Eq (3"}, {"Chapter": "1", "sentence_range": "2935-2938", "Text": "This is so, when the current leaves each cell from the positive\nelectrode If in the combination, the current leaves any cell from\nthe negative electrode, the emf of the cell enters the expression\nfor eeq with a negative sign, as in Eq (3 47)"}, {"Chapter": "1", "sentence_range": "2936-2939", "Text": "If in the combination, the current leaves any cell from\nthe negative electrode, the emf of the cell enters the expression\nfor eeq with a negative sign, as in Eq (3 47) Next, consider a parallel combination of the cells (Fig"}, {"Chapter": "1", "sentence_range": "2937-2940", "Text": "(3 47) Next, consider a parallel combination of the cells (Fig 3"}, {"Chapter": "1", "sentence_range": "2938-2941", "Text": "47) Next, consider a parallel combination of the cells (Fig 3 14)"}, {"Chapter": "1", "sentence_range": "2939-2942", "Text": "Next, consider a parallel combination of the cells (Fig 3 14) I1 and I2 are the currents leaving the positive electrodes of the\ncells"}, {"Chapter": "1", "sentence_range": "2940-2943", "Text": "3 14) I1 and I2 are the currents leaving the positive electrodes of the\ncells At the point B1, I1 and I2 flow in whereas the current I flows\nout"}, {"Chapter": "1", "sentence_range": "2941-2944", "Text": "14) I1 and I2 are the currents leaving the positive electrodes of the\ncells At the point B1, I1 and I2 flow in whereas the current I flows\nout Since as much charge flows in as out, we have\nI = I1 + I2\n(3"}, {"Chapter": "1", "sentence_range": "2942-2945", "Text": "I1 and I2 are the currents leaving the positive electrodes of the\ncells At the point B1, I1 and I2 flow in whereas the current I flows\nout Since as much charge flows in as out, we have\nI = I1 + I2\n(3 48)\nLet V (B1) and V (B2) be the potentials at B1 and B2, respectively"}, {"Chapter": "1", "sentence_range": "2943-2946", "Text": "At the point B1, I1 and I2 flow in whereas the current I flows\nout Since as much charge flows in as out, we have\nI = I1 + I2\n(3 48)\nLet V (B1) and V (B2) be the potentials at B1 and B2, respectively Then, considering the first cell, the potential difference across its\nterminals is V (B1) \u2013 V (B2)"}, {"Chapter": "1", "sentence_range": "2944-2947", "Text": "Since as much charge flows in as out, we have\nI = I1 + I2\n(3 48)\nLet V (B1) and V (B2) be the potentials at B1 and B2, respectively Then, considering the first cell, the potential difference across its\nterminals is V (B1) \u2013 V (B2) Hence, from Eq"}, {"Chapter": "1", "sentence_range": "2945-2948", "Text": "48)\nLet V (B1) and V (B2) be the potentials at B1 and B2, respectively Then, considering the first cell, the potential difference across its\nterminals is V (B1) \u2013 V (B2) Hence, from Eq (3"}, {"Chapter": "1", "sentence_range": "2946-2949", "Text": "Then, considering the first cell, the potential difference across its\nterminals is V (B1) \u2013 V (B2) Hence, from Eq (3 38)\n(\n)\n(\n)\n1\n2\n1\n1 1\n\u2013\n\u2013\nV\nV B\nV B\nI r\n\u03b5\n\u2261\n=\n(3"}, {"Chapter": "1", "sentence_range": "2947-2950", "Text": "Hence, from Eq (3 38)\n(\n)\n(\n)\n1\n2\n1\n1 1\n\u2013\n\u2013\nV\nV B\nV B\nI r\n\u03b5\n\u2261\n=\n(3 49)\nPoints B1 and B2 are connected exactly similarly to the second\ncell"}, {"Chapter": "1", "sentence_range": "2948-2951", "Text": "(3 38)\n(\n)\n(\n)\n1\n2\n1\n1 1\n\u2013\n\u2013\nV\nV B\nV B\nI r\n\u03b5\n\u2261\n=\n(3 49)\nPoints B1 and B2 are connected exactly similarly to the second\ncell Hence considering the second cell, we also have\n(\n)\n(\n)\n1\n2\n2\n2 2\n\u2013\n\u2013\nV\nV B\nV B\nI r\n\u03b5\n\u2261\n=\n(3"}, {"Chapter": "1", "sentence_range": "2949-2952", "Text": "38)\n(\n)\n(\n)\n1\n2\n1\n1 1\n\u2013\n\u2013\nV\nV B\nV B\nI r\n\u03b5\n\u2261\n=\n(3 49)\nPoints B1 and B2 are connected exactly similarly to the second\ncell Hence considering the second cell, we also have\n(\n)\n(\n)\n1\n2\n2\n2 2\n\u2013\n\u2013\nV\nV B\nV B\nI r\n\u03b5\n\u2261\n=\n(3 50)\nCombining the last three equations\n1\n2\n \n \n \n \nI\nI\nI\n=\n+\n    =\n+\n=\n+\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n+\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\u03b5\n\u03b5\n\u03b5\n\u03b5\n1\n1\n2\n2\n1\n1\n2\n2\n1\n2\n1\n1\n\u2013\n\u2013\n\u2013\nrV\nrV\nr\nr\nV\nr\nr\n(3"}, {"Chapter": "1", "sentence_range": "2950-2953", "Text": "49)\nPoints B1 and B2 are connected exactly similarly to the second\ncell Hence considering the second cell, we also have\n(\n)\n(\n)\n1\n2\n2\n2 2\n\u2013\n\u2013\nV\nV B\nV B\nI r\n\u03b5\n\u2261\n=\n(3 50)\nCombining the last three equations\n1\n2\n \n \n \n \nI\nI\nI\n=\n+\n    =\n+\n=\n+\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n+\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\u03b5\n\u03b5\n\u03b5\n\u03b5\n1\n1\n2\n2\n1\n1\n2\n2\n1\n2\n1\n1\n\u2013\n\u2013\n\u2013\nrV\nrV\nr\nr\nV\nr\nr\n(3 51)\nHence, V is given by,\n1 2\n2 1\n1 2\n1\n2\n1\n2\n\u2013\nr\nr\nr r\nV\nI\nr\nr\nr\nr\n\u03b5\n+\u03b5\n=\n+\n+\n(3"}, {"Chapter": "1", "sentence_range": "2951-2954", "Text": "Hence considering the second cell, we also have\n(\n)\n(\n)\n1\n2\n2\n2 2\n\u2013\n\u2013\nV\nV B\nV B\nI r\n\u03b5\n\u2261\n=\n(3 50)\nCombining the last three equations\n1\n2\n \n \n \n \nI\nI\nI\n=\n+\n    =\n+\n=\n+\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n+\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\u03b5\n\u03b5\n\u03b5\n\u03b5\n1\n1\n2\n2\n1\n1\n2\n2\n1\n2\n1\n1\n\u2013\n\u2013\n\u2013\nrV\nrV\nr\nr\nV\nr\nr\n(3 51)\nHence, V is given by,\n1 2\n2 1\n1 2\n1\n2\n1\n2\n\u2013\nr\nr\nr r\nV\nI\nr\nr\nr\nr\n\u03b5\n+\u03b5\n=\n+\n+\n(3 52)\nIf we want to replace the combination by a single cell, between B1 and\nB2, of emf eeq and internal resistance req, we would have\nV = eeq \u2013 I req\n(3"}, {"Chapter": "1", "sentence_range": "2952-2955", "Text": "50)\nCombining the last three equations\n1\n2\n \n \n \n \nI\nI\nI\n=\n+\n    =\n+\n=\n+\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n+\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\u03b5\n\u03b5\n\u03b5\n\u03b5\n1\n1\n2\n2\n1\n1\n2\n2\n1\n2\n1\n1\n\u2013\n\u2013\n\u2013\nrV\nrV\nr\nr\nV\nr\nr\n(3 51)\nHence, V is given by,\n1 2\n2 1\n1 2\n1\n2\n1\n2\n\u2013\nr\nr\nr r\nV\nI\nr\nr\nr\nr\n\u03b5\n+\u03b5\n=\n+\n+\n(3 52)\nIf we want to replace the combination by a single cell, between B1 and\nB2, of emf eeq and internal resistance req, we would have\nV = eeq \u2013 I req\n(3 53)\nThe last two equations should be the same and hence\n1 2\n2 1\n1\n2\neq\nr\nr\nr\nr\n\u03b5\n\u03b5\n\u03b5\n+\n=\n+\n(3"}, {"Chapter": "1", "sentence_range": "2953-2956", "Text": "51)\nHence, V is given by,\n1 2\n2 1\n1 2\n1\n2\n1\n2\n\u2013\nr\nr\nr r\nV\nI\nr\nr\nr\nr\n\u03b5\n+\u03b5\n=\n+\n+\n(3 52)\nIf we want to replace the combination by a single cell, between B1 and\nB2, of emf eeq and internal resistance req, we would have\nV = eeq \u2013 I req\n(3 53)\nThe last two equations should be the same and hence\n1 2\n2 1\n1\n2\neq\nr\nr\nr\nr\n\u03b5\n\u03b5\n\u03b5\n+\n=\n+\n(3 54)\n1 2\n1\n2\neq\nr r\nr\nr\nr\n=\n+\n(3"}, {"Chapter": "1", "sentence_range": "2954-2957", "Text": "52)\nIf we want to replace the combination by a single cell, between B1 and\nB2, of emf eeq and internal resistance req, we would have\nV = eeq \u2013 I req\n(3 53)\nThe last two equations should be the same and hence\n1 2\n2 1\n1\n2\neq\nr\nr\nr\nr\n\u03b5\n\u03b5\n\u03b5\n+\n=\n+\n(3 54)\n1 2\n1\n2\neq\nr r\nr\nr\nr\n=\n+\n(3 55)\nWe can put these equations in a simpler way,\nFIGURE 3"}, {"Chapter": "1", "sentence_range": "2955-2958", "Text": "53)\nThe last two equations should be the same and hence\n1 2\n2 1\n1\n2\neq\nr\nr\nr\nr\n\u03b5\n\u03b5\n\u03b5\n+\n=\n+\n(3 54)\n1 2\n1\n2\neq\nr r\nr\nr\nr\n=\n+\n(3 55)\nWe can put these equations in a simpler way,\nFIGURE 3 14 Two cells in\nparallel"}, {"Chapter": "1", "sentence_range": "2956-2959", "Text": "54)\n1 2\n1\n2\neq\nr r\nr\nr\nr\n=\n+\n(3 55)\nWe can put these equations in a simpler way,\nFIGURE 3 14 Two cells in\nparallel For connections\nacross A and C, the\ncombination can be\nreplaced by one cell of emf\neeq and internal resistances\nreq whose values are given in\nEqs"}, {"Chapter": "1", "sentence_range": "2957-2960", "Text": "55)\nWe can put these equations in a simpler way,\nFIGURE 3 14 Two cells in\nparallel For connections\nacross A and C, the\ncombination can be\nreplaced by one cell of emf\neeq and internal resistances\nreq whose values are given in\nEqs (3"}, {"Chapter": "1", "sentence_range": "2958-2961", "Text": "14 Two cells in\nparallel For connections\nacross A and C, the\ncombination can be\nreplaced by one cell of emf\neeq and internal resistances\nreq whose values are given in\nEqs (3 54) and (3"}, {"Chapter": "1", "sentence_range": "2959-2962", "Text": "For connections\nacross A and C, the\ncombination can be\nreplaced by one cell of emf\neeq and internal resistances\nreq whose values are given in\nEqs (3 54) and (3 55)"}, {"Chapter": "1", "sentence_range": "2960-2963", "Text": "(3 54) and (3 55) Rationalised 2023-24\nCurrent\nElectricity\n97\n1\n2\n1\n1\n1\nreq\nr\nr\n=\n+\n(3"}, {"Chapter": "1", "sentence_range": "2961-2964", "Text": "54) and (3 55) Rationalised 2023-24\nCurrent\nElectricity\n97\n1\n2\n1\n1\n1\nreq\nr\nr\n=\n+\n(3 56)\n1\n2\n1\n2\neq\nreq\nr\nr\n\u03b5\n\u03b5\n\u03b5\n=\n+\n(3"}, {"Chapter": "1", "sentence_range": "2962-2965", "Text": "55) Rationalised 2023-24\nCurrent\nElectricity\n97\n1\n2\n1\n1\n1\nreq\nr\nr\n=\n+\n(3 56)\n1\n2\n1\n2\neq\nreq\nr\nr\n\u03b5\n\u03b5\n\u03b5\n=\n+\n(3 57)\nIn Fig"}, {"Chapter": "1", "sentence_range": "2963-2966", "Text": "Rationalised 2023-24\nCurrent\nElectricity\n97\n1\n2\n1\n1\n1\nreq\nr\nr\n=\n+\n(3 56)\n1\n2\n1\n2\neq\nreq\nr\nr\n\u03b5\n\u03b5\n\u03b5\n=\n+\n(3 57)\nIn Fig (3"}, {"Chapter": "1", "sentence_range": "2964-2967", "Text": "56)\n1\n2\n1\n2\neq\nreq\nr\nr\n\u03b5\n\u03b5\n\u03b5\n=\n+\n(3 57)\nIn Fig (3 14), we had joined the positive terminals\ntogether and similarly the two negative ones, so that the\ncurrents I1, I2 flow out of positive terminals"}, {"Chapter": "1", "sentence_range": "2965-2968", "Text": "57)\nIn Fig (3 14), we had joined the positive terminals\ntogether and similarly the two negative ones, so that the\ncurrents I1, I2 flow out of positive terminals If the negative\nterminal of the second is connected to positive terminal\nof the first, Eqs"}, {"Chapter": "1", "sentence_range": "2966-2969", "Text": "(3 14), we had joined the positive terminals\ntogether and similarly the two negative ones, so that the\ncurrents I1, I2 flow out of positive terminals If the negative\nterminal of the second is connected to positive terminal\nof the first, Eqs (3"}, {"Chapter": "1", "sentence_range": "2967-2970", "Text": "14), we had joined the positive terminals\ntogether and similarly the two negative ones, so that the\ncurrents I1, I2 flow out of positive terminals If the negative\nterminal of the second is connected to positive terminal\nof the first, Eqs (3 56) and (3"}, {"Chapter": "1", "sentence_range": "2968-2971", "Text": "If the negative\nterminal of the second is connected to positive terminal\nof the first, Eqs (3 56) and (3 57) would still be valid with\ne 2 \u00ae \u2013e2\nEquations (3"}, {"Chapter": "1", "sentence_range": "2969-2972", "Text": "(3 56) and (3 57) would still be valid with\ne 2 \u00ae \u2013e2\nEquations (3 56) and (3"}, {"Chapter": "1", "sentence_range": "2970-2973", "Text": "56) and (3 57) would still be valid with\ne 2 \u00ae \u2013e2\nEquations (3 56) and (3 57) can be extended easily"}, {"Chapter": "1", "sentence_range": "2971-2974", "Text": "57) would still be valid with\ne 2 \u00ae \u2013e2\nEquations (3 56) and (3 57) can be extended easily If there are n cells of emf e1,"}, {"Chapter": "1", "sentence_range": "2972-2975", "Text": "56) and (3 57) can be extended easily If there are n cells of emf e1, en and of internal\nresistances r1,"}, {"Chapter": "1", "sentence_range": "2973-2976", "Text": "57) can be extended easily If there are n cells of emf e1, en and of internal\nresistances r1, rn respectively, connected in parallel, the\ncombination is equivalent to a single cell of emf eeq and\ninternal resistance req, such that\n1\n1\n1\n1\nr\nr\nr\neq\nn\n=\n+"}, {"Chapter": "1", "sentence_range": "2974-2977", "Text": "If there are n cells of emf e1, en and of internal\nresistances r1, rn respectively, connected in parallel, the\ncombination is equivalent to a single cell of emf eeq and\ninternal resistance req, such that\n1\n1\n1\n1\nr\nr\nr\neq\nn\n=\n+ +\n(3"}, {"Chapter": "1", "sentence_range": "2975-2978", "Text": "en and of internal\nresistances r1, rn respectively, connected in parallel, the\ncombination is equivalent to a single cell of emf eeq and\ninternal resistance req, such that\n1\n1\n1\n1\nr\nr\nr\neq\nn\n=\n+ +\n(3 58)\n\u03b5\n\u03b5\n\u03b5\neq\neq\nn\nn\nr\nr\nr\n=\n+\n+\n1\n1"}, {"Chapter": "1", "sentence_range": "2976-2979", "Text": "rn respectively, connected in parallel, the\ncombination is equivalent to a single cell of emf eeq and\ninternal resistance req, such that\n1\n1\n1\n1\nr\nr\nr\neq\nn\n=\n+ +\n(3 58)\n\u03b5\n\u03b5\n\u03b5\neq\neq\nn\nn\nr\nr\nr\n=\n+\n+\n1\n1 (3"}, {"Chapter": "1", "sentence_range": "2977-2980", "Text": "+\n(3 58)\n\u03b5\n\u03b5\n\u03b5\neq\neq\nn\nn\nr\nr\nr\n=\n+\n+\n1\n1 (3 59)\n3"}, {"Chapter": "1", "sentence_range": "2978-2981", "Text": "58)\n\u03b5\n\u03b5\n\u03b5\neq\neq\nn\nn\nr\nr\nr\n=\n+\n+\n1\n1 (3 59)\n3 12  KIRCHHOFF\u2019S RULES\nElectric circuits generally consist of a number of resistors\nand cells interconnected sometimes in a complicated way"}, {"Chapter": "1", "sentence_range": "2979-2982", "Text": "(3 59)\n3 12  KIRCHHOFF\u2019S RULES\nElectric circuits generally consist of a number of resistors\nand cells interconnected sometimes in a complicated way The formulae we have derived earlier for series and parallel combinations\nof resistors are not always sufficient to determine all the currents and\npotential differences in the circuit"}, {"Chapter": "1", "sentence_range": "2980-2983", "Text": "59)\n3 12  KIRCHHOFF\u2019S RULES\nElectric circuits generally consist of a number of resistors\nand cells interconnected sometimes in a complicated way The formulae we have derived earlier for series and parallel combinations\nof resistors are not always sufficient to determine all the currents and\npotential differences in the circuit Two rules, called Kirchhoff\u2019s rules,\nare very useful for analysis of electric circuits"}, {"Chapter": "1", "sentence_range": "2981-2984", "Text": "12  KIRCHHOFF\u2019S RULES\nElectric circuits generally consist of a number of resistors\nand cells interconnected sometimes in a complicated way The formulae we have derived earlier for series and parallel combinations\nof resistors are not always sufficient to determine all the currents and\npotential differences in the circuit Two rules, called Kirchhoff\u2019s rules,\nare very useful for analysis of electric circuits Given a circuit, we start by labelling currents in each resistor by a\nsymbol, say I, and a directed arrow to indicate that a current I flows\nalong the resistor in the direction indicated"}, {"Chapter": "1", "sentence_range": "2982-2985", "Text": "The formulae we have derived earlier for series and parallel combinations\nof resistors are not always sufficient to determine all the currents and\npotential differences in the circuit Two rules, called Kirchhoff\u2019s rules,\nare very useful for analysis of electric circuits Given a circuit, we start by labelling currents in each resistor by a\nsymbol, say I, and a directed arrow to indicate that a current I flows\nalong the resistor in the direction indicated If ultimately I is determined\nto be positive, the actual current in the resistor is in the direction of the\narrow"}, {"Chapter": "1", "sentence_range": "2983-2986", "Text": "Two rules, called Kirchhoff\u2019s rules,\nare very useful for analysis of electric circuits Given a circuit, we start by labelling currents in each resistor by a\nsymbol, say I, and a directed arrow to indicate that a current I flows\nalong the resistor in the direction indicated If ultimately I is determined\nto be positive, the actual current in the resistor is in the direction of the\narrow If I turns out to be negative, the current actually flows in a direction\nopposite to the arrow"}, {"Chapter": "1", "sentence_range": "2984-2987", "Text": "Given a circuit, we start by labelling currents in each resistor by a\nsymbol, say I, and a directed arrow to indicate that a current I flows\nalong the resistor in the direction indicated If ultimately I is determined\nto be positive, the actual current in the resistor is in the direction of the\narrow If I turns out to be negative, the current actually flows in a direction\nopposite to the arrow Similarly, for each source (i"}, {"Chapter": "1", "sentence_range": "2985-2988", "Text": "If ultimately I is determined\nto be positive, the actual current in the resistor is in the direction of the\narrow If I turns out to be negative, the current actually flows in a direction\nopposite to the arrow Similarly, for each source (i e"}, {"Chapter": "1", "sentence_range": "2986-2989", "Text": "If I turns out to be negative, the current actually flows in a direction\nopposite to the arrow Similarly, for each source (i e , cell or some other\nsource of electrical power) the positive and negative electrodes are labelled,\nas well as, a directed arrow with a symbol for the current flowing through\nthe cell"}, {"Chapter": "1", "sentence_range": "2987-2990", "Text": "Similarly, for each source (i e , cell or some other\nsource of electrical power) the positive and negative electrodes are labelled,\nas well as, a directed arrow with a symbol for the current flowing through\nthe cell This will tell us the potential difference, V = V (P) \u2013 V (N) = e \u2013 I r\n[Eq"}, {"Chapter": "1", "sentence_range": "2988-2991", "Text": "e , cell or some other\nsource of electrical power) the positive and negative electrodes are labelled,\nas well as, a directed arrow with a symbol for the current flowing through\nthe cell This will tell us the potential difference, V = V (P) \u2013 V (N) = e \u2013 I r\n[Eq (3"}, {"Chapter": "1", "sentence_range": "2989-2992", "Text": ", cell or some other\nsource of electrical power) the positive and negative electrodes are labelled,\nas well as, a directed arrow with a symbol for the current flowing through\nthe cell This will tell us the potential difference, V = V (P) \u2013 V (N) = e \u2013 I r\n[Eq (3 38) between the positive terminal P and the negative terminal N; I\nhere is the current flowing from N to P through the cell]"}, {"Chapter": "1", "sentence_range": "2990-2993", "Text": "This will tell us the potential difference, V = V (P) \u2013 V (N) = e \u2013 I r\n[Eq (3 38) between the positive terminal P and the negative terminal N; I\nhere is the current flowing from N to P through the cell] If, while labelling\nthe current I through the cell one goes from P to N, then of course\nV = e + I r\n(3"}, {"Chapter": "1", "sentence_range": "2991-2994", "Text": "(3 38) between the positive terminal P and the negative terminal N; I\nhere is the current flowing from N to P through the cell] If, while labelling\nthe current I through the cell one goes from P to N, then of course\nV = e + I r\n(3 60)\n(a)Having clarified labelling, we now state the rules and the proof:\nJunction rule: At any junction, the sum of the currents entering\nthe junction is equal to the sum of currents leaving the junction\n(Fig"}, {"Chapter": "1", "sentence_range": "2992-2995", "Text": "38) between the positive terminal P and the negative terminal N; I\nhere is the current flowing from N to P through the cell] If, while labelling\nthe current I through the cell one goes from P to N, then of course\nV = e + I r\n(3 60)\n(a)Having clarified labelling, we now state the rules and the proof:\nJunction rule: At any junction, the sum of the currents entering\nthe junction is equal to the sum of currents leaving the junction\n(Fig 3"}, {"Chapter": "1", "sentence_range": "2993-2996", "Text": "If, while labelling\nthe current I through the cell one goes from P to N, then of course\nV = e + I r\n(3 60)\n(a)Having clarified labelling, we now state the rules and the proof:\nJunction rule: At any junction, the sum of the currents entering\nthe junction is equal to the sum of currents leaving the junction\n(Fig 3 15)"}, {"Chapter": "1", "sentence_range": "2994-2997", "Text": "60)\n(a)Having clarified labelling, we now state the rules and the proof:\nJunction rule: At any junction, the sum of the currents entering\nthe junction is equal to the sum of currents leaving the junction\n(Fig 3 15) Gustav Robert Kirchhoff\n(1824 \u2013 1887) German\nphysicist, professor at\nHeidelberg \nand \nat\nBerlin"}, {"Chapter": "1", "sentence_range": "2995-2998", "Text": "3 15) Gustav Robert Kirchhoff\n(1824 \u2013 1887) German\nphysicist, professor at\nHeidelberg \nand \nat\nBerlin Mainly known for\nhis \ndevelopment \nof\nspectroscopy, he also\nmade many important\ncontributions to mathe-\nmatical physics, among\nthem, his first and\nsecond rules for circuits"}, {"Chapter": "1", "sentence_range": "2996-2999", "Text": "15) Gustav Robert Kirchhoff\n(1824 \u2013 1887) German\nphysicist, professor at\nHeidelberg \nand \nat\nBerlin Mainly known for\nhis \ndevelopment \nof\nspectroscopy, he also\nmade many important\ncontributions to mathe-\nmatical physics, among\nthem, his first and\nsecond rules for circuits GUSTAV ROBERT KIRCHHOFF (1824 \u2013\nRationalised 2023-24\nPhysics\n98\n EXAMPLE 3"}, {"Chapter": "1", "sentence_range": "2997-3000", "Text": "Gustav Robert Kirchhoff\n(1824 \u2013 1887) German\nphysicist, professor at\nHeidelberg \nand \nat\nBerlin Mainly known for\nhis \ndevelopment \nof\nspectroscopy, he also\nmade many important\ncontributions to mathe-\nmatical physics, among\nthem, his first and\nsecond rules for circuits GUSTAV ROBERT KIRCHHOFF (1824 \u2013\nRationalised 2023-24\nPhysics\n98\n EXAMPLE 3 5\nThis applies equally well if instead of a junction of\nseveral lines, we consider a point in a line"}, {"Chapter": "1", "sentence_range": "2998-3001", "Text": "Mainly known for\nhis \ndevelopment \nof\nspectroscopy, he also\nmade many important\ncontributions to mathe-\nmatical physics, among\nthem, his first and\nsecond rules for circuits GUSTAV ROBERT KIRCHHOFF (1824 \u2013\nRationalised 2023-24\nPhysics\n98\n EXAMPLE 3 5\nThis applies equally well if instead of a junction of\nseveral lines, we consider a point in a line The proof of this rule follows from the fact that\nwhen currents are steady, there is no accumulation\nof charges at any junction or at any point in a line"}, {"Chapter": "1", "sentence_range": "2999-3002", "Text": "GUSTAV ROBERT KIRCHHOFF (1824 \u2013\nRationalised 2023-24\nPhysics\n98\n EXAMPLE 3 5\nThis applies equally well if instead of a junction of\nseveral lines, we consider a point in a line The proof of this rule follows from the fact that\nwhen currents are steady, there is no accumulation\nof charges at any junction or at any point in a line Thus, the total current flowing in, (which is the rate\nat which charge flows into the junction), must equal\nthe total current flowing out"}, {"Chapter": "1", "sentence_range": "3000-3003", "Text": "5\nThis applies equally well if instead of a junction of\nseveral lines, we consider a point in a line The proof of this rule follows from the fact that\nwhen currents are steady, there is no accumulation\nof charges at any junction or at any point in a line Thus, the total current flowing in, (which is the rate\nat which charge flows into the junction), must equal\nthe total current flowing out (b)\nLoop rule: The algebraic sum of changes in\npotential around any closed loop involving\nresistors and cells in the loop is zero\n(Fig"}, {"Chapter": "1", "sentence_range": "3001-3004", "Text": "The proof of this rule follows from the fact that\nwhen currents are steady, there is no accumulation\nof charges at any junction or at any point in a line Thus, the total current flowing in, (which is the rate\nat which charge flows into the junction), must equal\nthe total current flowing out (b)\nLoop rule: The algebraic sum of changes in\npotential around any closed loop involving\nresistors and cells in the loop is zero\n(Fig 3"}, {"Chapter": "1", "sentence_range": "3002-3005", "Text": "Thus, the total current flowing in, (which is the rate\nat which charge flows into the junction), must equal\nthe total current flowing out (b)\nLoop rule: The algebraic sum of changes in\npotential around any closed loop involving\nresistors and cells in the loop is zero\n(Fig 3 15)"}, {"Chapter": "1", "sentence_range": "3003-3006", "Text": "(b)\nLoop rule: The algebraic sum of changes in\npotential around any closed loop involving\nresistors and cells in the loop is zero\n(Fig 3 15) This rule is also obvious, since electric potential is\ndependent on the location of the point"}, {"Chapter": "1", "sentence_range": "3004-3007", "Text": "3 15) This rule is also obvious, since electric potential is\ndependent on the location of the point Thus\nstarting with any point if we come back to the same\npoint, the total change must be zero"}, {"Chapter": "1", "sentence_range": "3005-3008", "Text": "15) This rule is also obvious, since electric potential is\ndependent on the location of the point Thus\nstarting with any point if we come back to the same\npoint, the total change must be zero In a closed\nloop, we do come back to the starting point and\nhence the rule"}, {"Chapter": "1", "sentence_range": "3006-3009", "Text": "This rule is also obvious, since electric potential is\ndependent on the location of the point Thus\nstarting with any point if we come back to the same\npoint, the total change must be zero In a closed\nloop, we do come back to the starting point and\nhence the rule FIGURE 3"}, {"Chapter": "1", "sentence_range": "3007-3010", "Text": "Thus\nstarting with any point if we come back to the same\npoint, the total change must be zero In a closed\nloop, we do come back to the starting point and\nhence the rule FIGURE 3 15 At junction a the current\nleaving is I1 + I2 and current entering is I3"}, {"Chapter": "1", "sentence_range": "3008-3011", "Text": "In a closed\nloop, we do come back to the starting point and\nhence the rule FIGURE 3 15 At junction a the current\nleaving is I1 + I2 and current entering is I3 The junction rule says I3 = I1 + I2"}, {"Chapter": "1", "sentence_range": "3009-3012", "Text": "FIGURE 3 15 At junction a the current\nleaving is I1 + I2 and current entering is I3 The junction rule says I3 = I1 + I2 At point\nh current entering is I1"}, {"Chapter": "1", "sentence_range": "3010-3013", "Text": "15 At junction a the current\nleaving is I1 + I2 and current entering is I3 The junction rule says I3 = I1 + I2 At point\nh current entering is I1 There is only one\ncurrent leaving h and by junction rule\nthat will also be I1"}, {"Chapter": "1", "sentence_range": "3011-3014", "Text": "The junction rule says I3 = I1 + I2 At point\nh current entering is I1 There is only one\ncurrent leaving h and by junction rule\nthat will also be I1 For the loops \u2018ahdcba\u2019\nand \u2018ahdefga\u2019, the loop rules give \u201330I1 \u2013\n41 I3 + 45 = 0 and \u201330I1 + 21 I2 \u2013 80 = 0"}, {"Chapter": "1", "sentence_range": "3012-3015", "Text": "At point\nh current entering is I1 There is only one\ncurrent leaving h and by junction rule\nthat will also be I1 For the loops \u2018ahdcba\u2019\nand \u2018ahdefga\u2019, the loop rules give \u201330I1 \u2013\n41 I3 + 45 = 0 and \u201330I1 + 21 I2 \u2013 80 = 0 Example 3"}, {"Chapter": "1", "sentence_range": "3013-3016", "Text": "There is only one\ncurrent leaving h and by junction rule\nthat will also be I1 For the loops \u2018ahdcba\u2019\nand \u2018ahdefga\u2019, the loop rules give \u201330I1 \u2013\n41 I3 + 45 = 0 and \u201330I1 + 21 I2 \u2013 80 = 0 Example 3 5 A battery of 10 V and negligible internal resistance is\nconnected across the diagonally opposite corners of a cubical network\nconsisting of 12 resistors each of resistance 1 W (Fig"}, {"Chapter": "1", "sentence_range": "3014-3017", "Text": "For the loops \u2018ahdcba\u2019\nand \u2018ahdefga\u2019, the loop rules give \u201330I1 \u2013\n41 I3 + 45 = 0 and \u201330I1 + 21 I2 \u2013 80 = 0 Example 3 5 A battery of 10 V and negligible internal resistance is\nconnected across the diagonally opposite corners of a cubical network\nconsisting of 12 resistors each of resistance 1 W (Fig 3"}, {"Chapter": "1", "sentence_range": "3015-3018", "Text": "Example 3 5 A battery of 10 V and negligible internal resistance is\nconnected across the diagonally opposite corners of a cubical network\nconsisting of 12 resistors each of resistance 1 W (Fig 3 16)"}, {"Chapter": "1", "sentence_range": "3016-3019", "Text": "5 A battery of 10 V and negligible internal resistance is\nconnected across the diagonally opposite corners of a cubical network\nconsisting of 12 resistors each of resistance 1 W (Fig 3 16) Determine\nthe equivalent resistance of the network and the current along each\nedge of the cube"}, {"Chapter": "1", "sentence_range": "3017-3020", "Text": "3 16) Determine\nthe equivalent resistance of the network and the current along each\nedge of the cube Z\nFIGURE 3"}, {"Chapter": "1", "sentence_range": "3018-3021", "Text": "16) Determine\nthe equivalent resistance of the network and the current along each\nedge of the cube Z\nFIGURE 3 16\nRationalised 2023-24\nCurrent\nElectricity\n99\n EXAMPLE 3"}, {"Chapter": "1", "sentence_range": "3019-3022", "Text": "Determine\nthe equivalent resistance of the network and the current along each\nedge of the cube Z\nFIGURE 3 16\nRationalised 2023-24\nCurrent\nElectricity\n99\n EXAMPLE 3 5\nSimilation for application of Kirchhoff\u2019s rules:\nhttp://www"}, {"Chapter": "1", "sentence_range": "3020-3023", "Text": "Z\nFIGURE 3 16\nRationalised 2023-24\nCurrent\nElectricity\n99\n EXAMPLE 3 5\nSimilation for application of Kirchhoff\u2019s rules:\nhttp://www phys"}, {"Chapter": "1", "sentence_range": "3021-3024", "Text": "16\nRationalised 2023-24\nCurrent\nElectricity\n99\n EXAMPLE 3 5\nSimilation for application of Kirchhoff\u2019s rules:\nhttp://www phys hawaii"}, {"Chapter": "1", "sentence_range": "3022-3025", "Text": "5\nSimilation for application of Kirchhoff\u2019s rules:\nhttp://www phys hawaii edu/~teb/optics/java/kirch3/\n EXAMPLE 3"}, {"Chapter": "1", "sentence_range": "3023-3026", "Text": "phys hawaii edu/~teb/optics/java/kirch3/\n EXAMPLE 3 6\nSolution  The network is not reducible to a simple series and parallel\ncombinations of resistors"}, {"Chapter": "1", "sentence_range": "3024-3027", "Text": "hawaii edu/~teb/optics/java/kirch3/\n EXAMPLE 3 6\nSolution  The network is not reducible to a simple series and parallel\ncombinations of resistors There is, however, a clear symmetry in the\nproblem which we can exploit to obtain the equivalent resistance of\nthe network"}, {"Chapter": "1", "sentence_range": "3025-3028", "Text": "edu/~teb/optics/java/kirch3/\n EXAMPLE 3 6\nSolution  The network is not reducible to a simple series and parallel\ncombinations of resistors There is, however, a clear symmetry in the\nproblem which we can exploit to obtain the equivalent resistance of\nthe network The paths AA\u00a2, AD and AB are obviously symmetrically placed in the\nnetwork"}, {"Chapter": "1", "sentence_range": "3026-3029", "Text": "6\nSolution  The network is not reducible to a simple series and parallel\ncombinations of resistors There is, however, a clear symmetry in the\nproblem which we can exploit to obtain the equivalent resistance of\nthe network The paths AA\u00a2, AD and AB are obviously symmetrically placed in the\nnetwork Thus, the current in each must be the same, say, I"}, {"Chapter": "1", "sentence_range": "3027-3030", "Text": "There is, however, a clear symmetry in the\nproblem which we can exploit to obtain the equivalent resistance of\nthe network The paths AA\u00a2, AD and AB are obviously symmetrically placed in the\nnetwork Thus, the current in each must be the same, say, I Further,\nat the corners A\u00a2, B and D, the incoming current I must split equally\ninto the two outgoing branches"}, {"Chapter": "1", "sentence_range": "3028-3031", "Text": "The paths AA\u00a2, AD and AB are obviously symmetrically placed in the\nnetwork Thus, the current in each must be the same, say, I Further,\nat the corners A\u00a2, B and D, the incoming current I must split equally\ninto the two outgoing branches In this manner, the current in all\nthe 12 edges of the cube are easily written down in terms of I, using\nKirchhoff\u2019s first rule and the symmetry in the problem"}, {"Chapter": "1", "sentence_range": "3029-3032", "Text": "Thus, the current in each must be the same, say, I Further,\nat the corners A\u00a2, B and D, the incoming current I must split equally\ninto the two outgoing branches In this manner, the current in all\nthe 12 edges of the cube are easily written down in terms of I, using\nKirchhoff\u2019s first rule and the symmetry in the problem Next take a closed loop, say, ABCC\u00a2EA, and apply Kirchhoff\u2019s second\nrule:\n\u2013IR \u2013 (1/2)IR \u2013 IR + e = 0\nwhere R is the resistance of each edge and e the emf of battery"}, {"Chapter": "1", "sentence_range": "3030-3033", "Text": "Further,\nat the corners A\u00a2, B and D, the incoming current I must split equally\ninto the two outgoing branches In this manner, the current in all\nthe 12 edges of the cube are easily written down in terms of I, using\nKirchhoff\u2019s first rule and the symmetry in the problem Next take a closed loop, say, ABCC\u00a2EA, and apply Kirchhoff\u2019s second\nrule:\n\u2013IR \u2013 (1/2)IR \u2013 IR + e = 0\nwhere R is the resistance of each edge and e the emf of battery Thus,\ne = 5\n2 I R\nThe equivalent resistance Req of the network is\n5\n3\n6\nReq\nR\n=\u03b5I\n=\nFor R = 1 W, Req = (5/6) W and for e = 10 V, the total current (= 3I) in\nthe network is\n3I = 10 V/(5/6) W = 12 A, i"}, {"Chapter": "1", "sentence_range": "3031-3034", "Text": "In this manner, the current in all\nthe 12 edges of the cube are easily written down in terms of I, using\nKirchhoff\u2019s first rule and the symmetry in the problem Next take a closed loop, say, ABCC\u00a2EA, and apply Kirchhoff\u2019s second\nrule:\n\u2013IR \u2013 (1/2)IR \u2013 IR + e = 0\nwhere R is the resistance of each edge and e the emf of battery Thus,\ne = 5\n2 I R\nThe equivalent resistance Req of the network is\n5\n3\n6\nReq\nR\n=\u03b5I\n=\nFor R = 1 W, Req = (5/6) W and for e = 10 V, the total current (= 3I) in\nthe network is\n3I = 10 V/(5/6) W = 12 A, i e"}, {"Chapter": "1", "sentence_range": "3032-3035", "Text": "Next take a closed loop, say, ABCC\u00a2EA, and apply Kirchhoff\u2019s second\nrule:\n\u2013IR \u2013 (1/2)IR \u2013 IR + e = 0\nwhere R is the resistance of each edge and e the emf of battery Thus,\ne = 5\n2 I R\nThe equivalent resistance Req of the network is\n5\n3\n6\nReq\nR\n=\u03b5I\n=\nFor R = 1 W, Req = (5/6) W and for e = 10 V, the total current (= 3I) in\nthe network is\n3I = 10 V/(5/6) W = 12 A, i e , I = 4 A\nThe current flowing in each edge can now be read off from the\nFig"}, {"Chapter": "1", "sentence_range": "3033-3036", "Text": "Thus,\ne = 5\n2 I R\nThe equivalent resistance Req of the network is\n5\n3\n6\nReq\nR\n=\u03b5I\n=\nFor R = 1 W, Req = (5/6) W and for e = 10 V, the total current (= 3I) in\nthe network is\n3I = 10 V/(5/6) W = 12 A, i e , I = 4 A\nThe current flowing in each edge can now be read off from the\nFig 3"}, {"Chapter": "1", "sentence_range": "3034-3037", "Text": "e , I = 4 A\nThe current flowing in each edge can now be read off from the\nFig 3 16"}, {"Chapter": "1", "sentence_range": "3035-3038", "Text": ", I = 4 A\nThe current flowing in each edge can now be read off from the\nFig 3 16 It should be noted that because of the symmetry of the network, the\ngreat power of Kirchhoff\u2019s rules has not been very apparent in Example 3"}, {"Chapter": "1", "sentence_range": "3036-3039", "Text": "3 16 It should be noted that because of the symmetry of the network, the\ngreat power of Kirchhoff\u2019s rules has not been very apparent in Example 3 5"}, {"Chapter": "1", "sentence_range": "3037-3040", "Text": "16 It should be noted that because of the symmetry of the network, the\ngreat power of Kirchhoff\u2019s rules has not been very apparent in Example 3 5 In a general network, there will be no such simplification due to symmetry,\nand only by application of Kirchhoff\u2019s rules to junctions and closed loops\n(as many as necessary to solve the unknowns in the network) can we\nhandle the problem"}, {"Chapter": "1", "sentence_range": "3038-3041", "Text": "It should be noted that because of the symmetry of the network, the\ngreat power of Kirchhoff\u2019s rules has not been very apparent in Example 3 5 In a general network, there will be no such simplification due to symmetry,\nand only by application of Kirchhoff\u2019s rules to junctions and closed loops\n(as many as necessary to solve the unknowns in the network) can we\nhandle the problem This will be illustrated in Example 3"}, {"Chapter": "1", "sentence_range": "3039-3042", "Text": "5 In a general network, there will be no such simplification due to symmetry,\nand only by application of Kirchhoff\u2019s rules to junctions and closed loops\n(as many as necessary to solve the unknowns in the network) can we\nhandle the problem This will be illustrated in Example 3 6"}, {"Chapter": "1", "sentence_range": "3040-3043", "Text": "In a general network, there will be no such simplification due to symmetry,\nand only by application of Kirchhoff\u2019s rules to junctions and closed loops\n(as many as necessary to solve the unknowns in the network) can we\nhandle the problem This will be illustrated in Example 3 6 Example 3"}, {"Chapter": "1", "sentence_range": "3041-3044", "Text": "This will be illustrated in Example 3 6 Example 3 6 Determine the current in each branch of the network\nshown in Fig"}, {"Chapter": "1", "sentence_range": "3042-3045", "Text": "6 Example 3 6 Determine the current in each branch of the network\nshown in Fig 3"}, {"Chapter": "1", "sentence_range": "3043-3046", "Text": "Example 3 6 Determine the current in each branch of the network\nshown in Fig 3 17"}, {"Chapter": "1", "sentence_range": "3044-3047", "Text": "6 Determine the current in each branch of the network\nshown in Fig 3 17 FIGURE 3"}, {"Chapter": "1", "sentence_range": "3045-3048", "Text": "3 17 FIGURE 3 17\nRationalised 2023-24\nPhysics\n100\n EXAMPLE 3"}, {"Chapter": "1", "sentence_range": "3046-3049", "Text": "17 FIGURE 3 17\nRationalised 2023-24\nPhysics\n100\n EXAMPLE 3 6\nSolution  Each branch of the network is assigned an unknown current\nto be determined by the application of Kirchhoff\u2019s rules"}, {"Chapter": "1", "sentence_range": "3047-3050", "Text": "FIGURE 3 17\nRationalised 2023-24\nPhysics\n100\n EXAMPLE 3 6\nSolution  Each branch of the network is assigned an unknown current\nto be determined by the application of Kirchhoff\u2019s rules To reduce\nthe number of unknowns at the outset, the first rule of Kirchhoff is\nused at every junction to assign the unknown current in each branch"}, {"Chapter": "1", "sentence_range": "3048-3051", "Text": "17\nRationalised 2023-24\nPhysics\n100\n EXAMPLE 3 6\nSolution  Each branch of the network is assigned an unknown current\nto be determined by the application of Kirchhoff\u2019s rules To reduce\nthe number of unknowns at the outset, the first rule of Kirchhoff is\nused at every junction to assign the unknown current in each branch We then have three unknowns I1, I2 and I3 which can be found by\napplying the second rule of Kirchhoff to three different closed loops"}, {"Chapter": "1", "sentence_range": "3049-3052", "Text": "6\nSolution  Each branch of the network is assigned an unknown current\nto be determined by the application of Kirchhoff\u2019s rules To reduce\nthe number of unknowns at the outset, the first rule of Kirchhoff is\nused at every junction to assign the unknown current in each branch We then have three unknowns I1, I2 and I3 which can be found by\napplying the second rule of Kirchhoff to three different closed loops Kirchhoff\u2019s second rule for the closed loop ADCA gives,\n10 \u2013 4(I1\u2013 I2) + 2(I2 + I3 \u2013 I1) \u2013 I1 = 0\n[3"}, {"Chapter": "1", "sentence_range": "3050-3053", "Text": "To reduce\nthe number of unknowns at the outset, the first rule of Kirchhoff is\nused at every junction to assign the unknown current in each branch We then have three unknowns I1, I2 and I3 which can be found by\napplying the second rule of Kirchhoff to three different closed loops Kirchhoff\u2019s second rule for the closed loop ADCA gives,\n10 \u2013 4(I1\u2013 I2) + 2(I2 + I3 \u2013 I1) \u2013 I1 = 0\n[3 61(a)]\nthat is, 7I1\u2013 6I2 \u2013 2I3 = 10\nFor the closed loop ABCA, we get\n10 \u2013 4I2\u2013 2 (I2 + I3) \u2013 I1 = 0\nthat is, I1 + 6I2 + 2I3 =10\n[3"}, {"Chapter": "1", "sentence_range": "3051-3054", "Text": "We then have three unknowns I1, I2 and I3 which can be found by\napplying the second rule of Kirchhoff to three different closed loops Kirchhoff\u2019s second rule for the closed loop ADCA gives,\n10 \u2013 4(I1\u2013 I2) + 2(I2 + I3 \u2013 I1) \u2013 I1 = 0\n[3 61(a)]\nthat is, 7I1\u2013 6I2 \u2013 2I3 = 10\nFor the closed loop ABCA, we get\n10 \u2013 4I2\u2013 2 (I2 + I3) \u2013 I1 = 0\nthat is, I1 + 6I2 + 2I3 =10\n[3 61(b)]\nFor the closed loop BCDEB, we get\n5 \u2013 2 (I2 + I3) \u2013 2 (I2 + I3 \u2013 I1) = 0\nthat is, 2I1 \u2013 4I2 \u2013 4I3 = \u20135\n[3"}, {"Chapter": "1", "sentence_range": "3052-3055", "Text": "Kirchhoff\u2019s second rule for the closed loop ADCA gives,\n10 \u2013 4(I1\u2013 I2) + 2(I2 + I3 \u2013 I1) \u2013 I1 = 0\n[3 61(a)]\nthat is, 7I1\u2013 6I2 \u2013 2I3 = 10\nFor the closed loop ABCA, we get\n10 \u2013 4I2\u2013 2 (I2 + I3) \u2013 I1 = 0\nthat is, I1 + 6I2 + 2I3 =10\n[3 61(b)]\nFor the closed loop BCDEB, we get\n5 \u2013 2 (I2 + I3) \u2013 2 (I2 + I3 \u2013 I1) = 0\nthat is, 2I1 \u2013 4I2 \u2013 4I3 = \u20135\n[3 61(c)]\nEquations (3"}, {"Chapter": "1", "sentence_range": "3053-3056", "Text": "61(a)]\nthat is, 7I1\u2013 6I2 \u2013 2I3 = 10\nFor the closed loop ABCA, we get\n10 \u2013 4I2\u2013 2 (I2 + I3) \u2013 I1 = 0\nthat is, I1 + 6I2 + 2I3 =10\n[3 61(b)]\nFor the closed loop BCDEB, we get\n5 \u2013 2 (I2 + I3) \u2013 2 (I2 + I3 \u2013 I1) = 0\nthat is, 2I1 \u2013 4I2 \u2013 4I3 = \u20135\n[3 61(c)]\nEquations (3 61 a, b, c) are three simultaneous equations in three\nunknowns"}, {"Chapter": "1", "sentence_range": "3054-3057", "Text": "61(b)]\nFor the closed loop BCDEB, we get\n5 \u2013 2 (I2 + I3) \u2013 2 (I2 + I3 \u2013 I1) = 0\nthat is, 2I1 \u2013 4I2 \u2013 4I3 = \u20135\n[3 61(c)]\nEquations (3 61 a, b, c) are three simultaneous equations in three\nunknowns These can be solved by the usual method to give\nI1 = 2"}, {"Chapter": "1", "sentence_range": "3055-3058", "Text": "61(c)]\nEquations (3 61 a, b, c) are three simultaneous equations in three\nunknowns These can be solved by the usual method to give\nI1 = 2 5A,   I2 = 5\n8  A,   I3 = 7\n18\n  A\nThe currents in the various branches of the network are\nAB : 5\n8  A,   CA : \n1\n2 2  A,   DEB : \n7\n1 8   A\nAD : \n7\n18\n A,   CD : 0 A,   BC : \n1\n2 2  A\nIt is easily verified that Kirchhoff\u2019s second rule applied to the\nremaining closed loops does not provide any additional independent\nequation, that is, the above values of currents satisfy the second\nrule for every closed loop of the network"}, {"Chapter": "1", "sentence_range": "3056-3059", "Text": "61 a, b, c) are three simultaneous equations in three\nunknowns These can be solved by the usual method to give\nI1 = 2 5A,   I2 = 5\n8  A,   I3 = 7\n18\n  A\nThe currents in the various branches of the network are\nAB : 5\n8  A,   CA : \n1\n2 2  A,   DEB : \n7\n1 8   A\nAD : \n7\n18\n A,   CD : 0 A,   BC : \n1\n2 2  A\nIt is easily verified that Kirchhoff\u2019s second rule applied to the\nremaining closed loops does not provide any additional independent\nequation, that is, the above values of currents satisfy the second\nrule for every closed loop of the network For example, the total voltage\ndrop over the closed loop BADEB\n5\n85\n4\n15\n8\n4\nV\nV\nV\n+\n\u00d7\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\u2212\n\u00d7\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7\nequal to zero, as required by Kirchhoff\u2019s second rule"}, {"Chapter": "1", "sentence_range": "3057-3060", "Text": "These can be solved by the usual method to give\nI1 = 2 5A,   I2 = 5\n8  A,   I3 = 7\n18\n  A\nThe currents in the various branches of the network are\nAB : 5\n8  A,   CA : \n1\n2 2  A,   DEB : \n7\n1 8   A\nAD : \n7\n18\n A,   CD : 0 A,   BC : \n1\n2 2  A\nIt is easily verified that Kirchhoff\u2019s second rule applied to the\nremaining closed loops does not provide any additional independent\nequation, that is, the above values of currents satisfy the second\nrule for every closed loop of the network For example, the total voltage\ndrop over the closed loop BADEB\n5\n85\n4\n15\n8\n4\nV\nV\nV\n+\n\u00d7\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\u2212\n\u00d7\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7\nequal to zero, as required by Kirchhoff\u2019s second rule 3"}, {"Chapter": "1", "sentence_range": "3058-3061", "Text": "5A,   I2 = 5\n8  A,   I3 = 7\n18\n  A\nThe currents in the various branches of the network are\nAB : 5\n8  A,   CA : \n1\n2 2  A,   DEB : \n7\n1 8   A\nAD : \n7\n18\n A,   CD : 0 A,   BC : \n1\n2 2  A\nIt is easily verified that Kirchhoff\u2019s second rule applied to the\nremaining closed loops does not provide any additional independent\nequation, that is, the above values of currents satisfy the second\nrule for every closed loop of the network For example, the total voltage\ndrop over the closed loop BADEB\n5\n85\n4\n15\n8\n4\nV\nV\nV\n+\n\u00d7\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\u2212\n\u00d7\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7\nequal to zero, as required by Kirchhoff\u2019s second rule 3 13  WHEATSTONE BRIDGE\nAs an application of Kirchhoff\u2019s rules consider the circuit shown in\nFig"}, {"Chapter": "1", "sentence_range": "3059-3062", "Text": "For example, the total voltage\ndrop over the closed loop BADEB\n5\n85\n4\n15\n8\n4\nV\nV\nV\n+\n\u00d7\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\u2212\n\u00d7\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7\nequal to zero, as required by Kirchhoff\u2019s second rule 3 13  WHEATSTONE BRIDGE\nAs an application of Kirchhoff\u2019s rules consider the circuit shown in\nFig 3"}, {"Chapter": "1", "sentence_range": "3060-3063", "Text": "3 13  WHEATSTONE BRIDGE\nAs an application of Kirchhoff\u2019s rules consider the circuit shown in\nFig 3 18, which is called the Wheatstone bridge"}, {"Chapter": "1", "sentence_range": "3061-3064", "Text": "13  WHEATSTONE BRIDGE\nAs an application of Kirchhoff\u2019s rules consider the circuit shown in\nFig 3 18, which is called the Wheatstone bridge The bridge has\nfour resistors R1, R2, R3 and R4"}, {"Chapter": "1", "sentence_range": "3062-3065", "Text": "3 18, which is called the Wheatstone bridge The bridge has\nfour resistors R1, R2, R3 and R4 Across one pair of diagonally opposite\npoints (A and C in the figure) a source is connected"}, {"Chapter": "1", "sentence_range": "3063-3066", "Text": "18, which is called the Wheatstone bridge The bridge has\nfour resistors R1, R2, R3 and R4 Across one pair of diagonally opposite\npoints (A and C in the figure) a source is connected This (i"}, {"Chapter": "1", "sentence_range": "3064-3067", "Text": "The bridge has\nfour resistors R1, R2, R3 and R4 Across one pair of diagonally opposite\npoints (A and C in the figure) a source is connected This (i e"}, {"Chapter": "1", "sentence_range": "3065-3068", "Text": "Across one pair of diagonally opposite\npoints (A and C in the figure) a source is connected This (i e , AC) is\ncalled the battery arm"}, {"Chapter": "1", "sentence_range": "3066-3069", "Text": "This (i e , AC) is\ncalled the battery arm Between the other two vertices, B and D, a\ngalvanometer G (which is a device to detect currents) is connected"}, {"Chapter": "1", "sentence_range": "3067-3070", "Text": "e , AC) is\ncalled the battery arm Between the other two vertices, B and D, a\ngalvanometer G (which is a device to detect currents) is connected This\nline, shown as BD in the figure, is called the galvanometer arm"}, {"Chapter": "1", "sentence_range": "3068-3071", "Text": ", AC) is\ncalled the battery arm Between the other two vertices, B and D, a\ngalvanometer G (which is a device to detect currents) is connected This\nline, shown as BD in the figure, is called the galvanometer arm For simplicity, we assume that the cell has no internal resistance"}, {"Chapter": "1", "sentence_range": "3069-3072", "Text": "Between the other two vertices, B and D, a\ngalvanometer G (which is a device to detect currents) is connected This\nline, shown as BD in the figure, is called the galvanometer arm For simplicity, we assume that the cell has no internal resistance In\ngeneral there will be currents flowing across all the resistors as well as a\ncurrent Ig through G"}, {"Chapter": "1", "sentence_range": "3070-3073", "Text": "This\nline, shown as BD in the figure, is called the galvanometer arm For simplicity, we assume that the cell has no internal resistance In\ngeneral there will be currents flowing across all the resistors as well as a\ncurrent Ig through G Of special interest, is the case of a balanced bridge\nwhere the resistors are such that Ig = 0"}, {"Chapter": "1", "sentence_range": "3071-3074", "Text": "For simplicity, we assume that the cell has no internal resistance In\ngeneral there will be currents flowing across all the resistors as well as a\ncurrent Ig through G Of special interest, is the case of a balanced bridge\nwhere the resistors are such that Ig = 0 We can easily get the balance\ncondition, such that there is no current through G"}, {"Chapter": "1", "sentence_range": "3072-3075", "Text": "In\ngeneral there will be currents flowing across all the resistors as well as a\ncurrent Ig through G Of special interest, is the case of a balanced bridge\nwhere the resistors are such that Ig = 0 We can easily get the balance\ncondition, such that there is no current through G In this case, the\nKirchhoff\u2019s junction rule applied to junctions D and B (see the figure)\nRationalised 2023-24\nCurrent\nElectricity\n101\nFIGURE 3"}, {"Chapter": "1", "sentence_range": "3073-3076", "Text": "Of special interest, is the case of a balanced bridge\nwhere the resistors are such that Ig = 0 We can easily get the balance\ncondition, such that there is no current through G In this case, the\nKirchhoff\u2019s junction rule applied to junctions D and B (see the figure)\nRationalised 2023-24\nCurrent\nElectricity\n101\nFIGURE 3 18\n EXAMPLE 3"}, {"Chapter": "1", "sentence_range": "3074-3077", "Text": "We can easily get the balance\ncondition, such that there is no current through G In this case, the\nKirchhoff\u2019s junction rule applied to junctions D and B (see the figure)\nRationalised 2023-24\nCurrent\nElectricity\n101\nFIGURE 3 18\n EXAMPLE 3 7\nimmediately gives us the relations I1 = I3 and I2 = I4"}, {"Chapter": "1", "sentence_range": "3075-3078", "Text": "In this case, the\nKirchhoff\u2019s junction rule applied to junctions D and B (see the figure)\nRationalised 2023-24\nCurrent\nElectricity\n101\nFIGURE 3 18\n EXAMPLE 3 7\nimmediately gives us the relations I1 = I3 and I2 = I4 Next, we apply\nKirchhoff\u2019s loop rule to closed loops ADBA and CBDC"}, {"Chapter": "1", "sentence_range": "3076-3079", "Text": "18\n EXAMPLE 3 7\nimmediately gives us the relations I1 = I3 and I2 = I4 Next, we apply\nKirchhoff\u2019s loop rule to closed loops ADBA and CBDC The first\nloop gives\n\u2013I1 R1 + 0 + I2 R2 = 0        (Ig = 0)  \n(3"}, {"Chapter": "1", "sentence_range": "3077-3080", "Text": "7\nimmediately gives us the relations I1 = I3 and I2 = I4 Next, we apply\nKirchhoff\u2019s loop rule to closed loops ADBA and CBDC The first\nloop gives\n\u2013I1 R1 + 0 + I2 R2 = 0        (Ig = 0)  \n(3 62)\nand the second loop gives, upon using I3 = I1, I4 = I2\nI2 R4 + 0 \u2013 I1 R3 = 0        \n(3"}, {"Chapter": "1", "sentence_range": "3078-3081", "Text": "Next, we apply\nKirchhoff\u2019s loop rule to closed loops ADBA and CBDC The first\nloop gives\n\u2013I1 R1 + 0 + I2 R2 = 0        (Ig = 0)  \n(3 62)\nand the second loop gives, upon using I3 = I1, I4 = I2\nI2 R4 + 0 \u2013 I1 R3 = 0        \n(3 63)\nFrom Eq"}, {"Chapter": "1", "sentence_range": "3079-3082", "Text": "The first\nloop gives\n\u2013I1 R1 + 0 + I2 R2 = 0        (Ig = 0)  \n(3 62)\nand the second loop gives, upon using I3 = I1, I4 = I2\nI2 R4 + 0 \u2013 I1 R3 = 0        \n(3 63)\nFrom Eq (3"}, {"Chapter": "1", "sentence_range": "3080-3083", "Text": "62)\nand the second loop gives, upon using I3 = I1, I4 = I2\nI2 R4 + 0 \u2013 I1 R3 = 0        \n(3 63)\nFrom Eq (3 62), we obtain,\n1\n2\n2\n1\nI\nR\nI\nR\n=\nwhereas from Eq"}, {"Chapter": "1", "sentence_range": "3081-3084", "Text": "63)\nFrom Eq (3 62), we obtain,\n1\n2\n2\n1\nI\nR\nI\nR\n=\nwhereas from Eq (3"}, {"Chapter": "1", "sentence_range": "3082-3085", "Text": "(3 62), we obtain,\n1\n2\n2\n1\nI\nR\nI\nR\n=\nwhereas from Eq (3 63), we obtain,\n1\n4\n2\n3\nI\nR\nI\nR\n=\nHence, we obtain the condition\n2\n4\n1\n3\nR\nR\nR\n=R\n[3"}, {"Chapter": "1", "sentence_range": "3083-3086", "Text": "62), we obtain,\n1\n2\n2\n1\nI\nR\nI\nR\n=\nwhereas from Eq (3 63), we obtain,\n1\n4\n2\n3\nI\nR\nI\nR\n=\nHence, we obtain the condition\n2\n4\n1\n3\nR\nR\nR\n=R\n[3 64(a)]\n This last equation relating the four resistors is called the balance\ncondition for the galvanometer to give zero or null deflection"}, {"Chapter": "1", "sentence_range": "3084-3087", "Text": "(3 63), we obtain,\n1\n4\n2\n3\nI\nR\nI\nR\n=\nHence, we obtain the condition\n2\n4\n1\n3\nR\nR\nR\n=R\n[3 64(a)]\n This last equation relating the four resistors is called the balance\ncondition for the galvanometer to give zero or null deflection The Wheatstone bridge and its balance condition provide a practical\nmethod for determination of an unknown resistance"}, {"Chapter": "1", "sentence_range": "3085-3088", "Text": "63), we obtain,\n1\n4\n2\n3\nI\nR\nI\nR\n=\nHence, we obtain the condition\n2\n4\n1\n3\nR\nR\nR\n=R\n[3 64(a)]\n This last equation relating the four resistors is called the balance\ncondition for the galvanometer to give zero or null deflection The Wheatstone bridge and its balance condition provide a practical\nmethod for determination of an unknown resistance Let us suppose we\nhave an unknown resistance, which we insert in the fourth arm; R4 is\nthus not known"}, {"Chapter": "1", "sentence_range": "3086-3089", "Text": "64(a)]\n This last equation relating the four resistors is called the balance\ncondition for the galvanometer to give zero or null deflection The Wheatstone bridge and its balance condition provide a practical\nmethod for determination of an unknown resistance Let us suppose we\nhave an unknown resistance, which we insert in the fourth arm; R4 is\nthus not known Keeping known resistances R1 and R2 in the first and\nsecond arm of the bridge, we go on varying R3 till the galvanometer shows\na null deflection"}, {"Chapter": "1", "sentence_range": "3087-3090", "Text": "The Wheatstone bridge and its balance condition provide a practical\nmethod for determination of an unknown resistance Let us suppose we\nhave an unknown resistance, which we insert in the fourth arm; R4 is\nthus not known Keeping known resistances R1 and R2 in the first and\nsecond arm of the bridge, we go on varying R3 till the galvanometer shows\na null deflection The bridge then is balanced, and from the balance\ncondition the value of the unknown resistance R4 is given by,\n2\n4\n3\n1\nR\nR\n=R R\n[3"}, {"Chapter": "1", "sentence_range": "3088-3091", "Text": "Let us suppose we\nhave an unknown resistance, which we insert in the fourth arm; R4 is\nthus not known Keeping known resistances R1 and R2 in the first and\nsecond arm of the bridge, we go on varying R3 till the galvanometer shows\na null deflection The bridge then is balanced, and from the balance\ncondition the value of the unknown resistance R4 is given by,\n2\n4\n3\n1\nR\nR\n=R R\n[3 64(b)]\nA practical device using this principle is called the meter bridge"}, {"Chapter": "1", "sentence_range": "3089-3092", "Text": "Keeping known resistances R1 and R2 in the first and\nsecond arm of the bridge, we go on varying R3 till the galvanometer shows\na null deflection The bridge then is balanced, and from the balance\ncondition the value of the unknown resistance R4 is given by,\n2\n4\n3\n1\nR\nR\n=R R\n[3 64(b)]\nA practical device using this principle is called the meter bridge Example 3"}, {"Chapter": "1", "sentence_range": "3090-3093", "Text": "The bridge then is balanced, and from the balance\ncondition the value of the unknown resistance R4 is given by,\n2\n4\n3\n1\nR\nR\n=R R\n[3 64(b)]\nA practical device using this principle is called the meter bridge Example 3 7 The four arms of a Wheatstone bridge (Fig"}, {"Chapter": "1", "sentence_range": "3091-3094", "Text": "64(b)]\nA practical device using this principle is called the meter bridge Example 3 7 The four arms of a Wheatstone bridge (Fig 3"}, {"Chapter": "1", "sentence_range": "3092-3095", "Text": "Example 3 7 The four arms of a Wheatstone bridge (Fig 3 19) have\nthe following resistances:\nAB = 100W, BC = 10W, CD = 5W, and DA = 60W"}, {"Chapter": "1", "sentence_range": "3093-3096", "Text": "7 The four arms of a Wheatstone bridge (Fig 3 19) have\nthe following resistances:\nAB = 100W, BC = 10W, CD = 5W, and DA = 60W FIGURE 3"}, {"Chapter": "1", "sentence_range": "3094-3097", "Text": "3 19) have\nthe following resistances:\nAB = 100W, BC = 10W, CD = 5W, and DA = 60W FIGURE 3 19\nRationalised 2023-24\nPhysics\n102\n EXAMPLE 3"}, {"Chapter": "1", "sentence_range": "3095-3098", "Text": "19) have\nthe following resistances:\nAB = 100W, BC = 10W, CD = 5W, and DA = 60W FIGURE 3 19\nRationalised 2023-24\nPhysics\n102\n EXAMPLE 3 7\n A galvanometer of 15W resistance is connected across BD"}, {"Chapter": "1", "sentence_range": "3096-3099", "Text": "FIGURE 3 19\nRationalised 2023-24\nPhysics\n102\n EXAMPLE 3 7\n A galvanometer of 15W resistance is connected across BD Calculate\nthe current through the galvanometer when a potential difference of\n10 V is maintained across AC"}, {"Chapter": "1", "sentence_range": "3097-3100", "Text": "19\nRationalised 2023-24\nPhysics\n102\n EXAMPLE 3 7\n A galvanometer of 15W resistance is connected across BD Calculate\nthe current through the galvanometer when a potential difference of\n10 V is maintained across AC Solution  Considering the mesh BADB, we have\n100I1 + 15Ig \u2013 60I2 = 0\nor   20I1 + 3Ig \u2013 12I2= 0\n[3"}, {"Chapter": "1", "sentence_range": "3098-3101", "Text": "7\n A galvanometer of 15W resistance is connected across BD Calculate\nthe current through the galvanometer when a potential difference of\n10 V is maintained across AC Solution  Considering the mesh BADB, we have\n100I1 + 15Ig \u2013 60I2 = 0\nor   20I1 + 3Ig \u2013 12I2= 0\n[3 65(a)]\nConsidering the mesh BCDB, we have\n10 (I1 \u2013 Ig) \u2013 15Ig \u2013 5 (I2 + Ig) = 0\n10I1 \u2013 30Ig \u20135I2 = 0\n2I1 \u2013 6Ig \u2013 I2 = 0\n[3"}, {"Chapter": "1", "sentence_range": "3099-3102", "Text": "Calculate\nthe current through the galvanometer when a potential difference of\n10 V is maintained across AC Solution  Considering the mesh BADB, we have\n100I1 + 15Ig \u2013 60I2 = 0\nor   20I1 + 3Ig \u2013 12I2= 0\n[3 65(a)]\nConsidering the mesh BCDB, we have\n10 (I1 \u2013 Ig) \u2013 15Ig \u2013 5 (I2 + Ig) = 0\n10I1 \u2013 30Ig \u20135I2 = 0\n2I1 \u2013 6Ig \u2013 I2 = 0\n[3 65(b)]\nConsidering the mesh ADCEA,\n60I2 + 5 (I2 + Ig) = 10\n65I2 + 5Ig = 10\n13I2 + Ig = 2\n[3"}, {"Chapter": "1", "sentence_range": "3100-3103", "Text": "Solution  Considering the mesh BADB, we have\n100I1 + 15Ig \u2013 60I2 = 0\nor   20I1 + 3Ig \u2013 12I2= 0\n[3 65(a)]\nConsidering the mesh BCDB, we have\n10 (I1 \u2013 Ig) \u2013 15Ig \u2013 5 (I2 + Ig) = 0\n10I1 \u2013 30Ig \u20135I2 = 0\n2I1 \u2013 6Ig \u2013 I2 = 0\n[3 65(b)]\nConsidering the mesh ADCEA,\n60I2 + 5 (I2 + Ig) = 10\n65I2 + 5Ig = 10\n13I2 + Ig = 2\n[3 65(c)]\nMultiplying Eq"}, {"Chapter": "1", "sentence_range": "3101-3104", "Text": "65(a)]\nConsidering the mesh BCDB, we have\n10 (I1 \u2013 Ig) \u2013 15Ig \u2013 5 (I2 + Ig) = 0\n10I1 \u2013 30Ig \u20135I2 = 0\n2I1 \u2013 6Ig \u2013 I2 = 0\n[3 65(b)]\nConsidering the mesh ADCEA,\n60I2 + 5 (I2 + Ig) = 10\n65I2 + 5Ig = 10\n13I2 + Ig = 2\n[3 65(c)]\nMultiplying Eq (3"}, {"Chapter": "1", "sentence_range": "3102-3105", "Text": "65(b)]\nConsidering the mesh ADCEA,\n60I2 + 5 (I2 + Ig) = 10\n65I2 + 5Ig = 10\n13I2 + Ig = 2\n[3 65(c)]\nMultiplying Eq (3 65b) by 10\n20I1 \u2013 60Ig \u2013 10I2 = 0\n[3"}, {"Chapter": "1", "sentence_range": "3103-3106", "Text": "65(c)]\nMultiplying Eq (3 65b) by 10\n20I1 \u2013 60Ig \u2013 10I2 = 0\n[3 65(d)]\nFrom Eqs"}, {"Chapter": "1", "sentence_range": "3104-3107", "Text": "(3 65b) by 10\n20I1 \u2013 60Ig \u2013 10I2 = 0\n[3 65(d)]\nFrom Eqs (3"}, {"Chapter": "1", "sentence_range": "3105-3108", "Text": "65b) by 10\n20I1 \u2013 60Ig \u2013 10I2 = 0\n[3 65(d)]\nFrom Eqs (3 65d) and (3"}, {"Chapter": "1", "sentence_range": "3106-3109", "Text": "65(d)]\nFrom Eqs (3 65d) and (3 65a) we have\n63Ig \u2013 2I2 = 0\nI2 = 31"}, {"Chapter": "1", "sentence_range": "3107-3110", "Text": "(3 65d) and (3 65a) we have\n63Ig \u2013 2I2 = 0\nI2 = 31 5Ig\n[3"}, {"Chapter": "1", "sentence_range": "3108-3111", "Text": "65d) and (3 65a) we have\n63Ig \u2013 2I2 = 0\nI2 = 31 5Ig\n[3 65(e)]\nSubstituting the value of I2 into Eq"}, {"Chapter": "1", "sentence_range": "3109-3112", "Text": "65a) we have\n63Ig \u2013 2I2 = 0\nI2 = 31 5Ig\n[3 65(e)]\nSubstituting the value of I2 into Eq [3"}, {"Chapter": "1", "sentence_range": "3110-3113", "Text": "5Ig\n[3 65(e)]\nSubstituting the value of I2 into Eq [3 65(c)], we get\n13 (31"}, {"Chapter": "1", "sentence_range": "3111-3114", "Text": "65(e)]\nSubstituting the value of I2 into Eq [3 65(c)], we get\n13 (31 5Ig ) + Ig = 2\n410"}, {"Chapter": "1", "sentence_range": "3112-3115", "Text": "[3 65(c)], we get\n13 (31 5Ig ) + Ig = 2\n410 5 Ig = 2\nIg = 4"}, {"Chapter": "1", "sentence_range": "3113-3116", "Text": "65(c)], we get\n13 (31 5Ig ) + Ig = 2\n410 5 Ig = 2\nIg = 4 87 mA"}, {"Chapter": "1", "sentence_range": "3114-3117", "Text": "5Ig ) + Ig = 2\n410 5 Ig = 2\nIg = 4 87 mA SUMMARY\n1"}, {"Chapter": "1", "sentence_range": "3115-3118", "Text": "5 Ig = 2\nIg = 4 87 mA SUMMARY\n1 Current through a given area of a conductor is the net charge passing\nper unit time through the area"}, {"Chapter": "1", "sentence_range": "3116-3119", "Text": "87 mA SUMMARY\n1 Current through a given area of a conductor is the net charge passing\nper unit time through the area 2"}, {"Chapter": "1", "sentence_range": "3117-3120", "Text": "SUMMARY\n1 Current through a given area of a conductor is the net charge passing\nper unit time through the area 2 To maintain a steady current, we must have a closed circuit in which\nan external agency moves electric charge from lower to higher potential\nenergy"}, {"Chapter": "1", "sentence_range": "3118-3121", "Text": "Current through a given area of a conductor is the net charge passing\nper unit time through the area 2 To maintain a steady current, we must have a closed circuit in which\nan external agency moves electric charge from lower to higher potential\nenergy The work done per unit charge by the source in taking the\ncharge from lower to higher potential energy (i"}, {"Chapter": "1", "sentence_range": "3119-3122", "Text": "2 To maintain a steady current, we must have a closed circuit in which\nan external agency moves electric charge from lower to higher potential\nenergy The work done per unit charge by the source in taking the\ncharge from lower to higher potential energy (i e"}, {"Chapter": "1", "sentence_range": "3120-3123", "Text": "To maintain a steady current, we must have a closed circuit in which\nan external agency moves electric charge from lower to higher potential\nenergy The work done per unit charge by the source in taking the\ncharge from lower to higher potential energy (i e , from one terminal\nof the source to the other) is called the electromotive force, or emf, of\nthe source"}, {"Chapter": "1", "sentence_range": "3121-3124", "Text": "The work done per unit charge by the source in taking the\ncharge from lower to higher potential energy (i e , from one terminal\nof the source to the other) is called the electromotive force, or emf, of\nthe source Note that the emf is not a force; it is the voltage difference\nbetween the two terminals of a source in open circuit"}, {"Chapter": "1", "sentence_range": "3122-3125", "Text": "e , from one terminal\nof the source to the other) is called the electromotive force, or emf, of\nthe source Note that the emf is not a force; it is the voltage difference\nbetween the two terminals of a source in open circuit 3"}, {"Chapter": "1", "sentence_range": "3123-3126", "Text": ", from one terminal\nof the source to the other) is called the electromotive force, or emf, of\nthe source Note that the emf is not a force; it is the voltage difference\nbetween the two terminals of a source in open circuit 3 Ohm\u2019s law: The electric current I flowing through a substance is\nproportional to the voltage V across its ends, i"}, {"Chapter": "1", "sentence_range": "3124-3127", "Text": "Note that the emf is not a force; it is the voltage difference\nbetween the two terminals of a source in open circuit 3 Ohm\u2019s law: The electric current I flowing through a substance is\nproportional to the voltage V across its ends, i e"}, {"Chapter": "1", "sentence_range": "3125-3128", "Text": "3 Ohm\u2019s law: The electric current I flowing through a substance is\nproportional to the voltage V across its ends, i e , V \u00b5 I or V = RI,\nwhere R is called the resistance of the substance"}, {"Chapter": "1", "sentence_range": "3126-3129", "Text": "Ohm\u2019s law: The electric current I flowing through a substance is\nproportional to the voltage V across its ends, i e , V \u00b5 I or V = RI,\nwhere R is called the resistance of the substance The unit of resistance\nis ohm: 1W = 1 V A\u20131"}, {"Chapter": "1", "sentence_range": "3127-3130", "Text": "e , V \u00b5 I or V = RI,\nwhere R is called the resistance of the substance The unit of resistance\nis ohm: 1W = 1 V A\u20131 Rationalised 2023-24\nCurrent\nElectricity\n103\n4"}, {"Chapter": "1", "sentence_range": "3128-3131", "Text": ", V \u00b5 I or V = RI,\nwhere R is called the resistance of the substance The unit of resistance\nis ohm: 1W = 1 V A\u20131 Rationalised 2023-24\nCurrent\nElectricity\n103\n4 The resistance R of a conductor depends on its length l and\ncross-sectional area A through the relation,\nl\nR\n\u03c1A\n=\nwhere r, called resistivity is a property of the material and depends on\ntemperature and pressure"}, {"Chapter": "1", "sentence_range": "3129-3132", "Text": "The unit of resistance\nis ohm: 1W = 1 V A\u20131 Rationalised 2023-24\nCurrent\nElectricity\n103\n4 The resistance R of a conductor depends on its length l and\ncross-sectional area A through the relation,\nl\nR\n\u03c1A\n=\nwhere r, called resistivity is a property of the material and depends on\ntemperature and pressure 5"}, {"Chapter": "1", "sentence_range": "3130-3133", "Text": "Rationalised 2023-24\nCurrent\nElectricity\n103\n4 The resistance R of a conductor depends on its length l and\ncross-sectional area A through the relation,\nl\nR\n\u03c1A\n=\nwhere r, called resistivity is a property of the material and depends on\ntemperature and pressure 5 Electrical resistivity of substances varies over a very wide range"}, {"Chapter": "1", "sentence_range": "3131-3134", "Text": "The resistance R of a conductor depends on its length l and\ncross-sectional area A through the relation,\nl\nR\n\u03c1A\n=\nwhere r, called resistivity is a property of the material and depends on\ntemperature and pressure 5 Electrical resistivity of substances varies over a very wide range Metals\nhave low resistivity, in the range of 10\u20138 W m to 10\u20136 W m"}, {"Chapter": "1", "sentence_range": "3132-3135", "Text": "5 Electrical resistivity of substances varies over a very wide range Metals\nhave low resistivity, in the range of 10\u20138 W m to 10\u20136 W m Insulators\nlike glass and rubber have 1022 to 1024 times greater resistivity"}, {"Chapter": "1", "sentence_range": "3133-3136", "Text": "Electrical resistivity of substances varies over a very wide range Metals\nhave low resistivity, in the range of 10\u20138 W m to 10\u20136 W m Insulators\nlike glass and rubber have 1022 to 1024 times greater resistivity Semiconductors like Si and Ge lie roughly in the middle range of\nresistivity on a logarithmic scale"}, {"Chapter": "1", "sentence_range": "3134-3137", "Text": "Metals\nhave low resistivity, in the range of 10\u20138 W m to 10\u20136 W m Insulators\nlike glass and rubber have 1022 to 1024 times greater resistivity Semiconductors like Si and Ge lie roughly in the middle range of\nresistivity on a logarithmic scale 6"}, {"Chapter": "1", "sentence_range": "3135-3138", "Text": "Insulators\nlike glass and rubber have 1022 to 1024 times greater resistivity Semiconductors like Si and Ge lie roughly in the middle range of\nresistivity on a logarithmic scale 6 In most substances, the carriers of current are electrons; in some\ncases, for example, ionic crystals and electrolytic liquids, positive and\nnegative ions carry the electric current"}, {"Chapter": "1", "sentence_range": "3136-3139", "Text": "Semiconductors like Si and Ge lie roughly in the middle range of\nresistivity on a logarithmic scale 6 In most substances, the carriers of current are electrons; in some\ncases, for example, ionic crystals and electrolytic liquids, positive and\nnegative ions carry the electric current 7"}, {"Chapter": "1", "sentence_range": "3137-3140", "Text": "6 In most substances, the carriers of current are electrons; in some\ncases, for example, ionic crystals and electrolytic liquids, positive and\nnegative ions carry the electric current 7 Current density j gives the amount of charge flowing per second per\nunit area normal to the flow,\nj = nq vd\nwhere n is the number density (number per unit volume) of charge\ncarriers each of charge q, and vd is the drift velocity of the charge\ncarriers"}, {"Chapter": "1", "sentence_range": "3138-3141", "Text": "In most substances, the carriers of current are electrons; in some\ncases, for example, ionic crystals and electrolytic liquids, positive and\nnegative ions carry the electric current 7 Current density j gives the amount of charge flowing per second per\nunit area normal to the flow,\nj = nq vd\nwhere n is the number density (number per unit volume) of charge\ncarriers each of charge q, and vd is the drift velocity of the charge\ncarriers For electrons q = \u2013 e"}, {"Chapter": "1", "sentence_range": "3139-3142", "Text": "7 Current density j gives the amount of charge flowing per second per\nunit area normal to the flow,\nj = nq vd\nwhere n is the number density (number per unit volume) of charge\ncarriers each of charge q, and vd is the drift velocity of the charge\ncarriers For electrons q = \u2013 e If j is normal to a cross-sectional area\nA and is constant over the area, the magnitude of the current I through\nthe area is nevd A"}, {"Chapter": "1", "sentence_range": "3140-3143", "Text": "Current density j gives the amount of charge flowing per second per\nunit area normal to the flow,\nj = nq vd\nwhere n is the number density (number per unit volume) of charge\ncarriers each of charge q, and vd is the drift velocity of the charge\ncarriers For electrons q = \u2013 e If j is normal to a cross-sectional area\nA and is constant over the area, the magnitude of the current I through\nthe area is nevd A 8"}, {"Chapter": "1", "sentence_range": "3141-3144", "Text": "For electrons q = \u2013 e If j is normal to a cross-sectional area\nA and is constant over the area, the magnitude of the current I through\nthe area is nevd A 8 Using E = V/l, I = nevd A, and Ohm\u2019s law, one obtains\n2\nd\neE\nne v\nm\n\u03c1m\n=\nThe proportionality between the force eE on the electrons in a metal\ndue to the external field E and the drift velocity vd (not acceleration)\ncan be understood, if we assume that the electrons suffer collisions\nwith ions in the metal, which deflect them randomly"}, {"Chapter": "1", "sentence_range": "3142-3145", "Text": "If j is normal to a cross-sectional area\nA and is constant over the area, the magnitude of the current I through\nthe area is nevd A 8 Using E = V/l, I = nevd A, and Ohm\u2019s law, one obtains\n2\nd\neE\nne v\nm\n\u03c1m\n=\nThe proportionality between the force eE on the electrons in a metal\ndue to the external field E and the drift velocity vd (not acceleration)\ncan be understood, if we assume that the electrons suffer collisions\nwith ions in the metal, which deflect them randomly If such collisions\noccur on an average at a time interval t,\nvd = at = eEt/m\nwhere a is the acceleration of the electron"}, {"Chapter": "1", "sentence_range": "3143-3146", "Text": "8 Using E = V/l, I = nevd A, and Ohm\u2019s law, one obtains\n2\nd\neE\nne v\nm\n\u03c1m\n=\nThe proportionality between the force eE on the electrons in a metal\ndue to the external field E and the drift velocity vd (not acceleration)\ncan be understood, if we assume that the electrons suffer collisions\nwith ions in the metal, which deflect them randomly If such collisions\noccur on an average at a time interval t,\nvd = at = eEt/m\nwhere a is the acceleration of the electron This gives\n2\nm\nne\n\u03c1\n\u03c4\n=\n9"}, {"Chapter": "1", "sentence_range": "3144-3147", "Text": "Using E = V/l, I = nevd A, and Ohm\u2019s law, one obtains\n2\nd\neE\nne v\nm\n\u03c1m\n=\nThe proportionality between the force eE on the electrons in a metal\ndue to the external field E and the drift velocity vd (not acceleration)\ncan be understood, if we assume that the electrons suffer collisions\nwith ions in the metal, which deflect them randomly If such collisions\noccur on an average at a time interval t,\nvd = at = eEt/m\nwhere a is the acceleration of the electron This gives\n2\nm\nne\n\u03c1\n\u03c4\n=\n9 In the temperature range in which resistivity increases linearly with\ntemperature, the temperature coefficient of resistivity a is defined as\nthe fractional increase in resistivity per unit increase in temperature"}, {"Chapter": "1", "sentence_range": "3145-3148", "Text": "If such collisions\noccur on an average at a time interval t,\nvd = at = eEt/m\nwhere a is the acceleration of the electron This gives\n2\nm\nne\n\u03c1\n\u03c4\n=\n9 In the temperature range in which resistivity increases linearly with\ntemperature, the temperature coefficient of resistivity a is defined as\nthe fractional increase in resistivity per unit increase in temperature 10"}, {"Chapter": "1", "sentence_range": "3146-3149", "Text": "This gives\n2\nm\nne\n\u03c1\n\u03c4\n=\n9 In the temperature range in which resistivity increases linearly with\ntemperature, the temperature coefficient of resistivity a is defined as\nthe fractional increase in resistivity per unit increase in temperature 10 Ohm\u2019s law is obeyed by many substances, but it is not a fundamental\nlaw of nature"}, {"Chapter": "1", "sentence_range": "3147-3150", "Text": "In the temperature range in which resistivity increases linearly with\ntemperature, the temperature coefficient of resistivity a is defined as\nthe fractional increase in resistivity per unit increase in temperature 10 Ohm\u2019s law is obeyed by many substances, but it is not a fundamental\nlaw of nature It fails if\n(a) V depends on I non-linearly"}, {"Chapter": "1", "sentence_range": "3148-3151", "Text": "10 Ohm\u2019s law is obeyed by many substances, but it is not a fundamental\nlaw of nature It fails if\n(a) V depends on I non-linearly (b) the relation between V and I depends on the sign of V for the same\nabsolute value of V"}, {"Chapter": "1", "sentence_range": "3149-3152", "Text": "Ohm\u2019s law is obeyed by many substances, but it is not a fundamental\nlaw of nature It fails if\n(a) V depends on I non-linearly (b) the relation between V and I depends on the sign of V for the same\nabsolute value of V (c)\nThe relation between V and I is non-unique"}, {"Chapter": "1", "sentence_range": "3150-3153", "Text": "It fails if\n(a) V depends on I non-linearly (b) the relation between V and I depends on the sign of V for the same\nabsolute value of V (c)\nThe relation between V and I is non-unique An example of (a) is when r increases with I (even if temperature is\nkept fixed)"}, {"Chapter": "1", "sentence_range": "3151-3154", "Text": "(b) the relation between V and I depends on the sign of V for the same\nabsolute value of V (c)\nThe relation between V and I is non-unique An example of (a) is when r increases with I (even if temperature is\nkept fixed) A rectifier combines features (a) and (b)"}, {"Chapter": "1", "sentence_range": "3152-3155", "Text": "(c)\nThe relation between V and I is non-unique An example of (a) is when r increases with I (even if temperature is\nkept fixed) A rectifier combines features (a) and (b) GaAs shows the\nfeature (c)"}, {"Chapter": "1", "sentence_range": "3153-3156", "Text": "An example of (a) is when r increases with I (even if temperature is\nkept fixed) A rectifier combines features (a) and (b) GaAs shows the\nfeature (c) 11"}, {"Chapter": "1", "sentence_range": "3154-3157", "Text": "A rectifier combines features (a) and (b) GaAs shows the\nfeature (c) 11 When a source of emf e is connected to an external resistance R, the\nvoltage Vext across R is given by\nVext = IR = \nR\nR\nr\n\u03b5\n+\nwhere r is the internal resistance of the source"}, {"Chapter": "1", "sentence_range": "3155-3158", "Text": "GaAs shows the\nfeature (c) 11 When a source of emf e is connected to an external resistance R, the\nvoltage Vext across R is given by\nVext = IR = \nR\nR\nr\n\u03b5\n+\nwhere r is the internal resistance of the source Rationalised 2023-24\nPhysics\n104\nPhysical Quantity\nSymbol\nDimensions\nUnit\nRemark\nElectric current\nI\n[A]\nA\nSI base unit\nCharge\nQ, q\n[T A]\nC\nVoltage, Electric\nV\n[M L\n2 T\n\u20133 A\n\u20131]\nV\nWork/charge\npotential difference\nElectromotive force\ne\n[M L\n2 T\n\u20133 A\n\u20131]\nV\nWork/charge\nResistance\nR\n[M L\n2 T\n\u20133 A\n\u20132]\nW\nR = V/I\nResistivity\nr\n[M L\n3 T\n\u20133 A\n\u20132]\nW m\nR = rl/A\nElectrical\ns\n[M\n\u20131 L\n\u20133 T\n3 A\n2]\nS\ns = 1/r\nconductivity\nElectric field\nE\n[M L T\n\u20133 A\n\u20131]\nV m\n\u20131\nElectric force\ncharge\nDrift speed\nvd\n[L T\n\u20131]\nm s\n\u20131\nvd\ne E\nm\n=\n\u03c4\nRelaxation time\nt\n[T]\ns\nCurrent density\nj\n[L\n\u20132 A]\nA m\n\u20132\ncurrent/area\nMobility\nm\n[M L\n3 T\n\u20134 A\n\u20131]\nm\n2 V\n\u20131s\n\u20131\nvd/\nE\n12"}, {"Chapter": "1", "sentence_range": "3156-3159", "Text": "11 When a source of emf e is connected to an external resistance R, the\nvoltage Vext across R is given by\nVext = IR = \nR\nR\nr\n\u03b5\n+\nwhere r is the internal resistance of the source Rationalised 2023-24\nPhysics\n104\nPhysical Quantity\nSymbol\nDimensions\nUnit\nRemark\nElectric current\nI\n[A]\nA\nSI base unit\nCharge\nQ, q\n[T A]\nC\nVoltage, Electric\nV\n[M L\n2 T\n\u20133 A\n\u20131]\nV\nWork/charge\npotential difference\nElectromotive force\ne\n[M L\n2 T\n\u20133 A\n\u20131]\nV\nWork/charge\nResistance\nR\n[M L\n2 T\n\u20133 A\n\u20132]\nW\nR = V/I\nResistivity\nr\n[M L\n3 T\n\u20133 A\n\u20132]\nW m\nR = rl/A\nElectrical\ns\n[M\n\u20131 L\n\u20133 T\n3 A\n2]\nS\ns = 1/r\nconductivity\nElectric field\nE\n[M L T\n\u20133 A\n\u20131]\nV m\n\u20131\nElectric force\ncharge\nDrift speed\nvd\n[L T\n\u20131]\nm s\n\u20131\nvd\ne E\nm\n=\n\u03c4\nRelaxation time\nt\n[T]\ns\nCurrent density\nj\n[L\n\u20132 A]\nA m\n\u20132\ncurrent/area\nMobility\nm\n[M L\n3 T\n\u20134 A\n\u20131]\nm\n2 V\n\u20131s\n\u20131\nvd/\nE\n12 Kirchhoff\u2019s Rules \u2013\n(a) Junction Rule: At any junction of circuit elements, the sum of\ncurrents entering the junction must equal the sum of currents\nleaving it"}, {"Chapter": "1", "sentence_range": "3157-3160", "Text": "When a source of emf e is connected to an external resistance R, the\nvoltage Vext across R is given by\nVext = IR = \nR\nR\nr\n\u03b5\n+\nwhere r is the internal resistance of the source Rationalised 2023-24\nPhysics\n104\nPhysical Quantity\nSymbol\nDimensions\nUnit\nRemark\nElectric current\nI\n[A]\nA\nSI base unit\nCharge\nQ, q\n[T A]\nC\nVoltage, Electric\nV\n[M L\n2 T\n\u20133 A\n\u20131]\nV\nWork/charge\npotential difference\nElectromotive force\ne\n[M L\n2 T\n\u20133 A\n\u20131]\nV\nWork/charge\nResistance\nR\n[M L\n2 T\n\u20133 A\n\u20132]\nW\nR = V/I\nResistivity\nr\n[M L\n3 T\n\u20133 A\n\u20132]\nW m\nR = rl/A\nElectrical\ns\n[M\n\u20131 L\n\u20133 T\n3 A\n2]\nS\ns = 1/r\nconductivity\nElectric field\nE\n[M L T\n\u20133 A\n\u20131]\nV m\n\u20131\nElectric force\ncharge\nDrift speed\nvd\n[L T\n\u20131]\nm s\n\u20131\nvd\ne E\nm\n=\n\u03c4\nRelaxation time\nt\n[T]\ns\nCurrent density\nj\n[L\n\u20132 A]\nA m\n\u20132\ncurrent/area\nMobility\nm\n[M L\n3 T\n\u20134 A\n\u20131]\nm\n2 V\n\u20131s\n\u20131\nvd/\nE\n12 Kirchhoff\u2019s Rules \u2013\n(a) Junction Rule: At any junction of circuit elements, the sum of\ncurrents entering the junction must equal the sum of currents\nleaving it (b) Loop Rule: The algebraic sum of changes in potential around any\nclosed loop must be zero"}, {"Chapter": "1", "sentence_range": "3158-3161", "Text": "Rationalised 2023-24\nPhysics\n104\nPhysical Quantity\nSymbol\nDimensions\nUnit\nRemark\nElectric current\nI\n[A]\nA\nSI base unit\nCharge\nQ, q\n[T A]\nC\nVoltage, Electric\nV\n[M L\n2 T\n\u20133 A\n\u20131]\nV\nWork/charge\npotential difference\nElectromotive force\ne\n[M L\n2 T\n\u20133 A\n\u20131]\nV\nWork/charge\nResistance\nR\n[M L\n2 T\n\u20133 A\n\u20132]\nW\nR = V/I\nResistivity\nr\n[M L\n3 T\n\u20133 A\n\u20132]\nW m\nR = rl/A\nElectrical\ns\n[M\n\u20131 L\n\u20133 T\n3 A\n2]\nS\ns = 1/r\nconductivity\nElectric field\nE\n[M L T\n\u20133 A\n\u20131]\nV m\n\u20131\nElectric force\ncharge\nDrift speed\nvd\n[L T\n\u20131]\nm s\n\u20131\nvd\ne E\nm\n=\n\u03c4\nRelaxation time\nt\n[T]\ns\nCurrent density\nj\n[L\n\u20132 A]\nA m\n\u20132\ncurrent/area\nMobility\nm\n[M L\n3 T\n\u20134 A\n\u20131]\nm\n2 V\n\u20131s\n\u20131\nvd/\nE\n12 Kirchhoff\u2019s Rules \u2013\n(a) Junction Rule: At any junction of circuit elements, the sum of\ncurrents entering the junction must equal the sum of currents\nleaving it (b) Loop Rule: The algebraic sum of changes in potential around any\nclosed loop must be zero 13"}, {"Chapter": "1", "sentence_range": "3159-3162", "Text": "Kirchhoff\u2019s Rules \u2013\n(a) Junction Rule: At any junction of circuit elements, the sum of\ncurrents entering the junction must equal the sum of currents\nleaving it (b) Loop Rule: The algebraic sum of changes in potential around any\nclosed loop must be zero 13 The Wheatstone bridge is an arrangement of four resistances \u2013 R1, R2,\nR3, R4 as shown in the text"}, {"Chapter": "1", "sentence_range": "3160-3163", "Text": "(b) Loop Rule: The algebraic sum of changes in potential around any\nclosed loop must be zero 13 The Wheatstone bridge is an arrangement of four resistances \u2013 R1, R2,\nR3, R4 as shown in the text The null-point condition is given by\n3\n1\n2\n4\nR\nRR\nR\n=\nusing which the value of one resistance can be determined, knowing\nthe other three resistances"}, {"Chapter": "1", "sentence_range": "3161-3164", "Text": "13 The Wheatstone bridge is an arrangement of four resistances \u2013 R1, R2,\nR3, R4 as shown in the text The null-point condition is given by\n3\n1\n2\n4\nR\nRR\nR\n=\nusing which the value of one resistance can be determined, knowing\nthe other three resistances POINTS TO PONDER\n1"}, {"Chapter": "1", "sentence_range": "3162-3165", "Text": "The Wheatstone bridge is an arrangement of four resistances \u2013 R1, R2,\nR3, R4 as shown in the text The null-point condition is given by\n3\n1\n2\n4\nR\nRR\nR\n=\nusing which the value of one resistance can be determined, knowing\nthe other three resistances POINTS TO PONDER\n1 Current is a scalar although we represent current with an arrow"}, {"Chapter": "1", "sentence_range": "3163-3166", "Text": "The null-point condition is given by\n3\n1\n2\n4\nR\nRR\nR\n=\nusing which the value of one resistance can be determined, knowing\nthe other three resistances POINTS TO PONDER\n1 Current is a scalar although we represent current with an arrow Currents do not obey the  law of vector addition"}, {"Chapter": "1", "sentence_range": "3164-3167", "Text": "POINTS TO PONDER\n1 Current is a scalar although we represent current with an arrow Currents do not obey the  law of vector addition That current is a\nscalar also follows from it\u2019s  definition"}, {"Chapter": "1", "sentence_range": "3165-3168", "Text": "Current is a scalar although we represent current with an arrow Currents do not obey the  law of vector addition That current is a\nscalar also follows from it\u2019s  definition The current I through an area\nof cross-section is given by the scalar product of two vectors:\nI =  j"}, {"Chapter": "1", "sentence_range": "3166-3169", "Text": "Currents do not obey the  law of vector addition That current is a\nscalar also follows from it\u2019s  definition The current I through an area\nof cross-section is given by the scalar product of two vectors:\nI =  j DS\nwhere j and DS are vectors"}, {"Chapter": "1", "sentence_range": "3167-3170", "Text": "That current is a\nscalar also follows from it\u2019s  definition The current I through an area\nof cross-section is given by the scalar product of two vectors:\nI =  j DS\nwhere j and DS are vectors Rationalised 2023-24\nCurrent\nElectricity\n105\n2"}, {"Chapter": "1", "sentence_range": "3168-3171", "Text": "The current I through an area\nof cross-section is given by the scalar product of two vectors:\nI =  j DS\nwhere j and DS are vectors Rationalised 2023-24\nCurrent\nElectricity\n105\n2 Refer to V-I curves of a resistor and a diode as drawn in the text"}, {"Chapter": "1", "sentence_range": "3169-3172", "Text": "DS\nwhere j and DS are vectors Rationalised 2023-24\nCurrent\nElectricity\n105\n2 Refer to V-I curves of a resistor and a diode as drawn in the text A\nresistor obeys Ohm\u2019s law while a diode does not"}, {"Chapter": "1", "sentence_range": "3170-3173", "Text": "Rationalised 2023-24\nCurrent\nElectricity\n105\n2 Refer to V-I curves of a resistor and a diode as drawn in the text A\nresistor obeys Ohm\u2019s law while a diode does not The assertion that\nV = IR is a statement of Ohm\u2019s law is not true"}, {"Chapter": "1", "sentence_range": "3171-3174", "Text": "Refer to V-I curves of a resistor and a diode as drawn in the text A\nresistor obeys Ohm\u2019s law while a diode does not The assertion that\nV = IR is a statement of Ohm\u2019s law is not true This equation defines\nresistance and it may be applied to all conducting devices whether\nthey obey Ohm\u2019s law or not"}, {"Chapter": "1", "sentence_range": "3172-3175", "Text": "A\nresistor obeys Ohm\u2019s law while a diode does not The assertion that\nV = IR is a statement of Ohm\u2019s law is not true This equation defines\nresistance and it may be applied to all conducting devices whether\nthey obey Ohm\u2019s law or not The Ohm\u2019s law asserts that the plot of I\nversus V is linear i"}, {"Chapter": "1", "sentence_range": "3173-3176", "Text": "The assertion that\nV = IR is a statement of Ohm\u2019s law is not true This equation defines\nresistance and it may be applied to all conducting devices whether\nthey obey Ohm\u2019s law or not The Ohm\u2019s law asserts that the plot of I\nversus V is linear i e"}, {"Chapter": "1", "sentence_range": "3174-3177", "Text": "This equation defines\nresistance and it may be applied to all conducting devices whether\nthey obey Ohm\u2019s law or not The Ohm\u2019s law asserts that the plot of I\nversus V is linear i e , R is independent of V"}, {"Chapter": "1", "sentence_range": "3175-3178", "Text": "The Ohm\u2019s law asserts that the plot of I\nversus V is linear i e , R is independent of V Equation E = r j leads to another statement of Ohm\u2019s law, i"}, {"Chapter": "1", "sentence_range": "3176-3179", "Text": "e , R is independent of V Equation E = r j leads to another statement of Ohm\u2019s law, i e"}, {"Chapter": "1", "sentence_range": "3177-3180", "Text": ", R is independent of V Equation E = r j leads to another statement of Ohm\u2019s law, i e , a\nconducting material obeys Ohm\u2019s law when the resistivity of the\nmaterial does not depend on the magnitude and direction of applied\nelectric field"}, {"Chapter": "1", "sentence_range": "3178-3181", "Text": "Equation E = r j leads to another statement of Ohm\u2019s law, i e , a\nconducting material obeys Ohm\u2019s law when the resistivity of the\nmaterial does not depend on the magnitude and direction of applied\nelectric field 3"}, {"Chapter": "1", "sentence_range": "3179-3182", "Text": "e , a\nconducting material obeys Ohm\u2019s law when the resistivity of the\nmaterial does not depend on the magnitude and direction of applied\nelectric field 3 Homogeneous conductors like silver or semiconductors like pure\ngermanium or germanium containing impurities obey Ohm\u2019s law within\nsome range of electric field values"}, {"Chapter": "1", "sentence_range": "3180-3183", "Text": ", a\nconducting material obeys Ohm\u2019s law when the resistivity of the\nmaterial does not depend on the magnitude and direction of applied\nelectric field 3 Homogeneous conductors like silver or semiconductors like pure\ngermanium or germanium containing impurities obey Ohm\u2019s law within\nsome range of electric field values If the field becomes too strong,\nthere are departures from Ohm\u2019s law in all cases"}, {"Chapter": "1", "sentence_range": "3181-3184", "Text": "3 Homogeneous conductors like silver or semiconductors like pure\ngermanium or germanium containing impurities obey Ohm\u2019s law within\nsome range of electric field values If the field becomes too strong,\nthere are departures from Ohm\u2019s law in all cases 4"}, {"Chapter": "1", "sentence_range": "3182-3185", "Text": "Homogeneous conductors like silver or semiconductors like pure\ngermanium or germanium containing impurities obey Ohm\u2019s law within\nsome range of electric field values If the field becomes too strong,\nthere are departures from Ohm\u2019s law in all cases 4 Motion of conduction electrons in electric field E is the sum of (i)\nmotion due to random collisions and (ii) that due to E"}, {"Chapter": "1", "sentence_range": "3183-3186", "Text": "If the field becomes too strong,\nthere are departures from Ohm\u2019s law in all cases 4 Motion of conduction electrons in electric field E is the sum of (i)\nmotion due to random collisions and (ii) that due to E The motion\ndue to random collisions averages to zero and does not contribute to\nvd (Chapter 11, Textbook of Class XI)"}, {"Chapter": "1", "sentence_range": "3184-3187", "Text": "4 Motion of conduction electrons in electric field E is the sum of (i)\nmotion due to random collisions and (ii) that due to E The motion\ndue to random collisions averages to zero and does not contribute to\nvd (Chapter 11, Textbook of Class XI) vd , thus is only due to applied\nelectric field on the electron"}, {"Chapter": "1", "sentence_range": "3185-3188", "Text": "Motion of conduction electrons in electric field E is the sum of (i)\nmotion due to random collisions and (ii) that due to E The motion\ndue to random collisions averages to zero and does not contribute to\nvd (Chapter 11, Textbook of Class XI) vd , thus is only due to applied\nelectric field on the electron 5"}, {"Chapter": "1", "sentence_range": "3186-3189", "Text": "The motion\ndue to random collisions averages to zero and does not contribute to\nvd (Chapter 11, Textbook of Class XI) vd , thus is only due to applied\nelectric field on the electron 5 The relation j = r v should be applied to each type of charge carriers\nseparately"}, {"Chapter": "1", "sentence_range": "3187-3190", "Text": "vd , thus is only due to applied\nelectric field on the electron 5 The relation j = r v should be applied to each type of charge carriers\nseparately In a conducting wire, the total current and charge density\narises from both positive and negative charges:\nj = r+ v+ + r\u2013 v\u2013\nrrrrr = r+ + r\u2013\nNow in a neutral wire carrying electric current,\nrrrrr+ = \u2013 r\u2013\nFurther, v+ ~ 0 which gives\nrrrrr = 0\nj = r\u2013 v\nThus, the relation j = r v does not apply to the total current charge\ndensity"}, {"Chapter": "1", "sentence_range": "3188-3191", "Text": "5 The relation j = r v should be applied to each type of charge carriers\nseparately In a conducting wire, the total current and charge density\narises from both positive and negative charges:\nj = r+ v+ + r\u2013 v\u2013\nrrrrr = r+ + r\u2013\nNow in a neutral wire carrying electric current,\nrrrrr+ = \u2013 r\u2013\nFurther, v+ ~ 0 which gives\nrrrrr = 0\nj = r\u2013 v\nThus, the relation j = r v does not apply to the total current charge\ndensity 6"}, {"Chapter": "1", "sentence_range": "3189-3192", "Text": "The relation j = r v should be applied to each type of charge carriers\nseparately In a conducting wire, the total current and charge density\narises from both positive and negative charges:\nj = r+ v+ + r\u2013 v\u2013\nrrrrr = r+ + r\u2013\nNow in a neutral wire carrying electric current,\nrrrrr+ = \u2013 r\u2013\nFurther, v+ ~ 0 which gives\nrrrrr = 0\nj = r\u2013 v\nThus, the relation j = r v does not apply to the total current charge\ndensity 6 Kirchhoff\u2019s junction rule is based on conservation of charge and the\noutgoing currents add up and are equal to incoming current at a\njunction"}, {"Chapter": "1", "sentence_range": "3190-3193", "Text": "In a conducting wire, the total current and charge density\narises from both positive and negative charges:\nj = r+ v+ + r\u2013 v\u2013\nrrrrr = r+ + r\u2013\nNow in a neutral wire carrying electric current,\nrrrrr+ = \u2013 r\u2013\nFurther, v+ ~ 0 which gives\nrrrrr = 0\nj = r\u2013 v\nThus, the relation j = r v does not apply to the total current charge\ndensity 6 Kirchhoff\u2019s junction rule is based on conservation of charge and the\noutgoing currents add up and are equal to incoming current at a\njunction Bending or reorienting the wire does not change the validity\nof Kirchhoff\u2019s junction rule"}, {"Chapter": "1", "sentence_range": "3191-3194", "Text": "6 Kirchhoff\u2019s junction rule is based on conservation of charge and the\noutgoing currents add up and are equal to incoming current at a\njunction Bending or reorienting the wire does not change the validity\nof Kirchhoff\u2019s junction rule EXERCISES\n3"}, {"Chapter": "1", "sentence_range": "3192-3195", "Text": "Kirchhoff\u2019s junction rule is based on conservation of charge and the\noutgoing currents add up and are equal to incoming current at a\njunction Bending or reorienting the wire does not change the validity\nof Kirchhoff\u2019s junction rule EXERCISES\n3 1\nThe storage battery of a car has an emf of 12 V"}, {"Chapter": "1", "sentence_range": "3193-3196", "Text": "Bending or reorienting the wire does not change the validity\nof Kirchhoff\u2019s junction rule EXERCISES\n3 1\nThe storage battery of a car has an emf of 12 V If the internal\nresistance of the battery is 0"}, {"Chapter": "1", "sentence_range": "3194-3197", "Text": "EXERCISES\n3 1\nThe storage battery of a car has an emf of 12 V If the internal\nresistance of the battery is 0 4 W, what is the maximum current\nthat can be drawn from the battery"}, {"Chapter": "1", "sentence_range": "3195-3198", "Text": "1\nThe storage battery of a car has an emf of 12 V If the internal\nresistance of the battery is 0 4 W, what is the maximum current\nthat can be drawn from the battery 3"}, {"Chapter": "1", "sentence_range": "3196-3199", "Text": "If the internal\nresistance of the battery is 0 4 W, what is the maximum current\nthat can be drawn from the battery 3 2\nA battery of emf 10 V and internal resistance 3 W is connected to a\nresistor"}, {"Chapter": "1", "sentence_range": "3197-3200", "Text": "4 W, what is the maximum current\nthat can be drawn from the battery 3 2\nA battery of emf 10 V and internal resistance 3 W is connected to a\nresistor If the current in the circuit is 0"}, {"Chapter": "1", "sentence_range": "3198-3201", "Text": "3 2\nA battery of emf 10 V and internal resistance 3 W is connected to a\nresistor If the current in the circuit is 0 5 A, what is the resistance\nof the resistor"}, {"Chapter": "1", "sentence_range": "3199-3202", "Text": "2\nA battery of emf 10 V and internal resistance 3 W is connected to a\nresistor If the current in the circuit is 0 5 A, what is the resistance\nof the resistor What is the terminal voltage of the battery when the\ncircuit is closed"}, {"Chapter": "1", "sentence_range": "3200-3203", "Text": "If the current in the circuit is 0 5 A, what is the resistance\nof the resistor What is the terminal voltage of the battery when the\ncircuit is closed 3"}, {"Chapter": "1", "sentence_range": "3201-3204", "Text": "5 A, what is the resistance\nof the resistor What is the terminal voltage of the battery when the\ncircuit is closed 3 3\nAt room temperature (27"}, {"Chapter": "1", "sentence_range": "3202-3205", "Text": "What is the terminal voltage of the battery when the\ncircuit is closed 3 3\nAt room temperature (27 0 \u00b0C) the resistance of a heating element\nis 100 W"}, {"Chapter": "1", "sentence_range": "3203-3206", "Text": "3 3\nAt room temperature (27 0 \u00b0C) the resistance of a heating element\nis 100 W What is the temperature of the element if the resistance is\nfound to be 117 W, given that the temperature coefficient of the\nmaterial of the resistor is 1"}, {"Chapter": "1", "sentence_range": "3204-3207", "Text": "3\nAt room temperature (27 0 \u00b0C) the resistance of a heating element\nis 100 W What is the temperature of the element if the resistance is\nfound to be 117 W, given that the temperature coefficient of the\nmaterial of the resistor is 1 70 \u00d7 10\u20134 \u00b0C\u20131"}, {"Chapter": "1", "sentence_range": "3205-3208", "Text": "0 \u00b0C) the resistance of a heating element\nis 100 W What is the temperature of the element if the resistance is\nfound to be 117 W, given that the temperature coefficient of the\nmaterial of the resistor is 1 70 \u00d7 10\u20134 \u00b0C\u20131 Rationalised 2023-24\nPhysics\n106\n3"}, {"Chapter": "1", "sentence_range": "3206-3209", "Text": "What is the temperature of the element if the resistance is\nfound to be 117 W, given that the temperature coefficient of the\nmaterial of the resistor is 1 70 \u00d7 10\u20134 \u00b0C\u20131 Rationalised 2023-24\nPhysics\n106\n3 4\nA negligibly small current is passed through a wire of length 15 m\nand uniform cross-section 6"}, {"Chapter": "1", "sentence_range": "3207-3210", "Text": "70 \u00d7 10\u20134 \u00b0C\u20131 Rationalised 2023-24\nPhysics\n106\n3 4\nA negligibly small current is passed through a wire of length 15 m\nand uniform cross-section 6 0 \u00d7 10\u20137 m2, and its resistance is\nmeasured to be 5"}, {"Chapter": "1", "sentence_range": "3208-3211", "Text": "Rationalised 2023-24\nPhysics\n106\n3 4\nA negligibly small current is passed through a wire of length 15 m\nand uniform cross-section 6 0 \u00d7 10\u20137 m2, and its resistance is\nmeasured to be 5 0 W"}, {"Chapter": "1", "sentence_range": "3209-3212", "Text": "4\nA negligibly small current is passed through a wire of length 15 m\nand uniform cross-section 6 0 \u00d7 10\u20137 m2, and its resistance is\nmeasured to be 5 0 W What is the resistivity of the material at the\ntemperature of the experiment"}, {"Chapter": "1", "sentence_range": "3210-3213", "Text": "0 \u00d7 10\u20137 m2, and its resistance is\nmeasured to be 5 0 W What is the resistivity of the material at the\ntemperature of the experiment 3"}, {"Chapter": "1", "sentence_range": "3211-3214", "Text": "0 W What is the resistivity of the material at the\ntemperature of the experiment 3 5\nA silver wire has a resistance of 2"}, {"Chapter": "1", "sentence_range": "3212-3215", "Text": "What is the resistivity of the material at the\ntemperature of the experiment 3 5\nA silver wire has a resistance of 2 1 W at 27"}, {"Chapter": "1", "sentence_range": "3213-3216", "Text": "3 5\nA silver wire has a resistance of 2 1 W at 27 5 \u00b0C, and a resistance\nof 2"}, {"Chapter": "1", "sentence_range": "3214-3217", "Text": "5\nA silver wire has a resistance of 2 1 W at 27 5 \u00b0C, and a resistance\nof 2 7 W at 100 \u00b0C"}, {"Chapter": "1", "sentence_range": "3215-3218", "Text": "1 W at 27 5 \u00b0C, and a resistance\nof 2 7 W at 100 \u00b0C Determine the temperature coefficient of\nresistivity of silver"}, {"Chapter": "1", "sentence_range": "3216-3219", "Text": "5 \u00b0C, and a resistance\nof 2 7 W at 100 \u00b0C Determine the temperature coefficient of\nresistivity of silver 3"}, {"Chapter": "1", "sentence_range": "3217-3220", "Text": "7 W at 100 \u00b0C Determine the temperature coefficient of\nresistivity of silver 3 6\nA heating element using nichrome connected to a 230 V supply\ndraws an initial current of 3"}, {"Chapter": "1", "sentence_range": "3218-3221", "Text": "Determine the temperature coefficient of\nresistivity of silver 3 6\nA heating element using nichrome connected to a 230 V supply\ndraws an initial current of 3 2 A which settles after a few seconds to\na steady value of 2"}, {"Chapter": "1", "sentence_range": "3219-3222", "Text": "3 6\nA heating element using nichrome connected to a 230 V supply\ndraws an initial current of 3 2 A which settles after a few seconds to\na steady value of 2 8 A"}, {"Chapter": "1", "sentence_range": "3220-3223", "Text": "6\nA heating element using nichrome connected to a 230 V supply\ndraws an initial current of 3 2 A which settles after a few seconds to\na steady value of 2 8 A What is the steady temperature of the heating\nelement if the room temperature is 27"}, {"Chapter": "1", "sentence_range": "3221-3224", "Text": "2 A which settles after a few seconds to\na steady value of 2 8 A What is the steady temperature of the heating\nelement if the room temperature is 27 0 \u00b0C"}, {"Chapter": "1", "sentence_range": "3222-3225", "Text": "8 A What is the steady temperature of the heating\nelement if the room temperature is 27 0 \u00b0C Temperature coefficient\nof resistance of nichrome averaged over the temperature range\ninvolved is 1"}, {"Chapter": "1", "sentence_range": "3223-3226", "Text": "What is the steady temperature of the heating\nelement if the room temperature is 27 0 \u00b0C Temperature coefficient\nof resistance of nichrome averaged over the temperature range\ninvolved is 1 70 \u00d7 10\u20134 \u00b0C\u20131"}, {"Chapter": "1", "sentence_range": "3224-3227", "Text": "0 \u00b0C Temperature coefficient\nof resistance of nichrome averaged over the temperature range\ninvolved is 1 70 \u00d7 10\u20134 \u00b0C\u20131 3"}, {"Chapter": "1", "sentence_range": "3225-3228", "Text": "Temperature coefficient\nof resistance of nichrome averaged over the temperature range\ninvolved is 1 70 \u00d7 10\u20134 \u00b0C\u20131 3 7\nDetermine the current in each branch of the network shown in\nFig"}, {"Chapter": "1", "sentence_range": "3226-3229", "Text": "70 \u00d7 10\u20134 \u00b0C\u20131 3 7\nDetermine the current in each branch of the network shown in\nFig 3"}, {"Chapter": "1", "sentence_range": "3227-3230", "Text": "3 7\nDetermine the current in each branch of the network shown in\nFig 3 20:\nFIGURE 3"}, {"Chapter": "1", "sentence_range": "3228-3231", "Text": "7\nDetermine the current in each branch of the network shown in\nFig 3 20:\nFIGURE 3 20\n3"}, {"Chapter": "1", "sentence_range": "3229-3232", "Text": "3 20:\nFIGURE 3 20\n3 8\nA storage battery of emf 8"}, {"Chapter": "1", "sentence_range": "3230-3233", "Text": "20:\nFIGURE 3 20\n3 8\nA storage battery of emf 8 0 V and internal resistance 0"}, {"Chapter": "1", "sentence_range": "3231-3234", "Text": "20\n3 8\nA storage battery of emf 8 0 V and internal resistance 0 5 W is being\ncharged by a 120 V dc supply using a series resistor of 15"}, {"Chapter": "1", "sentence_range": "3232-3235", "Text": "8\nA storage battery of emf 8 0 V and internal resistance 0 5 W is being\ncharged by a 120 V dc supply using a series resistor of 15 5 W"}, {"Chapter": "1", "sentence_range": "3233-3236", "Text": "0 V and internal resistance 0 5 W is being\ncharged by a 120 V dc supply using a series resistor of 15 5 W What\nis the terminal voltage of the battery during charging"}, {"Chapter": "1", "sentence_range": "3234-3237", "Text": "5 W is being\ncharged by a 120 V dc supply using a series resistor of 15 5 W What\nis the terminal voltage of the battery during charging What is the\npurpose of having a series resistor in the charging circuit"}, {"Chapter": "1", "sentence_range": "3235-3238", "Text": "5 W What\nis the terminal voltage of the battery during charging What is the\npurpose of having a series resistor in the charging circuit 3"}, {"Chapter": "1", "sentence_range": "3236-3239", "Text": "What\nis the terminal voltage of the battery during charging What is the\npurpose of having a series resistor in the charging circuit 3 9\nThe number density of  free electrons in a copper conductor\nestimated in Example 3"}, {"Chapter": "1", "sentence_range": "3237-3240", "Text": "What is the\npurpose of having a series resistor in the charging circuit 3 9\nThe number density of  free electrons in a copper conductor\nestimated in Example 3 1 is 8"}, {"Chapter": "1", "sentence_range": "3238-3241", "Text": "3 9\nThe number density of  free electrons in a copper conductor\nestimated in Example 3 1 is 8 5 \u00d7 1028 m\u20133"}, {"Chapter": "1", "sentence_range": "3239-3242", "Text": "9\nThe number density of  free electrons in a copper conductor\nestimated in Example 3 1 is 8 5 \u00d7 1028 m\u20133 How long does an electron\ntake to drift from one end of a wire 3"}, {"Chapter": "1", "sentence_range": "3240-3243", "Text": "1 is 8 5 \u00d7 1028 m\u20133 How long does an electron\ntake to drift from one end of a wire 3 0 m long to its other end"}, {"Chapter": "1", "sentence_range": "3241-3244", "Text": "5 \u00d7 1028 m\u20133 How long does an electron\ntake to drift from one end of a wire 3 0 m long to its other end The\narea of cross-section of the wire is 2"}, {"Chapter": "1", "sentence_range": "3242-3245", "Text": "How long does an electron\ntake to drift from one end of a wire 3 0 m long to its other end The\narea of cross-section of the wire is 2 0 \u00d7 10\u20136 m2 and it is carrying a\ncurrent of 3"}, {"Chapter": "1", "sentence_range": "3243-3246", "Text": "0 m long to its other end The\narea of cross-section of the wire is 2 0 \u00d7 10\u20136 m2 and it is carrying a\ncurrent of 3 0 A"}, {"Chapter": "1", "sentence_range": "3244-3247", "Text": "The\narea of cross-section of the wire is 2 0 \u00d7 10\u20136 m2 and it is carrying a\ncurrent of 3 0 A Rationalised 2023-24\n4"}, {"Chapter": "1", "sentence_range": "3245-3248", "Text": "0 \u00d7 10\u20136 m2 and it is carrying a\ncurrent of 3 0 A Rationalised 2023-24\n4 1  INTRODUCTION\nBoth Electricity and Magnetism have been known for more than 2000\nyears"}, {"Chapter": "1", "sentence_range": "3246-3249", "Text": "0 A Rationalised 2023-24\n4 1  INTRODUCTION\nBoth Electricity and Magnetism have been known for more than 2000\nyears However, it was only about 200 years ago, in 1820, that it was\nrealised that they were intimately related"}, {"Chapter": "1", "sentence_range": "3247-3250", "Text": "Rationalised 2023-24\n4 1  INTRODUCTION\nBoth Electricity and Magnetism have been known for more than 2000\nyears However, it was only about 200 years ago, in 1820, that it was\nrealised that they were intimately related During a lecture demonstration\nin the summer of 1820, Danish physicist Hans Christian Oersted noticed\nthat a current in a straight wire caused a noticeable deflection in a nearby\nmagnetic compass needle"}, {"Chapter": "1", "sentence_range": "3248-3251", "Text": "1  INTRODUCTION\nBoth Electricity and Magnetism have been known for more than 2000\nyears However, it was only about 200 years ago, in 1820, that it was\nrealised that they were intimately related During a lecture demonstration\nin the summer of 1820, Danish physicist Hans Christian Oersted noticed\nthat a current in a straight wire caused a noticeable deflection in a nearby\nmagnetic compass needle He investigated this phenomenon"}, {"Chapter": "1", "sentence_range": "3249-3252", "Text": "However, it was only about 200 years ago, in 1820, that it was\nrealised that they were intimately related During a lecture demonstration\nin the summer of 1820, Danish physicist Hans Christian Oersted noticed\nthat a current in a straight wire caused a noticeable deflection in a nearby\nmagnetic compass needle He investigated this phenomenon He found\nthat the alignment of the needle is tangential to an imaginary circle which\nhas the straight wire as its centre and has its plane perpendicular to the\nwire"}, {"Chapter": "1", "sentence_range": "3250-3253", "Text": "During a lecture demonstration\nin the summer of 1820, Danish physicist Hans Christian Oersted noticed\nthat a current in a straight wire caused a noticeable deflection in a nearby\nmagnetic compass needle He investigated this phenomenon He found\nthat the alignment of the needle is tangential to an imaginary circle which\nhas the straight wire as its centre and has its plane perpendicular to the\nwire This situation is depicted in Fig"}, {"Chapter": "1", "sentence_range": "3251-3254", "Text": "He investigated this phenomenon He found\nthat the alignment of the needle is tangential to an imaginary circle which\nhas the straight wire as its centre and has its plane perpendicular to the\nwire This situation is depicted in Fig 4"}, {"Chapter": "1", "sentence_range": "3252-3255", "Text": "He found\nthat the alignment of the needle is tangential to an imaginary circle which\nhas the straight wire as its centre and has its plane perpendicular to the\nwire This situation is depicted in Fig 4 1(a)"}, {"Chapter": "1", "sentence_range": "3253-3256", "Text": "This situation is depicted in Fig 4 1(a) It is noticeable when the\ncurrent is large and the needle sufficiently close to the wire so that the\nearth\u2019s magnetic field may be ignored"}, {"Chapter": "1", "sentence_range": "3254-3257", "Text": "4 1(a) It is noticeable when the\ncurrent is large and the needle sufficiently close to the wire so that the\nearth\u2019s magnetic field may be ignored Reversing the direction of the\ncurrent reverses the orientation of the needle [Fig"}, {"Chapter": "1", "sentence_range": "3255-3258", "Text": "1(a) It is noticeable when the\ncurrent is large and the needle sufficiently close to the wire so that the\nearth\u2019s magnetic field may be ignored Reversing the direction of the\ncurrent reverses the orientation of the needle [Fig 4"}, {"Chapter": "1", "sentence_range": "3256-3259", "Text": "It is noticeable when the\ncurrent is large and the needle sufficiently close to the wire so that the\nearth\u2019s magnetic field may be ignored Reversing the direction of the\ncurrent reverses the orientation of the needle [Fig 4 1(b)]"}, {"Chapter": "1", "sentence_range": "3257-3260", "Text": "Reversing the direction of the\ncurrent reverses the orientation of the needle [Fig 4 1(b)] The deflection\nincreases on increasing the current or bringing the needle closer to the\nwire"}, {"Chapter": "1", "sentence_range": "3258-3261", "Text": "4 1(b)] The deflection\nincreases on increasing the current or bringing the needle closer to the\nwire Iron filings sprinkled around the wire arrange themselves in\nconcentric circles with the wire as the centre [Fig"}, {"Chapter": "1", "sentence_range": "3259-3262", "Text": "1(b)] The deflection\nincreases on increasing the current or bringing the needle closer to the\nwire Iron filings sprinkled around the wire arrange themselves in\nconcentric circles with the wire as the centre [Fig 4"}, {"Chapter": "1", "sentence_range": "3260-3263", "Text": "The deflection\nincreases on increasing the current or bringing the needle closer to the\nwire Iron filings sprinkled around the wire arrange themselves in\nconcentric circles with the wire as the centre [Fig 4 1(c)]"}, {"Chapter": "1", "sentence_range": "3261-3264", "Text": "Iron filings sprinkled around the wire arrange themselves in\nconcentric circles with the wire as the centre [Fig 4 1(c)] Oersted\nconcluded that moving charges or currents produced a magnetic field\nin the surrounding space"}, {"Chapter": "1", "sentence_range": "3262-3265", "Text": "4 1(c)] Oersted\nconcluded that moving charges or currents produced a magnetic field\nin the surrounding space Following this, there was intense experimentation"}, {"Chapter": "1", "sentence_range": "3263-3266", "Text": "1(c)] Oersted\nconcluded that moving charges or currents produced a magnetic field\nin the surrounding space Following this, there was intense experimentation In 1864, the laws\nobeyed by electricity and magnetism were unified and formulated by\nChapter Four\nMOVING CHARGES\nAND MAGNETISM\nRationalised 2023-24\nPhysics\n108\nJames Maxwell who then realised that light was electromagnetic waves"}, {"Chapter": "1", "sentence_range": "3264-3267", "Text": "Oersted\nconcluded that moving charges or currents produced a magnetic field\nin the surrounding space Following this, there was intense experimentation In 1864, the laws\nobeyed by electricity and magnetism were unified and formulated by\nChapter Four\nMOVING CHARGES\nAND MAGNETISM\nRationalised 2023-24\nPhysics\n108\nJames Maxwell who then realised that light was electromagnetic waves Radio waves were discovered by Hertz, and produced by  J"}, {"Chapter": "1", "sentence_range": "3265-3268", "Text": "Following this, there was intense experimentation In 1864, the laws\nobeyed by electricity and magnetism were unified and formulated by\nChapter Four\nMOVING CHARGES\nAND MAGNETISM\nRationalised 2023-24\nPhysics\n108\nJames Maxwell who then realised that light was electromagnetic waves Radio waves were discovered by Hertz, and produced by  J C"}, {"Chapter": "1", "sentence_range": "3266-3269", "Text": "In 1864, the laws\nobeyed by electricity and magnetism were unified and formulated by\nChapter Four\nMOVING CHARGES\nAND MAGNETISM\nRationalised 2023-24\nPhysics\n108\nJames Maxwell who then realised that light was electromagnetic waves Radio waves were discovered by Hertz, and produced by  J C Bose and\nG"}, {"Chapter": "1", "sentence_range": "3267-3270", "Text": "Radio waves were discovered by Hertz, and produced by  J C Bose and\nG Marconi by the end of the 19th century"}, {"Chapter": "1", "sentence_range": "3268-3271", "Text": "C Bose and\nG Marconi by the end of the 19th century A  remarkable scientific and\ntechnological progress took place in the 20th century"}, {"Chapter": "1", "sentence_range": "3269-3272", "Text": "Bose and\nG Marconi by the end of the 19th century A  remarkable scientific and\ntechnological progress took place in the 20th century This was due to\nour increased understanding of electromagnetism and the invention of\ndevices for production, amplification, transmission and detection of\nelectromagnetic waves"}, {"Chapter": "1", "sentence_range": "3270-3273", "Text": "Marconi by the end of the 19th century A  remarkable scientific and\ntechnological progress took place in the 20th century This was due to\nour increased understanding of electromagnetism and the invention of\ndevices for production, amplification, transmission and detection of\nelectromagnetic waves In this chapter, we will see how magnetic field exerts\nforces on moving charged particles, like electrons, protons,\nand current-carrying wires"}, {"Chapter": "1", "sentence_range": "3271-3274", "Text": "A  remarkable scientific and\ntechnological progress took place in the 20th century This was due to\nour increased understanding of electromagnetism and the invention of\ndevices for production, amplification, transmission and detection of\nelectromagnetic waves In this chapter, we will see how magnetic field exerts\nforces on moving charged particles, like electrons, protons,\nand current-carrying wires We shall also learn how\ncurrents produce magnetic fields"}, {"Chapter": "1", "sentence_range": "3272-3275", "Text": "This was due to\nour increased understanding of electromagnetism and the invention of\ndevices for production, amplification, transmission and detection of\nelectromagnetic waves In this chapter, we will see how magnetic field exerts\nforces on moving charged particles, like electrons, protons,\nand current-carrying wires We shall also learn how\ncurrents produce magnetic fields We shall see how\nparticles can be accelerated to very high energies in a\ncyclotron"}, {"Chapter": "1", "sentence_range": "3273-3276", "Text": "In this chapter, we will see how magnetic field exerts\nforces on moving charged particles, like electrons, protons,\nand current-carrying wires We shall also learn how\ncurrents produce magnetic fields We shall see how\nparticles can be accelerated to very high energies in a\ncyclotron We shall study how currents and voltages are\ndetected by a galvanometer"}, {"Chapter": "1", "sentence_range": "3274-3277", "Text": "We shall also learn how\ncurrents produce magnetic fields We shall see how\nparticles can be accelerated to very high energies in a\ncyclotron We shall study how currents and voltages are\ndetected by a galvanometer In this and subsequent Chapter on magnetism,\nwe adopt the following convention: A current or a\nfield (electric or magnetic) emerging out of the plane of the\npaper is depicted by a dot (\u00a4)"}, {"Chapter": "1", "sentence_range": "3275-3278", "Text": "We shall see how\nparticles can be accelerated to very high energies in a\ncyclotron We shall study how currents and voltages are\ndetected by a galvanometer In this and subsequent Chapter on magnetism,\nwe adopt the following convention: A current or a\nfield (electric or magnetic) emerging out of the plane of the\npaper is depicted by a dot (\u00a4) A current or a field going\ninto the plane of the paper is depicted by a cross (\uf0c4 )*"}, {"Chapter": "1", "sentence_range": "3276-3279", "Text": "We shall study how currents and voltages are\ndetected by a galvanometer In this and subsequent Chapter on magnetism,\nwe adopt the following convention: A current or a\nfield (electric or magnetic) emerging out of the plane of the\npaper is depicted by a dot (\u00a4) A current or a field going\ninto the plane of the paper is depicted by a cross (\uf0c4 )* Figures"}, {"Chapter": "1", "sentence_range": "3277-3280", "Text": "In this and subsequent Chapter on magnetism,\nwe adopt the following convention: A current or a\nfield (electric or magnetic) emerging out of the plane of the\npaper is depicted by a dot (\u00a4) A current or a field going\ninto the plane of the paper is depicted by a cross (\uf0c4 )* Figures 4"}, {"Chapter": "1", "sentence_range": "3278-3281", "Text": "A current or a field going\ninto the plane of the paper is depicted by a cross (\uf0c4 )* Figures 4 1(a) and 4"}, {"Chapter": "1", "sentence_range": "3279-3282", "Text": "Figures 4 1(a) and 4 1(b) correspond to these two\nsituations, respectively"}, {"Chapter": "1", "sentence_range": "3280-3283", "Text": "4 1(a) and 4 1(b) correspond to these two\nsituations, respectively 4"}, {"Chapter": "1", "sentence_range": "3281-3284", "Text": "1(a) and 4 1(b) correspond to these two\nsituations, respectively 4 2  MAGNETIC FORCE\n4"}, {"Chapter": "1", "sentence_range": "3282-3285", "Text": "1(b) correspond to these two\nsituations, respectively 4 2  MAGNETIC FORCE\n4 2"}, {"Chapter": "1", "sentence_range": "3283-3286", "Text": "4 2  MAGNETIC FORCE\n4 2 1  Sources and fields\nBefore we introduce the concept of a magnetic field B, we\nshall recapitulate what we have learnt in Chapter 1 about\nthe electric field E"}, {"Chapter": "1", "sentence_range": "3284-3287", "Text": "2  MAGNETIC FORCE\n4 2 1  Sources and fields\nBefore we introduce the concept of a magnetic field B, we\nshall recapitulate what we have learnt in Chapter 1 about\nthe electric field E We have seen that the interaction\nbetween two charges can be considered in two stages"}, {"Chapter": "1", "sentence_range": "3285-3288", "Text": "2 1  Sources and fields\nBefore we introduce the concept of a magnetic field B, we\nshall recapitulate what we have learnt in Chapter 1 about\nthe electric field E We have seen that the interaction\nbetween two charges can be considered in two stages The charge Q, the source of the field, produces an electric\nfield E, where\nFIGURE 4"}, {"Chapter": "1", "sentence_range": "3286-3289", "Text": "1  Sources and fields\nBefore we introduce the concept of a magnetic field B, we\nshall recapitulate what we have learnt in Chapter 1 about\nthe electric field E We have seen that the interaction\nbetween two charges can be considered in two stages The charge Q, the source of the field, produces an electric\nfield E, where\nFIGURE 4 1 The magnetic field due to a straight long current-carrying\nwire"}, {"Chapter": "1", "sentence_range": "3287-3290", "Text": "We have seen that the interaction\nbetween two charges can be considered in two stages The charge Q, the source of the field, produces an electric\nfield E, where\nFIGURE 4 1 The magnetic field due to a straight long current-carrying\nwire The wire is perpendicular to the plane of the paper"}, {"Chapter": "1", "sentence_range": "3288-3291", "Text": "The charge Q, the source of the field, produces an electric\nfield E, where\nFIGURE 4 1 The magnetic field due to a straight long current-carrying\nwire The wire is perpendicular to the plane of the paper A ring of\ncompass needles surrounds the wire"}, {"Chapter": "1", "sentence_range": "3289-3292", "Text": "1 The magnetic field due to a straight long current-carrying\nwire The wire is perpendicular to the plane of the paper A ring of\ncompass needles surrounds the wire The orientation of the needles is\nshown when (a) the current emerges out of the plane of the paper,\n(b) the current moves into the plane of the paper"}, {"Chapter": "1", "sentence_range": "3290-3293", "Text": "The wire is perpendicular to the plane of the paper A ring of\ncompass needles surrounds the wire The orientation of the needles is\nshown when (a) the current emerges out of the plane of the paper,\n(b) the current moves into the plane of the paper (c) The arrangement of\niron filings around the wire"}, {"Chapter": "1", "sentence_range": "3291-3294", "Text": "A ring of\ncompass needles surrounds the wire The orientation of the needles is\nshown when (a) the current emerges out of the plane of the paper,\n(b) the current moves into the plane of the paper (c) The arrangement of\niron filings around the wire The darkened ends of the needle represent\nnorth poles"}, {"Chapter": "1", "sentence_range": "3292-3295", "Text": "The orientation of the needles is\nshown when (a) the current emerges out of the plane of the paper,\n(b) the current moves into the plane of the paper (c) The arrangement of\niron filings around the wire The darkened ends of the needle represent\nnorth poles The effect of the earth\u2019s magnetic field is neglected"}, {"Chapter": "1", "sentence_range": "3293-3296", "Text": "(c) The arrangement of\niron filings around the wire The darkened ends of the needle represent\nnorth poles The effect of the earth\u2019s magnetic field is neglected *\nA dot appears like the tip of an arrow pointed at you, a cross is like the feathered\ntail of an arrow moving away from you"}, {"Chapter": "1", "sentence_range": "3294-3297", "Text": "The darkened ends of the needle represent\nnorth poles The effect of the earth\u2019s magnetic field is neglected *\nA dot appears like the tip of an arrow pointed at you, a cross is like the feathered\ntail of an arrow moving away from you Hans Christian Oersted\n(1777\u20131851) \nDanish\nphysicist and chemist,\nprofessor at Copenhagen"}, {"Chapter": "1", "sentence_range": "3295-3298", "Text": "The effect of the earth\u2019s magnetic field is neglected *\nA dot appears like the tip of an arrow pointed at you, a cross is like the feathered\ntail of an arrow moving away from you Hans Christian Oersted\n(1777\u20131851) \nDanish\nphysicist and chemist,\nprofessor at Copenhagen He \nobserved \nthat \ncompass needle suffers aa\ndeflection when placed\nnear a wire carrying an\nelectric \ncurrent"}, {"Chapter": "1", "sentence_range": "3296-3299", "Text": "*\nA dot appears like the tip of an arrow pointed at you, a cross is like the feathered\ntail of an arrow moving away from you Hans Christian Oersted\n(1777\u20131851) \nDanish\nphysicist and chemist,\nprofessor at Copenhagen He \nobserved \nthat \ncompass needle suffers aa\ndeflection when placed\nnear a wire carrying an\nelectric \ncurrent This\ndiscovery gave the first\nempirical evidence of a\nconnection between electric\nand magnetic phenomena"}, {"Chapter": "1", "sentence_range": "3297-3300", "Text": "Hans Christian Oersted\n(1777\u20131851) \nDanish\nphysicist and chemist,\nprofessor at Copenhagen He \nobserved \nthat \ncompass needle suffers aa\ndeflection when placed\nnear a wire carrying an\nelectric \ncurrent This\ndiscovery gave the first\nempirical evidence of a\nconnection between electric\nand magnetic phenomena HANS CHRISTIAN OERSTED  (1777\u20131851)\nRationalised 2023-24\n109\nMoving Charges and\nMagnetism\nE = Q \u02c6r / (4pe0)r2\n(4"}, {"Chapter": "1", "sentence_range": "3298-3301", "Text": "He \nobserved \nthat \ncompass needle suffers aa\ndeflection when placed\nnear a wire carrying an\nelectric \ncurrent This\ndiscovery gave the first\nempirical evidence of a\nconnection between electric\nand magnetic phenomena HANS CHRISTIAN OERSTED  (1777\u20131851)\nRationalised 2023-24\n109\nMoving Charges and\nMagnetism\nE = Q \u02c6r / (4pe0)r2\n(4 1)\nwhere \u02c6r  is unit vector along r,  and  the field E is a vector\nfield"}, {"Chapter": "1", "sentence_range": "3299-3302", "Text": "This\ndiscovery gave the first\nempirical evidence of a\nconnection between electric\nand magnetic phenomena HANS CHRISTIAN OERSTED  (1777\u20131851)\nRationalised 2023-24\n109\nMoving Charges and\nMagnetism\nE = Q \u02c6r / (4pe0)r2\n(4 1)\nwhere \u02c6r  is unit vector along r,  and  the field E is a vector\nfield A charge q interacts with this field and experiences\na force F given by\nF  =  q  E   = q Q \u02c6r   / (4pe0) r 2\n(4"}, {"Chapter": "1", "sentence_range": "3300-3303", "Text": "HANS CHRISTIAN OERSTED  (1777\u20131851)\nRationalised 2023-24\n109\nMoving Charges and\nMagnetism\nE = Q \u02c6r / (4pe0)r2\n(4 1)\nwhere \u02c6r  is unit vector along r,  and  the field E is a vector\nfield A charge q interacts with this field and experiences\na force F given by\nF  =  q  E   = q Q \u02c6r   / (4pe0) r 2\n(4 2)\nAs pointed out in the Chapter 1, the field E is not just\nan artefact but has a physical role"}, {"Chapter": "1", "sentence_range": "3301-3304", "Text": "1)\nwhere \u02c6r  is unit vector along r,  and  the field E is a vector\nfield A charge q interacts with this field and experiences\na force F given by\nF  =  q  E   = q Q \u02c6r   / (4pe0) r 2\n(4 2)\nAs pointed out in the Chapter 1, the field E is not just\nan artefact but has a physical role It can  convey  energy\nand momentum and is not established instantaneously\nbut takes finite time to propagate"}, {"Chapter": "1", "sentence_range": "3302-3305", "Text": "A charge q interacts with this field and experiences\na force F given by\nF  =  q  E   = q Q \u02c6r   / (4pe0) r 2\n(4 2)\nAs pointed out in the Chapter 1, the field E is not just\nan artefact but has a physical role It can  convey  energy\nand momentum and is not established instantaneously\nbut takes finite time to propagate The concept of a field\nwas specially stressed by Faraday  and was incorporated\nby Maxwell in his unification of electricity and magnetism"}, {"Chapter": "1", "sentence_range": "3303-3306", "Text": "2)\nAs pointed out in the Chapter 1, the field E is not just\nan artefact but has a physical role It can  convey  energy\nand momentum and is not established instantaneously\nbut takes finite time to propagate The concept of a field\nwas specially stressed by Faraday  and was incorporated\nby Maxwell in his unification of electricity and magnetism In addition to depending on each point in space, it can\nalso vary with time, i"}, {"Chapter": "1", "sentence_range": "3304-3307", "Text": "It can  convey  energy\nand momentum and is not established instantaneously\nbut takes finite time to propagate The concept of a field\nwas specially stressed by Faraday  and was incorporated\nby Maxwell in his unification of electricity and magnetism In addition to depending on each point in space, it can\nalso vary with time, i e"}, {"Chapter": "1", "sentence_range": "3305-3308", "Text": "The concept of a field\nwas specially stressed by Faraday  and was incorporated\nby Maxwell in his unification of electricity and magnetism In addition to depending on each point in space, it can\nalso vary with time, i e , be a function of time"}, {"Chapter": "1", "sentence_range": "3306-3309", "Text": "In addition to depending on each point in space, it can\nalso vary with time, i e , be a function of time In our\ndiscussions in this chapter, we will assume that the fields\ndo not change with time"}, {"Chapter": "1", "sentence_range": "3307-3310", "Text": "e , be a function of time In our\ndiscussions in this chapter, we will assume that the fields\ndo not change with time The field at a particular point can be due to one or\nmore charges"}, {"Chapter": "1", "sentence_range": "3308-3311", "Text": ", be a function of time In our\ndiscussions in this chapter, we will assume that the fields\ndo not change with time The field at a particular point can be due to one or\nmore charges If there are more charges the fields add\nvectorially"}, {"Chapter": "1", "sentence_range": "3309-3312", "Text": "In our\ndiscussions in this chapter, we will assume that the fields\ndo not change with time The field at a particular point can be due to one or\nmore charges If there are more charges the fields add\nvectorially You have already learnt in Chapter 1 that this\nis called the principle of superposition"}, {"Chapter": "1", "sentence_range": "3310-3313", "Text": "The field at a particular point can be due to one or\nmore charges If there are more charges the fields add\nvectorially You have already learnt in Chapter 1 that this\nis called the principle of superposition Once the field is\nknown, the force on a test charge is given by Eq"}, {"Chapter": "1", "sentence_range": "3311-3314", "Text": "If there are more charges the fields add\nvectorially You have already learnt in Chapter 1 that this\nis called the principle of superposition Once the field is\nknown, the force on a test charge is given by Eq (4"}, {"Chapter": "1", "sentence_range": "3312-3315", "Text": "You have already learnt in Chapter 1 that this\nis called the principle of superposition Once the field is\nknown, the force on a test charge is given by Eq (4 2)"}, {"Chapter": "1", "sentence_range": "3313-3316", "Text": "Once the field is\nknown, the force on a test charge is given by Eq (4 2) Just as static charges produce an electric field, the\ncurrents or moving charges produce (in addition) a\nmagnetic field, denoted by B (r), again a vector field"}, {"Chapter": "1", "sentence_range": "3314-3317", "Text": "(4 2) Just as static charges produce an electric field, the\ncurrents or moving charges produce (in addition) a\nmagnetic field, denoted by B (r), again a vector field It\nhas several basic properties identical to the electric field"}, {"Chapter": "1", "sentence_range": "3315-3318", "Text": "2) Just as static charges produce an electric field, the\ncurrents or moving charges produce (in addition) a\nmagnetic field, denoted by B (r), again a vector field It\nhas several basic properties identical to the electric field It is defined at each point in space (and can in addition\ndepend on time)"}, {"Chapter": "1", "sentence_range": "3316-3319", "Text": "Just as static charges produce an electric field, the\ncurrents or moving charges produce (in addition) a\nmagnetic field, denoted by B (r), again a vector field It\nhas several basic properties identical to the electric field It is defined at each point in space (and can in addition\ndepend on time) Experimentally, it is found to obey the\nprinciple of superposition: the magnetic field of several\nsources is the vector addition of magnetic field of each\nindividual source"}, {"Chapter": "1", "sentence_range": "3317-3320", "Text": "It\nhas several basic properties identical to the electric field It is defined at each point in space (and can in addition\ndepend on time) Experimentally, it is found to obey the\nprinciple of superposition: the magnetic field of several\nsources is the vector addition of magnetic field of each\nindividual source 4"}, {"Chapter": "1", "sentence_range": "3318-3321", "Text": "It is defined at each point in space (and can in addition\ndepend on time) Experimentally, it is found to obey the\nprinciple of superposition: the magnetic field of several\nsources is the vector addition of magnetic field of each\nindividual source 4 2"}, {"Chapter": "1", "sentence_range": "3319-3322", "Text": "Experimentally, it is found to obey the\nprinciple of superposition: the magnetic field of several\nsources is the vector addition of magnetic field of each\nindividual source 4 2 2  Magnetic Field,  Lorentz  Force\nLet us suppose that there is  a point charge q (moving\nwith a velocity v and, located at r at a given time t) in\npresence of both the electric field E (r) and the magnetic\nfield B (r)"}, {"Chapter": "1", "sentence_range": "3320-3323", "Text": "4 2 2  Magnetic Field,  Lorentz  Force\nLet us suppose that there is  a point charge q (moving\nwith a velocity v and, located at r at a given time t) in\npresence of both the electric field E (r) and the magnetic\nfield B (r) The force on an electric charge q due to both\nof them can be written as\nF   = q [ E (r) +  v \u00d7 B (r)] \u00ba Felectric +Fmagnetic\n(4"}, {"Chapter": "1", "sentence_range": "3321-3324", "Text": "2 2  Magnetic Field,  Lorentz  Force\nLet us suppose that there is  a point charge q (moving\nwith a velocity v and, located at r at a given time t) in\npresence of both the electric field E (r) and the magnetic\nfield B (r) The force on an electric charge q due to both\nof them can be written as\nF   = q [ E (r) +  v \u00d7 B (r)] \u00ba Felectric +Fmagnetic\n(4 3)\nThis force was given first  by H"}, {"Chapter": "1", "sentence_range": "3322-3325", "Text": "2  Magnetic Field,  Lorentz  Force\nLet us suppose that there is  a point charge q (moving\nwith a velocity v and, located at r at a given time t) in\npresence of both the electric field E (r) and the magnetic\nfield B (r) The force on an electric charge q due to both\nof them can be written as\nF   = q [ E (r) +  v \u00d7 B (r)] \u00ba Felectric +Fmagnetic\n(4 3)\nThis force was given first  by H A"}, {"Chapter": "1", "sentence_range": "3323-3326", "Text": "The force on an electric charge q due to both\nof them can be written as\nF   = q [ E (r) +  v \u00d7 B (r)] \u00ba Felectric +Fmagnetic\n(4 3)\nThis force was given first  by H A Lorentz based on the extensive\nexperiments of Ampere and others"}, {"Chapter": "1", "sentence_range": "3324-3327", "Text": "3)\nThis force was given first  by H A Lorentz based on the extensive\nexperiments of Ampere and others It is called the Lorentz force"}, {"Chapter": "1", "sentence_range": "3325-3328", "Text": "A Lorentz based on the extensive\nexperiments of Ampere and others It is called the Lorentz force You\nhave already studied in detail the force due to the electric field"}, {"Chapter": "1", "sentence_range": "3326-3329", "Text": "Lorentz based on the extensive\nexperiments of Ampere and others It is called the Lorentz force You\nhave already studied in detail the force due to the electric field If we\nlook at the interaction with the magnetic field, we find the following\nfeatures"}, {"Chapter": "1", "sentence_range": "3327-3330", "Text": "It is called the Lorentz force You\nhave already studied in detail the force due to the electric field If we\nlook at the interaction with the magnetic field, we find the following\nfeatures (i)\nIt depends on q, v and B (charge of the particle, the velocity and the\nmagnetic field)"}, {"Chapter": "1", "sentence_range": "3328-3331", "Text": "You\nhave already studied in detail the force due to the electric field If we\nlook at the interaction with the magnetic field, we find the following\nfeatures (i)\nIt depends on q, v and B (charge of the particle, the velocity and the\nmagnetic field) Force on a negative charge is opposite to that on a\npositive charge"}, {"Chapter": "1", "sentence_range": "3329-3332", "Text": "If we\nlook at the interaction with the magnetic field, we find the following\nfeatures (i)\nIt depends on q, v and B (charge of the particle, the velocity and the\nmagnetic field) Force on a negative charge is opposite to that on a\npositive charge (ii) The magnetic force q [ v \u00d7 B ] includes a vector product of velocity\nand magnetic field"}, {"Chapter": "1", "sentence_range": "3330-3333", "Text": "(i)\nIt depends on q, v and B (charge of the particle, the velocity and the\nmagnetic field) Force on a negative charge is opposite to that on a\npositive charge (ii) The magnetic force q [ v \u00d7 B ] includes a vector product of velocity\nand magnetic field The vector product makes the force due to magnetic\nHENDRIK ANTOON LORENTZ (1853 \u2013 1928)\nHendrik Antoon Lorentz\n(1853 \u2013 1928) Dutch\ntheoretical \nphysicist,\nprofessor at Leiden"}, {"Chapter": "1", "sentence_range": "3331-3334", "Text": "Force on a negative charge is opposite to that on a\npositive charge (ii) The magnetic force q [ v \u00d7 B ] includes a vector product of velocity\nand magnetic field The vector product makes the force due to magnetic\nHENDRIK ANTOON LORENTZ (1853 \u2013 1928)\nHendrik Antoon Lorentz\n(1853 \u2013 1928) Dutch\ntheoretical \nphysicist,\nprofessor at Leiden He\ninvestigated \nthe\nrelationship \nbetween\nelectricity, magnetism, and\nmechanics"}, {"Chapter": "1", "sentence_range": "3332-3335", "Text": "(ii) The magnetic force q [ v \u00d7 B ] includes a vector product of velocity\nand magnetic field The vector product makes the force due to magnetic\nHENDRIK ANTOON LORENTZ (1853 \u2013 1928)\nHendrik Antoon Lorentz\n(1853 \u2013 1928) Dutch\ntheoretical \nphysicist,\nprofessor at Leiden He\ninvestigated \nthe\nrelationship \nbetween\nelectricity, magnetism, and\nmechanics In order to\nexplain the observed effect\nof magnetic fields on\nemitters of light (Zeeman\neffect), he postulated the\nexistence of electric charges\nin the atom, for which he\nwas awarded the Nobel Prize\nin 1902"}, {"Chapter": "1", "sentence_range": "3333-3336", "Text": "The vector product makes the force due to magnetic\nHENDRIK ANTOON LORENTZ (1853 \u2013 1928)\nHendrik Antoon Lorentz\n(1853 \u2013 1928) Dutch\ntheoretical \nphysicist,\nprofessor at Leiden He\ninvestigated \nthe\nrelationship \nbetween\nelectricity, magnetism, and\nmechanics In order to\nexplain the observed effect\nof magnetic fields on\nemitters of light (Zeeman\neffect), he postulated the\nexistence of electric charges\nin the atom, for which he\nwas awarded the Nobel Prize\nin 1902 He derived a set of\ntransformation equations\n(known \nafter \nhim, \nas\nLorentz \ntransformation\nequations) by some tangled\nmathematical arguments,\nbut he was not aware that\nthese equations hinge on a\nnew concept of space and\ntime"}, {"Chapter": "1", "sentence_range": "3334-3337", "Text": "He\ninvestigated \nthe\nrelationship \nbetween\nelectricity, magnetism, and\nmechanics In order to\nexplain the observed effect\nof magnetic fields on\nemitters of light (Zeeman\neffect), he postulated the\nexistence of electric charges\nin the atom, for which he\nwas awarded the Nobel Prize\nin 1902 He derived a set of\ntransformation equations\n(known \nafter \nhim, \nas\nLorentz \ntransformation\nequations) by some tangled\nmathematical arguments,\nbut he was not aware that\nthese equations hinge on a\nnew concept of space and\ntime Rationalised 2023-24\nPhysics\n110\nfield vanish (become zero) if  velocity and magnetic field are parallel\nor anti-parallel"}, {"Chapter": "1", "sentence_range": "3335-3338", "Text": "In order to\nexplain the observed effect\nof magnetic fields on\nemitters of light (Zeeman\neffect), he postulated the\nexistence of electric charges\nin the atom, for which he\nwas awarded the Nobel Prize\nin 1902 He derived a set of\ntransformation equations\n(known \nafter \nhim, \nas\nLorentz \ntransformation\nequations) by some tangled\nmathematical arguments,\nbut he was not aware that\nthese equations hinge on a\nnew concept of space and\ntime Rationalised 2023-24\nPhysics\n110\nfield vanish (become zero) if  velocity and magnetic field are parallel\nor anti-parallel The force acts in a (sideways) direction perpendicular\nto both the velocity and the magnetic field"}, {"Chapter": "1", "sentence_range": "3336-3339", "Text": "He derived a set of\ntransformation equations\n(known \nafter \nhim, \nas\nLorentz \ntransformation\nequations) by some tangled\nmathematical arguments,\nbut he was not aware that\nthese equations hinge on a\nnew concept of space and\ntime Rationalised 2023-24\nPhysics\n110\nfield vanish (become zero) if  velocity and magnetic field are parallel\nor anti-parallel The force acts in a (sideways) direction perpendicular\nto both the velocity and the magnetic field Its\ndirection  is given by the screw rule or right hand\nrule  for vector (or cross) product as illustrated\nin Fig"}, {"Chapter": "1", "sentence_range": "3337-3340", "Text": "Rationalised 2023-24\nPhysics\n110\nfield vanish (become zero) if  velocity and magnetic field are parallel\nor anti-parallel The force acts in a (sideways) direction perpendicular\nto both the velocity and the magnetic field Its\ndirection  is given by the screw rule or right hand\nrule  for vector (or cross) product as illustrated\nin Fig 4"}, {"Chapter": "1", "sentence_range": "3338-3341", "Text": "The force acts in a (sideways) direction perpendicular\nto both the velocity and the magnetic field Its\ndirection  is given by the screw rule or right hand\nrule  for vector (or cross) product as illustrated\nin Fig 4 2"}, {"Chapter": "1", "sentence_range": "3339-3342", "Text": "Its\ndirection  is given by the screw rule or right hand\nrule  for vector (or cross) product as illustrated\nin Fig 4 2 (iii)\nThe magnetic force is zero if  charge is not\nmoving (as then |v|= 0)"}, {"Chapter": "1", "sentence_range": "3340-3343", "Text": "4 2 (iii)\nThe magnetic force is zero if  charge is not\nmoving (as then |v|= 0) Only a moving\ncharge feels the magnetic force"}, {"Chapter": "1", "sentence_range": "3341-3344", "Text": "2 (iii)\nThe magnetic force is zero if  charge is not\nmoving (as then |v|= 0) Only a moving\ncharge feels the magnetic force The expression for the magnetic force helps\nus to define the unit of the magnetic field, if one\ntakes q, F and v, all to be unity in the force\nequation F = q [ v \u00d7 B] =q v B sin q \u02c6n ,  where q is\nthe angle between v and B [see  Fig"}, {"Chapter": "1", "sentence_range": "3342-3345", "Text": "(iii)\nThe magnetic force is zero if  charge is not\nmoving (as then |v|= 0) Only a moving\ncharge feels the magnetic force The expression for the magnetic force helps\nus to define the unit of the magnetic field, if one\ntakes q, F and v, all to be unity in the force\nequation F = q [ v \u00d7 B] =q v B sin q \u02c6n ,  where q is\nthe angle between v and B [see  Fig 4"}, {"Chapter": "1", "sentence_range": "3343-3346", "Text": "Only a moving\ncharge feels the magnetic force The expression for the magnetic force helps\nus to define the unit of the magnetic field, if one\ntakes q, F and v, all to be unity in the force\nequation F = q [ v \u00d7 B] =q v B sin q \u02c6n ,  where q is\nthe angle between v and B [see  Fig 4 2 (a)]"}, {"Chapter": "1", "sentence_range": "3344-3347", "Text": "The expression for the magnetic force helps\nus to define the unit of the magnetic field, if one\ntakes q, F and v, all to be unity in the force\nequation F = q [ v \u00d7 B] =q v B sin q \u02c6n ,  where q is\nthe angle between v and B [see  Fig 4 2 (a)] The\nmagnitude of magnetic field B is 1 SI unit, when\nthe force acting on a unit charge (1 C), moving\nperpendicular to B with a speed 1m/s, is one\nnewton"}, {"Chapter": "1", "sentence_range": "3345-3348", "Text": "4 2 (a)] The\nmagnitude of magnetic field B is 1 SI unit, when\nthe force acting on a unit charge (1 C), moving\nperpendicular to B with a speed 1m/s, is one\nnewton Dimensionally, we have [B] = [F/qv] and the unit\nof B are Newton second / (coulomb metre)"}, {"Chapter": "1", "sentence_range": "3346-3349", "Text": "2 (a)] The\nmagnitude of magnetic field B is 1 SI unit, when\nthe force acting on a unit charge (1 C), moving\nperpendicular to B with a speed 1m/s, is one\nnewton Dimensionally, we have [B] = [F/qv] and the unit\nof B are Newton second / (coulomb metre) This unit is called tesla (T)\nnamed after Nikola Tesla (1856 \u2013 1943)"}, {"Chapter": "1", "sentence_range": "3347-3350", "Text": "The\nmagnitude of magnetic field B is 1 SI unit, when\nthe force acting on a unit charge (1 C), moving\nperpendicular to B with a speed 1m/s, is one\nnewton Dimensionally, we have [B] = [F/qv] and the unit\nof B are Newton second / (coulomb metre) This unit is called tesla (T)\nnamed after Nikola Tesla (1856 \u2013 1943) Tesla is a rather large unit"}, {"Chapter": "1", "sentence_range": "3348-3351", "Text": "Dimensionally, we have [B] = [F/qv] and the unit\nof B are Newton second / (coulomb metre) This unit is called tesla (T)\nnamed after Nikola Tesla (1856 \u2013 1943) Tesla is a rather large unit A\nsmaller  unit (non-SI) called gauss (=10\u20134 tesla) is also often used"}, {"Chapter": "1", "sentence_range": "3349-3352", "Text": "This unit is called tesla (T)\nnamed after Nikola Tesla (1856 \u2013 1943) Tesla is a rather large unit A\nsmaller  unit (non-SI) called gauss (=10\u20134 tesla) is also often used The\nearth\u2019s magnetic field is about 3"}, {"Chapter": "1", "sentence_range": "3350-3353", "Text": "Tesla is a rather large unit A\nsmaller  unit (non-SI) called gauss (=10\u20134 tesla) is also often used The\nearth\u2019s magnetic field is about 3 6 \u00d7 10\u20135 T"}, {"Chapter": "1", "sentence_range": "3351-3354", "Text": "A\nsmaller  unit (non-SI) called gauss (=10\u20134 tesla) is also often used The\nearth\u2019s magnetic field is about 3 6 \u00d7 10\u20135 T 4"}, {"Chapter": "1", "sentence_range": "3352-3355", "Text": "The\nearth\u2019s magnetic field is about 3 6 \u00d7 10\u20135 T 4 2"}, {"Chapter": "1", "sentence_range": "3353-3356", "Text": "6 \u00d7 10\u20135 T 4 2 3  Magnetic force on a current-carrying conductor\nWe can extend the analysis for force due to magnetic field on a single\nmoving charge to a straight rod carrying current"}, {"Chapter": "1", "sentence_range": "3354-3357", "Text": "4 2 3  Magnetic force on a current-carrying conductor\nWe can extend the analysis for force due to magnetic field on a single\nmoving charge to a straight rod carrying current Consider a rod of a\nuniform cross-sectional area A and length l"}, {"Chapter": "1", "sentence_range": "3355-3358", "Text": "2 3  Magnetic force on a current-carrying conductor\nWe can extend the analysis for force due to magnetic field on a single\nmoving charge to a straight rod carrying current Consider a rod of a\nuniform cross-sectional area A and length l We shall assume one kind\nof mobile carriers as in a conductor (here electrons)"}, {"Chapter": "1", "sentence_range": "3356-3359", "Text": "3  Magnetic force on a current-carrying conductor\nWe can extend the analysis for force due to magnetic field on a single\nmoving charge to a straight rod carrying current Consider a rod of a\nuniform cross-sectional area A and length l We shall assume one kind\nof mobile carriers as in a conductor (here electrons) Let the number\ndensity of these mobile charge carriers in it be n"}, {"Chapter": "1", "sentence_range": "3357-3360", "Text": "Consider a rod of a\nuniform cross-sectional area A and length l We shall assume one kind\nof mobile carriers as in a conductor (here electrons) Let the number\ndensity of these mobile charge carriers in it be n Then the total number\nof mobile charge carriers in it is nlA"}, {"Chapter": "1", "sentence_range": "3358-3361", "Text": "We shall assume one kind\nof mobile carriers as in a conductor (here electrons) Let the number\ndensity of these mobile charge carriers in it be n Then the total number\nof mobile charge carriers in it is nlA For a steady current I in this\nconducting rod, we may assume that each mobile carrier has an average\ndrift velocity vd (see Chapter 3)"}, {"Chapter": "1", "sentence_range": "3359-3362", "Text": "Let the number\ndensity of these mobile charge carriers in it be n Then the total number\nof mobile charge carriers in it is nlA For a steady current I in this\nconducting rod, we may assume that each mobile carrier has an average\ndrift velocity vd (see Chapter 3) In the presence of an external magnetic\nfield B, the force on these carriers is:\nF = (nlA)q vd \u00b4\u00b4\u00b4\u00b4\u00b4 B\nwhere q is the value of the charge on  a carrier"}, {"Chapter": "1", "sentence_range": "3360-3363", "Text": "Then the total number\nof mobile charge carriers in it is nlA For a steady current I in this\nconducting rod, we may assume that each mobile carrier has an average\ndrift velocity vd (see Chapter 3) In the presence of an external magnetic\nfield B, the force on these carriers is:\nF = (nlA)q vd \u00b4\u00b4\u00b4\u00b4\u00b4 B\nwhere q is the value of the charge on  a carrier Now nq vd is the current\ndensity j and |(nq vd)|A is the current I (see Chapter 3 for the discussion\nof current and current density)"}, {"Chapter": "1", "sentence_range": "3361-3364", "Text": "For a steady current I in this\nconducting rod, we may assume that each mobile carrier has an average\ndrift velocity vd (see Chapter 3) In the presence of an external magnetic\nfield B, the force on these carriers is:\nF = (nlA)q vd \u00b4\u00b4\u00b4\u00b4\u00b4 B\nwhere q is the value of the charge on  a carrier Now nq vd is the current\ndensity j and |(nq vd)|A is the current I (see Chapter 3 for the discussion\nof current and current density) Thus,\nF = [(nq vd )lA] \u00d7 B = [ jAl ] \u00b4\u00b4\u00b4\u00b4\u00b4 B\n   = Il \u00b4\u00b4\u00b4\u00b4\u00b4 B\n(4"}, {"Chapter": "1", "sentence_range": "3362-3365", "Text": "In the presence of an external magnetic\nfield B, the force on these carriers is:\nF = (nlA)q vd \u00b4\u00b4\u00b4\u00b4\u00b4 B\nwhere q is the value of the charge on  a carrier Now nq vd is the current\ndensity j and |(nq vd)|A is the current I (see Chapter 3 for the discussion\nof current and current density) Thus,\nF = [(nq vd )lA] \u00d7 B = [ jAl ] \u00b4\u00b4\u00b4\u00b4\u00b4 B\n   = Il \u00b4\u00b4\u00b4\u00b4\u00b4 B\n(4 4)\nwhere l is a vector of magnitude l, the length of the rod, and with a direction\nidentical to the current I"}, {"Chapter": "1", "sentence_range": "3363-3366", "Text": "Now nq vd is the current\ndensity j and |(nq vd)|A is the current I (see Chapter 3 for the discussion\nof current and current density) Thus,\nF = [(nq vd )lA] \u00d7 B = [ jAl ] \u00b4\u00b4\u00b4\u00b4\u00b4 B\n   = Il \u00b4\u00b4\u00b4\u00b4\u00b4 B\n(4 4)\nwhere l is a vector of magnitude l, the length of the rod, and with a direction\nidentical to the current I Note that the current I is not a vector"}, {"Chapter": "1", "sentence_range": "3364-3367", "Text": "Thus,\nF = [(nq vd )lA] \u00d7 B = [ jAl ] \u00b4\u00b4\u00b4\u00b4\u00b4 B\n   = Il \u00b4\u00b4\u00b4\u00b4\u00b4 B\n(4 4)\nwhere l is a vector of magnitude l, the length of the rod, and with a direction\nidentical to the current I Note that the current I is not a vector In the last\nstep leading to Eq"}, {"Chapter": "1", "sentence_range": "3365-3368", "Text": "4)\nwhere l is a vector of magnitude l, the length of the rod, and with a direction\nidentical to the current I Note that the current I is not a vector In the last\nstep leading to Eq (4"}, {"Chapter": "1", "sentence_range": "3366-3369", "Text": "Note that the current I is not a vector In the last\nstep leading to Eq (4 4), we have transferred the vector sign from  j to l"}, {"Chapter": "1", "sentence_range": "3367-3370", "Text": "In the last\nstep leading to Eq (4 4), we have transferred the vector sign from  j to l Equation (4"}, {"Chapter": "1", "sentence_range": "3368-3371", "Text": "(4 4), we have transferred the vector sign from  j to l Equation (4 4) holds for a straight rod"}, {"Chapter": "1", "sentence_range": "3369-3372", "Text": "4), we have transferred the vector sign from  j to l Equation (4 4) holds for a straight rod In this equation, B is the\nexternal magnetic field"}, {"Chapter": "1", "sentence_range": "3370-3373", "Text": "Equation (4 4) holds for a straight rod In this equation, B is the\nexternal magnetic field It is not the field produced by the current-carrying\nrod"}, {"Chapter": "1", "sentence_range": "3371-3374", "Text": "4) holds for a straight rod In this equation, B is the\nexternal magnetic field It is not the field produced by the current-carrying\nrod If the wire has an arbitrary shape we can calculate the Lorentz force\non it by considering it as a collection of linear strips dlj and summing\nj\nj\nId\n\u00d7\nF\uf03d \uf0e5\nB\nl\nThis summation can be converted to an integral in most cases"}, {"Chapter": "1", "sentence_range": "3372-3375", "Text": "In this equation, B is the\nexternal magnetic field It is not the field produced by the current-carrying\nrod If the wire has an arbitrary shape we can calculate the Lorentz force\non it by considering it as a collection of linear strips dlj and summing\nj\nj\nId\n\u00d7\nF\uf03d \uf0e5\nB\nl\nThis summation can be converted to an integral in most cases FIGURE 4"}, {"Chapter": "1", "sentence_range": "3373-3376", "Text": "It is not the field produced by the current-carrying\nrod If the wire has an arbitrary shape we can calculate the Lorentz force\non it by considering it as a collection of linear strips dlj and summing\nj\nj\nId\n\u00d7\nF\uf03d \uf0e5\nB\nl\nThis summation can be converted to an integral in most cases FIGURE 4 2 The direction of the magnetic\nforce acting on a charged particle"}, {"Chapter": "1", "sentence_range": "3374-3377", "Text": "If the wire has an arbitrary shape we can calculate the Lorentz force\non it by considering it as a collection of linear strips dlj and summing\nj\nj\nId\n\u00d7\nF\uf03d \uf0e5\nB\nl\nThis summation can be converted to an integral in most cases FIGURE 4 2 The direction of the magnetic\nforce acting on a charged particle (a) The\nforce on a positively charged particle with\nvelocity v and making an angle q with the\nmagnetic field B is given by the right-hand\nrule"}, {"Chapter": "1", "sentence_range": "3375-3378", "Text": "FIGURE 4 2 The direction of the magnetic\nforce acting on a charged particle (a) The\nforce on a positively charged particle with\nvelocity v and making an angle q with the\nmagnetic field B is given by the right-hand\nrule (b) A moving charged particle q is\ndeflected in an opposite sense to \u2013q in the\npresence of magnetic field"}, {"Chapter": "1", "sentence_range": "3376-3379", "Text": "2 The direction of the magnetic\nforce acting on a charged particle (a) The\nforce on a positively charged particle with\nvelocity v and making an angle q with the\nmagnetic field B is given by the right-hand\nrule (b) A moving charged particle q is\ndeflected in an opposite sense to \u2013q in the\npresence of magnetic field Rationalised 2023-24\n111\nMoving Charges and\nMagnetism\n EXAMPLE 4"}, {"Chapter": "1", "sentence_range": "3377-3380", "Text": "(a) The\nforce on a positively charged particle with\nvelocity v and making an angle q with the\nmagnetic field B is given by the right-hand\nrule (b) A moving charged particle q is\ndeflected in an opposite sense to \u2013q in the\npresence of magnetic field Rationalised 2023-24\n111\nMoving Charges and\nMagnetism\n EXAMPLE 4 1\nExample 4"}, {"Chapter": "1", "sentence_range": "3378-3381", "Text": "(b) A moving charged particle q is\ndeflected in an opposite sense to \u2013q in the\npresence of magnetic field Rationalised 2023-24\n111\nMoving Charges and\nMagnetism\n EXAMPLE 4 1\nExample 4 1 A straight wire of mass 200 g and length 1"}, {"Chapter": "1", "sentence_range": "3379-3382", "Text": "Rationalised 2023-24\n111\nMoving Charges and\nMagnetism\n EXAMPLE 4 1\nExample 4 1 A straight wire of mass 200 g and length 1 5 m carries\na current of 2 A"}, {"Chapter": "1", "sentence_range": "3380-3383", "Text": "1\nExample 4 1 A straight wire of mass 200 g and length 1 5 m carries\na current of 2 A It is suspended in mid-air by a uniform horizontal\nmagnetic field B (Fig"}, {"Chapter": "1", "sentence_range": "3381-3384", "Text": "1 A straight wire of mass 200 g and length 1 5 m carries\na current of 2 A It is suspended in mid-air by a uniform horizontal\nmagnetic field B (Fig 4"}, {"Chapter": "1", "sentence_range": "3382-3385", "Text": "5 m carries\na current of 2 A It is suspended in mid-air by a uniform horizontal\nmagnetic field B (Fig 4 3)"}, {"Chapter": "1", "sentence_range": "3383-3386", "Text": "It is suspended in mid-air by a uniform horizontal\nmagnetic field B (Fig 4 3) What is the magnitude of the magnetic\nfield"}, {"Chapter": "1", "sentence_range": "3384-3387", "Text": "4 3) What is the magnitude of the magnetic\nfield FIGURE 4"}, {"Chapter": "1", "sentence_range": "3385-3388", "Text": "3) What is the magnitude of the magnetic\nfield FIGURE 4 3\nSolution  From Eq"}, {"Chapter": "1", "sentence_range": "3386-3389", "Text": "What is the magnitude of the magnetic\nfield FIGURE 4 3\nSolution  From Eq (4"}, {"Chapter": "1", "sentence_range": "3387-3390", "Text": "FIGURE 4 3\nSolution  From Eq (4 4), we find that there is an upward force F, of\nmagnitude IlB,"}, {"Chapter": "1", "sentence_range": "3388-3391", "Text": "3\nSolution  From Eq (4 4), we find that there is an upward force F, of\nmagnitude IlB, For mid-air suspension, this must be balanced by\nthe force due to gravity:\nm g = I lB\n  \nm g\nB\nI l\n=\n     \n0"}, {"Chapter": "1", "sentence_range": "3389-3392", "Text": "(4 4), we find that there is an upward force F, of\nmagnitude IlB, For mid-air suspension, this must be balanced by\nthe force due to gravity:\nm g = I lB\n  \nm g\nB\nI l\n=\n     \n0 2\n9"}, {"Chapter": "1", "sentence_range": "3390-3393", "Text": "4), we find that there is an upward force F, of\nmagnitude IlB, For mid-air suspension, this must be balanced by\nthe force due to gravity:\nm g = I lB\n  \nm g\nB\nI l\n=\n     \n0 2\n9 8\n0"}, {"Chapter": "1", "sentence_range": "3391-3394", "Text": "For mid-air suspension, this must be balanced by\nthe force due to gravity:\nm g = I lB\n  \nm g\nB\nI l\n=\n     \n0 2\n9 8\n0 65 T\n2\n\u00d71"}, {"Chapter": "1", "sentence_range": "3392-3395", "Text": "2\n9 8\n0 65 T\n2\n\u00d71 5\n=\n=\n\u00d7\nNote that it would have been sufficient to specify m/l, the mass per\nunit length of the wire"}, {"Chapter": "1", "sentence_range": "3393-3396", "Text": "8\n0 65 T\n2\n\u00d71 5\n=\n=\n\u00d7\nNote that it would have been sufficient to specify m/l, the mass per\nunit length of the wire The earth\u2019s magnetic field is approximately\n4 \u00d7 10\u20135 T and we have ignored it"}, {"Chapter": "1", "sentence_range": "3394-3397", "Text": "65 T\n2\n\u00d71 5\n=\n=\n\u00d7\nNote that it would have been sufficient to specify m/l, the mass per\nunit length of the wire The earth\u2019s magnetic field is approximately\n4 \u00d7 10\u20135 T and we have ignored it Example 4"}, {"Chapter": "1", "sentence_range": "3395-3398", "Text": "5\n=\n=\n\u00d7\nNote that it would have been sufficient to specify m/l, the mass per\nunit length of the wire The earth\u2019s magnetic field is approximately\n4 \u00d7 10\u20135 T and we have ignored it Example 4 2 If the magnetic field is parallel to the positive y-axis\nand the charged particle is moving along the positive x-axis (Fig"}, {"Chapter": "1", "sentence_range": "3396-3399", "Text": "The earth\u2019s magnetic field is approximately\n4 \u00d7 10\u20135 T and we have ignored it Example 4 2 If the magnetic field is parallel to the positive y-axis\nand the charged particle is moving along the positive x-axis (Fig 4"}, {"Chapter": "1", "sentence_range": "3397-3400", "Text": "Example 4 2 If the magnetic field is parallel to the positive y-axis\nand the charged particle is moving along the positive x-axis (Fig 4 4),\nwhich way would the Lorentz force be for (a) an electron (negative\ncharge), (b) a proton (positive charge)"}, {"Chapter": "1", "sentence_range": "3398-3401", "Text": "2 If the magnetic field is parallel to the positive y-axis\nand the charged particle is moving along the positive x-axis (Fig 4 4),\nwhich way would the Lorentz force be for (a) an electron (negative\ncharge), (b) a proton (positive charge) FIGURE 4"}, {"Chapter": "1", "sentence_range": "3399-3402", "Text": "4 4),\nwhich way would the Lorentz force be for (a) an electron (negative\ncharge), (b) a proton (positive charge) FIGURE 4 4\nSolution   The velocity v of particle is along the x-axis, while B, the\nmagnetic field is along the y-axis, so v \u00d7 B is along the z-axis (screw\nrule or right-hand thumb rule)"}, {"Chapter": "1", "sentence_range": "3400-3403", "Text": "4),\nwhich way would the Lorentz force be for (a) an electron (negative\ncharge), (b) a proton (positive charge) FIGURE 4 4\nSolution   The velocity v of particle is along the x-axis, while B, the\nmagnetic field is along the y-axis, so v \u00d7 B is along the z-axis (screw\nrule or right-hand thumb rule) So, (a) for electron it will be along \u2013z\naxis"}, {"Chapter": "1", "sentence_range": "3401-3404", "Text": "FIGURE 4 4\nSolution   The velocity v of particle is along the x-axis, while B, the\nmagnetic field is along the y-axis, so v \u00d7 B is along the z-axis (screw\nrule or right-hand thumb rule) So, (a) for electron it will be along \u2013z\naxis (b) for a positive charge (proton) the force is along +z axis"}, {"Chapter": "1", "sentence_range": "3402-3405", "Text": "4\nSolution   The velocity v of particle is along the x-axis, while B, the\nmagnetic field is along the y-axis, so v \u00d7 B is along the z-axis (screw\nrule or right-hand thumb rule) So, (a) for electron it will be along \u2013z\naxis (b) for a positive charge (proton) the force is along +z axis EXAMPLE 4"}, {"Chapter": "1", "sentence_range": "3403-3406", "Text": "So, (a) for electron it will be along \u2013z\naxis (b) for a positive charge (proton) the force is along +z axis EXAMPLE 4 2\nCharged particles moving in a magnetic field"}, {"Chapter": "1", "sentence_range": "3404-3407", "Text": "(b) for a positive charge (proton) the force is along +z axis EXAMPLE 4 2\nCharged particles moving in a magnetic field Interactive demonstration:\nhttp://www"}, {"Chapter": "1", "sentence_range": "3405-3408", "Text": "EXAMPLE 4 2\nCharged particles moving in a magnetic field Interactive demonstration:\nhttp://www phys"}, {"Chapter": "1", "sentence_range": "3406-3409", "Text": "2\nCharged particles moving in a magnetic field Interactive demonstration:\nhttp://www phys hawaii"}, {"Chapter": "1", "sentence_range": "3407-3410", "Text": "Interactive demonstration:\nhttp://www phys hawaii edu/~teb/optics/java/partmagn/index"}, {"Chapter": "1", "sentence_range": "3408-3411", "Text": "phys hawaii edu/~teb/optics/java/partmagn/index html\nRationalised 2023-24\nPhysics\n112\n4"}, {"Chapter": "1", "sentence_range": "3409-3412", "Text": "hawaii edu/~teb/optics/java/partmagn/index html\nRationalised 2023-24\nPhysics\n112\n4 3  MOTION IN A MAGNETIC FIELD\nWe will now consider, in greater detail, the motion of a charge moving in\na  magnetic field"}, {"Chapter": "1", "sentence_range": "3410-3413", "Text": "edu/~teb/optics/java/partmagn/index html\nRationalised 2023-24\nPhysics\n112\n4 3  MOTION IN A MAGNETIC FIELD\nWe will now consider, in greater detail, the motion of a charge moving in\na  magnetic field We have learnt  in Mechanics (see Class XI book, Chapter\n6) that a force on a particle does work if  the force has a component along\n(or opposed to) the direction of motion of the particle"}, {"Chapter": "1", "sentence_range": "3411-3414", "Text": "html\nRationalised 2023-24\nPhysics\n112\n4 3  MOTION IN A MAGNETIC FIELD\nWe will now consider, in greater detail, the motion of a charge moving in\na  magnetic field We have learnt  in Mechanics (see Class XI book, Chapter\n6) that a force on a particle does work if  the force has a component along\n(or opposed to) the direction of motion of the particle In the case of motion\nof a charge in a magnetic field, the magnetic force is perpendicular to the\nvelocity of the particle"}, {"Chapter": "1", "sentence_range": "3412-3415", "Text": "3  MOTION IN A MAGNETIC FIELD\nWe will now consider, in greater detail, the motion of a charge moving in\na  magnetic field We have learnt  in Mechanics (see Class XI book, Chapter\n6) that a force on a particle does work if  the force has a component along\n(or opposed to) the direction of motion of the particle In the case of motion\nof a charge in a magnetic field, the magnetic force is perpendicular to the\nvelocity of the particle So no work is done and no change in the magnitude\nof the velocity is produced (though the direction of momentum may be\nchanged)"}, {"Chapter": "1", "sentence_range": "3413-3416", "Text": "We have learnt  in Mechanics (see Class XI book, Chapter\n6) that a force on a particle does work if  the force has a component along\n(or opposed to) the direction of motion of the particle In the case of motion\nof a charge in a magnetic field, the magnetic force is perpendicular to the\nvelocity of the particle So no work is done and no change in the magnitude\nof the velocity is produced (though the direction of momentum may be\nchanged) [Notice that this is unlike the force due to an electric field, qE,\nwhich can have a component parallel (or antiparallel) to motion and thus\ncan transfer energy in addition to momentum"}, {"Chapter": "1", "sentence_range": "3414-3417", "Text": "In the case of motion\nof a charge in a magnetic field, the magnetic force is perpendicular to the\nvelocity of the particle So no work is done and no change in the magnitude\nof the velocity is produced (though the direction of momentum may be\nchanged) [Notice that this is unlike the force due to an electric field, qE,\nwhich can have a component parallel (or antiparallel) to motion and thus\ncan transfer energy in addition to momentum ]\nWe shall consider motion of a charged particle in a uniform magnetic\nfield"}, {"Chapter": "1", "sentence_range": "3415-3418", "Text": "So no work is done and no change in the magnitude\nof the velocity is produced (though the direction of momentum may be\nchanged) [Notice that this is unlike the force due to an electric field, qE,\nwhich can have a component parallel (or antiparallel) to motion and thus\ncan transfer energy in addition to momentum ]\nWe shall consider motion of a charged particle in a uniform magnetic\nfield First consider the case of v perpendicular to B"}, {"Chapter": "1", "sentence_range": "3416-3419", "Text": "[Notice that this is unlike the force due to an electric field, qE,\nwhich can have a component parallel (or antiparallel) to motion and thus\ncan transfer energy in addition to momentum ]\nWe shall consider motion of a charged particle in a uniform magnetic\nfield First consider the case of v perpendicular to B The\nperpendicular force, q v \u00d7 B, acts as a centripetal force and\nproduces a circular motion perpendicular to the magnetic field"}, {"Chapter": "1", "sentence_range": "3417-3420", "Text": "]\nWe shall consider motion of a charged particle in a uniform magnetic\nfield First consider the case of v perpendicular to B The\nperpendicular force, q v \u00d7 B, acts as a centripetal force and\nproduces a circular motion perpendicular to the magnetic field The particle will describe a circle if v and B are  perpendicular\nto each other (Fig"}, {"Chapter": "1", "sentence_range": "3418-3421", "Text": "First consider the case of v perpendicular to B The\nperpendicular force, q v \u00d7 B, acts as a centripetal force and\nproduces a circular motion perpendicular to the magnetic field The particle will describe a circle if v and B are  perpendicular\nto each other (Fig 4"}, {"Chapter": "1", "sentence_range": "3419-3422", "Text": "The\nperpendicular force, q v \u00d7 B, acts as a centripetal force and\nproduces a circular motion perpendicular to the magnetic field The particle will describe a circle if v and B are  perpendicular\nto each other (Fig 4 5)"}, {"Chapter": "1", "sentence_range": "3420-3423", "Text": "The particle will describe a circle if v and B are  perpendicular\nto each other (Fig 4 5) If velocity has a component along B, this component\nremains unchanged as the motion along the magnetic field will\nnot be affected by the magnetic field"}, {"Chapter": "1", "sentence_range": "3421-3424", "Text": "4 5) If velocity has a component along B, this component\nremains unchanged as the motion along the magnetic field will\nnot be affected by the magnetic field The motion in a plane\nperpendicular to B is as before a circular one, thereby producing\na helical motion (Fig"}, {"Chapter": "1", "sentence_range": "3422-3425", "Text": "5) If velocity has a component along B, this component\nremains unchanged as the motion along the magnetic field will\nnot be affected by the magnetic field The motion in a plane\nperpendicular to B is as before a circular one, thereby producing\na helical motion (Fig 4"}, {"Chapter": "1", "sentence_range": "3423-3426", "Text": "If velocity has a component along B, this component\nremains unchanged as the motion along the magnetic field will\nnot be affected by the magnetic field The motion in a plane\nperpendicular to B is as before a circular one, thereby producing\na helical motion (Fig 4 6)"}, {"Chapter": "1", "sentence_range": "3424-3427", "Text": "The motion in a plane\nperpendicular to B is as before a circular one, thereby producing\na helical motion (Fig 4 6) You have already learnt in earlier classes (See Class XI,\nChapter 4) that if r is the radius of the circular path of a particle,\nthen a force of  m v2 / r, acts perpendicular to the path towards\nthe centre of the circle, and is called the  centripetal force"}, {"Chapter": "1", "sentence_range": "3425-3428", "Text": "4 6) You have already learnt in earlier classes (See Class XI,\nChapter 4) that if r is the radius of the circular path of a particle,\nthen a force of  m v2 / r, acts perpendicular to the path towards\nthe centre of the circle, and is called the  centripetal force If the\nvelocity v is perpendicular to the magnetic field   B, the  magnetic\nforce is  perpendicular to both v and B and acts\nlike a centripetal force"}, {"Chapter": "1", "sentence_range": "3426-3429", "Text": "6) You have already learnt in earlier classes (See Class XI,\nChapter 4) that if r is the radius of the circular path of a particle,\nthen a force of  m v2 / r, acts perpendicular to the path towards\nthe centre of the circle, and is called the  centripetal force If the\nvelocity v is perpendicular to the magnetic field   B, the  magnetic\nforce is  perpendicular to both v and B and acts\nlike a centripetal force It has a magnitude q v\nB"}, {"Chapter": "1", "sentence_range": "3427-3430", "Text": "You have already learnt in earlier classes (See Class XI,\nChapter 4) that if r is the radius of the circular path of a particle,\nthen a force of  m v2 / r, acts perpendicular to the path towards\nthe centre of the circle, and is called the  centripetal force If the\nvelocity v is perpendicular to the magnetic field   B, the  magnetic\nforce is  perpendicular to both v and B and acts\nlike a centripetal force It has a magnitude q v\nB Equating the two expressions for centripetal\nforce,\nm v 2/r = q v B, which gives\nr =  m v / qB\n(4"}, {"Chapter": "1", "sentence_range": "3428-3431", "Text": "If the\nvelocity v is perpendicular to the magnetic field   B, the  magnetic\nforce is  perpendicular to both v and B and acts\nlike a centripetal force It has a magnitude q v\nB Equating the two expressions for centripetal\nforce,\nm v 2/r = q v B, which gives\nr =  m v / qB\n(4 5)\nfor the radius of the circle described by the\ncharged particle"}, {"Chapter": "1", "sentence_range": "3429-3432", "Text": "It has a magnitude q v\nB Equating the two expressions for centripetal\nforce,\nm v 2/r = q v B, which gives\nr =  m v / qB\n(4 5)\nfor the radius of the circle described by the\ncharged particle The larger the  momentum,  the\nlarger is the radius and bigger the circle\ndescribed"}, {"Chapter": "1", "sentence_range": "3430-3433", "Text": "Equating the two expressions for centripetal\nforce,\nm v 2/r = q v B, which gives\nr =  m v / qB\n(4 5)\nfor the radius of the circle described by the\ncharged particle The larger the  momentum,  the\nlarger is the radius and bigger the circle\ndescribed If w is the angular frequency, then   v\n= w  r"}, {"Chapter": "1", "sentence_range": "3431-3434", "Text": "5)\nfor the radius of the circle described by the\ncharged particle The larger the  momentum,  the\nlarger is the radius and bigger the circle\ndescribed If w is the angular frequency, then   v\n= w  r So,\nw  = 2p n =  q B/ m\n[4"}, {"Chapter": "1", "sentence_range": "3432-3435", "Text": "The larger the  momentum,  the\nlarger is the radius and bigger the circle\ndescribed If w is the angular frequency, then   v\n= w  r So,\nw  = 2p n =  q B/ m\n[4 6(a)]\nwhich is independent of the velocity or energy"}, {"Chapter": "1", "sentence_range": "3433-3436", "Text": "If w is the angular frequency, then   v\n= w  r So,\nw  = 2p n =  q B/ m\n[4 6(a)]\nwhich is independent of the velocity or energy Here n is the frequency of rotation"}, {"Chapter": "1", "sentence_range": "3434-3437", "Text": "So,\nw  = 2p n =  q B/ m\n[4 6(a)]\nwhich is independent of the velocity or energy Here n is the frequency of rotation The\nindependence of n from energy has important\napplication in the design of a cyclotron (see\nSection 4"}, {"Chapter": "1", "sentence_range": "3435-3438", "Text": "6(a)]\nwhich is independent of the velocity or energy Here n is the frequency of rotation The\nindependence of n from energy has important\napplication in the design of a cyclotron (see\nSection 4 4"}, {"Chapter": "1", "sentence_range": "3436-3439", "Text": "Here n is the frequency of rotation The\nindependence of n from energy has important\napplication in the design of a cyclotron (see\nSection 4 4 2)"}, {"Chapter": "1", "sentence_range": "3437-3440", "Text": "The\nindependence of n from energy has important\napplication in the design of a cyclotron (see\nSection 4 4 2) The time taken for one revolution is T= 2p/w\n\u00ba 1/n"}, {"Chapter": "1", "sentence_range": "3438-3441", "Text": "4 2) The time taken for one revolution is T= 2p/w\n\u00ba 1/n If there is a  component of the velocity\nparallel to the magnetic field (denoted by v||), it will make the particle\nFIGURE 4"}, {"Chapter": "1", "sentence_range": "3439-3442", "Text": "2) The time taken for one revolution is T= 2p/w\n\u00ba 1/n If there is a  component of the velocity\nparallel to the magnetic field (denoted by v||), it will make the particle\nFIGURE 4 5 Circular motion\nFIGURE 4"}, {"Chapter": "1", "sentence_range": "3440-3443", "Text": "The time taken for one revolution is T= 2p/w\n\u00ba 1/n If there is a  component of the velocity\nparallel to the magnetic field (denoted by v||), it will make the particle\nFIGURE 4 5 Circular motion\nFIGURE 4 6 Helical motion\nRationalised 2023-24\n113\nMoving Charges and\nMagnetism\n EXAMPLE 4"}, {"Chapter": "1", "sentence_range": "3441-3444", "Text": "If there is a  component of the velocity\nparallel to the magnetic field (denoted by v||), it will make the particle\nFIGURE 4 5 Circular motion\nFIGURE 4 6 Helical motion\nRationalised 2023-24\n113\nMoving Charges and\nMagnetism\n EXAMPLE 4 3\nExample  4"}, {"Chapter": "1", "sentence_range": "3442-3445", "Text": "5 Circular motion\nFIGURE 4 6 Helical motion\nRationalised 2023-24\n113\nMoving Charges and\nMagnetism\n EXAMPLE 4 3\nExample  4 3   What is the radius of the path of an electron (mass\n9 \u00d7 10-31 kg and charge 1"}, {"Chapter": "1", "sentence_range": "3443-3446", "Text": "6 Helical motion\nRationalised 2023-24\n113\nMoving Charges and\nMagnetism\n EXAMPLE 4 3\nExample  4 3   What is the radius of the path of an electron (mass\n9 \u00d7 10-31 kg and charge 1 6 \u00d7 10\u201319 C) moving at a speed of 3 \u00d7107 m/s in\na magnetic field of 6 \u00d7 10\u20134 T perpendicular to it"}, {"Chapter": "1", "sentence_range": "3444-3447", "Text": "3\nExample  4 3   What is the radius of the path of an electron (mass\n9 \u00d7 10-31 kg and charge 1 6 \u00d7 10\u201319 C) moving at a speed of 3 \u00d7107 m/s in\na magnetic field of 6 \u00d7 10\u20134 T perpendicular to it What is its\nfrequency"}, {"Chapter": "1", "sentence_range": "3445-3448", "Text": "3   What is the radius of the path of an electron (mass\n9 \u00d7 10-31 kg and charge 1 6 \u00d7 10\u201319 C) moving at a speed of 3 \u00d7107 m/s in\na magnetic field of 6 \u00d7 10\u20134 T perpendicular to it What is its\nfrequency Calculate its energy in  keV"}, {"Chapter": "1", "sentence_range": "3446-3449", "Text": "6 \u00d7 10\u201319 C) moving at a speed of 3 \u00d7107 m/s in\na magnetic field of 6 \u00d7 10\u20134 T perpendicular to it What is its\nfrequency Calculate its energy in  keV ( 1 eV = 1"}, {"Chapter": "1", "sentence_range": "3447-3450", "Text": "What is its\nfrequency Calculate its energy in  keV ( 1 eV = 1 6 \u00d7 10\u201319 J)"}, {"Chapter": "1", "sentence_range": "3448-3451", "Text": "Calculate its energy in  keV ( 1 eV = 1 6 \u00d7 10\u201319 J) Solution Using Eq"}, {"Chapter": "1", "sentence_range": "3449-3452", "Text": "( 1 eV = 1 6 \u00d7 10\u201319 J) Solution Using Eq (4"}, {"Chapter": "1", "sentence_range": "3450-3453", "Text": "6 \u00d7 10\u201319 J) Solution Using Eq (4 5) we find\nr  = m v / (qB)  = 9 \u00d710\u201331 kg \u00d7 3 \u00d7 107  m s\u20131 / ( 1"}, {"Chapter": "1", "sentence_range": "3451-3454", "Text": "Solution Using Eq (4 5) we find\nr  = m v / (qB)  = 9 \u00d710\u201331 kg \u00d7 3 \u00d7 107  m s\u20131 / ( 1 6 \u00d7 10\u201319 C \u00d7 6 \u00d7 10\u20134 T)\n                      = 28 \u00d7 10\u20132 m = 28 cm\nn = v / (2 pr) = 17\u00d7106 s\u20131   = 17\u00d7106  Hz   =17 MHz"}, {"Chapter": "1", "sentence_range": "3452-3455", "Text": "(4 5) we find\nr  = m v / (qB)  = 9 \u00d710\u201331 kg \u00d7 3 \u00d7 107  m s\u20131 / ( 1 6 \u00d7 10\u201319 C \u00d7 6 \u00d7 10\u20134 T)\n                      = 28 \u00d7 10\u20132 m = 28 cm\nn = v / (2 pr) = 17\u00d7106 s\u20131   = 17\u00d7106  Hz   =17 MHz E = (\u00bd )mv 2   = (\u00bd ) 9 \u00d7 10\u201331 kg \u00d7 9 \u00d7 1014 m2/s2  = 40"}, {"Chapter": "1", "sentence_range": "3453-3456", "Text": "5) we find\nr  = m v / (qB)  = 9 \u00d710\u201331 kg \u00d7 3 \u00d7 107  m s\u20131 / ( 1 6 \u00d7 10\u201319 C \u00d7 6 \u00d7 10\u20134 T)\n                      = 28 \u00d7 10\u20132 m = 28 cm\nn = v / (2 pr) = 17\u00d7106 s\u20131   = 17\u00d7106  Hz   =17 MHz E = (\u00bd )mv 2   = (\u00bd ) 9 \u00d7 10\u201331 kg \u00d7 9 \u00d7 1014 m2/s2  = 40 5 \u00d710\u201317 J\n  \u2248  4\u00d710\u201316 J = 2"}, {"Chapter": "1", "sentence_range": "3454-3457", "Text": "6 \u00d7 10\u201319 C \u00d7 6 \u00d7 10\u20134 T)\n                      = 28 \u00d7 10\u20132 m = 28 cm\nn = v / (2 pr) = 17\u00d7106 s\u20131   = 17\u00d7106  Hz   =17 MHz E = (\u00bd )mv 2   = (\u00bd ) 9 \u00d7 10\u201331 kg \u00d7 9 \u00d7 1014 m2/s2  = 40 5 \u00d710\u201317 J\n  \u2248  4\u00d710\u201316 J = 2 5 keV"}, {"Chapter": "1", "sentence_range": "3455-3458", "Text": "E = (\u00bd )mv 2   = (\u00bd ) 9 \u00d7 10\u201331 kg \u00d7 9 \u00d7 1014 m2/s2  = 40 5 \u00d710\u201317 J\n  \u2248  4\u00d710\u201316 J = 2 5 keV move along the field and the path of the particle would be a helical one\n(Fig"}, {"Chapter": "1", "sentence_range": "3456-3459", "Text": "5 \u00d710\u201317 J\n  \u2248  4\u00d710\u201316 J = 2 5 keV move along the field and the path of the particle would be a helical one\n(Fig 4"}, {"Chapter": "1", "sentence_range": "3457-3460", "Text": "5 keV move along the field and the path of the particle would be a helical one\n(Fig 4 6)"}, {"Chapter": "1", "sentence_range": "3458-3461", "Text": "move along the field and the path of the particle would be a helical one\n(Fig 4 6) The distance moved along the magnetic field  in one rotation is\ncalled pitch p"}, {"Chapter": "1", "sentence_range": "3459-3462", "Text": "4 6) The distance moved along the magnetic field  in one rotation is\ncalled pitch p Using Eq"}, {"Chapter": "1", "sentence_range": "3460-3463", "Text": "6) The distance moved along the magnetic field  in one rotation is\ncalled pitch p Using Eq [4"}, {"Chapter": "1", "sentence_range": "3461-3464", "Text": "The distance moved along the magnetic field  in one rotation is\ncalled pitch p Using Eq [4 6 (a)], we have\np =   v||T  =  2pm v|| / q B\n[4"}, {"Chapter": "1", "sentence_range": "3462-3465", "Text": "Using Eq [4 6 (a)], we have\np =   v||T  =  2pm v|| / q B\n[4 6(b)]\nThe radius of the circular component of motion is called the radius of\nthe helix"}, {"Chapter": "1", "sentence_range": "3463-3466", "Text": "[4 6 (a)], we have\np =   v||T  =  2pm v|| / q B\n[4 6(b)]\nThe radius of the circular component of motion is called the radius of\nthe helix 4"}, {"Chapter": "1", "sentence_range": "3464-3467", "Text": "6 (a)], we have\np =   v||T  =  2pm v|| / q B\n[4 6(b)]\nThe radius of the circular component of motion is called the radius of\nthe helix 4 4\nMAGNETIC FIELD DUE TO A CURRENT ELEMENT,\nBIOT-SAVART LAW\nAll magnetic fields that we know are due to currents (or moving\ncharges) and due to intrinsic magnetic moments of particles"}, {"Chapter": "1", "sentence_range": "3465-3468", "Text": "6(b)]\nThe radius of the circular component of motion is called the radius of\nthe helix 4 4\nMAGNETIC FIELD DUE TO A CURRENT ELEMENT,\nBIOT-SAVART LAW\nAll magnetic fields that we know are due to currents (or moving\ncharges) and due to intrinsic magnetic moments of particles Here, we shall study the relation between current and the\nmagnetic field it produces"}, {"Chapter": "1", "sentence_range": "3466-3469", "Text": "4 4\nMAGNETIC FIELD DUE TO A CURRENT ELEMENT,\nBIOT-SAVART LAW\nAll magnetic fields that we know are due to currents (or moving\ncharges) and due to intrinsic magnetic moments of particles Here, we shall study the relation between current and the\nmagnetic field it produces It is given by the Biot-Savart\u2019s  law"}, {"Chapter": "1", "sentence_range": "3467-3470", "Text": "4\nMAGNETIC FIELD DUE TO A CURRENT ELEMENT,\nBIOT-SAVART LAW\nAll magnetic fields that we know are due to currents (or moving\ncharges) and due to intrinsic magnetic moments of particles Here, we shall study the relation between current and the\nmagnetic field it produces It is given by the Biot-Savart\u2019s  law Fig"}, {"Chapter": "1", "sentence_range": "3468-3471", "Text": "Here, we shall study the relation between current and the\nmagnetic field it produces It is given by the Biot-Savart\u2019s  law Fig 4"}, {"Chapter": "1", "sentence_range": "3469-3472", "Text": "It is given by the Biot-Savart\u2019s  law Fig 4 7 shows a finite conductor XY carrying current I"}, {"Chapter": "1", "sentence_range": "3470-3473", "Text": "Fig 4 7 shows a finite conductor XY carrying current I Consider\nan infinitesimal element dl of the conductor"}, {"Chapter": "1", "sentence_range": "3471-3474", "Text": "4 7 shows a finite conductor XY carrying current I Consider\nan infinitesimal element dl of the conductor The magnetic field\ndB due to this element is to be determined at a point P which is at\na distance r from it"}, {"Chapter": "1", "sentence_range": "3472-3475", "Text": "7 shows a finite conductor XY carrying current I Consider\nan infinitesimal element dl of the conductor The magnetic field\ndB due to this element is to be determined at a point P which is at\na distance r from it Let q be the angle between dl and the\ndisplacement vector r"}, {"Chapter": "1", "sentence_range": "3473-3476", "Text": "Consider\nan infinitesimal element dl of the conductor The magnetic field\ndB due to this element is to be determined at a point P which is at\na distance r from it Let q be the angle between dl and the\ndisplacement vector r According to Biot-Savart\u2019s law, the\nmagnitude of the magnetic field dB is proportional to the current\nI, the element length |dl|, and inversely proportional to the square\nof the distance r"}, {"Chapter": "1", "sentence_range": "3474-3477", "Text": "The magnetic field\ndB due to this element is to be determined at a point P which is at\na distance r from it Let q be the angle between dl and the\ndisplacement vector r According to Biot-Savart\u2019s law, the\nmagnitude of the magnetic field dB is proportional to the current\nI, the element length |dl|, and inversely proportional to the square\nof the distance r Its direction* is perpendicular to the plane\ncontaining dl and r"}, {"Chapter": "1", "sentence_range": "3475-3478", "Text": "Let q be the angle between dl and the\ndisplacement vector r According to Biot-Savart\u2019s law, the\nmagnitude of the magnetic field dB is proportional to the current\nI, the element length |dl|, and inversely proportional to the square\nof the distance r Its direction* is perpendicular to the plane\ncontaining dl and r Thus, in vector notation,\nd\nI d\nr\nB\nr\n\u221d\n\u00d7\nl\n3\n           =\n\u00d7\n\u00b50\n3\n4\u03c0\nI d\nlr\nr\n [4"}, {"Chapter": "1", "sentence_range": "3476-3479", "Text": "According to Biot-Savart\u2019s law, the\nmagnitude of the magnetic field dB is proportional to the current\nI, the element length |dl|, and inversely proportional to the square\nof the distance r Its direction* is perpendicular to the plane\ncontaining dl and r Thus, in vector notation,\nd\nI d\nr\nB\nr\n\u221d\n\u00d7\nl\n3\n           =\n\u00d7\n\u00b50\n3\n4\u03c0\nI d\nlr\nr\n [4 11(a)]\nwhere m0/4p is a constant of proportionality"}, {"Chapter": "1", "sentence_range": "3477-3480", "Text": "Its direction* is perpendicular to the plane\ncontaining dl and r Thus, in vector notation,\nd\nI d\nr\nB\nr\n\u221d\n\u00d7\nl\n3\n           =\n\u00d7\n\u00b50\n3\n4\u03c0\nI d\nlr\nr\n [4 11(a)]\nwhere m0/4p is a constant of proportionality The above expression\nholds when the medium is vacuum"}, {"Chapter": "1", "sentence_range": "3478-3481", "Text": "Thus, in vector notation,\nd\nI d\nr\nB\nr\n\u221d\n\u00d7\nl\n3\n           =\n\u00d7\n\u00b50\n3\n4\u03c0\nI d\nlr\nr\n [4 11(a)]\nwhere m0/4p is a constant of proportionality The above expression\nholds when the medium is vacuum FIGURE 4"}, {"Chapter": "1", "sentence_range": "3479-3482", "Text": "11(a)]\nwhere m0/4p is a constant of proportionality The above expression\nholds when the medium is vacuum FIGURE 4 7 Illustration of\nthe Biot-Savart law"}, {"Chapter": "1", "sentence_range": "3480-3483", "Text": "The above expression\nholds when the medium is vacuum FIGURE 4 7 Illustration of\nthe Biot-Savart law The\ncurrent element I dl\nproduces a field dB at a\ndistance r"}, {"Chapter": "1", "sentence_range": "3481-3484", "Text": "FIGURE 4 7 Illustration of\nthe Biot-Savart law The\ncurrent element I dl\nproduces a field dB at a\ndistance r The \u00c4 sign\nindicates that the\nfield is perpendicular\nto the plane of this\npage and directed\ninto it"}, {"Chapter": "1", "sentence_range": "3482-3485", "Text": "7 Illustration of\nthe Biot-Savart law The\ncurrent element I dl\nproduces a field dB at a\ndistance r The \u00c4 sign\nindicates that the\nfield is perpendicular\nto the plane of this\npage and directed\ninto it *\nThe sense of  dl \u00d7 r is also given by the Right Hand Screw rule : Look at the\nplane containing vectors dl and r"}, {"Chapter": "1", "sentence_range": "3483-3486", "Text": "The\ncurrent element I dl\nproduces a field dB at a\ndistance r The \u00c4 sign\nindicates that the\nfield is perpendicular\nto the plane of this\npage and directed\ninto it *\nThe sense of  dl \u00d7 r is also given by the Right Hand Screw rule : Look at the\nplane containing vectors dl and r Imagine moving from the first vector towards\nsecond vector"}, {"Chapter": "1", "sentence_range": "3484-3487", "Text": "The \u00c4 sign\nindicates that the\nfield is perpendicular\nto the plane of this\npage and directed\ninto it *\nThe sense of  dl \u00d7 r is also given by the Right Hand Screw rule : Look at the\nplane containing vectors dl and r Imagine moving from the first vector towards\nsecond vector If the movement is anticlockwise, the resultant is towards you"}, {"Chapter": "1", "sentence_range": "3485-3488", "Text": "*\nThe sense of  dl \u00d7 r is also given by the Right Hand Screw rule : Look at the\nplane containing vectors dl and r Imagine moving from the first vector towards\nsecond vector If the movement is anticlockwise, the resultant is towards you If it is clockwise, the resultant is away from you"}, {"Chapter": "1", "sentence_range": "3486-3489", "Text": "Imagine moving from the first vector towards\nsecond vector If the movement is anticlockwise, the resultant is towards you If it is clockwise, the resultant is away from you Rationalised 2023-24\nPhysics\n114\n EXAMPLE 4"}, {"Chapter": "1", "sentence_range": "3487-3490", "Text": "If the movement is anticlockwise, the resultant is towards you If it is clockwise, the resultant is away from you Rationalised 2023-24\nPhysics\n114\n EXAMPLE 4 5\nThe magnitude of this field is,\n0\n2\nd sin\nd\n4\nI\nl\nr\n\u00b5\n\u03b8\n=\n\u03c0\nB\n[4"}, {"Chapter": "1", "sentence_range": "3488-3491", "Text": "If it is clockwise, the resultant is away from you Rationalised 2023-24\nPhysics\n114\n EXAMPLE 4 5\nThe magnitude of this field is,\n0\n2\nd sin\nd\n4\nI\nl\nr\n\u00b5\n\u03b8\n=\n\u03c0\nB\n[4 11(b)]\nwhere we have used the property of cross-product"}, {"Chapter": "1", "sentence_range": "3489-3492", "Text": "Rationalised 2023-24\nPhysics\n114\n EXAMPLE 4 5\nThe magnitude of this field is,\n0\n2\nd sin\nd\n4\nI\nl\nr\n\u00b5\n\u03b8\n=\n\u03c0\nB\n[4 11(b)]\nwhere we have used the property of cross-product Equation [4"}, {"Chapter": "1", "sentence_range": "3490-3493", "Text": "5\nThe magnitude of this field is,\n0\n2\nd sin\nd\n4\nI\nl\nr\n\u00b5\n\u03b8\n=\n\u03c0\nB\n[4 11(b)]\nwhere we have used the property of cross-product Equation [4 11 (a)]\nconstitutes our basic equation for the magnetic field"}, {"Chapter": "1", "sentence_range": "3491-3494", "Text": "11(b)]\nwhere we have used the property of cross-product Equation [4 11 (a)]\nconstitutes our basic equation for the magnetic field The proportionality\nconstant in SI units has the exact value,\n7\n0\n10\nTm/A\n\u00b54\n\u2212\n\u03c0=\n[4"}, {"Chapter": "1", "sentence_range": "3492-3495", "Text": "Equation [4 11 (a)]\nconstitutes our basic equation for the magnetic field The proportionality\nconstant in SI units has the exact value,\n7\n0\n10\nTm/A\n\u00b54\n\u2212\n\u03c0=\n[4 11(c)]\nWe call m0 the permeability of free space (or vacuum)"}, {"Chapter": "1", "sentence_range": "3493-3496", "Text": "11 (a)]\nconstitutes our basic equation for the magnetic field The proportionality\nconstant in SI units has the exact value,\n7\n0\n10\nTm/A\n\u00b54\n\u2212\n\u03c0=\n[4 11(c)]\nWe call m0 the permeability of free space (or vacuum) The Biot-Savart law for the magnetic field has certain similarities, as\nwell as, differences with the Coulomb\u2019s law for the electrostatic field"}, {"Chapter": "1", "sentence_range": "3494-3497", "Text": "The proportionality\nconstant in SI units has the exact value,\n7\n0\n10\nTm/A\n\u00b54\n\u2212\n\u03c0=\n[4 11(c)]\nWe call m0 the permeability of free space (or vacuum) The Biot-Savart law for the magnetic field has certain similarities, as\nwell as, differences with the Coulomb\u2019s law for the electrostatic field Some\nof these are:\n(i)\nBoth are long range, since both depend inversely on the square of\ndistance from the source to the point of interest"}, {"Chapter": "1", "sentence_range": "3495-3498", "Text": "11(c)]\nWe call m0 the permeability of free space (or vacuum) The Biot-Savart law for the magnetic field has certain similarities, as\nwell as, differences with the Coulomb\u2019s law for the electrostatic field Some\nof these are:\n(i)\nBoth are long range, since both depend inversely on the square of\ndistance from the source to the point of interest The principle of\nsuperposition applies to both fields"}, {"Chapter": "1", "sentence_range": "3496-3499", "Text": "The Biot-Savart law for the magnetic field has certain similarities, as\nwell as, differences with the Coulomb\u2019s law for the electrostatic field Some\nof these are:\n(i)\nBoth are long range, since both depend inversely on the square of\ndistance from the source to the point of interest The principle of\nsuperposition applies to both fields [In this connection, note that\nthe magnetic field is linear in the source I dl just as the electrostatic\nfield is linear in its source: the electric charge"}, {"Chapter": "1", "sentence_range": "3497-3500", "Text": "Some\nof these are:\n(i)\nBoth are long range, since both depend inversely on the square of\ndistance from the source to the point of interest The principle of\nsuperposition applies to both fields [In this connection, note that\nthe magnetic field is linear in the source I dl just as the electrostatic\nfield is linear in its source: the electric charge ]\n(ii) The electrostatic field is produced by a scalar source, namely, the electric\ncharge"}, {"Chapter": "1", "sentence_range": "3498-3501", "Text": "The principle of\nsuperposition applies to both fields [In this connection, note that\nthe magnetic field is linear in the source I dl just as the electrostatic\nfield is linear in its source: the electric charge ]\n(ii) The electrostatic field is produced by a scalar source, namely, the electric\ncharge The magnetic field is produced by a vector source I dl"}, {"Chapter": "1", "sentence_range": "3499-3502", "Text": "[In this connection, note that\nthe magnetic field is linear in the source I dl just as the electrostatic\nfield is linear in its source: the electric charge ]\n(ii) The electrostatic field is produced by a scalar source, namely, the electric\ncharge The magnetic field is produced by a vector source I dl (iii) The electrostatic field is along the displacement vector joining the\nsource and the field point"}, {"Chapter": "1", "sentence_range": "3500-3503", "Text": "]\n(ii) The electrostatic field is produced by a scalar source, namely, the electric\ncharge The magnetic field is produced by a vector source I dl (iii) The electrostatic field is along the displacement vector joining the\nsource and the field point The magnetic field is perpendicular to the\nplane containing the displacement vector r and the current element I dl"}, {"Chapter": "1", "sentence_range": "3501-3504", "Text": "The magnetic field is produced by a vector source I dl (iii) The electrostatic field is along the displacement vector joining the\nsource and the field point The magnetic field is perpendicular to the\nplane containing the displacement vector r and the current element I dl (iv) There is an angle dependence in the Biot-Savart law which is not\npresent  in the electrostatic case"}, {"Chapter": "1", "sentence_range": "3502-3505", "Text": "(iii) The electrostatic field is along the displacement vector joining the\nsource and the field point The magnetic field is perpendicular to the\nplane containing the displacement vector r and the current element I dl (iv) There is an angle dependence in the Biot-Savart law which is not\npresent  in the electrostatic case In Fig"}, {"Chapter": "1", "sentence_range": "3503-3506", "Text": "The magnetic field is perpendicular to the\nplane containing the displacement vector r and the current element I dl (iv) There is an angle dependence in the Biot-Savart law which is not\npresent  in the electrostatic case In Fig 4"}, {"Chapter": "1", "sentence_range": "3504-3507", "Text": "(iv) There is an angle dependence in the Biot-Savart law which is not\npresent  in the electrostatic case In Fig 4 7, the magnetic field at any\npoint in  the direction of dl (the dashed line) is zero"}, {"Chapter": "1", "sentence_range": "3505-3508", "Text": "In Fig 4 7, the magnetic field at any\npoint in  the direction of dl (the dashed line) is zero Along this line,\nq = 0, sin q = 0 and from Eq"}, {"Chapter": "1", "sentence_range": "3506-3509", "Text": "4 7, the magnetic field at any\npoint in  the direction of dl (the dashed line) is zero Along this line,\nq = 0, sin q = 0 and from Eq [4"}, {"Chapter": "1", "sentence_range": "3507-3510", "Text": "7, the magnetic field at any\npoint in  the direction of dl (the dashed line) is zero Along this line,\nq = 0, sin q = 0 and from Eq [4 11(a)], |dB| = 0"}, {"Chapter": "1", "sentence_range": "3508-3511", "Text": "Along this line,\nq = 0, sin q = 0 and from Eq [4 11(a)], |dB| = 0 There is an interesting  relation between e0, the permittivity of free\nspace; m0, the permeability of free space; and c, the speed of light in vacuum:\n(\n)\n0\n0\n0\n0\n4\n\u00b54\n\u03b5 \u00b5\n\u03b5\n \n \n=\n\u03c0\n \n \n \n\u03c0 \n \n(\n)\n7\n9\n1\n10\n9\n10\n\u2212\n \n \n=  \n \n \n \n\u00d7\n8 2\n2\n1\n1\n(3\n10 )\nc\n=\n=\n\u00d7\nWe will discuss this connection further in Chapter 8 on the\nelectromagnetic waves"}, {"Chapter": "1", "sentence_range": "3509-3512", "Text": "[4 11(a)], |dB| = 0 There is an interesting  relation between e0, the permittivity of free\nspace; m0, the permeability of free space; and c, the speed of light in vacuum:\n(\n)\n0\n0\n0\n0\n4\n\u00b54\n\u03b5 \u00b5\n\u03b5\n \n \n=\n\u03c0\n \n \n \n\u03c0 \n \n(\n)\n7\n9\n1\n10\n9\n10\n\u2212\n \n \n=  \n \n \n \n\u00d7\n8 2\n2\n1\n1\n(3\n10 )\nc\n=\n=\n\u00d7\nWe will discuss this connection further in Chapter 8 on the\nelectromagnetic waves Since the speed of light in vacuum is constant,\nthe product m0e0 is fixed in magnitude"}, {"Chapter": "1", "sentence_range": "3510-3513", "Text": "11(a)], |dB| = 0 There is an interesting  relation between e0, the permittivity of free\nspace; m0, the permeability of free space; and c, the speed of light in vacuum:\n(\n)\n0\n0\n0\n0\n4\n\u00b54\n\u03b5 \u00b5\n\u03b5\n \n \n=\n\u03c0\n \n \n \n\u03c0 \n \n(\n)\n7\n9\n1\n10\n9\n10\n\u2212\n \n \n=  \n \n \n \n\u00d7\n8 2\n2\n1\n1\n(3\n10 )\nc\n=\n=\n\u00d7\nWe will discuss this connection further in Chapter 8 on the\nelectromagnetic waves Since the speed of light in vacuum is constant,\nthe product m0e0 is fixed in magnitude Choosing the value of either e0 or\nm0, fixes the value of the other"}, {"Chapter": "1", "sentence_range": "3511-3514", "Text": "There is an interesting  relation between e0, the permittivity of free\nspace; m0, the permeability of free space; and c, the speed of light in vacuum:\n(\n)\n0\n0\n0\n0\n4\n\u00b54\n\u03b5 \u00b5\n\u03b5\n \n \n=\n\u03c0\n \n \n \n\u03c0 \n \n(\n)\n7\n9\n1\n10\n9\n10\n\u2212\n \n \n=  \n \n \n \n\u00d7\n8 2\n2\n1\n1\n(3\n10 )\nc\n=\n=\n\u00d7\nWe will discuss this connection further in Chapter 8 on the\nelectromagnetic waves Since the speed of light in vacuum is constant,\nthe product m0e0 is fixed in magnitude Choosing the value of either e0 or\nm0, fixes the value of the other In SI units, m0 is fixed to be equal to\n4p \u00d7 10\u20137\n in magnitude"}, {"Chapter": "1", "sentence_range": "3512-3515", "Text": "Since the speed of light in vacuum is constant,\nthe product m0e0 is fixed in magnitude Choosing the value of either e0 or\nm0, fixes the value of the other In SI units, m0 is fixed to be equal to\n4p \u00d7 10\u20137\n in magnitude Example 4"}, {"Chapter": "1", "sentence_range": "3513-3516", "Text": "Choosing the value of either e0 or\nm0, fixes the value of the other In SI units, m0 is fixed to be equal to\n4p \u00d7 10\u20137\n in magnitude Example 4 5 An element \n\uf044 \uf03d \uf044x i\u02c6\nl\n is placed at the origin and carries\na large current I = 10 A (Fig"}, {"Chapter": "1", "sentence_range": "3514-3517", "Text": "In SI units, m0 is fixed to be equal to\n4p \u00d7 10\u20137\n in magnitude Example 4 5 An element \n\uf044 \uf03d \uf044x i\u02c6\nl\n is placed at the origin and carries\na large current I = 10 A (Fig 4"}, {"Chapter": "1", "sentence_range": "3515-3518", "Text": "Example 4 5 An element \n\uf044 \uf03d \uf044x i\u02c6\nl\n is placed at the origin and carries\na large current I = 10 A (Fig 4 8)"}, {"Chapter": "1", "sentence_range": "3516-3519", "Text": "5 An element \n\uf044 \uf03d \uf044x i\u02c6\nl\n is placed at the origin and carries\na large current I = 10 A (Fig 4 8) What is the magnetic field on the y-\naxis at a distance of 0"}, {"Chapter": "1", "sentence_range": "3517-3520", "Text": "4 8) What is the magnetic field on the y-\naxis at a distance of 0 5 m"}, {"Chapter": "1", "sentence_range": "3518-3521", "Text": "8) What is the magnetic field on the y-\naxis at a distance of 0 5 m Dx = 1 cm"}, {"Chapter": "1", "sentence_range": "3519-3522", "Text": "What is the magnetic field on the y-\naxis at a distance of 0 5 m Dx = 1 cm FIGURE 4"}, {"Chapter": "1", "sentence_range": "3520-3523", "Text": "5 m Dx = 1 cm FIGURE 4 8\nRationalised 2023-24\n115\nMoving Charges and\nMagnetism\nSolution\n \n0\n2\nd sin\n|d\n|\n4\nI\nl\nr\n\u00b5\n\u03b8\n=\n\u03c0\nB\n [using Eq"}, {"Chapter": "1", "sentence_range": "3521-3524", "Text": "Dx = 1 cm FIGURE 4 8\nRationalised 2023-24\n115\nMoving Charges and\nMagnetism\nSolution\n \n0\n2\nd sin\n|d\n|\n4\nI\nl\nr\n\u00b5\n\u03b8\n=\n\u03c0\nB\n [using Eq (4"}, {"Chapter": "1", "sentence_range": "3522-3525", "Text": "FIGURE 4 8\nRationalised 2023-24\n115\nMoving Charges and\nMagnetism\nSolution\n \n0\n2\nd sin\n|d\n|\n4\nI\nl\nr\n\u00b5\n\u03b8\n=\n\u03c0\nB\n [using Eq (4 11)]\n2\nd\n10\nm\nl\nx\n\u2212\n= \u2206\n=\n, I = 10 A,  r = 0"}, {"Chapter": "1", "sentence_range": "3523-3526", "Text": "8\nRationalised 2023-24\n115\nMoving Charges and\nMagnetism\nSolution\n \n0\n2\nd sin\n|d\n|\n4\nI\nl\nr\n\u00b5\n\u03b8\n=\n\u03c0\nB\n [using Eq (4 11)]\n2\nd\n10\nm\nl\nx\n\u2212\n= \u2206\n=\n, I = 10 A,  r = 0 5 m = y, \n7\n0\nT m\n/4\n10\nA\n\u00b5\n\u2212\n\u03c0 =\nq = 90\u00b0 ; sin q = 1\n7\n2\n2\n10\n10\n10\nd\n25\n10\n\u2212\n\u2212\n\u2212\n\u00d7\n\u00d7\n=\n\u00d7\nB\n= 4 \u00d7 10\u20138 T\nThe direction of the field is in the +z-direction"}, {"Chapter": "1", "sentence_range": "3524-3527", "Text": "(4 11)]\n2\nd\n10\nm\nl\nx\n\u2212\n= \u2206\n=\n, I = 10 A,  r = 0 5 m = y, \n7\n0\nT m\n/4\n10\nA\n\u00b5\n\u2212\n\u03c0 =\nq = 90\u00b0 ; sin q = 1\n7\n2\n2\n10\n10\n10\nd\n25\n10\n\u2212\n\u2212\n\u2212\n\u00d7\n\u00d7\n=\n\u00d7\nB\n= 4 \u00d7 10\u20138 T\nThe direction of the field is in the +z-direction This is so since,\n\u02c6\n\u02c6\nd\n\uf03d \uf044\n\u00d7\ni \u00d7\nj\nx\ny\nr\nl\n(\n)\n\u02c6\n\u02c6\ny\nx\n=\n\u2206\ni \u00d7 j\n\u02c6\ny\nx\n=\n\u2206 k\nWe remind you of the following cyclic property of cross-products,\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n;\n;\n\u00d7\n=\n\u00d7\n=\n\u00d7\n=\ni\nj\nk\nj\nk\ni k\ni\nj\nNote that the field is small in magnitude"}, {"Chapter": "1", "sentence_range": "3525-3528", "Text": "11)]\n2\nd\n10\nm\nl\nx\n\u2212\n= \u2206\n=\n, I = 10 A,  r = 0 5 m = y, \n7\n0\nT m\n/4\n10\nA\n\u00b5\n\u2212\n\u03c0 =\nq = 90\u00b0 ; sin q = 1\n7\n2\n2\n10\n10\n10\nd\n25\n10\n\u2212\n\u2212\n\u2212\n\u00d7\n\u00d7\n=\n\u00d7\nB\n= 4 \u00d7 10\u20138 T\nThe direction of the field is in the +z-direction This is so since,\n\u02c6\n\u02c6\nd\n\uf03d \uf044\n\u00d7\ni \u00d7\nj\nx\ny\nr\nl\n(\n)\n\u02c6\n\u02c6\ny\nx\n=\n\u2206\ni \u00d7 j\n\u02c6\ny\nx\n=\n\u2206 k\nWe remind you of the following cyclic property of cross-products,\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n;\n;\n\u00d7\n=\n\u00d7\n=\n\u00d7\n=\ni\nj\nk\nj\nk\ni k\ni\nj\nNote that the field is small in magnitude In the next section, we shall use the Biot-Savart law to calculate the\nmagnetic field due to a circular loop"}, {"Chapter": "1", "sentence_range": "3526-3529", "Text": "5 m = y, \n7\n0\nT m\n/4\n10\nA\n\u00b5\n\u2212\n\u03c0 =\nq = 90\u00b0 ; sin q = 1\n7\n2\n2\n10\n10\n10\nd\n25\n10\n\u2212\n\u2212\n\u2212\n\u00d7\n\u00d7\n=\n\u00d7\nB\n= 4 \u00d7 10\u20138 T\nThe direction of the field is in the +z-direction This is so since,\n\u02c6\n\u02c6\nd\n\uf03d \uf044\n\u00d7\ni \u00d7\nj\nx\ny\nr\nl\n(\n)\n\u02c6\n\u02c6\ny\nx\n=\n\u2206\ni \u00d7 j\n\u02c6\ny\nx\n=\n\u2206 k\nWe remind you of the following cyclic property of cross-products,\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n;\n;\n\u00d7\n=\n\u00d7\n=\n\u00d7\n=\ni\nj\nk\nj\nk\ni k\ni\nj\nNote that the field is small in magnitude In the next section, we shall use the Biot-Savart law to calculate the\nmagnetic field due to a circular loop 4"}, {"Chapter": "1", "sentence_range": "3527-3530", "Text": "This is so since,\n\u02c6\n\u02c6\nd\n\uf03d \uf044\n\u00d7\ni \u00d7\nj\nx\ny\nr\nl\n(\n)\n\u02c6\n\u02c6\ny\nx\n=\n\u2206\ni \u00d7 j\n\u02c6\ny\nx\n=\n\u2206 k\nWe remind you of the following cyclic property of cross-products,\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n;\n;\n\u00d7\n=\n\u00d7\n=\n\u00d7\n=\ni\nj\nk\nj\nk\ni k\ni\nj\nNote that the field is small in magnitude In the next section, we shall use the Biot-Savart law to calculate the\nmagnetic field due to a circular loop 4 5 MAGNETIC FIELD ON THE AXIS OF A CIRCULAR\nCURRENT LOOP\nIn this section, we shall evaluate the magnetic field  due\nto a circular coil along its axis"}, {"Chapter": "1", "sentence_range": "3528-3531", "Text": "In the next section, we shall use the Biot-Savart law to calculate the\nmagnetic field due to a circular loop 4 5 MAGNETIC FIELD ON THE AXIS OF A CIRCULAR\nCURRENT LOOP\nIn this section, we shall evaluate the magnetic field  due\nto a circular coil along its axis The evaluation entails\nsumming up the effect of infinitesimal current elements\n(I dl) mentioned in the previous section"}, {"Chapter": "1", "sentence_range": "3529-3532", "Text": "4 5 MAGNETIC FIELD ON THE AXIS OF A CIRCULAR\nCURRENT LOOP\nIn this section, we shall evaluate the magnetic field  due\nto a circular coil along its axis The evaluation entails\nsumming up the effect of infinitesimal current elements\n(I dl) mentioned in the previous section We assume\nthat the current I is steady and that the evaluation is\ncarried out in free space (i"}, {"Chapter": "1", "sentence_range": "3530-3533", "Text": "5 MAGNETIC FIELD ON THE AXIS OF A CIRCULAR\nCURRENT LOOP\nIn this section, we shall evaluate the magnetic field  due\nto a circular coil along its axis The evaluation entails\nsumming up the effect of infinitesimal current elements\n(I dl) mentioned in the previous section We assume\nthat the current I is steady and that the evaluation is\ncarried out in free space (i e"}, {"Chapter": "1", "sentence_range": "3531-3534", "Text": "The evaluation entails\nsumming up the effect of infinitesimal current elements\n(I dl) mentioned in the previous section We assume\nthat the current I is steady and that the evaluation is\ncarried out in free space (i e , vacuum)"}, {"Chapter": "1", "sentence_range": "3532-3535", "Text": "We assume\nthat the current I is steady and that the evaluation is\ncarried out in free space (i e , vacuum) Fig"}, {"Chapter": "1", "sentence_range": "3533-3536", "Text": "e , vacuum) Fig 4"}, {"Chapter": "1", "sentence_range": "3534-3537", "Text": ", vacuum) Fig 4 9 depicts a circular loop carrying a steady\ncurrent I"}, {"Chapter": "1", "sentence_range": "3535-3538", "Text": "Fig 4 9 depicts a circular loop carrying a steady\ncurrent I The loop is placed in the y-z plane with its\ncentre at the origin O and has a radius R"}, {"Chapter": "1", "sentence_range": "3536-3539", "Text": "4 9 depicts a circular loop carrying a steady\ncurrent I The loop is placed in the y-z plane with its\ncentre at the origin O and has a radius R The x-axis is\nthe axis of the loop"}, {"Chapter": "1", "sentence_range": "3537-3540", "Text": "9 depicts a circular loop carrying a steady\ncurrent I The loop is placed in the y-z plane with its\ncentre at the origin O and has a radius R The x-axis is\nthe axis of the loop We wish to calculate the magnetic\nfield at the point P on this axis"}, {"Chapter": "1", "sentence_range": "3538-3541", "Text": "The loop is placed in the y-z plane with its\ncentre at the origin O and has a radius R The x-axis is\nthe axis of the loop We wish to calculate the magnetic\nfield at the point P on this axis Let x be the distance of\nP from the centre O of the loop"}, {"Chapter": "1", "sentence_range": "3539-3542", "Text": "The x-axis is\nthe axis of the loop We wish to calculate the magnetic\nfield at the point P on this axis Let x be the distance of\nP from the centre O of the loop Consider a conducting element dl of the loop"}, {"Chapter": "1", "sentence_range": "3540-3543", "Text": "We wish to calculate the magnetic\nfield at the point P on this axis Let x be the distance of\nP from the centre O of the loop Consider a conducting element dl of the loop This is\nshown in Fig"}, {"Chapter": "1", "sentence_range": "3541-3544", "Text": "Let x be the distance of\nP from the centre O of the loop Consider a conducting element dl of the loop This is\nshown in Fig 4"}, {"Chapter": "1", "sentence_range": "3542-3545", "Text": "Consider a conducting element dl of the loop This is\nshown in Fig 4 9"}, {"Chapter": "1", "sentence_range": "3543-3546", "Text": "This is\nshown in Fig 4 9 The magnitude dB of the magnetic\nfield due to dl is given by the Biot-Savart law [Eq"}, {"Chapter": "1", "sentence_range": "3544-3547", "Text": "4 9 The magnitude dB of the magnetic\nfield due to dl is given by the Biot-Savart law [Eq 4"}, {"Chapter": "1", "sentence_range": "3545-3548", "Text": "9 The magnitude dB of the magnetic\nfield due to dl is given by the Biot-Savart law [Eq 4 11(a)],\n0\n3\n\uf03d4\nI d\u00d7 r\ndB\nr\n\uf06d\n\uf070\nl\n(4"}, {"Chapter": "1", "sentence_range": "3546-3549", "Text": "The magnitude dB of the magnetic\nfield due to dl is given by the Biot-Savart law [Eq 4 11(a)],\n0\n3\n\uf03d4\nI d\u00d7 r\ndB\nr\n\uf06d\n\uf070\nl\n(4 12)\nNow r 2 = x 2 + R 2"}, {"Chapter": "1", "sentence_range": "3547-3550", "Text": "4 11(a)],\n0\n3\n\uf03d4\nI d\u00d7 r\ndB\nr\n\uf06d\n\uf070\nl\n(4 12)\nNow r 2 = x 2 + R 2 Further, any element of the loop\nwill be perpendicular to the displacement vector from\nthe element to the axial point"}, {"Chapter": "1", "sentence_range": "3548-3551", "Text": "11(a)],\n0\n3\n\uf03d4\nI d\u00d7 r\ndB\nr\n\uf06d\n\uf070\nl\n(4 12)\nNow r 2 = x 2 + R 2 Further, any element of the loop\nwill be perpendicular to the displacement vector from\nthe element to the axial point For example, the element dl in Fig"}, {"Chapter": "1", "sentence_range": "3549-3552", "Text": "12)\nNow r 2 = x 2 + R 2 Further, any element of the loop\nwill be perpendicular to the displacement vector from\nthe element to the axial point For example, the element dl in Fig 4"}, {"Chapter": "1", "sentence_range": "3550-3553", "Text": "Further, any element of the loop\nwill be perpendicular to the displacement vector from\nthe element to the axial point For example, the element dl in Fig 4 9 is\nin the y-z plane, whereas, the displacement vector r from dl to the axial\npoint P is in the x-y plane"}, {"Chapter": "1", "sentence_range": "3551-3554", "Text": "For example, the element dl in Fig 4 9 is\nin the y-z plane, whereas, the displacement vector r from dl to the axial\npoint P is in the x-y plane Hence |dl \u00d7 r|=r dl"}, {"Chapter": "1", "sentence_range": "3552-3555", "Text": "4 9 is\nin the y-z plane, whereas, the displacement vector r from dl to the axial\npoint P is in the x-y plane Hence |dl \u00d7 r|=r dl Thus,\n \n(\n)\n\u03c0\n0\n2\n2\nd\nd\n4\nI l\nB\nx\nR\n=\u00b5\n+\n(4"}, {"Chapter": "1", "sentence_range": "3553-3556", "Text": "9 is\nin the y-z plane, whereas, the displacement vector r from dl to the axial\npoint P is in the x-y plane Hence |dl \u00d7 r|=r dl Thus,\n \n(\n)\n\u03c0\n0\n2\n2\nd\nd\n4\nI l\nB\nx\nR\n=\u00b5\n+\n(4 13)\nFIGURE 4"}, {"Chapter": "1", "sentence_range": "3554-3557", "Text": "Hence |dl \u00d7 r|=r dl Thus,\n \n(\n)\n\u03c0\n0\n2\n2\nd\nd\n4\nI l\nB\nx\nR\n=\u00b5\n+\n(4 13)\nFIGURE 4 9 Magnetic field on the\naxis of a current carrying circular\nloop of radius R"}, {"Chapter": "1", "sentence_range": "3555-3558", "Text": "Thus,\n \n(\n)\n\u03c0\n0\n2\n2\nd\nd\n4\nI l\nB\nx\nR\n=\u00b5\n+\n(4 13)\nFIGURE 4 9 Magnetic field on the\naxis of a current carrying circular\nloop of radius R Shown are the\nmagnetic field dB (due to a line\nelement dl ) and its\ncomponents along and\nperpendicular to the axis"}, {"Chapter": "1", "sentence_range": "3556-3559", "Text": "13)\nFIGURE 4 9 Magnetic field on the\naxis of a current carrying circular\nloop of radius R Shown are the\nmagnetic field dB (due to a line\nelement dl ) and its\ncomponents along and\nperpendicular to the axis EXAMPLE 4"}, {"Chapter": "1", "sentence_range": "3557-3560", "Text": "9 Magnetic field on the\naxis of a current carrying circular\nloop of radius R Shown are the\nmagnetic field dB (due to a line\nelement dl ) and its\ncomponents along and\nperpendicular to the axis EXAMPLE 4 5\nRationalised 2023-24\nPhysics\n116\nThe direction of dB is shown in Fig"}, {"Chapter": "1", "sentence_range": "3558-3561", "Text": "Shown are the\nmagnetic field dB (due to a line\nelement dl ) and its\ncomponents along and\nperpendicular to the axis EXAMPLE 4 5\nRationalised 2023-24\nPhysics\n116\nThe direction of dB is shown in Fig 4"}, {"Chapter": "1", "sentence_range": "3559-3562", "Text": "EXAMPLE 4 5\nRationalised 2023-24\nPhysics\n116\nThe direction of dB is shown in Fig 4 9"}, {"Chapter": "1", "sentence_range": "3560-3563", "Text": "5\nRationalised 2023-24\nPhysics\n116\nThe direction of dB is shown in Fig 4 9 It is perpendicular to the\nplane formed by dl and r"}, {"Chapter": "1", "sentence_range": "3561-3564", "Text": "4 9 It is perpendicular to the\nplane formed by dl and r It has an x-component dBx and a component\nperpendicular to x-axis, dB^"}, {"Chapter": "1", "sentence_range": "3562-3565", "Text": "9 It is perpendicular to the\nplane formed by dl and r It has an x-component dBx and a component\nperpendicular to x-axis, dB^ When the components perpendicular to\nthe x-axis are summed over, they cancel out and we obtain a null result"}, {"Chapter": "1", "sentence_range": "3563-3566", "Text": "It is perpendicular to the\nplane formed by dl and r It has an x-component dBx and a component\nperpendicular to x-axis, dB^ When the components perpendicular to\nthe x-axis are summed over, they cancel out and we obtain a null result For example, the dB^ component due to dl is cancelled by the contribution\ndue to the diametrically opposite dl element, shown in\nFig"}, {"Chapter": "1", "sentence_range": "3564-3567", "Text": "It has an x-component dBx and a component\nperpendicular to x-axis, dB^ When the components perpendicular to\nthe x-axis are summed over, they cancel out and we obtain a null result For example, the dB^ component due to dl is cancelled by the contribution\ndue to the diametrically opposite dl element, shown in\nFig 4"}, {"Chapter": "1", "sentence_range": "3565-3568", "Text": "When the components perpendicular to\nthe x-axis are summed over, they cancel out and we obtain a null result For example, the dB^ component due to dl is cancelled by the contribution\ndue to the diametrically opposite dl element, shown in\nFig 4 9"}, {"Chapter": "1", "sentence_range": "3566-3569", "Text": "For example, the dB^ component due to dl is cancelled by the contribution\ndue to the diametrically opposite dl element, shown in\nFig 4 9 Thus, only the x-component survives"}, {"Chapter": "1", "sentence_range": "3567-3570", "Text": "4 9 Thus, only the x-component survives The net contribution along\nx-direction can be obtained by integrating dBx  = dB cos q over the loop"}, {"Chapter": "1", "sentence_range": "3568-3571", "Text": "9 Thus, only the x-component survives The net contribution along\nx-direction can be obtained by integrating dBx  = dB cos q over the loop For Fig"}, {"Chapter": "1", "sentence_range": "3569-3572", "Text": "Thus, only the x-component survives The net contribution along\nx-direction can be obtained by integrating dBx  = dB cos q over the loop For Fig 4"}, {"Chapter": "1", "sentence_range": "3570-3573", "Text": "The net contribution along\nx-direction can be obtained by integrating dBx  = dB cos q over the loop For Fig 4 9,\n2\n2 1/2\ncos\n(\n)\nR\nx\nR\n\u03b8 =\n+\n (4"}, {"Chapter": "1", "sentence_range": "3571-3574", "Text": "For Fig 4 9,\n2\n2 1/2\ncos\n(\n)\nR\nx\nR\n\u03b8 =\n+\n (4 14)\nFrom Eqs"}, {"Chapter": "1", "sentence_range": "3572-3575", "Text": "4 9,\n2\n2 1/2\ncos\n(\n)\nR\nx\nR\n\u03b8 =\n+\n (4 14)\nFrom Eqs (4"}, {"Chapter": "1", "sentence_range": "3573-3576", "Text": "9,\n2\n2 1/2\ncos\n(\n)\nR\nx\nR\n\u03b8 =\n+\n (4 14)\nFrom Eqs (4 13) and (4"}, {"Chapter": "1", "sentence_range": "3574-3577", "Text": "14)\nFrom Eqs (4 13) and (4 14),\n(\n)\n\u03c0\n0\n3/2\n2\n2\nd\nd\n4\nx\nI l\nR\nB\nx\nR\n=\u00b5\n+\nThe summation of elements dl over the loop yields 2pR, the\ncircumference of the loop"}, {"Chapter": "1", "sentence_range": "3575-3578", "Text": "(4 13) and (4 14),\n(\n)\n\u03c0\n0\n3/2\n2\n2\nd\nd\n4\nx\nI l\nR\nB\nx\nR\n=\u00b5\n+\nThe summation of elements dl over the loop yields 2pR, the\ncircumference of the loop Thus, the magnetic field at P due to entire\ncircular loop is\n(\n)\n2\n0\n3/2\n2\n2\n\u02c6\n\u02c6\n2\nx\nI R\nB\nx\nR\n\u00b5\n=\n=\n+\nB\ni\ni\n(4"}, {"Chapter": "1", "sentence_range": "3576-3579", "Text": "13) and (4 14),\n(\n)\n\u03c0\n0\n3/2\n2\n2\nd\nd\n4\nx\nI l\nR\nB\nx\nR\n=\u00b5\n+\nThe summation of elements dl over the loop yields 2pR, the\ncircumference of the loop Thus, the magnetic field at P due to entire\ncircular loop is\n(\n)\n2\n0\n3/2\n2\n2\n\u02c6\n\u02c6\n2\nx\nI R\nB\nx\nR\n\u00b5\n=\n=\n+\nB\ni\ni\n(4 15)\nAs a special case of the above result, we may obtain the field at the centre\nof the loop"}, {"Chapter": "1", "sentence_range": "3577-3580", "Text": "14),\n(\n)\n\u03c0\n0\n3/2\n2\n2\nd\nd\n4\nx\nI l\nR\nB\nx\nR\n=\u00b5\n+\nThe summation of elements dl over the loop yields 2pR, the\ncircumference of the loop Thus, the magnetic field at P due to entire\ncircular loop is\n(\n)\n2\n0\n3/2\n2\n2\n\u02c6\n\u02c6\n2\nx\nI R\nB\nx\nR\n\u00b5\n=\n=\n+\nB\ni\ni\n(4 15)\nAs a special case of the above result, we may obtain the field at the centre\nof the loop Here  x = 0, and we obtain,\n0\n0\n\u02c6\n2\nI\nR\n=\u00b5\nB\ni\n(4"}, {"Chapter": "1", "sentence_range": "3578-3581", "Text": "Thus, the magnetic field at P due to entire\ncircular loop is\n(\n)\n2\n0\n3/2\n2\n2\n\u02c6\n\u02c6\n2\nx\nI R\nB\nx\nR\n\u00b5\n=\n=\n+\nB\ni\ni\n(4 15)\nAs a special case of the above result, we may obtain the field at the centre\nof the loop Here  x = 0, and we obtain,\n0\n0\n\u02c6\n2\nI\nR\n=\u00b5\nB\ni\n(4 16)\nThe magnetic field lines due to a circular wire form closed loops and\nare shown in Fig"}, {"Chapter": "1", "sentence_range": "3579-3582", "Text": "15)\nAs a special case of the above result, we may obtain the field at the centre\nof the loop Here  x = 0, and we obtain,\n0\n0\n\u02c6\n2\nI\nR\n=\u00b5\nB\ni\n(4 16)\nThe magnetic field lines due to a circular wire form closed loops and\nare shown in Fig 4"}, {"Chapter": "1", "sentence_range": "3580-3583", "Text": "Here  x = 0, and we obtain,\n0\n0\n\u02c6\n2\nI\nR\n=\u00b5\nB\ni\n(4 16)\nThe magnetic field lines due to a circular wire form closed loops and\nare shown in Fig 4 10"}, {"Chapter": "1", "sentence_range": "3581-3584", "Text": "16)\nThe magnetic field lines due to a circular wire form closed loops and\nare shown in Fig 4 10 The direction of the magnetic field is given by\n(another) right-hand thumb rule stated below:\nCurl the palm of your right hand around the circular wire with the\nfingers pointing in the direction of the current"}, {"Chapter": "1", "sentence_range": "3582-3585", "Text": "4 10 The direction of the magnetic field is given by\n(another) right-hand thumb rule stated below:\nCurl the palm of your right hand around the circular wire with the\nfingers pointing in the direction of the current The right-hand thumb\ngives the direction of the magnetic field"}, {"Chapter": "1", "sentence_range": "3583-3586", "Text": "10 The direction of the magnetic field is given by\n(another) right-hand thumb rule stated below:\nCurl the palm of your right hand around the circular wire with the\nfingers pointing in the direction of the current The right-hand thumb\ngives the direction of the magnetic field FIGURE 4"}, {"Chapter": "1", "sentence_range": "3584-3587", "Text": "The direction of the magnetic field is given by\n(another) right-hand thumb rule stated below:\nCurl the palm of your right hand around the circular wire with the\nfingers pointing in the direction of the current The right-hand thumb\ngives the direction of the magnetic field FIGURE 4 10 The magnetic field lines for a current loop"}, {"Chapter": "1", "sentence_range": "3585-3588", "Text": "The right-hand thumb\ngives the direction of the magnetic field FIGURE 4 10 The magnetic field lines for a current loop The direction of\nthe field is given by the right-hand thumb rule described in the text"}, {"Chapter": "1", "sentence_range": "3586-3589", "Text": "FIGURE 4 10 The magnetic field lines for a current loop The direction of\nthe field is given by the right-hand thumb rule described in the text The\nupper side of the loop may be thought of as the north pole and the lower\nside as the south pole of a magnet"}, {"Chapter": "1", "sentence_range": "3587-3590", "Text": "10 The magnetic field lines for a current loop The direction of\nthe field is given by the right-hand thumb rule described in the text The\nupper side of the loop may be thought of as the north pole and the lower\nside as the south pole of a magnet Rationalised 2023-24\n117\nMoving Charges and\nMagnetism\n EXAMPLE 4"}, {"Chapter": "1", "sentence_range": "3588-3591", "Text": "The direction of\nthe field is given by the right-hand thumb rule described in the text The\nupper side of the loop may be thought of as the north pole and the lower\nside as the south pole of a magnet Rationalised 2023-24\n117\nMoving Charges and\nMagnetism\n EXAMPLE 4 6\nExample 4"}, {"Chapter": "1", "sentence_range": "3589-3592", "Text": "The\nupper side of the loop may be thought of as the north pole and the lower\nside as the south pole of a magnet Rationalised 2023-24\n117\nMoving Charges and\nMagnetism\n EXAMPLE 4 6\nExample 4 6 A straight wire carrying a current of 12 A is bent into a\nsemi-circular arc of radius 2"}, {"Chapter": "1", "sentence_range": "3590-3593", "Text": "Rationalised 2023-24\n117\nMoving Charges and\nMagnetism\n EXAMPLE 4 6\nExample 4 6 A straight wire carrying a current of 12 A is bent into a\nsemi-circular arc of radius 2 0 cm as shown in Fig"}, {"Chapter": "1", "sentence_range": "3591-3594", "Text": "6\nExample 4 6 A straight wire carrying a current of 12 A is bent into a\nsemi-circular arc of radius 2 0 cm as shown in Fig 4"}, {"Chapter": "1", "sentence_range": "3592-3595", "Text": "6 A straight wire carrying a current of 12 A is bent into a\nsemi-circular arc of radius 2 0 cm as shown in Fig 4 11(a)"}, {"Chapter": "1", "sentence_range": "3593-3596", "Text": "0 cm as shown in Fig 4 11(a) Consider\nthe magnetic field B at the centre of the arc"}, {"Chapter": "1", "sentence_range": "3594-3597", "Text": "4 11(a) Consider\nthe magnetic field B at the centre of the arc (a) What is the magnetic\nfield due to the straight segments"}, {"Chapter": "1", "sentence_range": "3595-3598", "Text": "11(a) Consider\nthe magnetic field B at the centre of the arc (a) What is the magnetic\nfield due to the straight segments (b) In what way the contribution\nto B from the semicircle differs from that of a circular loop and in\nwhat way does it resemble"}, {"Chapter": "1", "sentence_range": "3596-3599", "Text": "Consider\nthe magnetic field B at the centre of the arc (a) What is the magnetic\nfield due to the straight segments (b) In what way the contribution\nto B from the semicircle differs from that of a circular loop and in\nwhat way does it resemble (c) Would your answer be different if the\nwire were bent into a semi-circular arc of the same radius but in the\nopposite way as shown in Fig"}, {"Chapter": "1", "sentence_range": "3597-3600", "Text": "(a) What is the magnetic\nfield due to the straight segments (b) In what way the contribution\nto B from the semicircle differs from that of a circular loop and in\nwhat way does it resemble (c) Would your answer be different if the\nwire were bent into a semi-circular arc of the same radius but in the\nopposite way as shown in Fig 4"}, {"Chapter": "1", "sentence_range": "3598-3601", "Text": "(b) In what way the contribution\nto B from the semicircle differs from that of a circular loop and in\nwhat way does it resemble (c) Would your answer be different if the\nwire were bent into a semi-circular arc of the same radius but in the\nopposite way as shown in Fig 4 11(b)"}, {"Chapter": "1", "sentence_range": "3599-3602", "Text": "(c) Would your answer be different if the\nwire were bent into a semi-circular arc of the same radius but in the\nopposite way as shown in Fig 4 11(b) FIGURE 4"}, {"Chapter": "1", "sentence_range": "3600-3603", "Text": "4 11(b) FIGURE 4 11\nSolution\n(a) dl and r for each element of the straight segments are parallel"}, {"Chapter": "1", "sentence_range": "3601-3604", "Text": "11(b) FIGURE 4 11\nSolution\n(a) dl and r for each element of the straight segments are parallel Therefore, dl \u00d7 r = 0"}, {"Chapter": "1", "sentence_range": "3602-3605", "Text": "FIGURE 4 11\nSolution\n(a) dl and r for each element of the straight segments are parallel Therefore, dl \u00d7 r = 0 Straight segments do not contribute to\n|B|"}, {"Chapter": "1", "sentence_range": "3603-3606", "Text": "11\nSolution\n(a) dl and r for each element of the straight segments are parallel Therefore, dl \u00d7 r = 0 Straight segments do not contribute to\n|B| (b) For all segments of the semicircular arc, dl \u00d7 r are all parallel to\neach other (into the plane of the paper)"}, {"Chapter": "1", "sentence_range": "3604-3607", "Text": "Therefore, dl \u00d7 r = 0 Straight segments do not contribute to\n|B| (b) For all segments of the semicircular arc, dl \u00d7 r are all parallel to\neach other (into the plane of the paper) All such contributions\nadd up in magnitude"}, {"Chapter": "1", "sentence_range": "3605-3608", "Text": "Straight segments do not contribute to\n|B| (b) For all segments of the semicircular arc, dl \u00d7 r are all parallel to\neach other (into the plane of the paper) All such contributions\nadd up in magnitude Hence direction of B for a semicircular arc\nis given by the right-hand rule and magnitude is half that of a\ncircular loop"}, {"Chapter": "1", "sentence_range": "3606-3609", "Text": "(b) For all segments of the semicircular arc, dl \u00d7 r are all parallel to\neach other (into the plane of the paper) All such contributions\nadd up in magnitude Hence direction of B for a semicircular arc\nis given by the right-hand rule and magnitude is half that of a\ncircular loop Thus B is 1"}, {"Chapter": "1", "sentence_range": "3607-3610", "Text": "All such contributions\nadd up in magnitude Hence direction of B for a semicircular arc\nis given by the right-hand rule and magnitude is half that of a\ncircular loop Thus B is 1 9 \u00d7 10\u20134 T normal to the plane of the\npaper going into it"}, {"Chapter": "1", "sentence_range": "3608-3611", "Text": "Hence direction of B for a semicircular arc\nis given by the right-hand rule and magnitude is half that of a\ncircular loop Thus B is 1 9 \u00d7 10\u20134 T normal to the plane of the\npaper going into it (c) Same magnitude of B but opposite in direction to that in (b)"}, {"Chapter": "1", "sentence_range": "3609-3612", "Text": "Thus B is 1 9 \u00d7 10\u20134 T normal to the plane of the\npaper going into it (c) Same magnitude of B but opposite in direction to that in (b) Example 4"}, {"Chapter": "1", "sentence_range": "3610-3613", "Text": "9 \u00d7 10\u20134 T normal to the plane of the\npaper going into it (c) Same magnitude of B but opposite in direction to that in (b) Example 4 7 Consider a tightly wound 100 turn coil of radius 10 cm,\ncarrying a current of 1 A"}, {"Chapter": "1", "sentence_range": "3611-3614", "Text": "(c) Same magnitude of B but opposite in direction to that in (b) Example 4 7 Consider a tightly wound 100 turn coil of radius 10 cm,\ncarrying a current of 1 A What is the magnitude of the magnetic\nfield at the centre of the coil"}, {"Chapter": "1", "sentence_range": "3612-3615", "Text": "Example 4 7 Consider a tightly wound 100 turn coil of radius 10 cm,\ncarrying a current of 1 A What is the magnitude of the magnetic\nfield at the centre of the coil Solution Since the coil is tightly wound, we may take each circular\nelement to have the same radius R = 10 cm = 0"}, {"Chapter": "1", "sentence_range": "3613-3616", "Text": "7 Consider a tightly wound 100 turn coil of radius 10 cm,\ncarrying a current of 1 A What is the magnitude of the magnetic\nfield at the centre of the coil Solution Since the coil is tightly wound, we may take each circular\nelement to have the same radius R = 10 cm = 0 1 m"}, {"Chapter": "1", "sentence_range": "3614-3617", "Text": "What is the magnitude of the magnetic\nfield at the centre of the coil Solution Since the coil is tightly wound, we may take each circular\nelement to have the same radius R = 10 cm = 0 1 m The number of\nturns N = 100"}, {"Chapter": "1", "sentence_range": "3615-3618", "Text": "Solution Since the coil is tightly wound, we may take each circular\nelement to have the same radius R = 10 cm = 0 1 m The number of\nturns N = 100 The magnitude of the magnetic field is,\n\u20137\n2\n0\n\u20131\n4\n10\n10\n1\n2\n2\n10\nNI\nB\n\u00b5R\n\u03c0 \u00d7\n\u00d7\n\u00d7\n=\n=\n\u00d7\n4\n2\n10\u2212\n=\n\u03c0 \u00d7\n4\n6 28\n10\nT"}, {"Chapter": "1", "sentence_range": "3616-3619", "Text": "1 m The number of\nturns N = 100 The magnitude of the magnetic field is,\n\u20137\n2\n0\n\u20131\n4\n10\n10\n1\n2\n2\n10\nNI\nB\n\u00b5R\n\u03c0 \u00d7\n\u00d7\n\u00d7\n=\n=\n\u00d7\n4\n2\n10\u2212\n=\n\u03c0 \u00d7\n4\n6 28\n10\nT \u2212\n=\n\u00d7\n4"}, {"Chapter": "1", "sentence_range": "3617-3620", "Text": "The number of\nturns N = 100 The magnitude of the magnetic field is,\n\u20137\n2\n0\n\u20131\n4\n10\n10\n1\n2\n2\n10\nNI\nB\n\u00b5R\n\u03c0 \u00d7\n\u00d7\n\u00d7\n=\n=\n\u00d7\n4\n2\n10\u2212\n=\n\u03c0 \u00d7\n4\n6 28\n10\nT \u2212\n=\n\u00d7\n4 6  AMPERE\u2019S CIRCUITAL LAW\nThere is an alternative and appealing way in which the\nBiot-Savart law may be expressed"}, {"Chapter": "1", "sentence_range": "3618-3621", "Text": "The magnitude of the magnetic field is,\n\u20137\n2\n0\n\u20131\n4\n10\n10\n1\n2\n2\n10\nNI\nB\n\u00b5R\n\u03c0 \u00d7\n\u00d7\n\u00d7\n=\n=\n\u00d7\n4\n2\n10\u2212\n=\n\u03c0 \u00d7\n4\n6 28\n10\nT \u2212\n=\n\u00d7\n4 6  AMPERE\u2019S CIRCUITAL LAW\nThere is an alternative and appealing way in which the\nBiot-Savart law may be expressed Ampere\u2019s circuital law\nconsiders an open surface with a boundary (Fig"}, {"Chapter": "1", "sentence_range": "3619-3622", "Text": "\u2212\n=\n\u00d7\n4 6  AMPERE\u2019S CIRCUITAL LAW\nThere is an alternative and appealing way in which the\nBiot-Savart law may be expressed Ampere\u2019s circuital law\nconsiders an open surface with a boundary (Fig 4"}, {"Chapter": "1", "sentence_range": "3620-3623", "Text": "6  AMPERE\u2019S CIRCUITAL LAW\nThere is an alternative and appealing way in which the\nBiot-Savart law may be expressed Ampere\u2019s circuital law\nconsiders an open surface with a boundary (Fig 4 14)"}, {"Chapter": "1", "sentence_range": "3621-3624", "Text": "Ampere\u2019s circuital law\nconsiders an open surface with a boundary (Fig 4 14) The surface has current passing through it"}, {"Chapter": "1", "sentence_range": "3622-3625", "Text": "4 14) The surface has current passing through it We consider\nthe boundary to be made up of a number of small line\nelements"}, {"Chapter": "1", "sentence_range": "3623-3626", "Text": "14) The surface has current passing through it We consider\nthe boundary to be made up of a number of small line\nelements Consider one such element of length dl"}, {"Chapter": "1", "sentence_range": "3624-3627", "Text": "The surface has current passing through it We consider\nthe boundary to be made up of a number of small line\nelements Consider one such element of length dl We\ntake the value of the tangential component of the\nmagnetic field, Bt, at this element and multiply it by the\nFIGURE 4"}, {"Chapter": "1", "sentence_range": "3625-3628", "Text": "We consider\nthe boundary to be made up of a number of small line\nelements Consider one such element of length dl We\ntake the value of the tangential component of the\nmagnetic field, Bt, at this element and multiply it by the\nFIGURE 4 12\n EXAMPLE 4"}, {"Chapter": "1", "sentence_range": "3626-3629", "Text": "Consider one such element of length dl We\ntake the value of the tangential component of the\nmagnetic field, Bt, at this element and multiply it by the\nFIGURE 4 12\n EXAMPLE 4 7\nRationalised 2023-24\nPhysics\n118\nlength of that element dl [Note: Btdl=B"}, {"Chapter": "1", "sentence_range": "3627-3630", "Text": "We\ntake the value of the tangential component of the\nmagnetic field, Bt, at this element and multiply it by the\nFIGURE 4 12\n EXAMPLE 4 7\nRationalised 2023-24\nPhysics\n118\nlength of that element dl [Note: Btdl=B dl]"}, {"Chapter": "1", "sentence_range": "3628-3631", "Text": "12\n EXAMPLE 4 7\nRationalised 2023-24\nPhysics\n118\nlength of that element dl [Note: Btdl=B dl] All such\nproducts are added together"}, {"Chapter": "1", "sentence_range": "3629-3632", "Text": "7\nRationalised 2023-24\nPhysics\n118\nlength of that element dl [Note: Btdl=B dl] All such\nproducts are added together We consider the limit as the\nlengths of elements get smaller and their number gets\nlarger"}, {"Chapter": "1", "sentence_range": "3630-3633", "Text": "dl] All such\nproducts are added together We consider the limit as the\nlengths of elements get smaller and their number gets\nlarger The sum then tends to an integral"}, {"Chapter": "1", "sentence_range": "3631-3634", "Text": "All such\nproducts are added together We consider the limit as the\nlengths of elements get smaller and their number gets\nlarger The sum then tends to an integral Ampere\u2019s   law\nstates that this integral is  equal to m0 times the total\ncurrent passing through  the surface,  i"}, {"Chapter": "1", "sentence_range": "3632-3635", "Text": "We consider the limit as the\nlengths of elements get smaller and their number gets\nlarger The sum then tends to an integral Ampere\u2019s   law\nstates that this integral is  equal to m0 times the total\ncurrent passing through  the surface,  i e"}, {"Chapter": "1", "sentence_range": "3633-3636", "Text": "The sum then tends to an integral Ampere\u2019s   law\nstates that this integral is  equal to m0 times the total\ncurrent passing through  the surface,  i e ,\n\u201cB"}, {"Chapter": "1", "sentence_range": "3634-3637", "Text": "Ampere\u2019s   law\nstates that this integral is  equal to m0 times the total\ncurrent passing through  the surface,  i e ,\n\u201cB dl = m0I\n[4"}, {"Chapter": "1", "sentence_range": "3635-3638", "Text": "e ,\n\u201cB dl = m0I\n[4 17(a)]\nwhere I  is the total current through the surface"}, {"Chapter": "1", "sentence_range": "3636-3639", "Text": ",\n\u201cB dl = m0I\n[4 17(a)]\nwhere I  is the total current through the surface The\nintegral is taken over the closed loop coinciding with the\nboundary C of the surface"}, {"Chapter": "1", "sentence_range": "3637-3640", "Text": "dl = m0I\n[4 17(a)]\nwhere I  is the total current through the surface The\nintegral is taken over the closed loop coinciding with the\nboundary C of the surface The relation above involves a\nsign-convention, given by the right-hand rule"}, {"Chapter": "1", "sentence_range": "3638-3641", "Text": "17(a)]\nwhere I  is the total current through the surface The\nintegral is taken over the closed loop coinciding with the\nboundary C of the surface The relation above involves a\nsign-convention, given by the right-hand rule Let the\nfingers of the right-hand be curled in the sense the\nboundary is traversed in the loop integral \u201cB"}, {"Chapter": "1", "sentence_range": "3639-3642", "Text": "The\nintegral is taken over the closed loop coinciding with the\nboundary C of the surface The relation above involves a\nsign-convention, given by the right-hand rule Let the\nfingers of the right-hand be curled in the sense the\nboundary is traversed in the loop integral \u201cB dl"}, {"Chapter": "1", "sentence_range": "3640-3643", "Text": "The relation above involves a\nsign-convention, given by the right-hand rule Let the\nfingers of the right-hand be curled in the sense the\nboundary is traversed in the loop integral \u201cB dl Then\nthe direction of the thumb gives the sense in which the\ncurrent I  is regarded as positive"}, {"Chapter": "1", "sentence_range": "3641-3644", "Text": "Let the\nfingers of the right-hand be curled in the sense the\nboundary is traversed in the loop integral \u201cB dl Then\nthe direction of the thumb gives the sense in which the\ncurrent I  is regarded as positive For several applications, a much simplified version of\nEq"}, {"Chapter": "1", "sentence_range": "3642-3645", "Text": "dl Then\nthe direction of the thumb gives the sense in which the\ncurrent I  is regarded as positive For several applications, a much simplified version of\nEq [4"}, {"Chapter": "1", "sentence_range": "3643-3646", "Text": "Then\nthe direction of the thumb gives the sense in which the\ncurrent I  is regarded as positive For several applications, a much simplified version of\nEq [4 17(a)] proves sufficient"}, {"Chapter": "1", "sentence_range": "3644-3647", "Text": "For several applications, a much simplified version of\nEq [4 17(a)] proves sufficient We shall assume that, in\nsuch cases, it is possible to choose the loop (called\nan amperian loop) such that at each point of the\nloop, either\n(i)\nB is tangential to the loop and is a non-zero constant\nB, or\n(ii)\nB is normal to the loop, or\n(iii) B vanishes"}, {"Chapter": "1", "sentence_range": "3645-3648", "Text": "[4 17(a)] proves sufficient We shall assume that, in\nsuch cases, it is possible to choose the loop (called\nan amperian loop) such that at each point of the\nloop, either\n(i)\nB is tangential to the loop and is a non-zero constant\nB, or\n(ii)\nB is normal to the loop, or\n(iii) B vanishes Now, let L be the length (part) of the  loop for which B\nis tangential"}, {"Chapter": "1", "sentence_range": "3646-3649", "Text": "17(a)] proves sufficient We shall assume that, in\nsuch cases, it is possible to choose the loop (called\nan amperian loop) such that at each point of the\nloop, either\n(i)\nB is tangential to the loop and is a non-zero constant\nB, or\n(ii)\nB is normal to the loop, or\n(iii) B vanishes Now, let L be the length (part) of the  loop for which B\nis tangential Let Ie  be the current enclosed by the loop"}, {"Chapter": "1", "sentence_range": "3647-3650", "Text": "We shall assume that, in\nsuch cases, it is possible to choose the loop (called\nan amperian loop) such that at each point of the\nloop, either\n(i)\nB is tangential to the loop and is a non-zero constant\nB, or\n(ii)\nB is normal to the loop, or\n(iii) B vanishes Now, let L be the length (part) of the  loop for which B\nis tangential Let Ie  be the current enclosed by the loop Then, Eq"}, {"Chapter": "1", "sentence_range": "3648-3651", "Text": "Now, let L be the length (part) of the  loop for which B\nis tangential Let Ie  be the current enclosed by the loop Then, Eq (4"}, {"Chapter": "1", "sentence_range": "3649-3652", "Text": "Let Ie  be the current enclosed by the loop Then, Eq (4 17) reduces to,\nBL =m0Ie\n[4"}, {"Chapter": "1", "sentence_range": "3650-3653", "Text": "Then, Eq (4 17) reduces to,\nBL =m0Ie\n[4 17(b)]\nWhen there is a system with a symmetry such as for\na straight infinite current-carrying wire in Fig"}, {"Chapter": "1", "sentence_range": "3651-3654", "Text": "(4 17) reduces to,\nBL =m0Ie\n[4 17(b)]\nWhen there is a system with a symmetry such as for\na straight infinite current-carrying wire in Fig 4"}, {"Chapter": "1", "sentence_range": "3652-3655", "Text": "17) reduces to,\nBL =m0Ie\n[4 17(b)]\nWhen there is a system with a symmetry such as for\na straight infinite current-carrying wire in Fig 4 13, the\nAmpere\u2019s law enables an easy evaluation of the magnetic\nfield, much the same way Gauss\u2019 law helps in\ndetermination of the electric field"}, {"Chapter": "1", "sentence_range": "3653-3656", "Text": "17(b)]\nWhen there is a system with a symmetry such as for\na straight infinite current-carrying wire in Fig 4 13, the\nAmpere\u2019s law enables an easy evaluation of the magnetic\nfield, much the same way Gauss\u2019 law helps in\ndetermination of the electric field This is exhibited in the\nExample 4"}, {"Chapter": "1", "sentence_range": "3654-3657", "Text": "4 13, the\nAmpere\u2019s law enables an easy evaluation of the magnetic\nfield, much the same way Gauss\u2019 law helps in\ndetermination of the electric field This is exhibited in the\nExample 4 9 below"}, {"Chapter": "1", "sentence_range": "3655-3658", "Text": "13, the\nAmpere\u2019s law enables an easy evaluation of the magnetic\nfield, much the same way Gauss\u2019 law helps in\ndetermination of the electric field This is exhibited in the\nExample 4 9 below The boundary of the loop chosen is\na circle and magnetic field is tangential to the\ncircumference of the circle"}, {"Chapter": "1", "sentence_range": "3656-3659", "Text": "This is exhibited in the\nExample 4 9 below The boundary of the loop chosen is\na circle and magnetic field is tangential to the\ncircumference of the circle The law gives, for the left hand\nside of Eq"}, {"Chapter": "1", "sentence_range": "3657-3660", "Text": "9 below The boundary of the loop chosen is\na circle and magnetic field is tangential to the\ncircumference of the circle The law gives, for the left hand\nside of Eq [4"}, {"Chapter": "1", "sentence_range": "3658-3661", "Text": "The boundary of the loop chosen is\na circle and magnetic field is tangential to the\ncircumference of the circle The law gives, for the left hand\nside of Eq [4 17 (b)], B"}, {"Chapter": "1", "sentence_range": "3659-3662", "Text": "The law gives, for the left hand\nside of Eq [4 17 (b)], B 2pr"}, {"Chapter": "1", "sentence_range": "3660-3663", "Text": "[4 17 (b)], B 2pr We find that the magnetic\nfield at a distance r outside the wire is tangential and\ngiven by\nB \u00d7 2pr  =  m0 I,\nB = m0  I/ (2pr)\n(4"}, {"Chapter": "1", "sentence_range": "3661-3664", "Text": "17 (b)], B 2pr We find that the magnetic\nfield at a distance r outside the wire is tangential and\ngiven by\nB \u00d7 2pr  =  m0 I,\nB = m0  I/ (2pr)\n(4 18)\nThe above result for the infinite wire  is interesting\nfrom several points of view"}, {"Chapter": "1", "sentence_range": "3662-3665", "Text": "2pr We find that the magnetic\nfield at a distance r outside the wire is tangential and\ngiven by\nB \u00d7 2pr  =  m0 I,\nB = m0  I/ (2pr)\n(4 18)\nThe above result for the infinite wire  is interesting\nfrom several points of view ANDRE AMPERE (1775 \u20131836)\nAndre Ampere (1775 \u2013\n1836) Andre Marie Ampere\nwas a French physicist,\nmathematician and chemist\nwho founded the science of\nelectrodynamics"}, {"Chapter": "1", "sentence_range": "3663-3666", "Text": "We find that the magnetic\nfield at a distance r outside the wire is tangential and\ngiven by\nB \u00d7 2pr  =  m0 I,\nB = m0  I/ (2pr)\n(4 18)\nThe above result for the infinite wire  is interesting\nfrom several points of view ANDRE AMPERE (1775 \u20131836)\nAndre Ampere (1775 \u2013\n1836) Andre Marie Ampere\nwas a French physicist,\nmathematician and chemist\nwho founded the science of\nelectrodynamics Ampere\nwas \na \nchild \nprodigy\nwho mastered advanced\nmathematics by the age of\n12"}, {"Chapter": "1", "sentence_range": "3664-3667", "Text": "18)\nThe above result for the infinite wire  is interesting\nfrom several points of view ANDRE AMPERE (1775 \u20131836)\nAndre Ampere (1775 \u2013\n1836) Andre Marie Ampere\nwas a French physicist,\nmathematician and chemist\nwho founded the science of\nelectrodynamics Ampere\nwas \na \nchild \nprodigy\nwho mastered advanced\nmathematics by the age of\n12 Ampere grasped the\nsignificance of Oersted\u2019s\ndiscovery"}, {"Chapter": "1", "sentence_range": "3665-3668", "Text": "ANDRE AMPERE (1775 \u20131836)\nAndre Ampere (1775 \u2013\n1836) Andre Marie Ampere\nwas a French physicist,\nmathematician and chemist\nwho founded the science of\nelectrodynamics Ampere\nwas \na \nchild \nprodigy\nwho mastered advanced\nmathematics by the age of\n12 Ampere grasped the\nsignificance of Oersted\u2019s\ndiscovery He carried out a\nlarge series of experiments\nto explore the relationship\nbetween current electricity\nand magnetism"}, {"Chapter": "1", "sentence_range": "3666-3669", "Text": "Ampere\nwas \na \nchild \nprodigy\nwho mastered advanced\nmathematics by the age of\n12 Ampere grasped the\nsignificance of Oersted\u2019s\ndiscovery He carried out a\nlarge series of experiments\nto explore the relationship\nbetween current electricity\nand magnetism These\ninvestigations culminated\nin \n1827 \nwith \nthe\npublication \nof \nthe\n\u2018Mathematical Theory of\nElectrodynamic Pheno-\nmena Deduced Solely from\nExperiments\u2019"}, {"Chapter": "1", "sentence_range": "3667-3670", "Text": "Ampere grasped the\nsignificance of Oersted\u2019s\ndiscovery He carried out a\nlarge series of experiments\nto explore the relationship\nbetween current electricity\nand magnetism These\ninvestigations culminated\nin \n1827 \nwith \nthe\npublication \nof \nthe\n\u2018Mathematical Theory of\nElectrodynamic Pheno-\nmena Deduced Solely from\nExperiments\u2019 He hypo-\nthesised that all magnetic\nphenomena are due to\ncirculating \nelectric\ncurrents"}, {"Chapter": "1", "sentence_range": "3668-3671", "Text": "He carried out a\nlarge series of experiments\nto explore the relationship\nbetween current electricity\nand magnetism These\ninvestigations culminated\nin \n1827 \nwith \nthe\npublication \nof \nthe\n\u2018Mathematical Theory of\nElectrodynamic Pheno-\nmena Deduced Solely from\nExperiments\u2019 He hypo-\nthesised that all magnetic\nphenomena are due to\ncirculating \nelectric\ncurrents Ampere was\nhumble \nand \nabsent-\nminded"}, {"Chapter": "1", "sentence_range": "3669-3672", "Text": "These\ninvestigations culminated\nin \n1827 \nwith \nthe\npublication \nof \nthe\n\u2018Mathematical Theory of\nElectrodynamic Pheno-\nmena Deduced Solely from\nExperiments\u2019 He hypo-\nthesised that all magnetic\nphenomena are due to\ncirculating \nelectric\ncurrents Ampere was\nhumble \nand \nabsent-\nminded He once forgot an\ninvitation to dine with the\nEmperor Napoleon"}, {"Chapter": "1", "sentence_range": "3670-3673", "Text": "He hypo-\nthesised that all magnetic\nphenomena are due to\ncirculating \nelectric\ncurrents Ampere was\nhumble \nand \nabsent-\nminded He once forgot an\ninvitation to dine with the\nEmperor Napoleon He died\nof pneumonia at the age of\n61"}, {"Chapter": "1", "sentence_range": "3671-3674", "Text": "Ampere was\nhumble \nand \nabsent-\nminded He once forgot an\ninvitation to dine with the\nEmperor Napoleon He died\nof pneumonia at the age of\n61 His gravestone bears\nthe epitaph: Tandem Felix\n(Happy at last)"}, {"Chapter": "1", "sentence_range": "3672-3675", "Text": "He once forgot an\ninvitation to dine with the\nEmperor Napoleon He died\nof pneumonia at the age of\n61 His gravestone bears\nthe epitaph: Tandem Felix\n(Happy at last) (i)\nIt implies that the field at every point on a circle of\nradius r, (with the wire along the axis), is same in\nmagnitude"}, {"Chapter": "1", "sentence_range": "3673-3676", "Text": "He died\nof pneumonia at the age of\n61 His gravestone bears\nthe epitaph: Tandem Felix\n(Happy at last) (i)\nIt implies that the field at every point on a circle of\nradius r, (with the wire along the axis), is same in\nmagnitude In other words, the magnetic field\nRationalised 2023-24\n119\nMoving Charges and\nMagnetism\n EXAMPLE 4"}, {"Chapter": "1", "sentence_range": "3674-3677", "Text": "His gravestone bears\nthe epitaph: Tandem Felix\n(Happy at last) (i)\nIt implies that the field at every point on a circle of\nradius r, (with the wire along the axis), is same in\nmagnitude In other words, the magnetic field\nRationalised 2023-24\n119\nMoving Charges and\nMagnetism\n EXAMPLE 4 8\npossesses what is called a cylindrical symmetry"}, {"Chapter": "1", "sentence_range": "3675-3678", "Text": "(i)\nIt implies that the field at every point on a circle of\nradius r, (with the wire along the axis), is same in\nmagnitude In other words, the magnetic field\nRationalised 2023-24\n119\nMoving Charges and\nMagnetism\n EXAMPLE 4 8\npossesses what is called a cylindrical symmetry The field that\nnormally can depend on three coordinates depends only on one: r"}, {"Chapter": "1", "sentence_range": "3676-3679", "Text": "In other words, the magnetic field\nRationalised 2023-24\n119\nMoving Charges and\nMagnetism\n EXAMPLE 4 8\npossesses what is called a cylindrical symmetry The field that\nnormally can depend on three coordinates depends only on one: r Whenever there is symmetry, the solutions simplify"}, {"Chapter": "1", "sentence_range": "3677-3680", "Text": "8\npossesses what is called a cylindrical symmetry The field that\nnormally can depend on three coordinates depends only on one: r Whenever there is symmetry, the solutions simplify (ii)\nThe field direction at any point on this circle is tangential to it"}, {"Chapter": "1", "sentence_range": "3678-3681", "Text": "The field that\nnormally can depend on three coordinates depends only on one: r Whenever there is symmetry, the solutions simplify (ii)\nThe field direction at any point on this circle is tangential to it Thus, the lines of constant magnitude of magnetic field form\nconcentric circles"}, {"Chapter": "1", "sentence_range": "3679-3682", "Text": "Whenever there is symmetry, the solutions simplify (ii)\nThe field direction at any point on this circle is tangential to it Thus, the lines of constant magnitude of magnetic field form\nconcentric circles Notice now, in Fig"}, {"Chapter": "1", "sentence_range": "3680-3683", "Text": "(ii)\nThe field direction at any point on this circle is tangential to it Thus, the lines of constant magnitude of magnetic field form\nconcentric circles Notice now, in Fig 4"}, {"Chapter": "1", "sentence_range": "3681-3684", "Text": "Thus, the lines of constant magnitude of magnetic field form\nconcentric circles Notice now, in Fig 4 1(c), the iron filings form\nconcentric circles"}, {"Chapter": "1", "sentence_range": "3682-3685", "Text": "Notice now, in Fig 4 1(c), the iron filings form\nconcentric circles These lines called magnetic field lines form closed\nloops"}, {"Chapter": "1", "sentence_range": "3683-3686", "Text": "4 1(c), the iron filings form\nconcentric circles These lines called magnetic field lines form closed\nloops This is unlike the electrostatic field lines which originate\nfrom positive charges and end at negative charges"}, {"Chapter": "1", "sentence_range": "3684-3687", "Text": "1(c), the iron filings form\nconcentric circles These lines called magnetic field lines form closed\nloops This is unlike the electrostatic field lines which originate\nfrom positive charges and end at negative charges The expression\nfor the magnetic field of a straight wire provides a theoretical\njustification to Oersted\u2019s experiments"}, {"Chapter": "1", "sentence_range": "3685-3688", "Text": "These lines called magnetic field lines form closed\nloops This is unlike the electrostatic field lines which originate\nfrom positive charges and end at negative charges The expression\nfor the magnetic field of a straight wire provides a theoretical\njustification to Oersted\u2019s experiments (iii)\nAnother interesting point to note is that even though the wire is\ninfinite, the field due to it at a non-zero distance is not infinite"}, {"Chapter": "1", "sentence_range": "3686-3689", "Text": "This is unlike the electrostatic field lines which originate\nfrom positive charges and end at negative charges The expression\nfor the magnetic field of a straight wire provides a theoretical\njustification to Oersted\u2019s experiments (iii)\nAnother interesting point to note is that even though the wire is\ninfinite, the field due to it at a non-zero distance is not infinite It\ntends to blow up only when we come very close to the wire"}, {"Chapter": "1", "sentence_range": "3687-3690", "Text": "The expression\nfor the magnetic field of a straight wire provides a theoretical\njustification to Oersted\u2019s experiments (iii)\nAnother interesting point to note is that even though the wire is\ninfinite, the field due to it at a non-zero distance is not infinite It\ntends to blow up only when we come very close to the wire The\nfield is directly proportional to the current and inversely\nproportional to the distance from the (infinitely long) current source"}, {"Chapter": "1", "sentence_range": "3688-3691", "Text": "(iii)\nAnother interesting point to note is that even though the wire is\ninfinite, the field due to it at a non-zero distance is not infinite It\ntends to blow up only when we come very close to the wire The\nfield is directly proportional to the current and inversely\nproportional to the distance from the (infinitely long) current source (iv)\nThere exists a simple rule to determine the direction of the magnetic\nfield due to a long wire"}, {"Chapter": "1", "sentence_range": "3689-3692", "Text": "It\ntends to blow up only when we come very close to the wire The\nfield is directly proportional to the current and inversely\nproportional to the distance from the (infinitely long) current source (iv)\nThere exists a simple rule to determine the direction of the magnetic\nfield due to a long wire This rule, called the right-hand rule*, is:\nGrasp the wire in your right hand with your extended thumb pointing\nin the direction of the current"}, {"Chapter": "1", "sentence_range": "3690-3693", "Text": "The\nfield is directly proportional to the current and inversely\nproportional to the distance from the (infinitely long) current source (iv)\nThere exists a simple rule to determine the direction of the magnetic\nfield due to a long wire This rule, called the right-hand rule*, is:\nGrasp the wire in your right hand with your extended thumb pointing\nin the direction of the current Your fingers will curl around in the\ndirection of the magnetic field"}, {"Chapter": "1", "sentence_range": "3691-3694", "Text": "(iv)\nThere exists a simple rule to determine the direction of the magnetic\nfield due to a long wire This rule, called the right-hand rule*, is:\nGrasp the wire in your right hand with your extended thumb pointing\nin the direction of the current Your fingers will curl around in the\ndirection of the magnetic field Ampere\u2019s circuital law is not new in content from Biot-Savart law"}, {"Chapter": "1", "sentence_range": "3692-3695", "Text": "This rule, called the right-hand rule*, is:\nGrasp the wire in your right hand with your extended thumb pointing\nin the direction of the current Your fingers will curl around in the\ndirection of the magnetic field Ampere\u2019s circuital law is not new in content from Biot-Savart law Both relate the magnetic field and the current, and both express the same\nphysical consequences of a steady electrical current"}, {"Chapter": "1", "sentence_range": "3693-3696", "Text": "Your fingers will curl around in the\ndirection of the magnetic field Ampere\u2019s circuital law is not new in content from Biot-Savart law Both relate the magnetic field and the current, and both express the same\nphysical consequences of a steady electrical current Ampere\u2019s law is to\nBiot-Savart law, what Gauss\u2019s law is to Coulomb\u2019s law"}, {"Chapter": "1", "sentence_range": "3694-3697", "Text": "Ampere\u2019s circuital law is not new in content from Biot-Savart law Both relate the magnetic field and the current, and both express the same\nphysical consequences of a steady electrical current Ampere\u2019s law is to\nBiot-Savart law, what Gauss\u2019s law is to Coulomb\u2019s law Both, Ampere\u2019s\nand Gauss\u2019s law relate a physical quantity on the periphery or boundary\n(magnetic or electric field) to another physical quantity, namely, the source,\nin the interior (current or charge)"}, {"Chapter": "1", "sentence_range": "3695-3698", "Text": "Both relate the magnetic field and the current, and both express the same\nphysical consequences of a steady electrical current Ampere\u2019s law is to\nBiot-Savart law, what Gauss\u2019s law is to Coulomb\u2019s law Both, Ampere\u2019s\nand Gauss\u2019s law relate a physical quantity on the periphery or boundary\n(magnetic or electric field) to another physical quantity, namely, the source,\nin the interior (current or charge) We also note that Ampere\u2019s circuital\nlaw holds for steady currents which do not fluctuate with time"}, {"Chapter": "1", "sentence_range": "3696-3699", "Text": "Ampere\u2019s law is to\nBiot-Savart law, what Gauss\u2019s law is to Coulomb\u2019s law Both, Ampere\u2019s\nand Gauss\u2019s law relate a physical quantity on the periphery or boundary\n(magnetic or electric field) to another physical quantity, namely, the source,\nin the interior (current or charge) We also note that Ampere\u2019s circuital\nlaw holds for steady currents which do not fluctuate with time The\nfollowing example will help us understand what is meant by the term\nenclosed current"}, {"Chapter": "1", "sentence_range": "3697-3700", "Text": "Both, Ampere\u2019s\nand Gauss\u2019s law relate a physical quantity on the periphery or boundary\n(magnetic or electric field) to another physical quantity, namely, the source,\nin the interior (current or charge) We also note that Ampere\u2019s circuital\nlaw holds for steady currents which do not fluctuate with time The\nfollowing example will help us understand what is meant by the term\nenclosed current Example 4"}, {"Chapter": "1", "sentence_range": "3698-3701", "Text": "We also note that Ampere\u2019s circuital\nlaw holds for steady currents which do not fluctuate with time The\nfollowing example will help us understand what is meant by the term\nenclosed current Example 4 8 Figure 4"}, {"Chapter": "1", "sentence_range": "3699-3702", "Text": "The\nfollowing example will help us understand what is meant by the term\nenclosed current Example 4 8 Figure 4 13 shows a long straight wire of a circular\ncross-section (radius a) carrying steady current I"}, {"Chapter": "1", "sentence_range": "3700-3703", "Text": "Example 4 8 Figure 4 13 shows a long straight wire of a circular\ncross-section (radius a) carrying steady current I The current I is\nuniformly distributed across this cross-section"}, {"Chapter": "1", "sentence_range": "3701-3704", "Text": "8 Figure 4 13 shows a long straight wire of a circular\ncross-section (radius a) carrying steady current I The current I is\nuniformly distributed across this cross-section Calculate the\nmagnetic field in the region r < a and r > a"}, {"Chapter": "1", "sentence_range": "3702-3705", "Text": "13 shows a long straight wire of a circular\ncross-section (radius a) carrying steady current I The current I is\nuniformly distributed across this cross-section Calculate the\nmagnetic field in the region r < a and r > a FIGURE 4"}, {"Chapter": "1", "sentence_range": "3703-3706", "Text": "The current I is\nuniformly distributed across this cross-section Calculate the\nmagnetic field in the region r < a and r > a FIGURE 4 13\n*\nNote that there are two distinct right-hand rules: One which gives the direction\nof B on the axis of current-loop and the other which gives direction of B\nfor a straight conducting wire"}, {"Chapter": "1", "sentence_range": "3704-3707", "Text": "Calculate the\nmagnetic field in the region r < a and r > a FIGURE 4 13\n*\nNote that there are two distinct right-hand rules: One which gives the direction\nof B on the axis of current-loop and the other which gives direction of B\nfor a straight conducting wire Fingers and thumb play different roles in\nthe two"}, {"Chapter": "1", "sentence_range": "3705-3708", "Text": "FIGURE 4 13\n*\nNote that there are two distinct right-hand rules: One which gives the direction\nof B on the axis of current-loop and the other which gives direction of B\nfor a straight conducting wire Fingers and thumb play different roles in\nthe two Rationalised 2023-24\nPhysics\n120\n EXAMPLE 4"}, {"Chapter": "1", "sentence_range": "3706-3709", "Text": "13\n*\nNote that there are two distinct right-hand rules: One which gives the direction\nof B on the axis of current-loop and the other which gives direction of B\nfor a straight conducting wire Fingers and thumb play different roles in\nthe two Rationalised 2023-24\nPhysics\n120\n EXAMPLE 4 8\nSolution (a) Consider the case r > a"}, {"Chapter": "1", "sentence_range": "3707-3710", "Text": "Fingers and thumb play different roles in\nthe two Rationalised 2023-24\nPhysics\n120\n EXAMPLE 4 8\nSolution (a) Consider the case r > a The Amperian loop, labelled 2,\nis a circle concentric with the cross-section"}, {"Chapter": "1", "sentence_range": "3708-3711", "Text": "Rationalised 2023-24\nPhysics\n120\n EXAMPLE 4 8\nSolution (a) Consider the case r > a The Amperian loop, labelled 2,\nis a circle concentric with the cross-section For this loop,\nL  = 2 p r\nIe = Current enclosed by the loop = I\nThe result is the familiar expression for a long straight wire\nB (2p r) = m0I\n\u03c0 \n0\n2\nI\nB\nr\n=\u00b5\n[4"}, {"Chapter": "1", "sentence_range": "3709-3712", "Text": "8\nSolution (a) Consider the case r > a The Amperian loop, labelled 2,\nis a circle concentric with the cross-section For this loop,\nL  = 2 p r\nIe = Current enclosed by the loop = I\nThe result is the familiar expression for a long straight wire\nB (2p r) = m0I\n\u03c0 \n0\n2\nI\nB\nr\n=\u00b5\n[4 19(a)]\n1\nB\n\u221dr\n(r > a)\nNow the current enclosed Ie is not I, but is less than this value"}, {"Chapter": "1", "sentence_range": "3710-3713", "Text": "The Amperian loop, labelled 2,\nis a circle concentric with the cross-section For this loop,\nL  = 2 p r\nIe = Current enclosed by the loop = I\nThe result is the familiar expression for a long straight wire\nB (2p r) = m0I\n\u03c0 \n0\n2\nI\nB\nr\n=\u00b5\n[4 19(a)]\n1\nB\n\u221dr\n(r > a)\nNow the current enclosed Ie is not I, but is less than this value Since the current distribution is uniform, the current enclosed is,\nI\nI\nar\ne =\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n\u03c0\n\u03c0\n2\n2\n \n2\n2\naIr\n=\nUsing Ampere\u2019s law, \n\u03c0\n2\n0\n2\n(2\n)\nI r\nB\nr\na\n\u00b5\n=\nB\naI\nr\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\n\u00b50\n2\n2\ufffd\n[4"}, {"Chapter": "1", "sentence_range": "3711-3714", "Text": "For this loop,\nL  = 2 p r\nIe = Current enclosed by the loop = I\nThe result is the familiar expression for a long straight wire\nB (2p r) = m0I\n\u03c0 \n0\n2\nI\nB\nr\n=\u00b5\n[4 19(a)]\n1\nB\n\u221dr\n(r > a)\nNow the current enclosed Ie is not I, but is less than this value Since the current distribution is uniform, the current enclosed is,\nI\nI\nar\ne =\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n\u03c0\n\u03c0\n2\n2\n \n2\n2\naIr\n=\nUsing Ampere\u2019s law, \n\u03c0\n2\n0\n2\n(2\n)\nI r\nB\nr\na\n\u00b5\n=\nB\naI\nr\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\n\u00b50\n2\n2\ufffd\n[4 19(b)]\nB \u00b5 r      (r < a)\nFIGURE 4"}, {"Chapter": "1", "sentence_range": "3712-3715", "Text": "19(a)]\n1\nB\n\u221dr\n(r > a)\nNow the current enclosed Ie is not I, but is less than this value Since the current distribution is uniform, the current enclosed is,\nI\nI\nar\ne =\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n\u03c0\n\u03c0\n2\n2\n \n2\n2\naIr\n=\nUsing Ampere\u2019s law, \n\u03c0\n2\n0\n2\n(2\n)\nI r\nB\nr\na\n\u00b5\n=\nB\naI\nr\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\n\u00b50\n2\n2\ufffd\n[4 19(b)]\nB \u00b5 r      (r < a)\nFIGURE 4 14\nFigure (4"}, {"Chapter": "1", "sentence_range": "3713-3716", "Text": "Since the current distribution is uniform, the current enclosed is,\nI\nI\nar\ne =\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n\u03c0\n\u03c0\n2\n2\n \n2\n2\naIr\n=\nUsing Ampere\u2019s law, \n\u03c0\n2\n0\n2\n(2\n)\nI r\nB\nr\na\n\u00b5\n=\nB\naI\nr\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\n\u00b50\n2\n2\ufffd\n[4 19(b)]\nB \u00b5 r      (r < a)\nFIGURE 4 14\nFigure (4 14) shows a plot of the magnitude of B with distance r\nfrom the centre of the wire"}, {"Chapter": "1", "sentence_range": "3714-3717", "Text": "19(b)]\nB \u00b5 r      (r < a)\nFIGURE 4 14\nFigure (4 14) shows a plot of the magnitude of B with distance r\nfrom the centre of the wire The direction of the field is tangential to\nthe respective circular loop (1 or 2) and given by the right-hand\nrule described earlier in this section"}, {"Chapter": "1", "sentence_range": "3715-3718", "Text": "14\nFigure (4 14) shows a plot of the magnitude of B with distance r\nfrom the centre of the wire The direction of the field is tangential to\nthe respective circular loop (1 or 2) and given by the right-hand\nrule described earlier in this section This example possesses the required symmetry so that Ampere\u2019s\nlaw can be applied readily"}, {"Chapter": "1", "sentence_range": "3716-3719", "Text": "14) shows a plot of the magnitude of B with distance r\nfrom the centre of the wire The direction of the field is tangential to\nthe respective circular loop (1 or 2) and given by the right-hand\nrule described earlier in this section This example possesses the required symmetry so that Ampere\u2019s\nlaw can be applied readily It should be noted that while Ampere\u2019s circuital law holds for any\nloop, it may not always facilitate an evaluation of the magnetic field in\nevery case"}, {"Chapter": "1", "sentence_range": "3717-3720", "Text": "The direction of the field is tangential to\nthe respective circular loop (1 or 2) and given by the right-hand\nrule described earlier in this section This example possesses the required symmetry so that Ampere\u2019s\nlaw can be applied readily It should be noted that while Ampere\u2019s circuital law holds for any\nloop, it may not always facilitate an evaluation of the magnetic field in\nevery case For example, for the case of the circular loop discussed in\nSection 4"}, {"Chapter": "1", "sentence_range": "3718-3721", "Text": "This example possesses the required symmetry so that Ampere\u2019s\nlaw can be applied readily It should be noted that while Ampere\u2019s circuital law holds for any\nloop, it may not always facilitate an evaluation of the magnetic field in\nevery case For example, for the case of the circular loop discussed in\nSection 4 6, it cannot be applied to extract the simple expression\nB = m0I/2R [Eq"}, {"Chapter": "1", "sentence_range": "3719-3722", "Text": "It should be noted that while Ampere\u2019s circuital law holds for any\nloop, it may not always facilitate an evaluation of the magnetic field in\nevery case For example, for the case of the circular loop discussed in\nSection 4 6, it cannot be applied to extract the simple expression\nB = m0I/2R [Eq (4"}, {"Chapter": "1", "sentence_range": "3720-3723", "Text": "For example, for the case of the circular loop discussed in\nSection 4 6, it cannot be applied to extract the simple expression\nB = m0I/2R [Eq (4 16)] for the field at the centre of the loop"}, {"Chapter": "1", "sentence_range": "3721-3724", "Text": "6, it cannot be applied to extract the simple expression\nB = m0I/2R [Eq (4 16)] for the field at the centre of the loop However,\nthere exists a large number of situations of high symmetry where the law\ncan be conveniently applied"}, {"Chapter": "1", "sentence_range": "3722-3725", "Text": "(4 16)] for the field at the centre of the loop However,\nthere exists a large number of situations of high symmetry where the law\ncan be conveniently applied We shall use  it in the next section to calculate\np\nRationalised 2023-24\n121\nMoving Charges and\nMagnetism\nthe magnetic field produced by two commonly used and very useful\nmagnetic systems: the solenoid and the toroid"}, {"Chapter": "1", "sentence_range": "3723-3726", "Text": "16)] for the field at the centre of the loop However,\nthere exists a large number of situations of high symmetry where the law\ncan be conveniently applied We shall use  it in the next section to calculate\np\nRationalised 2023-24\n121\nMoving Charges and\nMagnetism\nthe magnetic field produced by two commonly used and very useful\nmagnetic systems: the solenoid and the toroid 4"}, {"Chapter": "1", "sentence_range": "3724-3727", "Text": "However,\nthere exists a large number of situations of high symmetry where the law\ncan be conveniently applied We shall use  it in the next section to calculate\np\nRationalised 2023-24\n121\nMoving Charges and\nMagnetism\nthe magnetic field produced by two commonly used and very useful\nmagnetic systems: the solenoid and the toroid 4 7  THE SOLENOID\nWe shall discuss a long solenoid"}, {"Chapter": "1", "sentence_range": "3725-3728", "Text": "We shall use  it in the next section to calculate\np\nRationalised 2023-24\n121\nMoving Charges and\nMagnetism\nthe magnetic field produced by two commonly used and very useful\nmagnetic systems: the solenoid and the toroid 4 7  THE SOLENOID\nWe shall discuss a long solenoid By long solenoid we mean that the\nsolenoid\u2019s length is large compared to its radius"}, {"Chapter": "1", "sentence_range": "3726-3729", "Text": "4 7  THE SOLENOID\nWe shall discuss a long solenoid By long solenoid we mean that the\nsolenoid\u2019s length is large compared to its radius It consists of a long wire\nwound in the form of a helix where the neighbouring turns are closely\nspaced"}, {"Chapter": "1", "sentence_range": "3727-3730", "Text": "7  THE SOLENOID\nWe shall discuss a long solenoid By long solenoid we mean that the\nsolenoid\u2019s length is large compared to its radius It consists of a long wire\nwound in the form of a helix where the neighbouring turns are closely\nspaced So each turn can be regarded as a circular loop"}, {"Chapter": "1", "sentence_range": "3728-3731", "Text": "By long solenoid we mean that the\nsolenoid\u2019s length is large compared to its radius It consists of a long wire\nwound in the form of a helix where the neighbouring turns are closely\nspaced So each turn can be regarded as a circular loop The net magnetic\nfield is the vector sum of the fields due to all the turns"}, {"Chapter": "1", "sentence_range": "3729-3732", "Text": "It consists of a long wire\nwound in the form of a helix where the neighbouring turns are closely\nspaced So each turn can be regarded as a circular loop The net magnetic\nfield is the vector sum of the fields due to all the turns Enamelled wires\nare used for winding so that turns are insulated from each other"}, {"Chapter": "1", "sentence_range": "3730-3733", "Text": "So each turn can be regarded as a circular loop The net magnetic\nfield is the vector sum of the fields due to all the turns Enamelled wires\nare used for winding so that turns are insulated from each other Figure 4"}, {"Chapter": "1", "sentence_range": "3731-3734", "Text": "The net magnetic\nfield is the vector sum of the fields due to all the turns Enamelled wires\nare used for winding so that turns are insulated from each other Figure 4 15 displays the magnetic field lines for a finite solenoid"}, {"Chapter": "1", "sentence_range": "3732-3735", "Text": "Enamelled wires\nare used for winding so that turns are insulated from each other Figure 4 15 displays the magnetic field lines for a finite solenoid We\nshow a section of this solenoid in an enlarged manner in Fig"}, {"Chapter": "1", "sentence_range": "3733-3736", "Text": "Figure 4 15 displays the magnetic field lines for a finite solenoid We\nshow a section of this solenoid in an enlarged manner in Fig 4"}, {"Chapter": "1", "sentence_range": "3734-3737", "Text": "15 displays the magnetic field lines for a finite solenoid We\nshow a section of this solenoid in an enlarged manner in Fig 4 15(a)"}, {"Chapter": "1", "sentence_range": "3735-3738", "Text": "We\nshow a section of this solenoid in an enlarged manner in Fig 4 15(a) Figure 4"}, {"Chapter": "1", "sentence_range": "3736-3739", "Text": "4 15(a) Figure 4 15(b) shows the entire finite solenoid with its magnetic field"}, {"Chapter": "1", "sentence_range": "3737-3740", "Text": "15(a) Figure 4 15(b) shows the entire finite solenoid with its magnetic field In\nFig"}, {"Chapter": "1", "sentence_range": "3738-3741", "Text": "Figure 4 15(b) shows the entire finite solenoid with its magnetic field In\nFig 4"}, {"Chapter": "1", "sentence_range": "3739-3742", "Text": "15(b) shows the entire finite solenoid with its magnetic field In\nFig 4 15(a), it is clear from the circular loops that the field  between two\nneighbouring turns vanishes"}, {"Chapter": "1", "sentence_range": "3740-3743", "Text": "In\nFig 4 15(a), it is clear from the circular loops that the field  between two\nneighbouring turns vanishes In Fig"}, {"Chapter": "1", "sentence_range": "3741-3744", "Text": "4 15(a), it is clear from the circular loops that the field  between two\nneighbouring turns vanishes In Fig 4"}, {"Chapter": "1", "sentence_range": "3742-3745", "Text": "15(a), it is clear from the circular loops that the field  between two\nneighbouring turns vanishes In Fig 4 15(b), we see that the field at the\ninterior mid-point P is uniform, strong and along the axis of the solenoid"}, {"Chapter": "1", "sentence_range": "3743-3746", "Text": "In Fig 4 15(b), we see that the field at the\ninterior mid-point P is uniform, strong and along the axis of the solenoid The field at the exterior mid-point Q is weak and moreover is along the\naxis of the solenoid with no perpendicular or normal component"}, {"Chapter": "1", "sentence_range": "3744-3747", "Text": "4 15(b), we see that the field at the\ninterior mid-point P is uniform, strong and along the axis of the solenoid The field at the exterior mid-point Q is weak and moreover is along the\naxis of the solenoid with no perpendicular or normal component As the\nFIGURE 4"}, {"Chapter": "1", "sentence_range": "3745-3748", "Text": "15(b), we see that the field at the\ninterior mid-point P is uniform, strong and along the axis of the solenoid The field at the exterior mid-point Q is weak and moreover is along the\naxis of the solenoid with no perpendicular or normal component As the\nFIGURE 4 15 (a) The magnetic field due to a section of the solenoid which has been\nstretched out for clarity"}, {"Chapter": "1", "sentence_range": "3746-3749", "Text": "The field at the exterior mid-point Q is weak and moreover is along the\naxis of the solenoid with no perpendicular or normal component As the\nFIGURE 4 15 (a) The magnetic field due to a section of the solenoid which has been\nstretched out for clarity Only the exterior semi-circular part is shown"}, {"Chapter": "1", "sentence_range": "3747-3750", "Text": "As the\nFIGURE 4 15 (a) The magnetic field due to a section of the solenoid which has been\nstretched out for clarity Only the exterior semi-circular part is shown Notice\nhow the circular loops between neighbouring turns tend to cancel"}, {"Chapter": "1", "sentence_range": "3748-3751", "Text": "15 (a) The magnetic field due to a section of the solenoid which has been\nstretched out for clarity Only the exterior semi-circular part is shown Notice\nhow the circular loops between neighbouring turns tend to cancel (b) The magnetic field of a finite solenoid"}, {"Chapter": "1", "sentence_range": "3749-3752", "Text": "Only the exterior semi-circular part is shown Notice\nhow the circular loops between neighbouring turns tend to cancel (b) The magnetic field of a finite solenoid FIGURE 4"}, {"Chapter": "1", "sentence_range": "3750-3753", "Text": "Notice\nhow the circular loops between neighbouring turns tend to cancel (b) The magnetic field of a finite solenoid FIGURE 4 16 The magnetic field of a very long solenoid"}, {"Chapter": "1", "sentence_range": "3751-3754", "Text": "(b) The magnetic field of a finite solenoid FIGURE 4 16 The magnetic field of a very long solenoid We consider a\nrectangular Amperian loop abcd to determine the field"}, {"Chapter": "1", "sentence_range": "3752-3755", "Text": "FIGURE 4 16 The magnetic field of a very long solenoid We consider a\nrectangular Amperian loop abcd to determine the field Rationalised 2023-24\nPhysics\n122\n EXAMPLE 4"}, {"Chapter": "1", "sentence_range": "3753-3756", "Text": "16 The magnetic field of a very long solenoid We consider a\nrectangular Amperian loop abcd to determine the field Rationalised 2023-24\nPhysics\n122\n EXAMPLE 4 9\nsolenoid is made longer it appears like a long cylindrical metal sheet"}, {"Chapter": "1", "sentence_range": "3754-3757", "Text": "We consider a\nrectangular Amperian loop abcd to determine the field Rationalised 2023-24\nPhysics\n122\n EXAMPLE 4 9\nsolenoid is made longer it appears like a long cylindrical metal sheet Figure 4"}, {"Chapter": "1", "sentence_range": "3755-3758", "Text": "Rationalised 2023-24\nPhysics\n122\n EXAMPLE 4 9\nsolenoid is made longer it appears like a long cylindrical metal sheet Figure 4 16 represents this idealised picture"}, {"Chapter": "1", "sentence_range": "3756-3759", "Text": "9\nsolenoid is made longer it appears like a long cylindrical metal sheet Figure 4 16 represents this idealised picture The field outside the solenoid\napproaches zero"}, {"Chapter": "1", "sentence_range": "3757-3760", "Text": "Figure 4 16 represents this idealised picture The field outside the solenoid\napproaches zero We shall assume that the field outside is zero"}, {"Chapter": "1", "sentence_range": "3758-3761", "Text": "16 represents this idealised picture The field outside the solenoid\napproaches zero We shall assume that the field outside is zero The field\ninside becomes everywhere parallel to the axis"}, {"Chapter": "1", "sentence_range": "3759-3762", "Text": "The field outside the solenoid\napproaches zero We shall assume that the field outside is zero The field\ninside becomes everywhere parallel to the axis Consider a rectangular Amperian loop abcd"}, {"Chapter": "1", "sentence_range": "3760-3763", "Text": "We shall assume that the field outside is zero The field\ninside becomes everywhere parallel to the axis Consider a rectangular Amperian loop abcd Along cd the field is zero\nas argued above"}, {"Chapter": "1", "sentence_range": "3761-3764", "Text": "The field\ninside becomes everywhere parallel to the axis Consider a rectangular Amperian loop abcd Along cd the field is zero\nas argued above Along transverse sections bc and ad, the field component\nis zero"}, {"Chapter": "1", "sentence_range": "3762-3765", "Text": "Consider a rectangular Amperian loop abcd Along cd the field is zero\nas argued above Along transverse sections bc and ad, the field component\nis zero Thus, these two sections  make no contribution"}, {"Chapter": "1", "sentence_range": "3763-3766", "Text": "Along cd the field is zero\nas argued above Along transverse sections bc and ad, the field component\nis zero Thus, these two sections  make no contribution Let the field along\nab be B"}, {"Chapter": "1", "sentence_range": "3764-3767", "Text": "Along transverse sections bc and ad, the field component\nis zero Thus, these two sections  make no contribution Let the field along\nab be B Thus, the relevant length of the Amperian loop is, L = h"}, {"Chapter": "1", "sentence_range": "3765-3768", "Text": "Thus, these two sections  make no contribution Let the field along\nab be B Thus, the relevant length of the Amperian loop is, L = h Let n be the number of turns per unit length, then the total number\nof turns is  nh"}, {"Chapter": "1", "sentence_range": "3766-3769", "Text": "Let the field along\nab be B Thus, the relevant length of the Amperian loop is, L = h Let n be the number of turns per unit length, then the total number\nof turns is  nh The enclosed current is,  Ie = I (n h), where I is the current\nin the solenoid"}, {"Chapter": "1", "sentence_range": "3767-3770", "Text": "Thus, the relevant length of the Amperian loop is, L = h Let n be the number of turns per unit length, then the total number\nof turns is  nh The enclosed current is,  Ie = I (n h), where I is the current\nin the solenoid From Ampere\u2019s circuital law [Eq"}, {"Chapter": "1", "sentence_range": "3768-3771", "Text": "Let n be the number of turns per unit length, then the total number\nof turns is  nh The enclosed current is,  Ie = I (n h), where I is the current\nin the solenoid From Ampere\u2019s circuital law [Eq 4"}, {"Chapter": "1", "sentence_range": "3769-3772", "Text": "The enclosed current is,  Ie = I (n h), where I is the current\nin the solenoid From Ampere\u2019s circuital law [Eq 4 17 (b)]\nBL =  m0Ie,    B h = m0I (n h)\nB = m0 n I\n(4"}, {"Chapter": "1", "sentence_range": "3770-3773", "Text": "From Ampere\u2019s circuital law [Eq 4 17 (b)]\nBL =  m0Ie,    B h = m0I (n h)\nB = m0 n I\n(4 20)\nThe direction of the field is given by the right-hand rule"}, {"Chapter": "1", "sentence_range": "3771-3774", "Text": "4 17 (b)]\nBL =  m0Ie,    B h = m0I (n h)\nB = m0 n I\n(4 20)\nThe direction of the field is given by the right-hand rule The solenoid\nis commonly used to obtain a uniform magnetic field"}, {"Chapter": "1", "sentence_range": "3772-3775", "Text": "17 (b)]\nBL =  m0Ie,    B h = m0I (n h)\nB = m0 n I\n(4 20)\nThe direction of the field is given by the right-hand rule The solenoid\nis commonly used to obtain a uniform magnetic field We shall see in the\nnext chapter that a large field is possible by inserting a soft iron core\ninside the solenoid"}, {"Chapter": "1", "sentence_range": "3773-3776", "Text": "20)\nThe direction of the field is given by the right-hand rule The solenoid\nis commonly used to obtain a uniform magnetic field We shall see in the\nnext chapter that a large field is possible by inserting a soft iron core\ninside the solenoid Example 4"}, {"Chapter": "1", "sentence_range": "3774-3777", "Text": "The solenoid\nis commonly used to obtain a uniform magnetic field We shall see in the\nnext chapter that a large field is possible by inserting a soft iron core\ninside the solenoid Example 4 9 A solenoid of length 0"}, {"Chapter": "1", "sentence_range": "3775-3778", "Text": "We shall see in the\nnext chapter that a large field is possible by inserting a soft iron core\ninside the solenoid Example 4 9 A solenoid of length 0 5 m has a radius of 1 cm and is\nmade up of 500 turns"}, {"Chapter": "1", "sentence_range": "3776-3779", "Text": "Example 4 9 A solenoid of length 0 5 m has a radius of 1 cm and is\nmade up of 500 turns It carries a current of 5 A"}, {"Chapter": "1", "sentence_range": "3777-3780", "Text": "9 A solenoid of length 0 5 m has a radius of 1 cm and is\nmade up of 500 turns It carries a current of 5 A What is the\nmagnitude of the magnetic field inside the solenoid"}, {"Chapter": "1", "sentence_range": "3778-3781", "Text": "5 m has a radius of 1 cm and is\nmade up of 500 turns It carries a current of 5 A What is the\nmagnitude of the magnetic field inside the solenoid Solution  The number of turns per unit length is,\n500\n1000\nn =0"}, {"Chapter": "1", "sentence_range": "3779-3782", "Text": "It carries a current of 5 A What is the\nmagnitude of the magnetic field inside the solenoid Solution  The number of turns per unit length is,\n500\n1000\nn =0 5\n=\n turns/m\nThe length l = 0"}, {"Chapter": "1", "sentence_range": "3780-3783", "Text": "What is the\nmagnitude of the magnetic field inside the solenoid Solution  The number of turns per unit length is,\n500\n1000\nn =0 5\n=\n turns/m\nThe length l = 0 5 m and radius r = 0"}, {"Chapter": "1", "sentence_range": "3781-3784", "Text": "Solution  The number of turns per unit length is,\n500\n1000\nn =0 5\n=\n turns/m\nThe length l = 0 5 m and radius r = 0 01 m"}, {"Chapter": "1", "sentence_range": "3782-3785", "Text": "5\n=\n turns/m\nThe length l = 0 5 m and radius r = 0 01 m Thus, l/a = 50 i"}, {"Chapter": "1", "sentence_range": "3783-3786", "Text": "5 m and radius r = 0 01 m Thus, l/a = 50 i e"}, {"Chapter": "1", "sentence_range": "3784-3787", "Text": "01 m Thus, l/a = 50 i e , l >> a"}, {"Chapter": "1", "sentence_range": "3785-3788", "Text": "Thus, l/a = 50 i e , l >> a Hence, we can use the long solenoid formula, namely, Eq"}, {"Chapter": "1", "sentence_range": "3786-3789", "Text": "e , l >> a Hence, we can use the long solenoid formula, namely, Eq (4"}, {"Chapter": "1", "sentence_range": "3787-3790", "Text": ", l >> a Hence, we can use the long solenoid formula, namely, Eq (4 20)\nB = m0n I\n   = 4p \u00d7 10\u20137 \u00d7 103 \u00d7 5\n   = 6"}, {"Chapter": "1", "sentence_range": "3788-3791", "Text": "Hence, we can use the long solenoid formula, namely, Eq (4 20)\nB = m0n I\n   = 4p \u00d7 10\u20137 \u00d7 103 \u00d7 5\n   = 6 28 \u00d7 10\u20133 T\nFIGURE 4"}, {"Chapter": "1", "sentence_range": "3789-3792", "Text": "(4 20)\nB = m0n I\n   = 4p \u00d7 10\u20137 \u00d7 103 \u00d7 5\n   = 6 28 \u00d7 10\u20133 T\nFIGURE 4 17 Two long straight\nparallel conductors carrying steady\ncurrents Ia and Ib and separated by a\ndistance d"}, {"Chapter": "1", "sentence_range": "3790-3793", "Text": "20)\nB = m0n I\n   = 4p \u00d7 10\u20137 \u00d7 103 \u00d7 5\n   = 6 28 \u00d7 10\u20133 T\nFIGURE 4 17 Two long straight\nparallel conductors carrying steady\ncurrents Ia and Ib and separated by a\ndistance d Ba is the magnetic field set\nup by conductor \u2018a\u2019 at conductor \u2018b\u2019"}, {"Chapter": "1", "sentence_range": "3791-3794", "Text": "28 \u00d7 10\u20133 T\nFIGURE 4 17 Two long straight\nparallel conductors carrying steady\ncurrents Ia and Ib and separated by a\ndistance d Ba is the magnetic field set\nup by conductor \u2018a\u2019 at conductor \u2018b\u2019 4"}, {"Chapter": "1", "sentence_range": "3792-3795", "Text": "17 Two long straight\nparallel conductors carrying steady\ncurrents Ia and Ib and separated by a\ndistance d Ba is the magnetic field set\nup by conductor \u2018a\u2019 at conductor \u2018b\u2019 4 8 FORCE BETWEEN TWO PARALLEL\nCURRENTS, THE AMPERE\nWe have learnt that there exists a magnetic field due to a\nconductor carrying a current which obeys the Biot-Savart\nlaw"}, {"Chapter": "1", "sentence_range": "3793-3796", "Text": "Ba is the magnetic field set\nup by conductor \u2018a\u2019 at conductor \u2018b\u2019 4 8 FORCE BETWEEN TWO PARALLEL\nCURRENTS, THE AMPERE\nWe have learnt that there exists a magnetic field due to a\nconductor carrying a current which obeys the Biot-Savart\nlaw Further, we have learnt that an external magnetic field\nwill exert a force on a current-carrying conductor"}, {"Chapter": "1", "sentence_range": "3794-3797", "Text": "4 8 FORCE BETWEEN TWO PARALLEL\nCURRENTS, THE AMPERE\nWe have learnt that there exists a magnetic field due to a\nconductor carrying a current which obeys the Biot-Savart\nlaw Further, we have learnt that an external magnetic field\nwill exert a force on a current-carrying conductor This\nfollows from the Lorentz force formula"}, {"Chapter": "1", "sentence_range": "3795-3798", "Text": "8 FORCE BETWEEN TWO PARALLEL\nCURRENTS, THE AMPERE\nWe have learnt that there exists a magnetic field due to a\nconductor carrying a current which obeys the Biot-Savart\nlaw Further, we have learnt that an external magnetic field\nwill exert a force on a current-carrying conductor This\nfollows from the Lorentz force formula Thus, it is logical\nto expect that two current-carrying conductors placed near\neach other will exert (magnetic) forces on each other"}, {"Chapter": "1", "sentence_range": "3796-3799", "Text": "Further, we have learnt that an external magnetic field\nwill exert a force on a current-carrying conductor This\nfollows from the Lorentz force formula Thus, it is logical\nto expect that two current-carrying conductors placed near\neach other will exert (magnetic) forces on each other In\nthe period 1820-25, Ampere studied the nature of this\nmagnetic force and its dependence on the magnitude of\nthe current, on the shape and size of the conductors, as\nwell as, the distances between the conductors"}, {"Chapter": "1", "sentence_range": "3797-3800", "Text": "This\nfollows from the Lorentz force formula Thus, it is logical\nto expect that two current-carrying conductors placed near\neach other will exert (magnetic) forces on each other In\nthe period 1820-25, Ampere studied the nature of this\nmagnetic force and its dependence on the magnitude of\nthe current, on the shape and size of the conductors, as\nwell as, the distances between the conductors In this\nsection, we shall take the simple example of two parallel\ncurrent- carrying conductors, which will perhaps help us\nto appreciate Ampere\u2019s painstaking work"}, {"Chapter": "1", "sentence_range": "3798-3801", "Text": "Thus, it is logical\nto expect that two current-carrying conductors placed near\neach other will exert (magnetic) forces on each other In\nthe period 1820-25, Ampere studied the nature of this\nmagnetic force and its dependence on the magnitude of\nthe current, on the shape and size of the conductors, as\nwell as, the distances between the conductors In this\nsection, we shall take the simple example of two parallel\ncurrent- carrying conductors, which will perhaps help us\nto appreciate Ampere\u2019s painstaking work Rationalised 2023-24\n123\nMoving Charges and\nMagnetism\nFigure 4"}, {"Chapter": "1", "sentence_range": "3799-3802", "Text": "In\nthe period 1820-25, Ampere studied the nature of this\nmagnetic force and its dependence on the magnitude of\nthe current, on the shape and size of the conductors, as\nwell as, the distances between the conductors In this\nsection, we shall take the simple example of two parallel\ncurrent- carrying conductors, which will perhaps help us\nto appreciate Ampere\u2019s painstaking work Rationalised 2023-24\n123\nMoving Charges and\nMagnetism\nFigure 4 17 shows two long parallel conductors a and b separated\nby a distance d and carrying (parallel) currents Ia and Ib, respectively"}, {"Chapter": "1", "sentence_range": "3800-3803", "Text": "In this\nsection, we shall take the simple example of two parallel\ncurrent- carrying conductors, which will perhaps help us\nto appreciate Ampere\u2019s painstaking work Rationalised 2023-24\n123\nMoving Charges and\nMagnetism\nFigure 4 17 shows two long parallel conductors a and b separated\nby a distance d and carrying (parallel) currents Ia and Ib, respectively The conductor \u2018a\u2019 produces,  the same magnetic field Ba at all points\nalong the conductor \u2018b\u2019"}, {"Chapter": "1", "sentence_range": "3801-3804", "Text": "Rationalised 2023-24\n123\nMoving Charges and\nMagnetism\nFigure 4 17 shows two long parallel conductors a and b separated\nby a distance d and carrying (parallel) currents Ia and Ib, respectively The conductor \u2018a\u2019 produces,  the same magnetic field Ba at all points\nalong the conductor \u2018b\u2019 The right-hand rule tells us that the direction of\nthis field is downwards (when the conductors are placed horizontally)"}, {"Chapter": "1", "sentence_range": "3802-3805", "Text": "17 shows two long parallel conductors a and b separated\nby a distance d and carrying (parallel) currents Ia and Ib, respectively The conductor \u2018a\u2019 produces,  the same magnetic field Ba at all points\nalong the conductor \u2018b\u2019 The right-hand rule tells us that the direction of\nthis field is downwards (when the conductors are placed horizontally) Its magnitude is given by Eq"}, {"Chapter": "1", "sentence_range": "3803-3806", "Text": "The conductor \u2018a\u2019 produces,  the same magnetic field Ba at all points\nalong the conductor \u2018b\u2019 The right-hand rule tells us that the direction of\nthis field is downwards (when the conductors are placed horizontally) Its magnitude is given by Eq [4"}, {"Chapter": "1", "sentence_range": "3804-3807", "Text": "The right-hand rule tells us that the direction of\nthis field is downwards (when the conductors are placed horizontally) Its magnitude is given by Eq [4 19(a)] or from Ampere\u2019s circuital law,\n0\n2\na\na\nI\nB\nd\n=\u00b5\n\u03c0\nThe conductor \u2018b\u2019 carrying a current Ib will experience a sideways\nforce due to the  field Ba"}, {"Chapter": "1", "sentence_range": "3805-3808", "Text": "Its magnitude is given by Eq [4 19(a)] or from Ampere\u2019s circuital law,\n0\n2\na\na\nI\nB\nd\n=\u00b5\n\u03c0\nThe conductor \u2018b\u2019 carrying a current Ib will experience a sideways\nforce due to the  field Ba The direction of this force is towards the\nconductor \u2018a\u2019 (Verify this)"}, {"Chapter": "1", "sentence_range": "3806-3809", "Text": "[4 19(a)] or from Ampere\u2019s circuital law,\n0\n2\na\na\nI\nB\nd\n=\u00b5\n\u03c0\nThe conductor \u2018b\u2019 carrying a current Ib will experience a sideways\nforce due to the  field Ba The direction of this force is towards the\nconductor \u2018a\u2019 (Verify this) We label this force as Fba, the force on a\nsegment L of \u2018b\u2019 due to \u2018a\u2019"}, {"Chapter": "1", "sentence_range": "3807-3810", "Text": "19(a)] or from Ampere\u2019s circuital law,\n0\n2\na\na\nI\nB\nd\n=\u00b5\n\u03c0\nThe conductor \u2018b\u2019 carrying a current Ib will experience a sideways\nforce due to the  field Ba The direction of this force is towards the\nconductor \u2018a\u2019 (Verify this) We label this force as Fba, the force on a\nsegment L of \u2018b\u2019 due to \u2018a\u2019 The magnitude of this force is given by\nEq"}, {"Chapter": "1", "sentence_range": "3808-3811", "Text": "The direction of this force is towards the\nconductor \u2018a\u2019 (Verify this) We label this force as Fba, the force on a\nsegment L of \u2018b\u2019 due to \u2018a\u2019 The magnitude of this force is given by\nEq (4"}, {"Chapter": "1", "sentence_range": "3809-3812", "Text": "We label this force as Fba, the force on a\nsegment L of \u2018b\u2019 due to \u2018a\u2019 The magnitude of this force is given by\nEq (4 4),\nFba = Ib L Ba\n      \n0\n2\na\nI Ib\nL\nd\n=\u00b5\n\u03c0\n(4"}, {"Chapter": "1", "sentence_range": "3810-3813", "Text": "The magnitude of this force is given by\nEq (4 4),\nFba = Ib L Ba\n      \n0\n2\na\nI Ib\nL\nd\n=\u00b5\n\u03c0\n(4 23)\nIt is of course possible to compute the force on \u2018a\u2019 due to \u2018b\u2019"}, {"Chapter": "1", "sentence_range": "3811-3814", "Text": "(4 4),\nFba = Ib L Ba\n      \n0\n2\na\nI Ib\nL\nd\n=\u00b5\n\u03c0\n(4 23)\nIt is of course possible to compute the force on \u2018a\u2019 due to \u2018b\u2019 From\nconsiderations similar to above we can find the force Fab, on a segment of\nlength L of \u2018a\u2019 due to the current in \u2018b\u2019"}, {"Chapter": "1", "sentence_range": "3812-3815", "Text": "4),\nFba = Ib L Ba\n      \n0\n2\na\nI Ib\nL\nd\n=\u00b5\n\u03c0\n(4 23)\nIt is of course possible to compute the force on \u2018a\u2019 due to \u2018b\u2019 From\nconsiderations similar to above we can find the force Fab, on a segment of\nlength L of \u2018a\u2019 due to the current in \u2018b\u2019 It  is equal in magnitude to Fba,\nand directed towards \u2018b\u2019"}, {"Chapter": "1", "sentence_range": "3813-3816", "Text": "23)\nIt is of course possible to compute the force on \u2018a\u2019 due to \u2018b\u2019 From\nconsiderations similar to above we can find the force Fab, on a segment of\nlength L of \u2018a\u2019 due to the current in \u2018b\u2019 It  is equal in magnitude to Fba,\nand directed towards \u2018b\u2019 Thus,\nFba = \u2013Fab\n(4"}, {"Chapter": "1", "sentence_range": "3814-3817", "Text": "From\nconsiderations similar to above we can find the force Fab, on a segment of\nlength L of \u2018a\u2019 due to the current in \u2018b\u2019 It  is equal in magnitude to Fba,\nand directed towards \u2018b\u2019 Thus,\nFba = \u2013Fab\n(4 24)\nNote that this is consistent with Newton\u2019s third Law"}, {"Chapter": "1", "sentence_range": "3815-3818", "Text": "It  is equal in magnitude to Fba,\nand directed towards \u2018b\u2019 Thus,\nFba = \u2013Fab\n(4 24)\nNote that this is consistent with Newton\u2019s third Law Thus, at least for\nparallel conductors and steady currents, we have shown that the\nBiot-Savart law and the Lorentz force yield results in accordance with\nNewton\u2019s third Law*"}, {"Chapter": "1", "sentence_range": "3816-3819", "Text": "Thus,\nFba = \u2013Fab\n(4 24)\nNote that this is consistent with Newton\u2019s third Law Thus, at least for\nparallel conductors and steady currents, we have shown that the\nBiot-Savart law and the Lorentz force yield results in accordance with\nNewton\u2019s third Law* We have seen from above that currents flowing in the same direction\nattract each other"}, {"Chapter": "1", "sentence_range": "3817-3820", "Text": "24)\nNote that this is consistent with Newton\u2019s third Law Thus, at least for\nparallel conductors and steady currents, we have shown that the\nBiot-Savart law and the Lorentz force yield results in accordance with\nNewton\u2019s third Law* We have seen from above that currents flowing in the same direction\nattract each other One can show that oppositely directed currents repel\neach other"}, {"Chapter": "1", "sentence_range": "3818-3821", "Text": "Thus, at least for\nparallel conductors and steady currents, we have shown that the\nBiot-Savart law and the Lorentz force yield results in accordance with\nNewton\u2019s third Law* We have seen from above that currents flowing in the same direction\nattract each other One can show that oppositely directed currents repel\neach other Thus,\nParallel currents attract, and antiparallel currents repel"}, {"Chapter": "1", "sentence_range": "3819-3822", "Text": "We have seen from above that currents flowing in the same direction\nattract each other One can show that oppositely directed currents repel\neach other Thus,\nParallel currents attract, and antiparallel currents repel This rule is the opposite of what we find in electrostatics"}, {"Chapter": "1", "sentence_range": "3820-3823", "Text": "One can show that oppositely directed currents repel\neach other Thus,\nParallel currents attract, and antiparallel currents repel This rule is the opposite of what we find in electrostatics Like (same\nsign) charges repel each other, but like (parallel) currents attract each\nother"}, {"Chapter": "1", "sentence_range": "3821-3824", "Text": "Thus,\nParallel currents attract, and antiparallel currents repel This rule is the opposite of what we find in electrostatics Like (same\nsign) charges repel each other, but like (parallel) currents attract each\nother Let fba represent the magnitude of the force Fba per unit length"}, {"Chapter": "1", "sentence_range": "3822-3825", "Text": "This rule is the opposite of what we find in electrostatics Like (same\nsign) charges repel each other, but like (parallel) currents attract each\nother Let fba represent the magnitude of the force Fba per unit length Then,\nfrom Eq"}, {"Chapter": "1", "sentence_range": "3823-3826", "Text": "Like (same\nsign) charges repel each other, but like (parallel) currents attract each\nother Let fba represent the magnitude of the force Fba per unit length Then,\nfrom Eq (4"}, {"Chapter": "1", "sentence_range": "3824-3827", "Text": "Let fba represent the magnitude of the force Fba per unit length Then,\nfrom Eq (4 23),\n\u03c0\n0\n2\na\nb\nba\nI I\nf\nd\n=\u00b5\n(4"}, {"Chapter": "1", "sentence_range": "3825-3828", "Text": "Then,\nfrom Eq (4 23),\n\u03c0\n0\n2\na\nb\nba\nI I\nf\nd\n=\u00b5\n(4 25)\nThe above expression is used to define the ampere (A), which is one of\nthe seven SI base units"}, {"Chapter": "1", "sentence_range": "3826-3829", "Text": "(4 23),\n\u03c0\n0\n2\na\nb\nba\nI I\nf\nd\n=\u00b5\n(4 25)\nThe above expression is used to define the ampere (A), which is one of\nthe seven SI base units *\nIt turns out that when we have time-dependent currents and/or charges in\nmotion, Newton\u2019s third law may not hold for forces between charges and/or\nconductors"}, {"Chapter": "1", "sentence_range": "3827-3830", "Text": "23),\n\u03c0\n0\n2\na\nb\nba\nI I\nf\nd\n=\u00b5\n(4 25)\nThe above expression is used to define the ampere (A), which is one of\nthe seven SI base units *\nIt turns out that when we have time-dependent currents and/or charges in\nmotion, Newton\u2019s third law may not hold for forces between charges and/or\nconductors An essential consequence of the Newton\u2019s third law in mechanics\nis conservation of momentum of an isolated system"}, {"Chapter": "1", "sentence_range": "3828-3831", "Text": "25)\nThe above expression is used to define the ampere (A), which is one of\nthe seven SI base units *\nIt turns out that when we have time-dependent currents and/or charges in\nmotion, Newton\u2019s third law may not hold for forces between charges and/or\nconductors An essential consequence of the Newton\u2019s third law in mechanics\nis conservation of momentum of an isolated system This, however, holds even\nfor the case of time-dependent situations with electromagnetic fields, provided\nthe momentum carried by fields is also taken into account"}, {"Chapter": "1", "sentence_range": "3829-3832", "Text": "*\nIt turns out that when we have time-dependent currents and/or charges in\nmotion, Newton\u2019s third law may not hold for forces between charges and/or\nconductors An essential consequence of the Newton\u2019s third law in mechanics\nis conservation of momentum of an isolated system This, however, holds even\nfor the case of time-dependent situations with electromagnetic fields, provided\nthe momentum carried by fields is also taken into account Rationalised 2023-24\nPhysics\n124\n EXAMPLE 4"}, {"Chapter": "1", "sentence_range": "3830-3833", "Text": "An essential consequence of the Newton\u2019s third law in mechanics\nis conservation of momentum of an isolated system This, however, holds even\nfor the case of time-dependent situations with electromagnetic fields, provided\nthe momentum carried by fields is also taken into account Rationalised 2023-24\nPhysics\n124\n EXAMPLE 4 10\nThe ampere is the value of that steady current which, when maintained\nin each of the two very long, straight, parallel conductors of negligible\ncross-section, and placed one metre apart in vacuum, would produce\non  each of these conductors a force equal to  2 \u00d7 10\u20137 newtons per metre\nof length"}, {"Chapter": "1", "sentence_range": "3831-3834", "Text": "This, however, holds even\nfor the case of time-dependent situations with electromagnetic fields, provided\nthe momentum carried by fields is also taken into account Rationalised 2023-24\nPhysics\n124\n EXAMPLE 4 10\nThe ampere is the value of that steady current which, when maintained\nin each of the two very long, straight, parallel conductors of negligible\ncross-section, and placed one metre apart in vacuum, would produce\non  each of these conductors a force equal to  2 \u00d7 10\u20137 newtons per metre\nof length This definition of the ampere was adopted in 1946"}, {"Chapter": "1", "sentence_range": "3832-3835", "Text": "Rationalised 2023-24\nPhysics\n124\n EXAMPLE 4 10\nThe ampere is the value of that steady current which, when maintained\nin each of the two very long, straight, parallel conductors of negligible\ncross-section, and placed one metre apart in vacuum, would produce\non  each of these conductors a force equal to  2 \u00d7 10\u20137 newtons per metre\nof length This definition of the ampere was adopted in 1946 It is a theoretical\ndefinition"}, {"Chapter": "1", "sentence_range": "3833-3836", "Text": "10\nThe ampere is the value of that steady current which, when maintained\nin each of the two very long, straight, parallel conductors of negligible\ncross-section, and placed one metre apart in vacuum, would produce\non  each of these conductors a force equal to  2 \u00d7 10\u20137 newtons per metre\nof length This definition of the ampere was adopted in 1946 It is a theoretical\ndefinition In practice, one must eliminate the effect of the earth\u2019s magnetic\nfield and substitute very long wires by multiturn coils of appropriate\ngeometries"}, {"Chapter": "1", "sentence_range": "3834-3837", "Text": "This definition of the ampere was adopted in 1946 It is a theoretical\ndefinition In practice, one must eliminate the effect of the earth\u2019s magnetic\nfield and substitute very long wires by multiturn coils of appropriate\ngeometries An instrument called the current balance is used to measure\nthis mechanical force"}, {"Chapter": "1", "sentence_range": "3835-3838", "Text": "It is a theoretical\ndefinition In practice, one must eliminate the effect of the earth\u2019s magnetic\nfield and substitute very long wires by multiturn coils of appropriate\ngeometries An instrument called the current balance is used to measure\nthis mechanical force The SI unit of charge, namely, the coulomb, can now be defined in\nterms of the ampere"}, {"Chapter": "1", "sentence_range": "3836-3839", "Text": "In practice, one must eliminate the effect of the earth\u2019s magnetic\nfield and substitute very long wires by multiturn coils of appropriate\ngeometries An instrument called the current balance is used to measure\nthis mechanical force The SI unit of charge, namely, the coulomb, can now be defined in\nterms of the ampere When a steady current of 1A is set up in a conductor, the quantity of\ncharge that flows through its cross-section in 1s is one coulomb (1C)"}, {"Chapter": "1", "sentence_range": "3837-3840", "Text": "An instrument called the current balance is used to measure\nthis mechanical force The SI unit of charge, namely, the coulomb, can now be defined in\nterms of the ampere When a steady current of 1A is set up in a conductor, the quantity of\ncharge that flows through its cross-section in 1s is one coulomb (1C) Example 4"}, {"Chapter": "1", "sentence_range": "3838-3841", "Text": "The SI unit of charge, namely, the coulomb, can now be defined in\nterms of the ampere When a steady current of 1A is set up in a conductor, the quantity of\ncharge that flows through its cross-section in 1s is one coulomb (1C) Example 4 10 The horizontal component of the earth\u2019s magnetic field\nat a certain place is 3"}, {"Chapter": "1", "sentence_range": "3839-3842", "Text": "When a steady current of 1A is set up in a conductor, the quantity of\ncharge that flows through its cross-section in 1s is one coulomb (1C) Example 4 10 The horizontal component of the earth\u2019s magnetic field\nat a certain place is 3 0 \u00d710\u20135 T and the direction of the field is from\nthe geographic south to the geographic north"}, {"Chapter": "1", "sentence_range": "3840-3843", "Text": "Example 4 10 The horizontal component of the earth\u2019s magnetic field\nat a certain place is 3 0 \u00d710\u20135 T and the direction of the field is from\nthe geographic south to the geographic north A very long straight\nconductor is carrying a steady current of 1A"}, {"Chapter": "1", "sentence_range": "3841-3844", "Text": "10 The horizontal component of the earth\u2019s magnetic field\nat a certain place is 3 0 \u00d710\u20135 T and the direction of the field is from\nthe geographic south to the geographic north A very long straight\nconductor is carrying a steady current of 1A What is the force per\nunit length on it when it is placed on a horizontal table and the\ndirection of the current is (a) east to west; (b) south to north"}, {"Chapter": "1", "sentence_range": "3842-3845", "Text": "0 \u00d710\u20135 T and the direction of the field is from\nthe geographic south to the geographic north A very long straight\nconductor is carrying a steady current of 1A What is the force per\nunit length on it when it is placed on a horizontal table and the\ndirection of the current is (a) east to west; (b) south to north Solution  F = Il \u00d7 B\nF = IlB sinq\nThe force per unit length is\nf = F/l = I B sinq\n(a) When the current is flowing from east to west,\nq = 90\u00b0\nHence,\nf = I B\n  = 1 \u00d7 3 \u00d7 10\u20135 = 3 \u00d7 10\u20135 N m\u20131\nThis is larger than the value 2\u00d710\u20137 Nm\u20131 quoted in the definition\nof the ampere"}, {"Chapter": "1", "sentence_range": "3843-3846", "Text": "A very long straight\nconductor is carrying a steady current of 1A What is the force per\nunit length on it when it is placed on a horizontal table and the\ndirection of the current is (a) east to west; (b) south to north Solution  F = Il \u00d7 B\nF = IlB sinq\nThe force per unit length is\nf = F/l = I B sinq\n(a) When the current is flowing from east to west,\nq = 90\u00b0\nHence,\nf = I B\n  = 1 \u00d7 3 \u00d7 10\u20135 = 3 \u00d7 10\u20135 N m\u20131\nThis is larger than the value 2\u00d710\u20137 Nm\u20131 quoted in the definition\nof the ampere Hence it is important to eliminate the effect of the\nearth\u2019s magnetic field and other stray fields while standardising\nthe ampere"}, {"Chapter": "1", "sentence_range": "3844-3847", "Text": "What is the force per\nunit length on it when it is placed on a horizontal table and the\ndirection of the current is (a) east to west; (b) south to north Solution  F = Il \u00d7 B\nF = IlB sinq\nThe force per unit length is\nf = F/l = I B sinq\n(a) When the current is flowing from east to west,\nq = 90\u00b0\nHence,\nf = I B\n  = 1 \u00d7 3 \u00d7 10\u20135 = 3 \u00d7 10\u20135 N m\u20131\nThis is larger than the value 2\u00d710\u20137 Nm\u20131 quoted in the definition\nof the ampere Hence it is important to eliminate the effect of the\nearth\u2019s magnetic field and other stray fields while standardising\nthe ampere The direction of the force is downwards"}, {"Chapter": "1", "sentence_range": "3845-3848", "Text": "Solution  F = Il \u00d7 B\nF = IlB sinq\nThe force per unit length is\nf = F/l = I B sinq\n(a) When the current is flowing from east to west,\nq = 90\u00b0\nHence,\nf = I B\n  = 1 \u00d7 3 \u00d7 10\u20135 = 3 \u00d7 10\u20135 N m\u20131\nThis is larger than the value 2\u00d710\u20137 Nm\u20131 quoted in the definition\nof the ampere Hence it is important to eliminate the effect of the\nearth\u2019s magnetic field and other stray fields while standardising\nthe ampere The direction of the force is downwards This direction may be\nobtained by the directional property of cross product of vectors"}, {"Chapter": "1", "sentence_range": "3846-3849", "Text": "Hence it is important to eliminate the effect of the\nearth\u2019s magnetic field and other stray fields while standardising\nthe ampere The direction of the force is downwards This direction may be\nobtained by the directional property of cross product of vectors (b) When the current is flowing from south to north,\nq = 0o\nf = 0\nHence there is no force on the conductor"}, {"Chapter": "1", "sentence_range": "3847-3850", "Text": "The direction of the force is downwards This direction may be\nobtained by the directional property of cross product of vectors (b) When the current is flowing from south to north,\nq = 0o\nf = 0\nHence there is no force on the conductor 4"}, {"Chapter": "1", "sentence_range": "3848-3851", "Text": "This direction may be\nobtained by the directional property of cross product of vectors (b) When the current is flowing from south to north,\nq = 0o\nf = 0\nHence there is no force on the conductor 4 9  TORQUE ON CURRENT LOOP, MAGNETIC DIPOLE\n4"}, {"Chapter": "1", "sentence_range": "3849-3852", "Text": "(b) When the current is flowing from south to north,\nq = 0o\nf = 0\nHence there is no force on the conductor 4 9  TORQUE ON CURRENT LOOP, MAGNETIC DIPOLE\n4 9"}, {"Chapter": "1", "sentence_range": "3850-3853", "Text": "4 9  TORQUE ON CURRENT LOOP, MAGNETIC DIPOLE\n4 9 1\nTorque on a rectangular current loop in a uniform\nmagnetic field\nWe now show that a rectangular loop carrying a steady current I and\nplaced in a uniform magnetic field experiences a torque"}, {"Chapter": "1", "sentence_range": "3851-3854", "Text": "9  TORQUE ON CURRENT LOOP, MAGNETIC DIPOLE\n4 9 1\nTorque on a rectangular current loop in a uniform\nmagnetic field\nWe now show that a rectangular loop carrying a steady current I and\nplaced in a uniform magnetic field experiences a torque It does not\nexperience a net force"}, {"Chapter": "1", "sentence_range": "3852-3855", "Text": "9 1\nTorque on a rectangular current loop in a uniform\nmagnetic field\nWe now show that a rectangular loop carrying a steady current I and\nplaced in a uniform magnetic field experiences a torque It does not\nexperience a net force This behaviour is analogous to that of electric\ndipole in a uniform electric field (Section 1"}, {"Chapter": "1", "sentence_range": "3853-3856", "Text": "1\nTorque on a rectangular current loop in a uniform\nmagnetic field\nWe now show that a rectangular loop carrying a steady current I and\nplaced in a uniform magnetic field experiences a torque It does not\nexperience a net force This behaviour is analogous to that of electric\ndipole in a uniform electric field (Section 1 12)"}, {"Chapter": "1", "sentence_range": "3854-3857", "Text": "It does not\nexperience a net force This behaviour is analogous to that of electric\ndipole in a uniform electric field (Section 1 12) Rationalised 2023-24\n125\nMoving Charges and\nMagnetism\nWe first consider the simple case when the\nrectangular loop is placed such that the uniform\nmagnetic field B is in the plane of the loop"}, {"Chapter": "1", "sentence_range": "3855-3858", "Text": "This behaviour is analogous to that of electric\ndipole in a uniform electric field (Section 1 12) Rationalised 2023-24\n125\nMoving Charges and\nMagnetism\nWe first consider the simple case when the\nrectangular loop is placed such that the uniform\nmagnetic field B is in the plane of the loop This is\nillustrated in Fig"}, {"Chapter": "1", "sentence_range": "3856-3859", "Text": "12) Rationalised 2023-24\n125\nMoving Charges and\nMagnetism\nWe first consider the simple case when the\nrectangular loop is placed such that the uniform\nmagnetic field B is in the plane of the loop This is\nillustrated in Fig 4"}, {"Chapter": "1", "sentence_range": "3857-3860", "Text": "Rationalised 2023-24\n125\nMoving Charges and\nMagnetism\nWe first consider the simple case when the\nrectangular loop is placed such that the uniform\nmagnetic field B is in the plane of the loop This is\nillustrated in Fig 4 18(a)"}, {"Chapter": "1", "sentence_range": "3858-3861", "Text": "This is\nillustrated in Fig 4 18(a) The field exerts no force on the two arms AD and BC\nof the loop"}, {"Chapter": "1", "sentence_range": "3859-3862", "Text": "4 18(a) The field exerts no force on the two arms AD and BC\nof the loop It is perpendicular to the arm AB of the loop\nand exerts a force F1 on it which is directed into the\nplane of the loop"}, {"Chapter": "1", "sentence_range": "3860-3863", "Text": "18(a) The field exerts no force on the two arms AD and BC\nof the loop It is perpendicular to the arm AB of the loop\nand exerts a force F1 on it which is directed into the\nplane of the loop Its magnitude is,\nF1 = I b B\nSimilarly, it exerts a force F2 on the arm CD and F2\nis directed out of the plane of the paper"}, {"Chapter": "1", "sentence_range": "3861-3864", "Text": "The field exerts no force on the two arms AD and BC\nof the loop It is perpendicular to the arm AB of the loop\nand exerts a force F1 on it which is directed into the\nplane of the loop Its magnitude is,\nF1 = I b B\nSimilarly, it exerts a force F2 on the arm CD and F2\nis directed out of the plane of the paper F2 = I b B = F1\nThus, the net force on the loop is zero"}, {"Chapter": "1", "sentence_range": "3862-3865", "Text": "It is perpendicular to the arm AB of the loop\nand exerts a force F1 on it which is directed into the\nplane of the loop Its magnitude is,\nF1 = I b B\nSimilarly, it exerts a force F2 on the arm CD and F2\nis directed out of the plane of the paper F2 = I b B = F1\nThus, the net force on the loop is zero There is a\ntorque on the loop due to the pair of forces F1 and F2"}, {"Chapter": "1", "sentence_range": "3863-3866", "Text": "Its magnitude is,\nF1 = I b B\nSimilarly, it exerts a force F2 on the arm CD and F2\nis directed out of the plane of the paper F2 = I b B = F1\nThus, the net force on the loop is zero There is a\ntorque on the loop due to the pair of forces F1 and F2 Figure 4"}, {"Chapter": "1", "sentence_range": "3864-3867", "Text": "F2 = I b B = F1\nThus, the net force on the loop is zero There is a\ntorque on the loop due to the pair of forces F1 and F2 Figure 4 18(b) shows a view of the loop from the AD\nend"}, {"Chapter": "1", "sentence_range": "3865-3868", "Text": "There is a\ntorque on the loop due to the pair of forces F1 and F2 Figure 4 18(b) shows a view of the loop from the AD\nend It shows that the torque on the loop tends to rotate\nit anticlockwise"}, {"Chapter": "1", "sentence_range": "3866-3869", "Text": "Figure 4 18(b) shows a view of the loop from the AD\nend It shows that the torque on the loop tends to rotate\nit anticlockwise This torque is (in magnitude),\n1\n2\n2\n2\na\na\nF\nF\n\u03c4 =\n+\n(\n)\n2\n2\na\na\nIbB\nIbB\nI ab B\n=\n+\n=\n    = I A B\n(4"}, {"Chapter": "1", "sentence_range": "3867-3870", "Text": "18(b) shows a view of the loop from the AD\nend It shows that the torque on the loop tends to rotate\nit anticlockwise This torque is (in magnitude),\n1\n2\n2\n2\na\na\nF\nF\n\u03c4 =\n+\n(\n)\n2\n2\na\na\nIbB\nIbB\nI ab B\n=\n+\n=\n    = I A B\n(4 26)\nwhere A = ab is the area of the rectangle"}, {"Chapter": "1", "sentence_range": "3868-3871", "Text": "It shows that the torque on the loop tends to rotate\nit anticlockwise This torque is (in magnitude),\n1\n2\n2\n2\na\na\nF\nF\n\u03c4 =\n+\n(\n)\n2\n2\na\na\nIbB\nIbB\nI ab B\n=\n+\n=\n    = I A B\n(4 26)\nwhere A = ab is the area of the rectangle We next consider the case when the plane of the loop,\nis not along the magnetic field, but makes an angle with\nit"}, {"Chapter": "1", "sentence_range": "3869-3872", "Text": "This torque is (in magnitude),\n1\n2\n2\n2\na\na\nF\nF\n\u03c4 =\n+\n(\n)\n2\n2\na\na\nIbB\nIbB\nI ab B\n=\n+\n=\n    = I A B\n(4 26)\nwhere A = ab is the area of the rectangle We next consider the case when the plane of the loop,\nis not along the magnetic field, but makes an angle with\nit We take the angle between the field and the normal to\nthe coil to be angle q (The previous case corresponds to\nq = p/2)"}, {"Chapter": "1", "sentence_range": "3870-3873", "Text": "26)\nwhere A = ab is the area of the rectangle We next consider the case when the plane of the loop,\nis not along the magnetic field, but makes an angle with\nit We take the angle between the field and the normal to\nthe coil to be angle q (The previous case corresponds to\nq = p/2) Figure 4"}, {"Chapter": "1", "sentence_range": "3871-3874", "Text": "We next consider the case when the plane of the loop,\nis not along the magnetic field, but makes an angle with\nit We take the angle between the field and the normal to\nthe coil to be angle q (The previous case corresponds to\nq = p/2) Figure 4 19 illustrates this general case"}, {"Chapter": "1", "sentence_range": "3872-3875", "Text": "We take the angle between the field and the normal to\nthe coil to be angle q (The previous case corresponds to\nq = p/2) Figure 4 19 illustrates this general case The forces on the arms BC and DA are equal, opposite, and act along\nthe axis of the coil, which connects the centres of mass of BC and DA"}, {"Chapter": "1", "sentence_range": "3873-3876", "Text": "Figure 4 19 illustrates this general case The forces on the arms BC and DA are equal, opposite, and act along\nthe axis of the coil, which connects the centres of mass of BC and DA Being collinear along the axis they cancel each other, resulting in no net\nforce or torque"}, {"Chapter": "1", "sentence_range": "3874-3877", "Text": "19 illustrates this general case The forces on the arms BC and DA are equal, opposite, and act along\nthe axis of the coil, which connects the centres of mass of BC and DA Being collinear along the axis they cancel each other, resulting in no net\nforce or torque The forces on arms AB and CD are F1 and F2"}, {"Chapter": "1", "sentence_range": "3875-3878", "Text": "The forces on the arms BC and DA are equal, opposite, and act along\nthe axis of the coil, which connects the centres of mass of BC and DA Being collinear along the axis they cancel each other, resulting in no net\nforce or torque The forces on arms AB and CD are F1 and F2 They too\nare equal and opposite, with magnitude,\nF1 = F2 = I b B\nBut they are not collinear"}, {"Chapter": "1", "sentence_range": "3876-3879", "Text": "Being collinear along the axis they cancel each other, resulting in no net\nforce or torque The forces on arms AB and CD are F1 and F2 They too\nare equal and opposite, with magnitude,\nF1 = F2 = I b B\nBut they are not collinear This results in a couple as before"}, {"Chapter": "1", "sentence_range": "3877-3880", "Text": "The forces on arms AB and CD are F1 and F2 They too\nare equal and opposite, with magnitude,\nF1 = F2 = I b B\nBut they are not collinear This results in a couple as before The\ntorque is, however, less than the earlier case when plane of loop was\nalong the magnetic field"}, {"Chapter": "1", "sentence_range": "3878-3881", "Text": "They too\nare equal and opposite, with magnitude,\nF1 = F2 = I b B\nBut they are not collinear This results in a couple as before The\ntorque is, however, less than the earlier case when plane of loop was\nalong the magnetic field This is because the perpendicular distance\nbetween the forces of the couple has decreased"}, {"Chapter": "1", "sentence_range": "3879-3882", "Text": "This results in a couple as before The\ntorque is, however, less than the earlier case when plane of loop was\nalong the magnetic field This is because the perpendicular distance\nbetween the forces of the couple has decreased Figure 4"}, {"Chapter": "1", "sentence_range": "3880-3883", "Text": "The\ntorque is, however, less than the earlier case when plane of loop was\nalong the magnetic field This is because the perpendicular distance\nbetween the forces of the couple has decreased Figure 4 19(b) is a view\nof the arrangement from the AD end and it illustrates these two forces\nconstituting a couple"}, {"Chapter": "1", "sentence_range": "3881-3884", "Text": "This is because the perpendicular distance\nbetween the forces of the couple has decreased Figure 4 19(b) is a view\nof the arrangement from the AD end and it illustrates these two forces\nconstituting a couple The magnitude of the torque on the loop is,\n1\n2\nsin\nsin\n2\n2\na\na\nF\nF\n\u03c4\n\u03b8\n\u03b8\n=\n+\n= I ab B sin q\n   = I A B sin q\n(4"}, {"Chapter": "1", "sentence_range": "3882-3885", "Text": "Figure 4 19(b) is a view\nof the arrangement from the AD end and it illustrates these two forces\nconstituting a couple The magnitude of the torque on the loop is,\n1\n2\nsin\nsin\n2\n2\na\na\nF\nF\n\u03c4\n\u03b8\n\u03b8\n=\n+\n= I ab B sin q\n   = I A B sin q\n(4 27)\nFIGURE 4"}, {"Chapter": "1", "sentence_range": "3883-3886", "Text": "19(b) is a view\nof the arrangement from the AD end and it illustrates these two forces\nconstituting a couple The magnitude of the torque on the loop is,\n1\n2\nsin\nsin\n2\n2\na\na\nF\nF\n\u03c4\n\u03b8\n\u03b8\n=\n+\n= I ab B sin q\n   = I A B sin q\n(4 27)\nFIGURE 4 18 (a) A rectangular\ncurrent-carrying coil in uniform\nmagnetic field"}, {"Chapter": "1", "sentence_range": "3884-3887", "Text": "The magnitude of the torque on the loop is,\n1\n2\nsin\nsin\n2\n2\na\na\nF\nF\n\u03c4\n\u03b8\n\u03b8\n=\n+\n= I ab B sin q\n   = I A B sin q\n(4 27)\nFIGURE 4 18 (a) A rectangular\ncurrent-carrying coil in uniform\nmagnetic field The magnetic moment\nm points downwards"}, {"Chapter": "1", "sentence_range": "3885-3888", "Text": "27)\nFIGURE 4 18 (a) A rectangular\ncurrent-carrying coil in uniform\nmagnetic field The magnetic moment\nm points downwards The torque ttttt is\nalong the axis and tends to rotate the\ncoil anticlockwise"}, {"Chapter": "1", "sentence_range": "3886-3889", "Text": "18 (a) A rectangular\ncurrent-carrying coil in uniform\nmagnetic field The magnetic moment\nm points downwards The torque ttttt is\nalong the axis and tends to rotate the\ncoil anticlockwise (b) The couple\nacting on the coil"}, {"Chapter": "1", "sentence_range": "3887-3890", "Text": "The magnetic moment\nm points downwards The torque ttttt is\nalong the axis and tends to rotate the\ncoil anticlockwise (b) The couple\nacting on the coil Rationalised 2023-24\nPhysics\n126\n EXAMPLE 4"}, {"Chapter": "1", "sentence_range": "3888-3891", "Text": "The torque ttttt is\nalong the axis and tends to rotate the\ncoil anticlockwise (b) The couple\nacting on the coil Rationalised 2023-24\nPhysics\n126\n EXAMPLE 4 11\nAs q \u00e0 0, the perpendicular distance between\nthe forces of the couple also approaches zero"}, {"Chapter": "1", "sentence_range": "3889-3892", "Text": "(b) The couple\nacting on the coil Rationalised 2023-24\nPhysics\n126\n EXAMPLE 4 11\nAs q \u00e0 0, the perpendicular distance between\nthe forces of the couple also approaches zero This\nmakes the forces collinear and the net force and\ntorque zero"}, {"Chapter": "1", "sentence_range": "3890-3893", "Text": "Rationalised 2023-24\nPhysics\n126\n EXAMPLE 4 11\nAs q \u00e0 0, the perpendicular distance between\nthe forces of the couple also approaches zero This\nmakes the forces collinear and the net force and\ntorque zero The torques in Eqs"}, {"Chapter": "1", "sentence_range": "3891-3894", "Text": "11\nAs q \u00e0 0, the perpendicular distance between\nthe forces of the couple also approaches zero This\nmakes the forces collinear and the net force and\ntorque zero The torques in Eqs (4"}, {"Chapter": "1", "sentence_range": "3892-3895", "Text": "This\nmakes the forces collinear and the net force and\ntorque zero The torques in Eqs (4 26) and (4"}, {"Chapter": "1", "sentence_range": "3893-3896", "Text": "The torques in Eqs (4 26) and (4 27)\ncan be expressed as vector product of the magnetic\nmoment of the coil and the magnetic field"}, {"Chapter": "1", "sentence_range": "3894-3897", "Text": "(4 26) and (4 27)\ncan be expressed as vector product of the magnetic\nmoment of the coil and the magnetic field We define the magnetic moment of the current\nloop as,\nm = I A\n(4"}, {"Chapter": "1", "sentence_range": "3895-3898", "Text": "26) and (4 27)\ncan be expressed as vector product of the magnetic\nmoment of the coil and the magnetic field We define the magnetic moment of the current\nloop as,\nm = I A\n(4 28)\nwhere the direction of the area vector A is given by\nthe right-hand thumb rule and is directed into\nthe plane of the paper in Fig"}, {"Chapter": "1", "sentence_range": "3896-3899", "Text": "27)\ncan be expressed as vector product of the magnetic\nmoment of the coil and the magnetic field We define the magnetic moment of the current\nloop as,\nm = I A\n(4 28)\nwhere the direction of the area vector A is given by\nthe right-hand thumb rule and is directed into\nthe plane of the paper in Fig 4"}, {"Chapter": "1", "sentence_range": "3897-3900", "Text": "We define the magnetic moment of the current\nloop as,\nm = I A\n(4 28)\nwhere the direction of the area vector A is given by\nthe right-hand thumb rule and is directed into\nthe plane of the paper in Fig 4 18"}, {"Chapter": "1", "sentence_range": "3898-3901", "Text": "28)\nwhere the direction of the area vector A is given by\nthe right-hand thumb rule and is directed into\nthe plane of the paper in Fig 4 18 Then as the\nangle between m and B is q ,  Eqs"}, {"Chapter": "1", "sentence_range": "3899-3902", "Text": "4 18 Then as the\nangle between m and B is q ,  Eqs (4"}, {"Chapter": "1", "sentence_range": "3900-3903", "Text": "18 Then as the\nangle between m and B is q ,  Eqs (4 26) and (4"}, {"Chapter": "1", "sentence_range": "3901-3904", "Text": "Then as the\nangle between m and B is q ,  Eqs (4 26) and (4 27)\ncan be expressed by one expression\n(4"}, {"Chapter": "1", "sentence_range": "3902-3905", "Text": "(4 26) and (4 27)\ncan be expressed by one expression\n(4 29)\nThis  is analogous to the electrostatic case\n(Electric dipole of dipole moment pe in an electric\nfield E)"}, {"Chapter": "1", "sentence_range": "3903-3906", "Text": "26) and (4 27)\ncan be expressed by one expression\n(4 29)\nThis  is analogous to the electrostatic case\n(Electric dipole of dipole moment pe in an electric\nfield E) \u03c4\u03c4\n= p\u00d7\u00d7\nE\ne\nAs is clear from Eq"}, {"Chapter": "1", "sentence_range": "3904-3907", "Text": "27)\ncan be expressed by one expression\n(4 29)\nThis  is analogous to the electrostatic case\n(Electric dipole of dipole moment pe in an electric\nfield E) \u03c4\u03c4\n= p\u00d7\u00d7\nE\ne\nAs is clear from Eq (4"}, {"Chapter": "1", "sentence_range": "3905-3908", "Text": "29)\nThis  is analogous to the electrostatic case\n(Electric dipole of dipole moment pe in an electric\nfield E) \u03c4\u03c4\n= p\u00d7\u00d7\nE\ne\nAs is clear from Eq (4 28), the dimensions of the\nmagnetic moment are [A][L2] and its unit is Am2"}, {"Chapter": "1", "sentence_range": "3906-3909", "Text": "\u03c4\u03c4\n= p\u00d7\u00d7\nE\ne\nAs is clear from Eq (4 28), the dimensions of the\nmagnetic moment are [A][L2] and its unit is Am2 From Eq"}, {"Chapter": "1", "sentence_range": "3907-3910", "Text": "(4 28), the dimensions of the\nmagnetic moment are [A][L2] and its unit is Am2 From Eq (4"}, {"Chapter": "1", "sentence_range": "3908-3911", "Text": "28), the dimensions of the\nmagnetic moment are [A][L2] and its unit is Am2 From Eq (4 29), we see that the torque ttttt\nvanishes when m is either parallel or antiparallel\nto the magnetic field B"}, {"Chapter": "1", "sentence_range": "3909-3912", "Text": "From Eq (4 29), we see that the torque ttttt\nvanishes when m is either parallel or antiparallel\nto the magnetic field B This indicates a state of\nequilibrium as there is no torque on the coil (this\nalso applies to any object with a magnetic moment\nm)"}, {"Chapter": "1", "sentence_range": "3910-3913", "Text": "(4 29), we see that the torque ttttt\nvanishes when m is either parallel or antiparallel\nto the magnetic field B This indicates a state of\nequilibrium as there is no torque on the coil (this\nalso applies to any object with a magnetic moment\nm) When m and B are parallel the equilibrium is\na stable one"}, {"Chapter": "1", "sentence_range": "3911-3914", "Text": "29), we see that the torque ttttt\nvanishes when m is either parallel or antiparallel\nto the magnetic field B This indicates a state of\nequilibrium as there is no torque on the coil (this\nalso applies to any object with a magnetic moment\nm) When m and B are parallel the equilibrium is\na stable one Any small rotation of the coil\nproduces a torque which brings it back to its original position"}, {"Chapter": "1", "sentence_range": "3912-3915", "Text": "This indicates a state of\nequilibrium as there is no torque on the coil (this\nalso applies to any object with a magnetic moment\nm) When m and B are parallel the equilibrium is\na stable one Any small rotation of the coil\nproduces a torque which brings it back to its original position When\nthey are antiparallel, the equilibrium is unstable as any rotation produces\na torque which increases with the amount of rotation"}, {"Chapter": "1", "sentence_range": "3913-3916", "Text": "When m and B are parallel the equilibrium is\na stable one Any small rotation of the coil\nproduces a torque which brings it back to its original position When\nthey are antiparallel, the equilibrium is unstable as any rotation produces\na torque which increases with the amount of rotation The presence of\nthis torque is also the reason why a small magnet or any magnetic dipole\naligns itself with the external magnetic field"}, {"Chapter": "1", "sentence_range": "3914-3917", "Text": "Any small rotation of the coil\nproduces a torque which brings it back to its original position When\nthey are antiparallel, the equilibrium is unstable as any rotation produces\na torque which increases with the amount of rotation The presence of\nthis torque is also the reason why a small magnet or any magnetic dipole\naligns itself with the external magnetic field If the loop has N closely wound turns, the expression for torque, Eq"}, {"Chapter": "1", "sentence_range": "3915-3918", "Text": "When\nthey are antiparallel, the equilibrium is unstable as any rotation produces\na torque which increases with the amount of rotation The presence of\nthis torque is also the reason why a small magnet or any magnetic dipole\naligns itself with the external magnetic field If the loop has N closely wound turns, the expression for torque, Eq (4"}, {"Chapter": "1", "sentence_range": "3916-3919", "Text": "The presence of\nthis torque is also the reason why a small magnet or any magnetic dipole\naligns itself with the external magnetic field If the loop has N closely wound turns, the expression for torque, Eq (4 29), still holds, with\nm = N I A\n(4"}, {"Chapter": "1", "sentence_range": "3917-3920", "Text": "If the loop has N closely wound turns, the expression for torque, Eq (4 29), still holds, with\nm = N I A\n(4 30)\nExample 4"}, {"Chapter": "1", "sentence_range": "3918-3921", "Text": "(4 29), still holds, with\nm = N I A\n(4 30)\nExample 4 11 A 100 turn closely wound circular coil of radius 10 cm\ncarries a current of 3"}, {"Chapter": "1", "sentence_range": "3919-3922", "Text": "29), still holds, with\nm = N I A\n(4 30)\nExample 4 11 A 100 turn closely wound circular coil of radius 10 cm\ncarries a current of 3 2 A"}, {"Chapter": "1", "sentence_range": "3920-3923", "Text": "30)\nExample 4 11 A 100 turn closely wound circular coil of radius 10 cm\ncarries a current of 3 2 A (a) What is the field at the centre of the\ncoil"}, {"Chapter": "1", "sentence_range": "3921-3924", "Text": "11 A 100 turn closely wound circular coil of radius 10 cm\ncarries a current of 3 2 A (a) What is the field at the centre of the\ncoil (b) What is the magnetic moment of this coil"}, {"Chapter": "1", "sentence_range": "3922-3925", "Text": "2 A (a) What is the field at the centre of the\ncoil (b) What is the magnetic moment of this coil The coil is placed in a vertical plane and is free to rotate about a\nhorizontal axis which coincides with its diameter"}, {"Chapter": "1", "sentence_range": "3923-3926", "Text": "(a) What is the field at the centre of the\ncoil (b) What is the magnetic moment of this coil The coil is placed in a vertical plane and is free to rotate about a\nhorizontal axis which coincides with its diameter A uniform magnetic\nfield of 2T in the horizontal direction exists such that initially the axis\nof the coil is in the direction of the field"}, {"Chapter": "1", "sentence_range": "3924-3927", "Text": "(b) What is the magnetic moment of this coil The coil is placed in a vertical plane and is free to rotate about a\nhorizontal axis which coincides with its diameter A uniform magnetic\nfield of 2T in the horizontal direction exists such that initially the axis\nof the coil is in the direction of the field The coil rotates through an\nangle of 90\u00b0 under the influence of the magnetic field"}, {"Chapter": "1", "sentence_range": "3925-3928", "Text": "The coil is placed in a vertical plane and is free to rotate about a\nhorizontal axis which coincides with its diameter A uniform magnetic\nfield of 2T in the horizontal direction exists such that initially the axis\nof the coil is in the direction of the field The coil rotates through an\nangle of 90\u00b0 under the influence of the magnetic field (c) What are the\nmagnitudes of the torques on the coil in the initial and final position"}, {"Chapter": "1", "sentence_range": "3926-3929", "Text": "A uniform magnetic\nfield of 2T in the horizontal direction exists such that initially the axis\nof the coil is in the direction of the field The coil rotates through an\nangle of 90\u00b0 under the influence of the magnetic field (c) What are the\nmagnitudes of the torques on the coil in the initial and final position (d) What is the angular speed acquired by the coil when it has rotated\nby 90\u00b0"}, {"Chapter": "1", "sentence_range": "3927-3930", "Text": "The coil rotates through an\nangle of 90\u00b0 under the influence of the magnetic field (c) What are the\nmagnitudes of the torques on the coil in the initial and final position (d) What is the angular speed acquired by the coil when it has rotated\nby 90\u00b0 The moment of inertia of the coil is 0"}, {"Chapter": "1", "sentence_range": "3928-3931", "Text": "(c) What are the\nmagnitudes of the torques on the coil in the initial and final position (d) What is the angular speed acquired by the coil when it has rotated\nby 90\u00b0 The moment of inertia of the coil is 0 1 kg m2"}, {"Chapter": "1", "sentence_range": "3929-3932", "Text": "(d) What is the angular speed acquired by the coil when it has rotated\nby 90\u00b0 The moment of inertia of the coil is 0 1 kg m2 FIGURE 4"}, {"Chapter": "1", "sentence_range": "3930-3933", "Text": "The moment of inertia of the coil is 0 1 kg m2 FIGURE 4 19 (a) The area vector of the loop\nABCD makes an arbitrary angle q with\nthe magnetic field"}, {"Chapter": "1", "sentence_range": "3931-3934", "Text": "1 kg m2 FIGURE 4 19 (a) The area vector of the loop\nABCD makes an arbitrary angle q with\nthe magnetic field (b) Top view of\nthe loop"}, {"Chapter": "1", "sentence_range": "3932-3935", "Text": "FIGURE 4 19 (a) The area vector of the loop\nABCD makes an arbitrary angle q with\nthe magnetic field (b) Top view of\nthe loop The forces F1 and F2 acting\non the arms AB and CD\nare indicated"}, {"Chapter": "1", "sentence_range": "3933-3936", "Text": "19 (a) The area vector of the loop\nABCD makes an arbitrary angle q with\nthe magnetic field (b) Top view of\nthe loop The forces F1 and F2 acting\non the arms AB and CD\nare indicated Rationalised 2023-24\n127\nMoving Charges and\nMagnetism\nEXAMPLE 4"}, {"Chapter": "1", "sentence_range": "3934-3937", "Text": "(b) Top view of\nthe loop The forces F1 and F2 acting\non the arms AB and CD\nare indicated Rationalised 2023-24\n127\nMoving Charges and\nMagnetism\nEXAMPLE 4 12\nEXAMPLE 4"}, {"Chapter": "1", "sentence_range": "3935-3938", "Text": "The forces F1 and F2 acting\non the arms AB and CD\nare indicated Rationalised 2023-24\n127\nMoving Charges and\nMagnetism\nEXAMPLE 4 12\nEXAMPLE 4 11\nSolution\n(a) From Eq"}, {"Chapter": "1", "sentence_range": "3936-3939", "Text": "Rationalised 2023-24\n127\nMoving Charges and\nMagnetism\nEXAMPLE 4 12\nEXAMPLE 4 11\nSolution\n(a) From Eq (4"}, {"Chapter": "1", "sentence_range": "3937-3940", "Text": "12\nEXAMPLE 4 11\nSolution\n(a) From Eq (4 16)\nB\n= \u00b50RNI\n2\nHere, N = 100; I = 3"}, {"Chapter": "1", "sentence_range": "3938-3941", "Text": "11\nSolution\n(a) From Eq (4 16)\nB\n= \u00b50RNI\n2\nHere, N = 100; I = 3 2 A, and R = 0"}, {"Chapter": "1", "sentence_range": "3939-3942", "Text": "(4 16)\nB\n= \u00b50RNI\n2\nHere, N = 100; I = 3 2 A, and R = 0 1 m"}, {"Chapter": "1", "sentence_range": "3940-3943", "Text": "16)\nB\n= \u00b50RNI\n2\nHere, N = 100; I = 3 2 A, and R = 0 1 m Hence,\n =\n\u00d7\n\u00d7\n\u00d7\n\u2212\n\u2212\n4\n10\n10\n2\n10\n5\n1\n     (using p \u00b4 3"}, {"Chapter": "1", "sentence_range": "3941-3944", "Text": "2 A, and R = 0 1 m Hence,\n =\n\u00d7\n\u00d7\n\u00d7\n\u2212\n\u2212\n4\n10\n10\n2\n10\n5\n1\n     (using p \u00b4 3 2 = 10)\n    = 2 \u00d7 10\u20133 T\nThe direction is given by the right-hand thumb rule"}, {"Chapter": "1", "sentence_range": "3942-3945", "Text": "1 m Hence,\n =\n\u00d7\n\u00d7\n\u00d7\n\u2212\n\u2212\n4\n10\n10\n2\n10\n5\n1\n     (using p \u00b4 3 2 = 10)\n    = 2 \u00d7 10\u20133 T\nThe direction is given by the right-hand thumb rule (b) The magnetic moment is given by Eq"}, {"Chapter": "1", "sentence_range": "3943-3946", "Text": "Hence,\n =\n\u00d7\n\u00d7\n\u00d7\n\u2212\n\u2212\n4\n10\n10\n2\n10\n5\n1\n     (using p \u00b4 3 2 = 10)\n    = 2 \u00d7 10\u20133 T\nThe direction is given by the right-hand thumb rule (b) The magnetic moment is given by Eq (4"}, {"Chapter": "1", "sentence_range": "3944-3947", "Text": "2 = 10)\n    = 2 \u00d7 10\u20133 T\nThe direction is given by the right-hand thumb rule (b) The magnetic moment is given by Eq (4 30),\nm = N I A = N I p r2 = 100 \u00d7 3"}, {"Chapter": "1", "sentence_range": "3945-3948", "Text": "(b) The magnetic moment is given by Eq (4 30),\nm = N I A = N I p r2 = 100 \u00d7 3 2 \u00d7 3"}, {"Chapter": "1", "sentence_range": "3946-3949", "Text": "(4 30),\nm = N I A = N I p r2 = 100 \u00d7 3 2 \u00d7 3 14 \u00d7 10\u20132 = 10 A m2\nThe direction is once again given by the right-hand thumb rule"}, {"Chapter": "1", "sentence_range": "3947-3950", "Text": "30),\nm = N I A = N I p r2 = 100 \u00d7 3 2 \u00d7 3 14 \u00d7 10\u20132 = 10 A m2\nThe direction is once again given by the right-hand thumb rule (c) t = m \u00d7 B    [from Eq"}, {"Chapter": "1", "sentence_range": "3948-3951", "Text": "2 \u00d7 3 14 \u00d7 10\u20132 = 10 A m2\nThe direction is once again given by the right-hand thumb rule (c) t = m \u00d7 B    [from Eq (4"}, {"Chapter": "1", "sentence_range": "3949-3952", "Text": "14 \u00d7 10\u20132 = 10 A m2\nThe direction is once again given by the right-hand thumb rule (c) t = m \u00d7 B    [from Eq (4 29)]\n   = m B sin q\nInitially, q = 0"}, {"Chapter": "1", "sentence_range": "3950-3953", "Text": "(c) t = m \u00d7 B    [from Eq (4 29)]\n   = m B sin q\nInitially, q = 0 Thus, initial torque ti = 0"}, {"Chapter": "1", "sentence_range": "3951-3954", "Text": "(4 29)]\n   = m B sin q\nInitially, q = 0 Thus, initial torque ti = 0 Finally, q = p/2 (or 90\u00ba)"}, {"Chapter": "1", "sentence_range": "3952-3955", "Text": "29)]\n   = m B sin q\nInitially, q = 0 Thus, initial torque ti = 0 Finally, q = p/2 (or 90\u00ba) Thus, final torque tf = m B = 10 \u00b4 2 = 20 N m"}, {"Chapter": "1", "sentence_range": "3953-3956", "Text": "Thus, initial torque ti = 0 Finally, q = p/2 (or 90\u00ba) Thus, final torque tf = m B = 10 \u00b4 2 = 20 N m (d)  From Newton\u2019s second law,\nI  \nwhere I  is the moment of inertia of the coil"}, {"Chapter": "1", "sentence_range": "3954-3957", "Text": "Finally, q = p/2 (or 90\u00ba) Thus, final torque tf = m B = 10 \u00b4 2 = 20 N m (d)  From Newton\u2019s second law,\nI  \nwhere I  is the moment of inertia of the coil From chain rule,\nd\nd\nd\nd\nd\nd\nd\nd\n\uf03d\n\uf03d\nt\nt\n\uf077\n\uf077\n\uf071\n\uf077 \uf077\n\uf071\n\uf071\nUsing this,\nI  \nd\nsin\nd\n\uf03d m B\n\uf077\n\uf077\n\uf071\n\uf071\nIntegrating from q = 0 to q = p/2,\nExample 4"}, {"Chapter": "1", "sentence_range": "3955-3958", "Text": "Thus, final torque tf = m B = 10 \u00b4 2 = 20 N m (d)  From Newton\u2019s second law,\nI  \nwhere I  is the moment of inertia of the coil From chain rule,\nd\nd\nd\nd\nd\nd\nd\nd\n\uf03d\n\uf03d\nt\nt\n\uf077\n\uf077\n\uf071\n\uf077 \uf077\n\uf071\n\uf071\nUsing this,\nI  \nd\nsin\nd\n\uf03d m B\n\uf077\n\uf077\n\uf071\n\uf071\nIntegrating from q = 0 to q = p/2,\nExample 4 12\n(a) A current-carrying circular loop lies on a smooth horizontal plane"}, {"Chapter": "1", "sentence_range": "3956-3959", "Text": "(d)  From Newton\u2019s second law,\nI  \nwhere I  is the moment of inertia of the coil From chain rule,\nd\nd\nd\nd\nd\nd\nd\nd\n\uf03d\n\uf03d\nt\nt\n\uf077\n\uf077\n\uf071\n\uf077 \uf077\n\uf071\n\uf071\nUsing this,\nI  \nd\nsin\nd\n\uf03d m B\n\uf077\n\uf077\n\uf071\n\uf071\nIntegrating from q = 0 to q = p/2,\nExample 4 12\n(a) A current-carrying circular loop lies on a smooth horizontal plane Can a uniform magnetic field be set up in such a manner that\nthe loop turns around itself (i"}, {"Chapter": "1", "sentence_range": "3957-3960", "Text": "From chain rule,\nd\nd\nd\nd\nd\nd\nd\nd\n\uf03d\n\uf03d\nt\nt\n\uf077\n\uf077\n\uf071\n\uf077 \uf077\n\uf071\n\uf071\nUsing this,\nI  \nd\nsin\nd\n\uf03d m B\n\uf077\n\uf077\n\uf071\n\uf071\nIntegrating from q = 0 to q = p/2,\nExample 4 12\n(a) A current-carrying circular loop lies on a smooth horizontal plane Can a uniform magnetic field be set up in such a manner that\nthe loop turns around itself (i e"}, {"Chapter": "1", "sentence_range": "3958-3961", "Text": "12\n(a) A current-carrying circular loop lies on a smooth horizontal plane Can a uniform magnetic field be set up in such a manner that\nthe loop turns around itself (i e , turns about the vertical axis)"}, {"Chapter": "1", "sentence_range": "3959-3962", "Text": "Can a uniform magnetic field be set up in such a manner that\nthe loop turns around itself (i e , turns about the vertical axis) (b) A current-carrying circular loop is located in a uniform external\nmagnetic field"}, {"Chapter": "1", "sentence_range": "3960-3963", "Text": "e , turns about the vertical axis) (b) A current-carrying circular loop is located in a uniform external\nmagnetic field If the loop is\nfree to turn, what is its orientation\nof stable equilibrium"}, {"Chapter": "1", "sentence_range": "3961-3964", "Text": ", turns about the vertical axis) (b) A current-carrying circular loop is located in a uniform external\nmagnetic field If the loop is\nfree to turn, what is its orientation\nof stable equilibrium Show that in this orientation, the flux of\nRationalised 2023-24\nPhysics\n128\n EXAMPLE 4"}, {"Chapter": "1", "sentence_range": "3962-3965", "Text": "(b) A current-carrying circular loop is located in a uniform external\nmagnetic field If the loop is\nfree to turn, what is its orientation\nof stable equilibrium Show that in this orientation, the flux of\nRationalised 2023-24\nPhysics\n128\n EXAMPLE 4 12\nthe total field (external field + field produced by the loop) is\nmaximum"}, {"Chapter": "1", "sentence_range": "3963-3966", "Text": "If the loop is\nfree to turn, what is its orientation\nof stable equilibrium Show that in this orientation, the flux of\nRationalised 2023-24\nPhysics\n128\n EXAMPLE 4 12\nthe total field (external field + field produced by the loop) is\nmaximum (c) A loop of irregular shape carrying current is located in an external\nmagnetic field"}, {"Chapter": "1", "sentence_range": "3964-3967", "Text": "Show that in this orientation, the flux of\nRationalised 2023-24\nPhysics\n128\n EXAMPLE 4 12\nthe total field (external field + field produced by the loop) is\nmaximum (c) A loop of irregular shape carrying current is located in an external\nmagnetic field If the wire is flexible, why does it change to a\ncircular shape"}, {"Chapter": "1", "sentence_range": "3965-3968", "Text": "12\nthe total field (external field + field produced by the loop) is\nmaximum (c) A loop of irregular shape carrying current is located in an external\nmagnetic field If the wire is flexible, why does it change to a\ncircular shape Solution\n(a) No, because that would require ttttt to be in the vertical direction"}, {"Chapter": "1", "sentence_range": "3966-3969", "Text": "(c) A loop of irregular shape carrying current is located in an external\nmagnetic field If the wire is flexible, why does it change to a\ncircular shape Solution\n(a) No, because that would require ttttt to be in the vertical direction But ttttt = I A \u00d7 B, and since A of the horizontal loop is in the vertical\ndirection, t would be in the plane of the loop for any B"}, {"Chapter": "1", "sentence_range": "3967-3970", "Text": "If the wire is flexible, why does it change to a\ncircular shape Solution\n(a) No, because that would require ttttt to be in the vertical direction But ttttt = I A \u00d7 B, and since A of the horizontal loop is in the vertical\ndirection, t would be in the plane of the loop for any B (b) Orientation of stable equilibrium is one where the area vector A\nof the loop is in the direction of external magnetic field"}, {"Chapter": "1", "sentence_range": "3968-3971", "Text": "Solution\n(a) No, because that would require ttttt to be in the vertical direction But ttttt = I A \u00d7 B, and since A of the horizontal loop is in the vertical\ndirection, t would be in the plane of the loop for any B (b) Orientation of stable equilibrium is one where the area vector A\nof the loop is in the direction of external magnetic field In this\norientation, the magnetic field produced by the loop is in the same\ndirection as external field, both normal to the  plane of the loop,\nthus giving rise to maximum flux of the total field"}, {"Chapter": "1", "sentence_range": "3969-3972", "Text": "But ttttt = I A \u00d7 B, and since A of the horizontal loop is in the vertical\ndirection, t would be in the plane of the loop for any B (b) Orientation of stable equilibrium is one where the area vector A\nof the loop is in the direction of external magnetic field In this\norientation, the magnetic field produced by the loop is in the same\ndirection as external field, both normal to the  plane of the loop,\nthus giving rise to maximum flux of the total field (c) It assumes circular shape with its plane normal to the field to\nmaximise flux, since for a given perimeter, a circle encloses greater\narea than any other shape"}, {"Chapter": "1", "sentence_range": "3970-3973", "Text": "(b) Orientation of stable equilibrium is one where the area vector A\nof the loop is in the direction of external magnetic field In this\norientation, the magnetic field produced by the loop is in the same\ndirection as external field, both normal to the  plane of the loop,\nthus giving rise to maximum flux of the total field (c) It assumes circular shape with its plane normal to the field to\nmaximise flux, since for a given perimeter, a circle encloses greater\narea than any other shape 4"}, {"Chapter": "1", "sentence_range": "3971-3974", "Text": "In this\norientation, the magnetic field produced by the loop is in the same\ndirection as external field, both normal to the  plane of the loop,\nthus giving rise to maximum flux of the total field (c) It assumes circular shape with its plane normal to the field to\nmaximise flux, since for a given perimeter, a circle encloses greater\narea than any other shape 4 9"}, {"Chapter": "1", "sentence_range": "3972-3975", "Text": "(c) It assumes circular shape with its plane normal to the field to\nmaximise flux, since for a given perimeter, a circle encloses greater\narea than any other shape 4 9 2  Circular current loop as a magnetic dipole\nIn this section, we shall consider the elementary magnetic element: the\ncurrent loop"}, {"Chapter": "1", "sentence_range": "3973-3976", "Text": "4 9 2  Circular current loop as a magnetic dipole\nIn this section, we shall consider the elementary magnetic element: the\ncurrent loop We shall show that the magnetic field (at large distances)\ndue to current in a circular current loop is very similar in behaviour to\nthe electric field of an electric dipole"}, {"Chapter": "1", "sentence_range": "3974-3977", "Text": "9 2  Circular current loop as a magnetic dipole\nIn this section, we shall consider the elementary magnetic element: the\ncurrent loop We shall show that the magnetic field (at large distances)\ndue to current in a circular current loop is very similar in behaviour to\nthe electric field of an electric dipole In Section 4"}, {"Chapter": "1", "sentence_range": "3975-3978", "Text": "2  Circular current loop as a magnetic dipole\nIn this section, we shall consider the elementary magnetic element: the\ncurrent loop We shall show that the magnetic field (at large distances)\ndue to current in a circular current loop is very similar in behaviour to\nthe electric field of an electric dipole In Section 4 6, we have evaluated\nthe magnetic field on the axis of a circular loop, of a radius R, carrying a\nsteady current I"}, {"Chapter": "1", "sentence_range": "3976-3979", "Text": "We shall show that the magnetic field (at large distances)\ndue to current in a circular current loop is very similar in behaviour to\nthe electric field of an electric dipole In Section 4 6, we have evaluated\nthe magnetic field on the axis of a circular loop, of a radius R, carrying a\nsteady current I The magnitude of this field is [(Eq"}, {"Chapter": "1", "sentence_range": "3977-3980", "Text": "In Section 4 6, we have evaluated\nthe magnetic field on the axis of a circular loop, of a radius R, carrying a\nsteady current I The magnitude of this field is [(Eq (4"}, {"Chapter": "1", "sentence_range": "3978-3981", "Text": "6, we have evaluated\nthe magnetic field on the axis of a circular loop, of a radius R, carrying a\nsteady current I The magnitude of this field is [(Eq (4 15)],\n(\n)\n2\n0\n3/2\n2\n2\n2\n\u00b5\n=\n+\nI R\nB\nx\nR\nand its direction is along the axis and given by the right-hand thumb\nrule (Fig"}, {"Chapter": "1", "sentence_range": "3979-3982", "Text": "The magnitude of this field is [(Eq (4 15)],\n(\n)\n2\n0\n3/2\n2\n2\n2\n\u00b5\n=\n+\nI R\nB\nx\nR\nand its direction is along the axis and given by the right-hand thumb\nrule (Fig 4"}, {"Chapter": "1", "sentence_range": "3980-3983", "Text": "(4 15)],\n(\n)\n2\n0\n3/2\n2\n2\n2\n\u00b5\n=\n+\nI R\nB\nx\nR\nand its direction is along the axis and given by the right-hand thumb\nrule (Fig 4 12)"}, {"Chapter": "1", "sentence_range": "3981-3984", "Text": "15)],\n(\n)\n2\n0\n3/2\n2\n2\n2\n\u00b5\n=\n+\nI R\nB\nx\nR\nand its direction is along the axis and given by the right-hand thumb\nrule (Fig 4 12) Here, x is the distance along the axis from the centre of\nthe loop"}, {"Chapter": "1", "sentence_range": "3982-3985", "Text": "4 12) Here, x is the distance along the axis from the centre of\nthe loop For x >> R, we may drop the R2 term in the denominator"}, {"Chapter": "1", "sentence_range": "3983-3986", "Text": "12) Here, x is the distance along the axis from the centre of\nthe loop For x >> R, we may drop the R2 term in the denominator Thus,\n2\n0\n3\n2\nIR\nB\nx\n\u00b5\n=\nNote that the area of the loop A = pR2"}, {"Chapter": "1", "sentence_range": "3984-3987", "Text": "Here, x is the distance along the axis from the centre of\nthe loop For x >> R, we may drop the R2 term in the denominator Thus,\n2\n0\n3\n2\nIR\nB\nx\n\u00b5\n=\nNote that the area of the loop A = pR2 Thus,\n0\n3\n2\nIA\nB\nx\n=\u00b5\n\u03c0\nAs earlier, we define the magnetic moment m to have a magnitude IA,\nm  = I A"}, {"Chapter": "1", "sentence_range": "3985-3988", "Text": "For x >> R, we may drop the R2 term in the denominator Thus,\n2\n0\n3\n2\nIR\nB\nx\n\u00b5\n=\nNote that the area of the loop A = pR2 Thus,\n0\n3\n2\nIA\nB\nx\n=\u00b5\n\u03c0\nAs earlier, we define the magnetic moment m to have a magnitude IA,\nm  = I A Hence,\nB\n\u2243 \u00b50m\n3\n2 \u03c0x\n    \n\u03c0\n0\n3\n2\n4\nx\n=\u00b5\nm\n[4"}, {"Chapter": "1", "sentence_range": "3986-3989", "Text": "Thus,\n2\n0\n3\n2\nIR\nB\nx\n\u00b5\n=\nNote that the area of the loop A = pR2 Thus,\n0\n3\n2\nIA\nB\nx\n=\u00b5\n\u03c0\nAs earlier, we define the magnetic moment m to have a magnitude IA,\nm  = I A Hence,\nB\n\u2243 \u00b50m\n3\n2 \u03c0x\n    \n\u03c0\n0\n3\n2\n4\nx\n=\u00b5\nm\n[4 31(a)]\nThe expression of Eq"}, {"Chapter": "1", "sentence_range": "3987-3990", "Text": "Thus,\n0\n3\n2\nIA\nB\nx\n=\u00b5\n\u03c0\nAs earlier, we define the magnetic moment m to have a magnitude IA,\nm  = I A Hence,\nB\n\u2243 \u00b50m\n3\n2 \u03c0x\n    \n\u03c0\n0\n3\n2\n4\nx\n=\u00b5\nm\n[4 31(a)]\nThe expression of Eq [4"}, {"Chapter": "1", "sentence_range": "3988-3991", "Text": "Hence,\nB\n\u2243 \u00b50m\n3\n2 \u03c0x\n    \n\u03c0\n0\n3\n2\n4\nx\n=\u00b5\nm\n[4 31(a)]\nThe expression of Eq [4 31(a)] is very similar to an expression obtained\nearlier for the electric field of a dipole"}, {"Chapter": "1", "sentence_range": "3989-3992", "Text": "31(a)]\nThe expression of Eq [4 31(a)] is very similar to an expression obtained\nearlier for the electric field of a dipole The similarity may be seen if we\nsubstitute,\n \n0\n1/0\n\u00b5\n\u03b5\n\u2192\nRationalised 2023-24\n129\nMoving Charges and\nMagnetism\ne\nm\u2192\np  (electrostatic dipole)\n \nB\u2192\nE   (electrostatic field)\nWe then obtain,\n3\n0\n2\n4\ne\n\u03b5x\n=\n\u03c0\np\nE\nwhich is precisely the field for an electric dipole at a point on its axis"}, {"Chapter": "1", "sentence_range": "3990-3993", "Text": "[4 31(a)] is very similar to an expression obtained\nearlier for the electric field of a dipole The similarity may be seen if we\nsubstitute,\n \n0\n1/0\n\u00b5\n\u03b5\n\u2192\nRationalised 2023-24\n129\nMoving Charges and\nMagnetism\ne\nm\u2192\np  (electrostatic dipole)\n \nB\u2192\nE   (electrostatic field)\nWe then obtain,\n3\n0\n2\n4\ne\n\u03b5x\n=\n\u03c0\np\nE\nwhich is precisely the field for an electric dipole at a point on its axis considered in Chapter 1, Section 1"}, {"Chapter": "1", "sentence_range": "3991-3994", "Text": "31(a)] is very similar to an expression obtained\nearlier for the electric field of a dipole The similarity may be seen if we\nsubstitute,\n \n0\n1/0\n\u00b5\n\u03b5\n\u2192\nRationalised 2023-24\n129\nMoving Charges and\nMagnetism\ne\nm\u2192\np  (electrostatic dipole)\n \nB\u2192\nE   (electrostatic field)\nWe then obtain,\n3\n0\n2\n4\ne\n\u03b5x\n=\n\u03c0\np\nE\nwhich is precisely the field for an electric dipole at a point on its axis considered in Chapter 1, Section 1 10 [Eq"}, {"Chapter": "1", "sentence_range": "3992-3995", "Text": "The similarity may be seen if we\nsubstitute,\n \n0\n1/0\n\u00b5\n\u03b5\n\u2192\nRationalised 2023-24\n129\nMoving Charges and\nMagnetism\ne\nm\u2192\np  (electrostatic dipole)\n \nB\u2192\nE   (electrostatic field)\nWe then obtain,\n3\n0\n2\n4\ne\n\u03b5x\n=\n\u03c0\np\nE\nwhich is precisely the field for an electric dipole at a point on its axis considered in Chapter 1, Section 1 10 [Eq (1"}, {"Chapter": "1", "sentence_range": "3993-3996", "Text": "considered in Chapter 1, Section 1 10 [Eq (1 20)]"}, {"Chapter": "1", "sentence_range": "3994-3997", "Text": "10 [Eq (1 20)] It can be shown that the above analogy can be carried further"}, {"Chapter": "1", "sentence_range": "3995-3998", "Text": "(1 20)] It can be shown that the above analogy can be carried further We\nhad found in Chapter 1 that the electric field on the perpendicular bisector\nof the dipole is given by [See Eq"}, {"Chapter": "1", "sentence_range": "3996-3999", "Text": "20)] It can be shown that the above analogy can be carried further We\nhad found in Chapter 1 that the electric field on the perpendicular bisector\nof the dipole is given by [See Eq (1"}, {"Chapter": "1", "sentence_range": "3997-4000", "Text": "It can be shown that the above analogy can be carried further We\nhad found in Chapter 1 that the electric field on the perpendicular bisector\nof the dipole is given by [See Eq (1 21)],\nE \u2243\npe\nx\n4\n0\n3\n\u03c0\u03b5\nwhere x is the distance from the dipole"}, {"Chapter": "1", "sentence_range": "3998-4001", "Text": "We\nhad found in Chapter 1 that the electric field on the perpendicular bisector\nof the dipole is given by [See Eq (1 21)],\nE \u2243\npe\nx\n4\n0\n3\n\u03c0\u03b5\nwhere x is the distance from the dipole If we replace p \u00e0 m and \n0\n1/0\n\u00b5\n\u03b5\n\u2192\nin the above expression, we obtain the result for B for a point in the\nplane of the loop at a distance x  from the centre"}, {"Chapter": "1", "sentence_range": "3999-4002", "Text": "(1 21)],\nE \u2243\npe\nx\n4\n0\n3\n\u03c0\u03b5\nwhere x is the distance from the dipole If we replace p \u00e0 m and \n0\n1/0\n\u00b5\n\u03b5\n\u2192\nin the above expression, we obtain the result for B for a point in the\nplane of the loop at a distance x  from the centre For  x >>R,\nB\nm\n\u2243 \u00b50\n4\u03c0 x3\nx\nR\n;\n>>\n[4"}, {"Chapter": "1", "sentence_range": "4000-4003", "Text": "21)],\nE \u2243\npe\nx\n4\n0\n3\n\u03c0\u03b5\nwhere x is the distance from the dipole If we replace p \u00e0 m and \n0\n1/0\n\u00b5\n\u03b5\n\u2192\nin the above expression, we obtain the result for B for a point in the\nplane of the loop at a distance x  from the centre For  x >>R,\nB\nm\n\u2243 \u00b50\n4\u03c0 x3\nx\nR\n;\n>>\n[4 31(b)]\nThe results given by Eqs"}, {"Chapter": "1", "sentence_range": "4001-4004", "Text": "If we replace p \u00e0 m and \n0\n1/0\n\u00b5\n\u03b5\n\u2192\nin the above expression, we obtain the result for B for a point in the\nplane of the loop at a distance x  from the centre For  x >>R,\nB\nm\n\u2243 \u00b50\n4\u03c0 x3\nx\nR\n;\n>>\n[4 31(b)]\nThe results given by Eqs [4"}, {"Chapter": "1", "sentence_range": "4002-4005", "Text": "For  x >>R,\nB\nm\n\u2243 \u00b50\n4\u03c0 x3\nx\nR\n;\n>>\n[4 31(b)]\nThe results given by Eqs [4 31(a)] and [4"}, {"Chapter": "1", "sentence_range": "4003-4006", "Text": "31(b)]\nThe results given by Eqs [4 31(a)] and [4 31(b)] become exact for a\npoint magnetic dipole"}, {"Chapter": "1", "sentence_range": "4004-4007", "Text": "[4 31(a)] and [4 31(b)] become exact for a\npoint magnetic dipole The results obtained above can be shown to apply to any planar loop:\na planar current loop is equivalent to a magnetic dipole of dipole moment\nm = I A, which is the analogue of electric dipole moment p"}, {"Chapter": "1", "sentence_range": "4005-4008", "Text": "31(a)] and [4 31(b)] become exact for a\npoint magnetic dipole The results obtained above can be shown to apply to any planar loop:\na planar current loop is equivalent to a magnetic dipole of dipole moment\nm = I A, which is the analogue of electric dipole moment p Note, however,\na fundamental difference: an electric dipole is built up of two elementary\nunits \u2014 the charges (or electric monopoles)"}, {"Chapter": "1", "sentence_range": "4006-4009", "Text": "31(b)] become exact for a\npoint magnetic dipole The results obtained above can be shown to apply to any planar loop:\na planar current loop is equivalent to a magnetic dipole of dipole moment\nm = I A, which is the analogue of electric dipole moment p Note, however,\na fundamental difference: an electric dipole is built up of two elementary\nunits \u2014 the charges (or electric monopoles) In magnetism, a magnetic\ndipole (or a current loop) is the most elementary element"}, {"Chapter": "1", "sentence_range": "4007-4010", "Text": "The results obtained above can be shown to apply to any planar loop:\na planar current loop is equivalent to a magnetic dipole of dipole moment\nm = I A, which is the analogue of electric dipole moment p Note, however,\na fundamental difference: an electric dipole is built up of two elementary\nunits \u2014 the charges (or electric monopoles) In magnetism, a magnetic\ndipole (or a current loop) is the most elementary element The equivalent\nof electric charges, i"}, {"Chapter": "1", "sentence_range": "4008-4011", "Text": "Note, however,\na fundamental difference: an electric dipole is built up of two elementary\nunits \u2014 the charges (or electric monopoles) In magnetism, a magnetic\ndipole (or a current loop) is the most elementary element The equivalent\nof electric charges, i e"}, {"Chapter": "1", "sentence_range": "4009-4012", "Text": "In magnetism, a magnetic\ndipole (or a current loop) is the most elementary element The equivalent\nof electric charges, i e , magnetic monopoles, are not known to exist"}, {"Chapter": "1", "sentence_range": "4010-4013", "Text": "The equivalent\nof electric charges, i e , magnetic monopoles, are not known to exist We have shown that a current loop (i) produces a magnetic field and\nbehaves like a magnetic dipole at large distances, and\n(ii) is subject to torque like a magnetic needle"}, {"Chapter": "1", "sentence_range": "4011-4014", "Text": "e , magnetic monopoles, are not known to exist We have shown that a current loop (i) produces a magnetic field and\nbehaves like a magnetic dipole at large distances, and\n(ii) is subject to torque like a magnetic needle This led Ampere to suggest\nthat all magnetism is due to circulating currents"}, {"Chapter": "1", "sentence_range": "4012-4015", "Text": ", magnetic monopoles, are not known to exist We have shown that a current loop (i) produces a magnetic field and\nbehaves like a magnetic dipole at large distances, and\n(ii) is subject to torque like a magnetic needle This led Ampere to suggest\nthat all magnetism is due to circulating currents This seems to be partly\ntrue and no magnetic monopoles have been seen so far"}, {"Chapter": "1", "sentence_range": "4013-4016", "Text": "We have shown that a current loop (i) produces a magnetic field and\nbehaves like a magnetic dipole at large distances, and\n(ii) is subject to torque like a magnetic needle This led Ampere to suggest\nthat all magnetism is due to circulating currents This seems to be partly\ntrue and no magnetic monopoles have been seen so far However,\nelementary particles such as an electron or a proton also carry an intrinsic\nmagnetic moment, not accounted by circulating currents"}, {"Chapter": "1", "sentence_range": "4014-4017", "Text": "This led Ampere to suggest\nthat all magnetism is due to circulating currents This seems to be partly\ntrue and no magnetic monopoles have been seen so far However,\nelementary particles such as an electron or a proton also carry an intrinsic\nmagnetic moment, not accounted by circulating currents 4"}, {"Chapter": "1", "sentence_range": "4015-4018", "Text": "This seems to be partly\ntrue and no magnetic monopoles have been seen so far However,\nelementary particles such as an electron or a proton also carry an intrinsic\nmagnetic moment, not accounted by circulating currents 4 10  THE MOVING COIL GALVANOMETER\nCurrents and voltages in circuits have been discussed extensively in\nChapters 3"}, {"Chapter": "1", "sentence_range": "4016-4019", "Text": "However,\nelementary particles such as an electron or a proton also carry an intrinsic\nmagnetic moment, not accounted by circulating currents 4 10  THE MOVING COIL GALVANOMETER\nCurrents and voltages in circuits have been discussed extensively in\nChapters 3 But how do we measure them"}, {"Chapter": "1", "sentence_range": "4017-4020", "Text": "4 10  THE MOVING COIL GALVANOMETER\nCurrents and voltages in circuits have been discussed extensively in\nChapters 3 But how do we measure them How do we claim that\ncurrent in a circuit is 1"}, {"Chapter": "1", "sentence_range": "4018-4021", "Text": "10  THE MOVING COIL GALVANOMETER\nCurrents and voltages in circuits have been discussed extensively in\nChapters 3 But how do we measure them How do we claim that\ncurrent in a circuit is 1 5 A or the voltage drop across a resistor is 1"}, {"Chapter": "1", "sentence_range": "4019-4022", "Text": "But how do we measure them How do we claim that\ncurrent in a circuit is 1 5 A or the voltage drop across a resistor is 1 2 V"}, {"Chapter": "1", "sentence_range": "4020-4023", "Text": "How do we claim that\ncurrent in a circuit is 1 5 A or the voltage drop across a resistor is 1 2 V Figure 4"}, {"Chapter": "1", "sentence_range": "4021-4024", "Text": "5 A or the voltage drop across a resistor is 1 2 V Figure 4 20 exhibits a very useful instrument for this purpose: the moving\ncoil galvanometer (MCG)"}, {"Chapter": "1", "sentence_range": "4022-4025", "Text": "2 V Figure 4 20 exhibits a very useful instrument for this purpose: the moving\ncoil galvanometer (MCG) It is a device whose principle can be understood\non the basis of our discussion in Section 4"}, {"Chapter": "1", "sentence_range": "4023-4026", "Text": "Figure 4 20 exhibits a very useful instrument for this purpose: the moving\ncoil galvanometer (MCG) It is a device whose principle can be understood\non the basis of our discussion in Section 4 10"}, {"Chapter": "1", "sentence_range": "4024-4027", "Text": "20 exhibits a very useful instrument for this purpose: the moving\ncoil galvanometer (MCG) It is a device whose principle can be understood\non the basis of our discussion in Section 4 10 The galvanometer consists of a coil, with many turns, free to rotate\nabout a fixed axis (Fig"}, {"Chapter": "1", "sentence_range": "4025-4028", "Text": "It is a device whose principle can be understood\non the basis of our discussion in Section 4 10 The galvanometer consists of a coil, with many turns, free to rotate\nabout a fixed axis (Fig 4"}, {"Chapter": "1", "sentence_range": "4026-4029", "Text": "10 The galvanometer consists of a coil, with many turns, free to rotate\nabout a fixed axis (Fig 4 20), in a uniform radial magnetic field"}, {"Chapter": "1", "sentence_range": "4027-4030", "Text": "The galvanometer consists of a coil, with many turns, free to rotate\nabout a fixed axis (Fig 4 20), in a uniform radial magnetic field There is\na cylindrical soft iron core which not only makes the field radial but also\nincreases the strength of the magnetic field"}, {"Chapter": "1", "sentence_range": "4028-4031", "Text": "4 20), in a uniform radial magnetic field There is\na cylindrical soft iron core which not only makes the field radial but also\nincreases the strength of the magnetic field When a current flows through\nthe coil, a torque acts on it"}, {"Chapter": "1", "sentence_range": "4029-4032", "Text": "20), in a uniform radial magnetic field There is\na cylindrical soft iron core which not only makes the field radial but also\nincreases the strength of the magnetic field When a current flows through\nthe coil, a torque acts on it This torque is given by Eq"}, {"Chapter": "1", "sentence_range": "4030-4033", "Text": "There is\na cylindrical soft iron core which not only makes the field radial but also\nincreases the strength of the magnetic field When a current flows through\nthe coil, a torque acts on it This torque is given by Eq (4"}, {"Chapter": "1", "sentence_range": "4031-4034", "Text": "When a current flows through\nthe coil, a torque acts on it This torque is given by Eq (4 26) to be\nt = NI AB\nRationalised 2023-24\nPhysics\n130\nwhere the symbols have their usual meaning"}, {"Chapter": "1", "sentence_range": "4032-4035", "Text": "This torque is given by Eq (4 26) to be\nt = NI AB\nRationalised 2023-24\nPhysics\n130\nwhere the symbols have their usual meaning Since\nthe field is radial by design, we have taken sin q = 1 in\nthe above expression for the torque"}, {"Chapter": "1", "sentence_range": "4033-4036", "Text": "(4 26) to be\nt = NI AB\nRationalised 2023-24\nPhysics\n130\nwhere the symbols have their usual meaning Since\nthe field is radial by design, we have taken sin q = 1 in\nthe above expression for the torque The magnetic\ntorque NIAB tends to rotate the coil"}, {"Chapter": "1", "sentence_range": "4034-4037", "Text": "26) to be\nt = NI AB\nRationalised 2023-24\nPhysics\n130\nwhere the symbols have their usual meaning Since\nthe field is radial by design, we have taken sin q = 1 in\nthe above expression for the torque The magnetic\ntorque NIAB tends to rotate the coil A spring Sp\nprovides a counter torque kf that balances the\nmagnetic torque NIAB; resulting in a steady angular\ndeflection f"}, {"Chapter": "1", "sentence_range": "4035-4038", "Text": "Since\nthe field is radial by design, we have taken sin q = 1 in\nthe above expression for the torque The magnetic\ntorque NIAB tends to rotate the coil A spring Sp\nprovides a counter torque kf that balances the\nmagnetic torque NIAB; resulting in a steady angular\ndeflection f In equilibrium\nkf = NI AB\nwhere k is the torsional constant of the spring; i"}, {"Chapter": "1", "sentence_range": "4036-4039", "Text": "The magnetic\ntorque NIAB tends to rotate the coil A spring Sp\nprovides a counter torque kf that balances the\nmagnetic torque NIAB; resulting in a steady angular\ndeflection f In equilibrium\nkf = NI AB\nwhere k is the torsional constant of the spring; i e"}, {"Chapter": "1", "sentence_range": "4037-4040", "Text": "A spring Sp\nprovides a counter torque kf that balances the\nmagnetic torque NIAB; resulting in a steady angular\ndeflection f In equilibrium\nkf = NI AB\nwhere k is the torsional constant of the spring; i e the\nrestoring torque per unit twist"}, {"Chapter": "1", "sentence_range": "4038-4041", "Text": "In equilibrium\nkf = NI AB\nwhere k is the torsional constant of the spring; i e the\nrestoring torque per unit twist The deflection f is\nindicated on the scale by a pointer attached to the\nspring"}, {"Chapter": "1", "sentence_range": "4039-4042", "Text": "e the\nrestoring torque per unit twist The deflection f is\nindicated on the scale by a pointer attached to the\nspring We have\n\u03c6 = \uf8eb\n\uf8ed\uf8ec\nNAB\uf8f8\uf8f7\uf8f6\nk\nI\n(4"}, {"Chapter": "1", "sentence_range": "4040-4043", "Text": "the\nrestoring torque per unit twist The deflection f is\nindicated on the scale by a pointer attached to the\nspring We have\n\u03c6 = \uf8eb\n\uf8ed\uf8ec\nNAB\uf8f8\uf8f7\uf8f6\nk\nI\n(4 38)\nThe quantity in brackets is a constant for a given\ngalvanometer"}, {"Chapter": "1", "sentence_range": "4041-4044", "Text": "The deflection f is\nindicated on the scale by a pointer attached to the\nspring We have\n\u03c6 = \uf8eb\n\uf8ed\uf8ec\nNAB\uf8f8\uf8f7\uf8f6\nk\nI\n(4 38)\nThe quantity in brackets is a constant for a given\ngalvanometer The galvanometer can be used in a number of ways"}, {"Chapter": "1", "sentence_range": "4042-4045", "Text": "We have\n\u03c6 = \uf8eb\n\uf8ed\uf8ec\nNAB\uf8f8\uf8f7\uf8f6\nk\nI\n(4 38)\nThe quantity in brackets is a constant for a given\ngalvanometer The galvanometer can be used in a number of ways It can be used as a detector to check if a current is\nflowing in the circuit"}, {"Chapter": "1", "sentence_range": "4043-4046", "Text": "38)\nThe quantity in brackets is a constant for a given\ngalvanometer The galvanometer can be used in a number of ways It can be used as a detector to check if a current is\nflowing in the circuit We have come across this usage\nin the Wheatstone\u2019s bridge arrangement"}, {"Chapter": "1", "sentence_range": "4044-4047", "Text": "The galvanometer can be used in a number of ways It can be used as a detector to check if a current is\nflowing in the circuit We have come across this usage\nin the Wheatstone\u2019s bridge arrangement In this usage\nthe neutral position of the pointer (when no current is\nflowing through the galvanometer) is in the middle of\nthe scale and not at the left end as shown in Fig"}, {"Chapter": "1", "sentence_range": "4045-4048", "Text": "It can be used as a detector to check if a current is\nflowing in the circuit We have come across this usage\nin the Wheatstone\u2019s bridge arrangement In this usage\nthe neutral position of the pointer (when no current is\nflowing through the galvanometer) is in the middle of\nthe scale and not at the left end as shown in Fig 4"}, {"Chapter": "1", "sentence_range": "4046-4049", "Text": "We have come across this usage\nin the Wheatstone\u2019s bridge arrangement In this usage\nthe neutral position of the pointer (when no current is\nflowing through the galvanometer) is in the middle of\nthe scale and not at the left end as shown in Fig 4 20"}, {"Chapter": "1", "sentence_range": "4047-4050", "Text": "In this usage\nthe neutral position of the pointer (when no current is\nflowing through the galvanometer) is in the middle of\nthe scale and not at the left end as shown in Fig 4 20 Depending on the direction of the current, the pointer\u2019s\ndeflection is either to the right or the left"}, {"Chapter": "1", "sentence_range": "4048-4051", "Text": "4 20 Depending on the direction of the current, the pointer\u2019s\ndeflection is either to the right or the left The galvanometer cannot as such be used as an\nammeter to measure the value of the current in a given circuit"}, {"Chapter": "1", "sentence_range": "4049-4052", "Text": "20 Depending on the direction of the current, the pointer\u2019s\ndeflection is either to the right or the left The galvanometer cannot as such be used as an\nammeter to measure the value of the current in a given circuit This is for\ntwo reasons: (i) Galvanometer is a very sensitive device, it gives a full-\nscale deflection for a current of the order of mA"}, {"Chapter": "1", "sentence_range": "4050-4053", "Text": "Depending on the direction of the current, the pointer\u2019s\ndeflection is either to the right or the left The galvanometer cannot as such be used as an\nammeter to measure the value of the current in a given circuit This is for\ntwo reasons: (i) Galvanometer is a very sensitive device, it gives a full-\nscale deflection for a current of the order of mA (ii) For measuring currents,\nthe galvanometer has to be connected in series, and as it has a large\nresistance, this will change the value of the current in the circuit"}, {"Chapter": "1", "sentence_range": "4051-4054", "Text": "The galvanometer cannot as such be used as an\nammeter to measure the value of the current in a given circuit This is for\ntwo reasons: (i) Galvanometer is a very sensitive device, it gives a full-\nscale deflection for a current of the order of mA (ii) For measuring currents,\nthe galvanometer has to be connected in series, and as it has a large\nresistance, this will change the value of the current in the circuit To\novercome these difficulties, one attaches a small resistance rs, called shunt\nresistance, in parallel with the galvanometer coil; so that most of the\ncurrent passes through the shunt"}, {"Chapter": "1", "sentence_range": "4052-4055", "Text": "This is for\ntwo reasons: (i) Galvanometer is a very sensitive device, it gives a full-\nscale deflection for a current of the order of mA (ii) For measuring currents,\nthe galvanometer has to be connected in series, and as it has a large\nresistance, this will change the value of the current in the circuit To\novercome these difficulties, one attaches a small resistance rs, called shunt\nresistance, in parallel with the galvanometer coil; so that most of the\ncurrent passes through the shunt The resistance of this arrangement is,\nRG rs / (RG  + rs)   \u2243  rs         if    RG >>  rs\nIf rs has small value, in relation to the resistance of the rest of the\ncircuit Rc, the effect of introducing the measuring instrument is also small\nand negligible"}, {"Chapter": "1", "sentence_range": "4053-4056", "Text": "(ii) For measuring currents,\nthe galvanometer has to be connected in series, and as it has a large\nresistance, this will change the value of the current in the circuit To\novercome these difficulties, one attaches a small resistance rs, called shunt\nresistance, in parallel with the galvanometer coil; so that most of the\ncurrent passes through the shunt The resistance of this arrangement is,\nRG rs / (RG  + rs)   \u2243  rs         if    RG >>  rs\nIf rs has small value, in relation to the resistance of the rest of the\ncircuit Rc, the effect of introducing the measuring instrument is also small\nand negligible This arrangement is schematically shown in Fig"}, {"Chapter": "1", "sentence_range": "4054-4057", "Text": "To\novercome these difficulties, one attaches a small resistance rs, called shunt\nresistance, in parallel with the galvanometer coil; so that most of the\ncurrent passes through the shunt The resistance of this arrangement is,\nRG rs / (RG  + rs)   \u2243  rs         if    RG >>  rs\nIf rs has small value, in relation to the resistance of the rest of the\ncircuit Rc, the effect of introducing the measuring instrument is also small\nand negligible This arrangement is schematically shown in Fig 4"}, {"Chapter": "1", "sentence_range": "4055-4058", "Text": "The resistance of this arrangement is,\nRG rs / (RG  + rs)   \u2243  rs         if    RG >>  rs\nIf rs has small value, in relation to the resistance of the rest of the\ncircuit Rc, the effect of introducing the measuring instrument is also small\nand negligible This arrangement is schematically shown in Fig 4 21"}, {"Chapter": "1", "sentence_range": "4056-4059", "Text": "This arrangement is schematically shown in Fig 4 21 The scale of this ammeter is calibrated and then graduated to read off\nthe current value with ease"}, {"Chapter": "1", "sentence_range": "4057-4060", "Text": "4 21 The scale of this ammeter is calibrated and then graduated to read off\nthe current value with ease We define the current sensitivity of the\ngalvanometer as the deflection per unit current"}, {"Chapter": "1", "sentence_range": "4058-4061", "Text": "21 The scale of this ammeter is calibrated and then graduated to read off\nthe current value with ease We define the current sensitivity of the\ngalvanometer as the deflection per unit current From Eq"}, {"Chapter": "1", "sentence_range": "4059-4062", "Text": "The scale of this ammeter is calibrated and then graduated to read off\nthe current value with ease We define the current sensitivity of the\ngalvanometer as the deflection per unit current From Eq (4"}, {"Chapter": "1", "sentence_range": "4060-4063", "Text": "We define the current sensitivity of the\ngalvanometer as the deflection per unit current From Eq (4 38) this\ncurrent sensitivity is,\nNAB\nI\nk\n\u03c6 =\n(4"}, {"Chapter": "1", "sentence_range": "4061-4064", "Text": "From Eq (4 38) this\ncurrent sensitivity is,\nNAB\nI\nk\n\u03c6 =\n(4 39)\nA convenient way for the manufacturer  to increase the sensitivity is\nto increase the number of turns N"}, {"Chapter": "1", "sentence_range": "4062-4065", "Text": "(4 38) this\ncurrent sensitivity is,\nNAB\nI\nk\n\u03c6 =\n(4 39)\nA convenient way for the manufacturer  to increase the sensitivity is\nto increase the number of turns N We choose galvanometers having\nsensitivities of value, required by our experiment"}, {"Chapter": "1", "sentence_range": "4063-4066", "Text": "38) this\ncurrent sensitivity is,\nNAB\nI\nk\n\u03c6 =\n(4 39)\nA convenient way for the manufacturer  to increase the sensitivity is\nto increase the number of turns N We choose galvanometers having\nsensitivities of value, required by our experiment FIGURE 4"}, {"Chapter": "1", "sentence_range": "4064-4067", "Text": "39)\nA convenient way for the manufacturer  to increase the sensitivity is\nto increase the number of turns N We choose galvanometers having\nsensitivities of value, required by our experiment FIGURE 4 20 The moving coil\ngalvanometer"}, {"Chapter": "1", "sentence_range": "4065-4068", "Text": "We choose galvanometers having\nsensitivities of value, required by our experiment FIGURE 4 20 The moving coil\ngalvanometer Its elements are\ndescribed in the text"}, {"Chapter": "1", "sentence_range": "4066-4069", "Text": "FIGURE 4 20 The moving coil\ngalvanometer Its elements are\ndescribed in the text Depending on\nthe requirement, this device can be\nused as a current detector or for\nmeasuring the value of the current\n(ammeter) or voltage (voltmeter)"}, {"Chapter": "1", "sentence_range": "4067-4070", "Text": "20 The moving coil\ngalvanometer Its elements are\ndescribed in the text Depending on\nthe requirement, this device can be\nused as a current detector or for\nmeasuring the value of the current\n(ammeter) or voltage (voltmeter) FIGURE 4"}, {"Chapter": "1", "sentence_range": "4068-4071", "Text": "Its elements are\ndescribed in the text Depending on\nthe requirement, this device can be\nused as a current detector or for\nmeasuring the value of the current\n(ammeter) or voltage (voltmeter) FIGURE 4 21\nConversion of a\ngalvanometer (G) to\nan ammeter by the\nintroduction of a\nshunt resistance rs of\nvery small value in\nparallel"}, {"Chapter": "1", "sentence_range": "4069-4072", "Text": "Depending on\nthe requirement, this device can be\nused as a current detector or for\nmeasuring the value of the current\n(ammeter) or voltage (voltmeter) FIGURE 4 21\nConversion of a\ngalvanometer (G) to\nan ammeter by the\nintroduction of a\nshunt resistance rs of\nvery small value in\nparallel Rationalised 2023-24\n131\nMoving Charges and\nMagnetism\nThe galvanometer can also be used as a voltmeter to measure the\nvoltage across a given section of the circuit"}, {"Chapter": "1", "sentence_range": "4070-4073", "Text": "FIGURE 4 21\nConversion of a\ngalvanometer (G) to\nan ammeter by the\nintroduction of a\nshunt resistance rs of\nvery small value in\nparallel Rationalised 2023-24\n131\nMoving Charges and\nMagnetism\nThe galvanometer can also be used as a voltmeter to measure the\nvoltage across a given section of the circuit For this it must be connected\nin parallel with that section of the circuit"}, {"Chapter": "1", "sentence_range": "4071-4074", "Text": "21\nConversion of a\ngalvanometer (G) to\nan ammeter by the\nintroduction of a\nshunt resistance rs of\nvery small value in\nparallel Rationalised 2023-24\n131\nMoving Charges and\nMagnetism\nThe galvanometer can also be used as a voltmeter to measure the\nvoltage across a given section of the circuit For this it must be connected\nin parallel with that section of the circuit Further, it must draw a very\nsmall current, otherwise the voltage measurement will disturb the original\nset up by an amount which is very large"}, {"Chapter": "1", "sentence_range": "4072-4075", "Text": "Rationalised 2023-24\n131\nMoving Charges and\nMagnetism\nThe galvanometer can also be used as a voltmeter to measure the\nvoltage across a given section of the circuit For this it must be connected\nin parallel with that section of the circuit Further, it must draw a very\nsmall current, otherwise the voltage measurement will disturb the original\nset up by an amount which is very large Usually we like to keep the\ndisturbance due to the measuring device below one per cent"}, {"Chapter": "1", "sentence_range": "4073-4076", "Text": "For this it must be connected\nin parallel with that section of the circuit Further, it must draw a very\nsmall current, otherwise the voltage measurement will disturb the original\nset up by an amount which is very large Usually we like to keep the\ndisturbance due to the measuring device below one per cent To ensure\nthis, a large resistance R is connected in series with the galvanometer"}, {"Chapter": "1", "sentence_range": "4074-4077", "Text": "Further, it must draw a very\nsmall current, otherwise the voltage measurement will disturb the original\nset up by an amount which is very large Usually we like to keep the\ndisturbance due to the measuring device below one per cent To ensure\nthis, a large resistance R is connected in series with the galvanometer This arrangement is schematically depicted in Fig"}, {"Chapter": "1", "sentence_range": "4075-4078", "Text": "Usually we like to keep the\ndisturbance due to the measuring device below one per cent To ensure\nthis, a large resistance R is connected in series with the galvanometer This arrangement is schematically depicted in Fig 4"}, {"Chapter": "1", "sentence_range": "4076-4079", "Text": "To ensure\nthis, a large resistance R is connected in series with the galvanometer This arrangement is schematically depicted in Fig 4 22"}, {"Chapter": "1", "sentence_range": "4077-4080", "Text": "This arrangement is schematically depicted in Fig 4 22 Note that the\nresistance of the voltmeter is now,\nRG + R \u2243 R :  large\nThe scale of the voltmeter is calibrated to read off the voltage value\nwith ease"}, {"Chapter": "1", "sentence_range": "4078-4081", "Text": "4 22 Note that the\nresistance of the voltmeter is now,\nRG + R \u2243 R :  large\nThe scale of the voltmeter is calibrated to read off the voltage value\nwith ease We define the voltage sensitivity as the deflection per unit\nvoltage"}, {"Chapter": "1", "sentence_range": "4079-4082", "Text": "22 Note that the\nresistance of the voltmeter is now,\nRG + R \u2243 R :  large\nThe scale of the voltmeter is calibrated to read off the voltage value\nwith ease We define the voltage sensitivity as the deflection per unit\nvoltage From Eq"}, {"Chapter": "1", "sentence_range": "4080-4083", "Text": "Note that the\nresistance of the voltmeter is now,\nRG + R \u2243 R :  large\nThe scale of the voltmeter is calibrated to read off the voltage value\nwith ease We define the voltage sensitivity as the deflection per unit\nvoltage From Eq (4"}, {"Chapter": "1", "sentence_range": "4081-4084", "Text": "We define the voltage sensitivity as the deflection per unit\nvoltage From Eq (4 38),\nV\u03c6\nNAB\nk\nVI\nNAB\nk\nR\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\n1\n(4"}, {"Chapter": "1", "sentence_range": "4082-4085", "Text": "From Eq (4 38),\nV\u03c6\nNAB\nk\nVI\nNAB\nk\nR\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\n1\n(4 40)\nAn interesting point to note is that increasing the current sensitivity\nmay not necessarily increase the voltage sensitivity"}, {"Chapter": "1", "sentence_range": "4083-4086", "Text": "(4 38),\nV\u03c6\nNAB\nk\nVI\nNAB\nk\nR\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\n1\n(4 40)\nAn interesting point to note is that increasing the current sensitivity\nmay not necessarily increase the voltage sensitivity Let us take Eq"}, {"Chapter": "1", "sentence_range": "4084-4087", "Text": "38),\nV\u03c6\nNAB\nk\nVI\nNAB\nk\nR\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\n1\n(4 40)\nAn interesting point to note is that increasing the current sensitivity\nmay not necessarily increase the voltage sensitivity Let us take Eq (4"}, {"Chapter": "1", "sentence_range": "4085-4088", "Text": "40)\nAn interesting point to note is that increasing the current sensitivity\nmay not necessarily increase the voltage sensitivity Let us take Eq (4 39)\nwhich provides a measure of current sensitivity"}, {"Chapter": "1", "sentence_range": "4086-4089", "Text": "Let us take Eq (4 39)\nwhich provides a measure of current sensitivity If N \u00ae 2N, i"}, {"Chapter": "1", "sentence_range": "4087-4090", "Text": "(4 39)\nwhich provides a measure of current sensitivity If N \u00ae 2N, i e"}, {"Chapter": "1", "sentence_range": "4088-4091", "Text": "39)\nwhich provides a measure of current sensitivity If N \u00ae 2N, i e , we double\nthe number of turns, then\n2\nI\nI\n\u03c6\n\u03c6\n\u2192\nThus, the current sensitivity doubles"}, {"Chapter": "1", "sentence_range": "4089-4092", "Text": "If N \u00ae 2N, i e , we double\nthe number of turns, then\n2\nI\nI\n\u03c6\n\u03c6\n\u2192\nThus, the current sensitivity doubles However, the resistance of the\ngalvanometer is also likely to double, since it is proportional to the length\nof the wire"}, {"Chapter": "1", "sentence_range": "4090-4093", "Text": "e , we double\nthe number of turns, then\n2\nI\nI\n\u03c6\n\u03c6\n\u2192\nThus, the current sensitivity doubles However, the resistance of the\ngalvanometer is also likely to double, since it is proportional to the length\nof the wire In Eq"}, {"Chapter": "1", "sentence_range": "4091-4094", "Text": ", we double\nthe number of turns, then\n2\nI\nI\n\u03c6\n\u03c6\n\u2192\nThus, the current sensitivity doubles However, the resistance of the\ngalvanometer is also likely to double, since it is proportional to the length\nof the wire In Eq (4"}, {"Chapter": "1", "sentence_range": "4092-4095", "Text": "However, the resistance of the\ngalvanometer is also likely to double, since it is proportional to the length\nof the wire In Eq (4 40), N \u00ae2N, and R \u00ae2R, thus the voltage sensitivity,\nV\nV\n\u03c6\n\u03c6\n\u2192\nremains unchanged"}, {"Chapter": "1", "sentence_range": "4093-4096", "Text": "In Eq (4 40), N \u00ae2N, and R \u00ae2R, thus the voltage sensitivity,\nV\nV\n\u03c6\n\u03c6\n\u2192\nremains unchanged So in general, the modification needed for conversion\nof a galvanometer to an ammeter  will be different from what is needed for\nconverting it into a voltmeter"}, {"Chapter": "1", "sentence_range": "4094-4097", "Text": "(4 40), N \u00ae2N, and R \u00ae2R, thus the voltage sensitivity,\nV\nV\n\u03c6\n\u03c6\n\u2192\nremains unchanged So in general, the modification needed for conversion\nof a galvanometer to an ammeter  will be different from what is needed for\nconverting it into a voltmeter Example 4"}, {"Chapter": "1", "sentence_range": "4095-4098", "Text": "40), N \u00ae2N, and R \u00ae2R, thus the voltage sensitivity,\nV\nV\n\u03c6\n\u03c6\n\u2192\nremains unchanged So in general, the modification needed for conversion\nof a galvanometer to an ammeter  will be different from what is needed for\nconverting it into a voltmeter Example 4 13 In the circuit (Fig"}, {"Chapter": "1", "sentence_range": "4096-4099", "Text": "So in general, the modification needed for conversion\nof a galvanometer to an ammeter  will be different from what is needed for\nconverting it into a voltmeter Example 4 13 In the circuit (Fig 4"}, {"Chapter": "1", "sentence_range": "4097-4100", "Text": "Example 4 13 In the circuit (Fig 4 23) the current is to be\nmeasured"}, {"Chapter": "1", "sentence_range": "4098-4101", "Text": "13 In the circuit (Fig 4 23) the current is to be\nmeasured What is the value of the current if the ammeter shown\n(a) is a galvanometer with a resistance RG = 60"}, {"Chapter": "1", "sentence_range": "4099-4102", "Text": "4 23) the current is to be\nmeasured What is the value of the current if the ammeter shown\n(a) is a galvanometer with a resistance RG = 60 00 W; (b) is a\ngalvanometer described in (a) but converted to an ammeter by a\nshunt resistance rs = 0"}, {"Chapter": "1", "sentence_range": "4100-4103", "Text": "23) the current is to be\nmeasured What is the value of the current if the ammeter shown\n(a) is a galvanometer with a resistance RG = 60 00 W; (b) is a\ngalvanometer described in (a) but converted to an ammeter by a\nshunt resistance rs = 0 02 W; (c) is an ideal ammeter with zero\nresistance"}, {"Chapter": "1", "sentence_range": "4101-4104", "Text": "What is the value of the current if the ammeter shown\n(a) is a galvanometer with a resistance RG = 60 00 W; (b) is a\ngalvanometer described in (a) but converted to an ammeter by a\nshunt resistance rs = 0 02 W; (c) is an ideal ammeter with zero\nresistance FIGURE 4"}, {"Chapter": "1", "sentence_range": "4102-4105", "Text": "00 W; (b) is a\ngalvanometer described in (a) but converted to an ammeter by a\nshunt resistance rs = 0 02 W; (c) is an ideal ammeter with zero\nresistance FIGURE 4 23\nFIGURE 4"}, {"Chapter": "1", "sentence_range": "4103-4106", "Text": "02 W; (c) is an ideal ammeter with zero\nresistance FIGURE 4 23\nFIGURE 4 22\nConversion of a\ngalvanometer (G) to a\nvoltmeter by the\nintroduction of a\nresistance R of large\nvalue in series"}, {"Chapter": "1", "sentence_range": "4104-4107", "Text": "FIGURE 4 23\nFIGURE 4 22\nConversion of a\ngalvanometer (G) to a\nvoltmeter by the\nintroduction of a\nresistance R of large\nvalue in series EXAMPLE 4"}, {"Chapter": "1", "sentence_range": "4105-4108", "Text": "23\nFIGURE 4 22\nConversion of a\ngalvanometer (G) to a\nvoltmeter by the\nintroduction of a\nresistance R of large\nvalue in series EXAMPLE 4 13\nRationalised 2023-24\nPhysics\n132\nSUMMARY\n1"}, {"Chapter": "1", "sentence_range": "4106-4109", "Text": "22\nConversion of a\ngalvanometer (G) to a\nvoltmeter by the\nintroduction of a\nresistance R of large\nvalue in series EXAMPLE 4 13\nRationalised 2023-24\nPhysics\n132\nSUMMARY\n1 The total force on a charge q moving with velocity v in the presence of\nmagnetic and electric fields B and E, respectively is called the Lorentz\nforce"}, {"Chapter": "1", "sentence_range": "4107-4110", "Text": "EXAMPLE 4 13\nRationalised 2023-24\nPhysics\n132\nSUMMARY\n1 The total force on a charge q moving with velocity v in the presence of\nmagnetic and electric fields B and E, respectively is called the Lorentz\nforce It is given by the expression:\nF = q (v \u00d7 B + E)\nThe magnetic force q (v \u00d7 B) is normal to v and work done by it is zero"}, {"Chapter": "1", "sentence_range": "4108-4111", "Text": "13\nRationalised 2023-24\nPhysics\n132\nSUMMARY\n1 The total force on a charge q moving with velocity v in the presence of\nmagnetic and electric fields B and E, respectively is called the Lorentz\nforce It is given by the expression:\nF = q (v \u00d7 B + E)\nThe magnetic force q (v \u00d7 B) is normal to v and work done by it is zero 2"}, {"Chapter": "1", "sentence_range": "4109-4112", "Text": "The total force on a charge q moving with velocity v in the presence of\nmagnetic and electric fields B and E, respectively is called the Lorentz\nforce It is given by the expression:\nF = q (v \u00d7 B + E)\nThe magnetic force q (v \u00d7 B) is normal to v and work done by it is zero 2 A straight conductor of length l and carrying a steady current I\nexperiences a force F in a uniform external magnetic field B,\nF = I l \u00d7 B\nwhere|l| = l and the direction of l is given by the direction of the\ncurrent"}, {"Chapter": "1", "sentence_range": "4110-4113", "Text": "It is given by the expression:\nF = q (v \u00d7 B + E)\nThe magnetic force q (v \u00d7 B) is normal to v and work done by it is zero 2 A straight conductor of length l and carrying a steady current I\nexperiences a force F in a uniform external magnetic field B,\nF = I l \u00d7 B\nwhere|l| = l and the direction of l is given by the direction of the\ncurrent 3"}, {"Chapter": "1", "sentence_range": "4111-4114", "Text": "2 A straight conductor of length l and carrying a steady current I\nexperiences a force F in a uniform external magnetic field B,\nF = I l \u00d7 B\nwhere|l| = l and the direction of l is given by the direction of the\ncurrent 3 In a uniform magnetic field B, a charge q executes a circular orbit in\na plane normal to B"}, {"Chapter": "1", "sentence_range": "4112-4115", "Text": "A straight conductor of length l and carrying a steady current I\nexperiences a force F in a uniform external magnetic field B,\nF = I l \u00d7 B\nwhere|l| = l and the direction of l is given by the direction of the\ncurrent 3 In a uniform magnetic field B, a charge q executes a circular orbit in\na plane normal to B Its frequency of uniform circular motion is called\nthe cyclotron frequency and is given by:\n2\nc\nq B\nm\n\u03bd =\n\u03c0\nThis frequency is independent of the particle\u2019s speed and radius"}, {"Chapter": "1", "sentence_range": "4113-4116", "Text": "3 In a uniform magnetic field B, a charge q executes a circular orbit in\na plane normal to B Its frequency of uniform circular motion is called\nthe cyclotron frequency and is given by:\n2\nc\nq B\nm\n\u03bd =\n\u03c0\nThis frequency is independent of the particle\u2019s speed and radius This\nfact is exploited in a machine, the cyclotron, which is used to\naccelerate charged particles"}, {"Chapter": "1", "sentence_range": "4114-4117", "Text": "In a uniform magnetic field B, a charge q executes a circular orbit in\na plane normal to B Its frequency of uniform circular motion is called\nthe cyclotron frequency and is given by:\n2\nc\nq B\nm\n\u03bd =\n\u03c0\nThis frequency is independent of the particle\u2019s speed and radius This\nfact is exploited in a machine, the cyclotron, which is used to\naccelerate charged particles 4"}, {"Chapter": "1", "sentence_range": "4115-4118", "Text": "Its frequency of uniform circular motion is called\nthe cyclotron frequency and is given by:\n2\nc\nq B\nm\n\u03bd =\n\u03c0\nThis frequency is independent of the particle\u2019s speed and radius This\nfact is exploited in a machine, the cyclotron, which is used to\naccelerate charged particles 4 The Biot-Savart law asserts that the magnetic field dB due to an\nelement dl carrying a steady current I at a point P at a distance r from\nthe current element is:\n0\nd3\nd\n4\nI\nr\n\u00b5\n\u00d7\n=\n\u03c0\nl\nr\nB\nTo obtain the total field at P, we must integrate this vector expression\nover the entire length of the conductor"}, {"Chapter": "1", "sentence_range": "4116-4119", "Text": "This\nfact is exploited in a machine, the cyclotron, which is used to\naccelerate charged particles 4 The Biot-Savart law asserts that the magnetic field dB due to an\nelement dl carrying a steady current I at a point P at a distance r from\nthe current element is:\n0\nd3\nd\n4\nI\nr\n\u00b5\n\u00d7\n=\n\u03c0\nl\nr\nB\nTo obtain the total field at P, we must integrate this vector expression\nover the entire length of the conductor 5"}, {"Chapter": "1", "sentence_range": "4117-4120", "Text": "4 The Biot-Savart law asserts that the magnetic field dB due to an\nelement dl carrying a steady current I at a point P at a distance r from\nthe current element is:\n0\nd3\nd\n4\nI\nr\n\u00b5\n\u00d7\n=\n\u03c0\nl\nr\nB\nTo obtain the total field at P, we must integrate this vector expression\nover the entire length of the conductor 5 The magnitude of the magnetic field due to a circular coil of radius R\ncarrying a current I at an axial distance x from the centre is\nEXAMPLE 4"}, {"Chapter": "1", "sentence_range": "4118-4121", "Text": "The Biot-Savart law asserts that the magnetic field dB due to an\nelement dl carrying a steady current I at a point P at a distance r from\nthe current element is:\n0\nd3\nd\n4\nI\nr\n\u00b5\n\u00d7\n=\n\u03c0\nl\nr\nB\nTo obtain the total field at P, we must integrate this vector expression\nover the entire length of the conductor 5 The magnitude of the magnetic field due to a circular coil of radius R\ncarrying a current I at an axial distance x from the centre is\nEXAMPLE 4 13\nSolution\n(a) Total resistance in the circuit is,\n3\n63\nRG\n+\n=\n\u2126"}, {"Chapter": "1", "sentence_range": "4119-4122", "Text": "5 The magnitude of the magnetic field due to a circular coil of radius R\ncarrying a current I at an axial distance x from the centre is\nEXAMPLE 4 13\nSolution\n(a) Total resistance in the circuit is,\n3\n63\nRG\n+\n=\n\u2126 Hence, I = 3/63 = 0"}, {"Chapter": "1", "sentence_range": "4120-4123", "Text": "The magnitude of the magnetic field due to a circular coil of radius R\ncarrying a current I at an axial distance x from the centre is\nEXAMPLE 4 13\nSolution\n(a) Total resistance in the circuit is,\n3\n63\nRG\n+\n=\n\u2126 Hence, I = 3/63 = 0 048 A"}, {"Chapter": "1", "sentence_range": "4121-4124", "Text": "13\nSolution\n(a) Total resistance in the circuit is,\n3\n63\nRG\n+\n=\n\u2126 Hence, I = 3/63 = 0 048 A (b) Resistance of the galvanometer converted to an ammeter is,\nR\nr\nR\nr\nG\ns\nG\n+s\n=\n+\u00d7\n60\n0 02\n60\n\u21260 02"}, {"Chapter": "1", "sentence_range": "4122-4125", "Text": "Hence, I = 3/63 = 0 048 A (b) Resistance of the galvanometer converted to an ammeter is,\nR\nr\nR\nr\nG\ns\nG\n+s\n=\n+\u00d7\n60\n0 02\n60\n\u21260 02 \u2126\u2126\n("}, {"Chapter": "1", "sentence_range": "4123-4126", "Text": "048 A (b) Resistance of the galvanometer converted to an ammeter is,\nR\nr\nR\nr\nG\ns\nG\n+s\n=\n+\u00d7\n60\n0 02\n60\n\u21260 02 \u2126\u2126\n( )\n \u2243 0"}, {"Chapter": "1", "sentence_range": "4124-4127", "Text": "(b) Resistance of the galvanometer converted to an ammeter is,\nR\nr\nR\nr\nG\ns\nG\n+s\n=\n+\u00d7\n60\n0 02\n60\n\u21260 02 \u2126\u2126\n( )\n \u2243 0 02W\nTotal resistance in the circuit is,\n0"}, {"Chapter": "1", "sentence_range": "4125-4128", "Text": "\u2126\u2126\n( )\n \u2243 0 02W\nTotal resistance in the circuit is,\n0 02\n3\n3"}, {"Chapter": "1", "sentence_range": "4126-4129", "Text": ")\n \u2243 0 02W\nTotal resistance in the circuit is,\n0 02\n3\n3 02\n\u2126 +\n\u2126 =\n\u2126"}, {"Chapter": "1", "sentence_range": "4127-4130", "Text": "02W\nTotal resistance in the circuit is,\n0 02\n3\n3 02\n\u2126 +\n\u2126 =\n\u2126 Hence, I = 3/3"}, {"Chapter": "1", "sentence_range": "4128-4131", "Text": "02\n3\n3 02\n\u2126 +\n\u2126 =\n\u2126 Hence, I = 3/3 02 = 0"}, {"Chapter": "1", "sentence_range": "4129-4132", "Text": "02\n\u2126 +\n\u2126 =\n\u2126 Hence, I = 3/3 02 = 0 99 A"}, {"Chapter": "1", "sentence_range": "4130-4133", "Text": "Hence, I = 3/3 02 = 0 99 A (c) For the ideal ammeter with zero resistance,\n I =  3/3 = 1"}, {"Chapter": "1", "sentence_range": "4131-4134", "Text": "02 = 0 99 A (c) For the ideal ammeter with zero resistance,\n I =  3/3 = 1 00 A\nRationalised 2023-24\n133\nMoving Charges and\nMagnetism\n2\n0\n2\n2 3/2\n2(\n)\nIR\nB\nx\nR\n\u00b5\n=\n+\nAt the centre this reduces to\n0\n2\nI\nB\nR\n\u00b5\n=\n6"}, {"Chapter": "1", "sentence_range": "4132-4135", "Text": "99 A (c) For the ideal ammeter with zero resistance,\n I =  3/3 = 1 00 A\nRationalised 2023-24\n133\nMoving Charges and\nMagnetism\n2\n0\n2\n2 3/2\n2(\n)\nIR\nB\nx\nR\n\u00b5\n=\n+\nAt the centre this reduces to\n0\n2\nI\nB\nR\n\u00b5\n=\n6 Ampere\u2019s Circuital Law:  Let an open surface S be bounded by a loop\nC"}, {"Chapter": "1", "sentence_range": "4133-4136", "Text": "(c) For the ideal ammeter with zero resistance,\n I =  3/3 = 1 00 A\nRationalised 2023-24\n133\nMoving Charges and\nMagnetism\n2\n0\n2\n2 3/2\n2(\n)\nIR\nB\nx\nR\n\u00b5\n=\n+\nAt the centre this reduces to\n0\n2\nI\nB\nR\n\u00b5\n=\n6 Ampere\u2019s Circuital Law:  Let an open surface S be bounded by a loop\nC Then the Ampere\u2019s law states that \nB"}, {"Chapter": "1", "sentence_range": "4134-4137", "Text": "00 A\nRationalised 2023-24\n133\nMoving Charges and\nMagnetism\n2\n0\n2\n2 3/2\n2(\n)\nIR\nB\nx\nR\n\u00b5\n=\n+\nAt the centre this reduces to\n0\n2\nI\nB\nR\n\u00b5\n=\n6 Ampere\u2019s Circuital Law:  Let an open surface S be bounded by a loop\nC Then the Ampere\u2019s law states that \nB dl\nI\n=\n\u222b\n\u00b50\nC\ufffd\n where I refers to\nthe current passing through S"}, {"Chapter": "1", "sentence_range": "4135-4138", "Text": "Ampere\u2019s Circuital Law:  Let an open surface S be bounded by a loop\nC Then the Ampere\u2019s law states that \nB dl\nI\n=\n\u222b\n\u00b50\nC\ufffd\n where I refers to\nthe current passing through S The sign of I is determined from the\nright-hand rule"}, {"Chapter": "1", "sentence_range": "4136-4139", "Text": "Then the Ampere\u2019s law states that \nB dl\nI\n=\n\u222b\n\u00b50\nC\ufffd\n where I refers to\nthe current passing through S The sign of I is determined from the\nright-hand rule We have discussed a simplified form of this law"}, {"Chapter": "1", "sentence_range": "4137-4140", "Text": "dl\nI\n=\n\u222b\n\u00b50\nC\ufffd\n where I refers to\nthe current passing through S The sign of I is determined from the\nright-hand rule We have discussed a simplified form of this law If B\nis directed along the tangent to every point on the perimeter L of a\nclosed curve and is constant in magnitude along perimeter then,\nBL = m0 Ie\nwhere Ie is the net current enclosed by the closed circuit"}, {"Chapter": "1", "sentence_range": "4138-4141", "Text": "The sign of I is determined from the\nright-hand rule We have discussed a simplified form of this law If B\nis directed along the tangent to every point on the perimeter L of a\nclosed curve and is constant in magnitude along perimeter then,\nBL = m0 Ie\nwhere Ie is the net current enclosed by the closed circuit 7"}, {"Chapter": "1", "sentence_range": "4139-4142", "Text": "We have discussed a simplified form of this law If B\nis directed along the tangent to every point on the perimeter L of a\nclosed curve and is constant in magnitude along perimeter then,\nBL = m0 Ie\nwhere Ie is the net current enclosed by the closed circuit 7 The magnitude of the magnetic field at a distance R from a long,\nstraight wire carrying a current I is given by:\n\u03c0\n0\n2\nI\nB\nR\n\u00b5\n=\nThe field lines are circles concentric with the wire"}, {"Chapter": "1", "sentence_range": "4140-4143", "Text": "If B\nis directed along the tangent to every point on the perimeter L of a\nclosed curve and is constant in magnitude along perimeter then,\nBL = m0 Ie\nwhere Ie is the net current enclosed by the closed circuit 7 The magnitude of the magnetic field at a distance R from a long,\nstraight wire carrying a current I is given by:\n\u03c0\n0\n2\nI\nB\nR\n\u00b5\n=\nThe field lines are circles concentric with the wire 8"}, {"Chapter": "1", "sentence_range": "4141-4144", "Text": "7 The magnitude of the magnetic field at a distance R from a long,\nstraight wire carrying a current I is given by:\n\u03c0\n0\n2\nI\nB\nR\n\u00b5\n=\nThe field lines are circles concentric with the wire 8 The magnitude of the field B inside a long solenoid carrying a current\nI is\nB = m0nI\nwhere n is the number of turns per unit length"}, {"Chapter": "1", "sentence_range": "4142-4145", "Text": "The magnitude of the magnetic field at a distance R from a long,\nstraight wire carrying a current I is given by:\n\u03c0\n0\n2\nI\nB\nR\n\u00b5\n=\nThe field lines are circles concentric with the wire 8 The magnitude of the field B inside a long solenoid carrying a current\nI is\nB = m0nI\nwhere n is the number of turns per unit length where N is the total number of turns and r is the average radius"}, {"Chapter": "1", "sentence_range": "4143-4146", "Text": "8 The magnitude of the field B inside a long solenoid carrying a current\nI is\nB = m0nI\nwhere n is the number of turns per unit length where N is the total number of turns and r is the average radius 9"}, {"Chapter": "1", "sentence_range": "4144-4147", "Text": "The magnitude of the field B inside a long solenoid carrying a current\nI is\nB = m0nI\nwhere n is the number of turns per unit length where N is the total number of turns and r is the average radius 9 Parallel currents attract and anti-parallel currents repel"}, {"Chapter": "1", "sentence_range": "4145-4148", "Text": "where N is the total number of turns and r is the average radius 9 Parallel currents attract and anti-parallel currents repel 10"}, {"Chapter": "1", "sentence_range": "4146-4149", "Text": "9 Parallel currents attract and anti-parallel currents repel 10 A planar loop carrying a current I, having N closely wound turns, and\nan area A possesses a magnetic moment m where,\nm = N I A\nand the direction of m is given by the right-hand thumb rule : curl\nthe palm of your right hand along the loop with the fingers pointing\nin the direction of the current"}, {"Chapter": "1", "sentence_range": "4147-4150", "Text": "Parallel currents attract and anti-parallel currents repel 10 A planar loop carrying a current I, having N closely wound turns, and\nan area A possesses a magnetic moment m where,\nm = N I A\nand the direction of m is given by the right-hand thumb rule : curl\nthe palm of your right hand along the loop with the fingers pointing\nin the direction of the current The thumb sticking out gives the\ndirection of m (and A)\nWhen this loop is placed in a uniform magnetic field B, the force F on\nit is:  F = 0\nAnd the torque on it is,\nt = m \u00d7 B\nIn a moving coil galvanometer, this torque is balanced by a counter-\ntorque due to a spring, yielding\nkf = NI AB\nwhere f  is the equilibrium deflection and k the torsion constant of\nthe spring"}, {"Chapter": "1", "sentence_range": "4148-4151", "Text": "10 A planar loop carrying a current I, having N closely wound turns, and\nan area A possesses a magnetic moment m where,\nm = N I A\nand the direction of m is given by the right-hand thumb rule : curl\nthe palm of your right hand along the loop with the fingers pointing\nin the direction of the current The thumb sticking out gives the\ndirection of m (and A)\nWhen this loop is placed in a uniform magnetic field B, the force F on\nit is:  F = 0\nAnd the torque on it is,\nt = m \u00d7 B\nIn a moving coil galvanometer, this torque is balanced by a counter-\ntorque due to a spring, yielding\nkf = NI AB\nwhere f  is the equilibrium deflection and k the torsion constant of\nthe spring 11"}, {"Chapter": "1", "sentence_range": "4149-4152", "Text": "A planar loop carrying a current I, having N closely wound turns, and\nan area A possesses a magnetic moment m where,\nm = N I A\nand the direction of m is given by the right-hand thumb rule : curl\nthe palm of your right hand along the loop with the fingers pointing\nin the direction of the current The thumb sticking out gives the\ndirection of m (and A)\nWhen this loop is placed in a uniform magnetic field B, the force F on\nit is:  F = 0\nAnd the torque on it is,\nt = m \u00d7 B\nIn a moving coil galvanometer, this torque is balanced by a counter-\ntorque due to a spring, yielding\nkf = NI AB\nwhere f  is the equilibrium deflection and k the torsion constant of\nthe spring 11 A moving coil galvanometer can be converted into a ammeter by\nintroducing a shunt resistance rs, of small value in parallel"}, {"Chapter": "1", "sentence_range": "4150-4153", "Text": "The thumb sticking out gives the\ndirection of m (and A)\nWhen this loop is placed in a uniform magnetic field B, the force F on\nit is:  F = 0\nAnd the torque on it is,\nt = m \u00d7 B\nIn a moving coil galvanometer, this torque is balanced by a counter-\ntorque due to a spring, yielding\nkf = NI AB\nwhere f  is the equilibrium deflection and k the torsion constant of\nthe spring 11 A moving coil galvanometer can be converted into a ammeter by\nintroducing a shunt resistance rs, of small value in parallel It can be\nconverted into a voltmeter by introducing a resistance of a large value\nin series"}, {"Chapter": "1", "sentence_range": "4151-4154", "Text": "11 A moving coil galvanometer can be converted into a ammeter by\nintroducing a shunt resistance rs, of small value in parallel It can be\nconverted into a voltmeter by introducing a resistance of a large value\nin series Rationalised 2023-24\nPhysics\n134\nPhysical Quantity\nSymbol\nNature\nDimensions\nUnits\nRemarks\nPermeability of free\nm0\nScalar\n[MLT \u20132A\u20132]\nT m A\u20131\n4p \u00b4 10\u20137 T m A\u20131\nspace\nMagnetic Field\nB\nVector\n[M T \u20132A\u20131]\nT (telsa)\nMagnetic Moment\nm\nVector\n[L2A]\nA m2 or J/T\nTorsion Constant\nk\n  Scalar\n[M L2T \u20132]\n    N m rad\u20131\nAppears in MCG\nPOINTS TO PONDER\n1"}, {"Chapter": "1", "sentence_range": "4152-4155", "Text": "A moving coil galvanometer can be converted into a ammeter by\nintroducing a shunt resistance rs, of small value in parallel It can be\nconverted into a voltmeter by introducing a resistance of a large value\nin series Rationalised 2023-24\nPhysics\n134\nPhysical Quantity\nSymbol\nNature\nDimensions\nUnits\nRemarks\nPermeability of free\nm0\nScalar\n[MLT \u20132A\u20132]\nT m A\u20131\n4p \u00b4 10\u20137 T m A\u20131\nspace\nMagnetic Field\nB\nVector\n[M T \u20132A\u20131]\nT (telsa)\nMagnetic Moment\nm\nVector\n[L2A]\nA m2 or J/T\nTorsion Constant\nk\n  Scalar\n[M L2T \u20132]\n    N m rad\u20131\nAppears in MCG\nPOINTS TO PONDER\n1 Electrostatic field lines originate at a positive charge and terminate at a\nnegative charge or fade at infinity"}, {"Chapter": "1", "sentence_range": "4153-4156", "Text": "It can be\nconverted into a voltmeter by introducing a resistance of a large value\nin series Rationalised 2023-24\nPhysics\n134\nPhysical Quantity\nSymbol\nNature\nDimensions\nUnits\nRemarks\nPermeability of free\nm0\nScalar\n[MLT \u20132A\u20132]\nT m A\u20131\n4p \u00b4 10\u20137 T m A\u20131\nspace\nMagnetic Field\nB\nVector\n[M T \u20132A\u20131]\nT (telsa)\nMagnetic Moment\nm\nVector\n[L2A]\nA m2 or J/T\nTorsion Constant\nk\n  Scalar\n[M L2T \u20132]\n    N m rad\u20131\nAppears in MCG\nPOINTS TO PONDER\n1 Electrostatic field lines originate at a positive charge and terminate at a\nnegative charge or fade at infinity Magnetic field lines always form\nclosed loops"}, {"Chapter": "1", "sentence_range": "4154-4157", "Text": "Rationalised 2023-24\nPhysics\n134\nPhysical Quantity\nSymbol\nNature\nDimensions\nUnits\nRemarks\nPermeability of free\nm0\nScalar\n[MLT \u20132A\u20132]\nT m A\u20131\n4p \u00b4 10\u20137 T m A\u20131\nspace\nMagnetic Field\nB\nVector\n[M T \u20132A\u20131]\nT (telsa)\nMagnetic Moment\nm\nVector\n[L2A]\nA m2 or J/T\nTorsion Constant\nk\n  Scalar\n[M L2T \u20132]\n    N m rad\u20131\nAppears in MCG\nPOINTS TO PONDER\n1 Electrostatic field lines originate at a positive charge and terminate at a\nnegative charge or fade at infinity Magnetic field lines always form\nclosed loops 2"}, {"Chapter": "1", "sentence_range": "4155-4158", "Text": "Electrostatic field lines originate at a positive charge and terminate at a\nnegative charge or fade at infinity Magnetic field lines always form\nclosed loops 2 The discussion in this Chapter holds only for steady currents which do\nnot vary with time"}, {"Chapter": "1", "sentence_range": "4156-4159", "Text": "Magnetic field lines always form\nclosed loops 2 The discussion in this Chapter holds only for steady currents which do\nnot vary with time When currents vary with time Newton\u2019s third law is valid only if momentum\ncarried by the electromagnetic field is taken into account"}, {"Chapter": "1", "sentence_range": "4157-4160", "Text": "2 The discussion in this Chapter holds only for steady currents which do\nnot vary with time When currents vary with time Newton\u2019s third law is valid only if momentum\ncarried by the electromagnetic field is taken into account 3"}, {"Chapter": "1", "sentence_range": "4158-4161", "Text": "The discussion in this Chapter holds only for steady currents which do\nnot vary with time When currents vary with time Newton\u2019s third law is valid only if momentum\ncarried by the electromagnetic field is taken into account 3 Recall the expression for the Lorentz force,\nF = q (v \u00d7 B + E)\nThis velocity dependent force has occupied the attention of some of the\ngreatest scientific thinkers"}, {"Chapter": "1", "sentence_range": "4159-4162", "Text": "When currents vary with time Newton\u2019s third law is valid only if momentum\ncarried by the electromagnetic field is taken into account 3 Recall the expression for the Lorentz force,\nF = q (v \u00d7 B + E)\nThis velocity dependent force has occupied the attention of some of the\ngreatest scientific thinkers If one switches to a frame with instantaneous\nvelocity v, the magnetic part of the force vanishes"}, {"Chapter": "1", "sentence_range": "4160-4163", "Text": "3 Recall the expression for the Lorentz force,\nF = q (v \u00d7 B + E)\nThis velocity dependent force has occupied the attention of some of the\ngreatest scientific thinkers If one switches to a frame with instantaneous\nvelocity v, the magnetic part of the force vanishes The motion of the\ncharged particle is then explained by arguing that there exists an\nappropriate electric field in the new frame"}, {"Chapter": "1", "sentence_range": "4161-4164", "Text": "Recall the expression for the Lorentz force,\nF = q (v \u00d7 B + E)\nThis velocity dependent force has occupied the attention of some of the\ngreatest scientific thinkers If one switches to a frame with instantaneous\nvelocity v, the magnetic part of the force vanishes The motion of the\ncharged particle is then explained by arguing that there exists an\nappropriate electric field in the new frame We shall not discuss the\ndetails of this mechanism"}, {"Chapter": "1", "sentence_range": "4162-4165", "Text": "If one switches to a frame with instantaneous\nvelocity v, the magnetic part of the force vanishes The motion of the\ncharged particle is then explained by arguing that there exists an\nappropriate electric field in the new frame We shall not discuss the\ndetails of this mechanism However, we stress that the resolution of this\nparadox implies that electricity and magnetism are linked phenomena\n(electromagnetism) and that the Lorentz force expression does not imply\na universal preferred frame of reference in nature"}, {"Chapter": "1", "sentence_range": "4163-4166", "Text": "The motion of the\ncharged particle is then explained by arguing that there exists an\nappropriate electric field in the new frame We shall not discuss the\ndetails of this mechanism However, we stress that the resolution of this\nparadox implies that electricity and magnetism are linked phenomena\n(electromagnetism) and that the Lorentz force expression does not imply\na universal preferred frame of reference in nature 4"}, {"Chapter": "1", "sentence_range": "4164-4167", "Text": "We shall not discuss the\ndetails of this mechanism However, we stress that the resolution of this\nparadox implies that electricity and magnetism are linked phenomena\n(electromagnetism) and that the Lorentz force expression does not imply\na universal preferred frame of reference in nature 4 Ampere\u2019s Circuital law is not independent of the Biot-Savart law"}, {"Chapter": "1", "sentence_range": "4165-4168", "Text": "However, we stress that the resolution of this\nparadox implies that electricity and magnetism are linked phenomena\n(electromagnetism) and that the Lorentz force expression does not imply\na universal preferred frame of reference in nature 4 Ampere\u2019s Circuital law is not independent of the Biot-Savart law It\ncan be derived from the Biot-Savart law"}, {"Chapter": "1", "sentence_range": "4166-4169", "Text": "4 Ampere\u2019s Circuital law is not independent of the Biot-Savart law It\ncan be derived from the Biot-Savart law Its relationship to the\nBiot-Savart law is similar to the relationship between Gauss\u2019s law and\nCoulomb\u2019s law"}, {"Chapter": "1", "sentence_range": "4167-4170", "Text": "Ampere\u2019s Circuital law is not independent of the Biot-Savart law It\ncan be derived from the Biot-Savart law Its relationship to the\nBiot-Savart law is similar to the relationship between Gauss\u2019s law and\nCoulomb\u2019s law EXERCISES\n4"}, {"Chapter": "1", "sentence_range": "4168-4171", "Text": "It\ncan be derived from the Biot-Savart law Its relationship to the\nBiot-Savart law is similar to the relationship between Gauss\u2019s law and\nCoulomb\u2019s law EXERCISES\n4 1\nA circular coil of wire consisting of 100 turns, each of radius 8"}, {"Chapter": "1", "sentence_range": "4169-4172", "Text": "Its relationship to the\nBiot-Savart law is similar to the relationship between Gauss\u2019s law and\nCoulomb\u2019s law EXERCISES\n4 1\nA circular coil of wire consisting of 100 turns, each of radius 8 0 cm\ncarries a current of 0"}, {"Chapter": "1", "sentence_range": "4170-4173", "Text": "EXERCISES\n4 1\nA circular coil of wire consisting of 100 turns, each of radius 8 0 cm\ncarries a current of 0 40 A"}, {"Chapter": "1", "sentence_range": "4171-4174", "Text": "1\nA circular coil of wire consisting of 100 turns, each of radius 8 0 cm\ncarries a current of 0 40 A What is the magnitude of the magnetic\nfield B at the centre of the coil"}, {"Chapter": "1", "sentence_range": "4172-4175", "Text": "0 cm\ncarries a current of 0 40 A What is the magnitude of the magnetic\nfield B at the centre of the coil 4"}, {"Chapter": "1", "sentence_range": "4173-4176", "Text": "40 A What is the magnitude of the magnetic\nfield B at the centre of the coil 4 2\nA long straight wire carries a current of 35 A"}, {"Chapter": "1", "sentence_range": "4174-4177", "Text": "What is the magnitude of the magnetic\nfield B at the centre of the coil 4 2\nA long straight wire carries a current of 35 A What is the magnitude\nof the field B at a point 20 cm from the wire"}, {"Chapter": "1", "sentence_range": "4175-4178", "Text": "4 2\nA long straight wire carries a current of 35 A What is the magnitude\nof the field B at a point 20 cm from the wire 4"}, {"Chapter": "1", "sentence_range": "4176-4179", "Text": "2\nA long straight wire carries a current of 35 A What is the magnitude\nof the field B at a point 20 cm from the wire 4 3\nA long straight wire in the horizontal plane carries a current of 50 A\nin north to south direction"}, {"Chapter": "1", "sentence_range": "4177-4180", "Text": "What is the magnitude\nof the field B at a point 20 cm from the wire 4 3\nA long straight wire in the horizontal plane carries a current of 50 A\nin north to south direction Give the magnitude and direction of B\nat a point 2"}, {"Chapter": "1", "sentence_range": "4178-4181", "Text": "4 3\nA long straight wire in the horizontal plane carries a current of 50 A\nin north to south direction Give the magnitude and direction of B\nat a point 2 5 m east of the wire"}, {"Chapter": "1", "sentence_range": "4179-4182", "Text": "3\nA long straight wire in the horizontal plane carries a current of 50 A\nin north to south direction Give the magnitude and direction of B\nat a point 2 5 m east of the wire Rationalised 2023-24\n135\nMoving Charges and\nMagnetism\n4"}, {"Chapter": "1", "sentence_range": "4180-4183", "Text": "Give the magnitude and direction of B\nat a point 2 5 m east of the wire Rationalised 2023-24\n135\nMoving Charges and\nMagnetism\n4 4\nA horizontal overhead power line carries a current of 90 A in east to\nwest direction"}, {"Chapter": "1", "sentence_range": "4181-4184", "Text": "5 m east of the wire Rationalised 2023-24\n135\nMoving Charges and\nMagnetism\n4 4\nA horizontal overhead power line carries a current of 90 A in east to\nwest direction What is the magnitude and direction of the magnetic\nfield due to the current 1"}, {"Chapter": "1", "sentence_range": "4182-4185", "Text": "Rationalised 2023-24\n135\nMoving Charges and\nMagnetism\n4 4\nA horizontal overhead power line carries a current of 90 A in east to\nwest direction What is the magnitude and direction of the magnetic\nfield due to the current 1 5 m below the line"}, {"Chapter": "1", "sentence_range": "4183-4186", "Text": "4\nA horizontal overhead power line carries a current of 90 A in east to\nwest direction What is the magnitude and direction of the magnetic\nfield due to the current 1 5 m below the line 4"}, {"Chapter": "1", "sentence_range": "4184-4187", "Text": "What is the magnitude and direction of the magnetic\nfield due to the current 1 5 m below the line 4 5\nWhat is the magnitude of magnetic force per unit length on a wire\ncarrying a current of 8 A and making an angle of 30\u00ba with the\ndirection of a uniform magnetic field of  0"}, {"Chapter": "1", "sentence_range": "4185-4188", "Text": "5 m below the line 4 5\nWhat is the magnitude of magnetic force per unit length on a wire\ncarrying a current of 8 A and making an angle of 30\u00ba with the\ndirection of a uniform magnetic field of  0 15 T"}, {"Chapter": "1", "sentence_range": "4186-4189", "Text": "4 5\nWhat is the magnitude of magnetic force per unit length on a wire\ncarrying a current of 8 A and making an angle of 30\u00ba with the\ndirection of a uniform magnetic field of  0 15 T 4"}, {"Chapter": "1", "sentence_range": "4187-4190", "Text": "5\nWhat is the magnitude of magnetic force per unit length on a wire\ncarrying a current of 8 A and making an angle of 30\u00ba with the\ndirection of a uniform magnetic field of  0 15 T 4 6\nA 3"}, {"Chapter": "1", "sentence_range": "4188-4191", "Text": "15 T 4 6\nA 3 0 cm wire carrying a current of 10 A is placed inside a solenoid\nperpendicular to its axis"}, {"Chapter": "1", "sentence_range": "4189-4192", "Text": "4 6\nA 3 0 cm wire carrying a current of 10 A is placed inside a solenoid\nperpendicular to its axis The magnetic field inside the solenoid is\ngiven to be 0"}, {"Chapter": "1", "sentence_range": "4190-4193", "Text": "6\nA 3 0 cm wire carrying a current of 10 A is placed inside a solenoid\nperpendicular to its axis The magnetic field inside the solenoid is\ngiven to be 0 27 T"}, {"Chapter": "1", "sentence_range": "4191-4194", "Text": "0 cm wire carrying a current of 10 A is placed inside a solenoid\nperpendicular to its axis The magnetic field inside the solenoid is\ngiven to be 0 27 T What is the magnetic force on the wire"}, {"Chapter": "1", "sentence_range": "4192-4195", "Text": "The magnetic field inside the solenoid is\ngiven to be 0 27 T What is the magnetic force on the wire 4"}, {"Chapter": "1", "sentence_range": "4193-4196", "Text": "27 T What is the magnetic force on the wire 4 7\nTwo long and parallel straight wires A and B carrying currents of\n8"}, {"Chapter": "1", "sentence_range": "4194-4197", "Text": "What is the magnetic force on the wire 4 7\nTwo long and parallel straight wires A and B carrying currents of\n8 0 A and 5"}, {"Chapter": "1", "sentence_range": "4195-4198", "Text": "4 7\nTwo long and parallel straight wires A and B carrying currents of\n8 0 A and 5 0 A in the same direction are separated by a distance of\n4"}, {"Chapter": "1", "sentence_range": "4196-4199", "Text": "7\nTwo long and parallel straight wires A and B carrying currents of\n8 0 A and 5 0 A in the same direction are separated by a distance of\n4 0 cm"}, {"Chapter": "1", "sentence_range": "4197-4200", "Text": "0 A and 5 0 A in the same direction are separated by a distance of\n4 0 cm Estimate the force on a 10 cm section of wire A"}, {"Chapter": "1", "sentence_range": "4198-4201", "Text": "0 A in the same direction are separated by a distance of\n4 0 cm Estimate the force on a 10 cm section of wire A 4"}, {"Chapter": "1", "sentence_range": "4199-4202", "Text": "0 cm Estimate the force on a 10 cm section of wire A 4 8\nA closely wound solenoid 80 cm long has 5 layers of windings of 400\nturns each"}, {"Chapter": "1", "sentence_range": "4200-4203", "Text": "Estimate the force on a 10 cm section of wire A 4 8\nA closely wound solenoid 80 cm long has 5 layers of windings of 400\nturns each The diameter of the solenoid is 1"}, {"Chapter": "1", "sentence_range": "4201-4204", "Text": "4 8\nA closely wound solenoid 80 cm long has 5 layers of windings of 400\nturns each The diameter of the solenoid is 1 8 cm"}, {"Chapter": "1", "sentence_range": "4202-4205", "Text": "8\nA closely wound solenoid 80 cm long has 5 layers of windings of 400\nturns each The diameter of the solenoid is 1 8 cm If the current\ncarried is 8"}, {"Chapter": "1", "sentence_range": "4203-4206", "Text": "The diameter of the solenoid is 1 8 cm If the current\ncarried is 8 0 A, estimate the magnitude of B inside the solenoid\nnear its centre"}, {"Chapter": "1", "sentence_range": "4204-4207", "Text": "8 cm If the current\ncarried is 8 0 A, estimate the magnitude of B inside the solenoid\nnear its centre 4"}, {"Chapter": "1", "sentence_range": "4205-4208", "Text": "If the current\ncarried is 8 0 A, estimate the magnitude of B inside the solenoid\nnear its centre 4 9\nA square coil of side 10 cm consists of 20 turns and carries a current\nof 12 A"}, {"Chapter": "1", "sentence_range": "4206-4209", "Text": "0 A, estimate the magnitude of B inside the solenoid\nnear its centre 4 9\nA square coil of side 10 cm consists of 20 turns and carries a current\nof 12 A The coil is suspended vertically and the normal to the plane\nof the coil makes an angle of 30\u00ba with the direction of a uniform\nhorizontal magnetic field of magnitude 0"}, {"Chapter": "1", "sentence_range": "4207-4210", "Text": "4 9\nA square coil of side 10 cm consists of 20 turns and carries a current\nof 12 A The coil is suspended vertically and the normal to the plane\nof the coil makes an angle of 30\u00ba with the direction of a uniform\nhorizontal magnetic field of magnitude 0 80 T"}, {"Chapter": "1", "sentence_range": "4208-4211", "Text": "9\nA square coil of side 10 cm consists of 20 turns and carries a current\nof 12 A The coil is suspended vertically and the normal to the plane\nof the coil makes an angle of 30\u00ba with the direction of a uniform\nhorizontal magnetic field of magnitude 0 80 T What is the magnitude\nof torque experienced by the coil"}, {"Chapter": "1", "sentence_range": "4209-4212", "Text": "The coil is suspended vertically and the normal to the plane\nof the coil makes an angle of 30\u00ba with the direction of a uniform\nhorizontal magnetic field of magnitude 0 80 T What is the magnitude\nof torque experienced by the coil 4"}, {"Chapter": "1", "sentence_range": "4210-4213", "Text": "80 T What is the magnitude\nof torque experienced by the coil 4 10\nTwo moving coil meters, M1 and M2 have the following particulars:\nR1 = 10 W,  N1 = 30,\nA1 = 3"}, {"Chapter": "1", "sentence_range": "4211-4214", "Text": "What is the magnitude\nof torque experienced by the coil 4 10\nTwo moving coil meters, M1 and M2 have the following particulars:\nR1 = 10 W,  N1 = 30,\nA1 = 3 6 \u00d7 10\u20133 m2, B1 = 0"}, {"Chapter": "1", "sentence_range": "4212-4215", "Text": "4 10\nTwo moving coil meters, M1 and M2 have the following particulars:\nR1 = 10 W,  N1 = 30,\nA1 = 3 6 \u00d7 10\u20133 m2, B1 = 0 25 T\nR2 = 14 W,  N2 = 42,\nA2 = 1"}, {"Chapter": "1", "sentence_range": "4213-4216", "Text": "10\nTwo moving coil meters, M1 and M2 have the following particulars:\nR1 = 10 W,  N1 = 30,\nA1 = 3 6 \u00d7 10\u20133 m2, B1 = 0 25 T\nR2 = 14 W,  N2 = 42,\nA2 = 1 8 \u00d7 10\u20133 m2, B2 = 0"}, {"Chapter": "1", "sentence_range": "4214-4217", "Text": "6 \u00d7 10\u20133 m2, B1 = 0 25 T\nR2 = 14 W,  N2 = 42,\nA2 = 1 8 \u00d7 10\u20133 m2, B2 = 0 50 T\n(The spring constants are identical for the two meters)"}, {"Chapter": "1", "sentence_range": "4215-4218", "Text": "25 T\nR2 = 14 W,  N2 = 42,\nA2 = 1 8 \u00d7 10\u20133 m2, B2 = 0 50 T\n(The spring constants are identical for the two meters) Determine the ratio of (a) current sensitivity and (b) voltage\nsensitivity of M2 and M1"}, {"Chapter": "1", "sentence_range": "4216-4219", "Text": "8 \u00d7 10\u20133 m2, B2 = 0 50 T\n(The spring constants are identical for the two meters) Determine the ratio of (a) current sensitivity and (b) voltage\nsensitivity of M2 and M1 4"}, {"Chapter": "1", "sentence_range": "4217-4220", "Text": "50 T\n(The spring constants are identical for the two meters) Determine the ratio of (a) current sensitivity and (b) voltage\nsensitivity of M2 and M1 4 11\nIn a chamber, a uniform magnetic field of 6"}, {"Chapter": "1", "sentence_range": "4218-4221", "Text": "Determine the ratio of (a) current sensitivity and (b) voltage\nsensitivity of M2 and M1 4 11\nIn a chamber, a uniform magnetic field of 6 5 G (1 G = 10\u20134 T) is\nmaintained"}, {"Chapter": "1", "sentence_range": "4219-4222", "Text": "4 11\nIn a chamber, a uniform magnetic field of 6 5 G (1 G = 10\u20134 T) is\nmaintained An electron is shot into the field with a speed of\n4"}, {"Chapter": "1", "sentence_range": "4220-4223", "Text": "11\nIn a chamber, a uniform magnetic field of 6 5 G (1 G = 10\u20134 T) is\nmaintained An electron is shot into the field with a speed of\n4 8 \u00d7 106 m s\u20131 normal to the field"}, {"Chapter": "1", "sentence_range": "4221-4224", "Text": "5 G (1 G = 10\u20134 T) is\nmaintained An electron is shot into the field with a speed of\n4 8 \u00d7 106 m s\u20131 normal to the field Explain why the  path of the\nelectron is a circle"}, {"Chapter": "1", "sentence_range": "4222-4225", "Text": "An electron is shot into the field with a speed of\n4 8 \u00d7 106 m s\u20131 normal to the field Explain why the  path of the\nelectron is a circle Determine the radius of the circular orbit"}, {"Chapter": "1", "sentence_range": "4223-4226", "Text": "8 \u00d7 106 m s\u20131 normal to the field Explain why the  path of the\nelectron is a circle Determine the radius of the circular orbit (e = 1"}, {"Chapter": "1", "sentence_range": "4224-4227", "Text": "Explain why the  path of the\nelectron is a circle Determine the radius of the circular orbit (e = 1 5 \u00d7 10\u201319 C, me = 9"}, {"Chapter": "1", "sentence_range": "4225-4228", "Text": "Determine the radius of the circular orbit (e = 1 5 \u00d7 10\u201319 C, me = 9 1\u00d710\u201331 kg)\n4"}, {"Chapter": "1", "sentence_range": "4226-4229", "Text": "(e = 1 5 \u00d7 10\u201319 C, me = 9 1\u00d710\u201331 kg)\n4 12\nIn Exercise 4"}, {"Chapter": "1", "sentence_range": "4227-4230", "Text": "5 \u00d7 10\u201319 C, me = 9 1\u00d710\u201331 kg)\n4 12\nIn Exercise 4 11 obtain the frequency of revolution of the electron in\nits circular orbit"}, {"Chapter": "1", "sentence_range": "4228-4231", "Text": "1\u00d710\u201331 kg)\n4 12\nIn Exercise 4 11 obtain the frequency of revolution of the electron in\nits circular orbit Does the answer depend on the speed of the\nelectron"}, {"Chapter": "1", "sentence_range": "4229-4232", "Text": "12\nIn Exercise 4 11 obtain the frequency of revolution of the electron in\nits circular orbit Does the answer depend on the speed of the\nelectron Explain"}, {"Chapter": "1", "sentence_range": "4230-4233", "Text": "11 obtain the frequency of revolution of the electron in\nits circular orbit Does the answer depend on the speed of the\nelectron Explain 4"}, {"Chapter": "1", "sentence_range": "4231-4234", "Text": "Does the answer depend on the speed of the\nelectron Explain 4 13\n(a) A circular coil of 30 turns and radius 8"}, {"Chapter": "1", "sentence_range": "4232-4235", "Text": "Explain 4 13\n(a) A circular coil of 30 turns and radius 8 0 cm carrying a current\nof 6"}, {"Chapter": "1", "sentence_range": "4233-4236", "Text": "4 13\n(a) A circular coil of 30 turns and radius 8 0 cm carrying a current\nof 6 0 A is suspended vertically in a uniform horizontal magnetic\nfield of magnitude 1"}, {"Chapter": "1", "sentence_range": "4234-4237", "Text": "13\n(a) A circular coil of 30 turns and radius 8 0 cm carrying a current\nof 6 0 A is suspended vertically in a uniform horizontal magnetic\nfield of magnitude 1 0 T"}, {"Chapter": "1", "sentence_range": "4235-4238", "Text": "0 cm carrying a current\nof 6 0 A is suspended vertically in a uniform horizontal magnetic\nfield of magnitude 1 0 T The field lines make an angle of 60\u00b0\nwith the normal of the coil"}, {"Chapter": "1", "sentence_range": "4236-4239", "Text": "0 A is suspended vertically in a uniform horizontal magnetic\nfield of magnitude 1 0 T The field lines make an angle of 60\u00b0\nwith the normal of the coil Calculate the magnitude of the\ncounter torque that must be applied to prevent the coil from\nturning"}, {"Chapter": "1", "sentence_range": "4237-4240", "Text": "0 T The field lines make an angle of 60\u00b0\nwith the normal of the coil Calculate the magnitude of the\ncounter torque that must be applied to prevent the coil from\nturning (b) Would your answer change, if the circular coil in (a) were replaced\nby a planar coil of some irregular shape that encloses the same\narea"}, {"Chapter": "1", "sentence_range": "4238-4241", "Text": "The field lines make an angle of 60\u00b0\nwith the normal of the coil Calculate the magnitude of the\ncounter torque that must be applied to prevent the coil from\nturning (b) Would your answer change, if the circular coil in (a) were replaced\nby a planar coil of some irregular shape that encloses the same\narea (All other particulars are also unaltered"}, {"Chapter": "1", "sentence_range": "4239-4242", "Text": "Calculate the magnitude of the\ncounter torque that must be applied to prevent the coil from\nturning (b) Would your answer change, if the circular coil in (a) were replaced\nby a planar coil of some irregular shape that encloses the same\narea (All other particulars are also unaltered )\nRationalised 2023-24\nPhysics\n136\n5"}, {"Chapter": "1", "sentence_range": "4240-4243", "Text": "(b) Would your answer change, if the circular coil in (a) were replaced\nby a planar coil of some irregular shape that encloses the same\narea (All other particulars are also unaltered )\nRationalised 2023-24\nPhysics\n136\n5 1  INTRODUCTION\nMagnetic phenomena are universal in nature"}, {"Chapter": "1", "sentence_range": "4241-4244", "Text": "(All other particulars are also unaltered )\nRationalised 2023-24\nPhysics\n136\n5 1  INTRODUCTION\nMagnetic phenomena are universal in nature Vast, distant galaxies, the\ntiny invisible atoms, humans and beasts all are permeated through and\nthrough with a host of magnetic fields from a variety of sources"}, {"Chapter": "1", "sentence_range": "4242-4245", "Text": ")\nRationalised 2023-24\nPhysics\n136\n5 1  INTRODUCTION\nMagnetic phenomena are universal in nature Vast, distant galaxies, the\ntiny invisible atoms, humans and beasts all are permeated through and\nthrough with a host of magnetic fields from a variety of sources The earth\u2019s\nmagnetism predates human evolution"}, {"Chapter": "1", "sentence_range": "4243-4246", "Text": "1  INTRODUCTION\nMagnetic phenomena are universal in nature Vast, distant galaxies, the\ntiny invisible atoms, humans and beasts all are permeated through and\nthrough with a host of magnetic fields from a variety of sources The earth\u2019s\nmagnetism predates human evolution The word magnet is derived from\nthe name of an island in Greece called magnesia where magnetic ore\ndeposits were found, as early as 600 BC"}, {"Chapter": "1", "sentence_range": "4244-4247", "Text": "Vast, distant galaxies, the\ntiny invisible atoms, humans and beasts all are permeated through and\nthrough with a host of magnetic fields from a variety of sources The earth\u2019s\nmagnetism predates human evolution The word magnet is derived from\nthe name of an island in Greece called magnesia where magnetic ore\ndeposits were found, as early as 600 BC In the previous chapter we have learned that moving charges or electric\ncurrents produce magnetic fields"}, {"Chapter": "1", "sentence_range": "4245-4248", "Text": "The earth\u2019s\nmagnetism predates human evolution The word magnet is derived from\nthe name of an island in Greece called magnesia where magnetic ore\ndeposits were found, as early as 600 BC In the previous chapter we have learned that moving charges or electric\ncurrents produce magnetic fields This discovery, which was made in the\nearly part of the nineteenth century is credited to Oersted, Ampere, Biot\nand Savart, among others"}, {"Chapter": "1", "sentence_range": "4246-4249", "Text": "The word magnet is derived from\nthe name of an island in Greece called magnesia where magnetic ore\ndeposits were found, as early as 600 BC In the previous chapter we have learned that moving charges or electric\ncurrents produce magnetic fields This discovery, which was made in the\nearly part of the nineteenth century is credited to Oersted, Ampere, Biot\nand Savart, among others In the present chapter, we take a look at magnetism as a subject in its\nown right"}, {"Chapter": "1", "sentence_range": "4247-4250", "Text": "In the previous chapter we have learned that moving charges or electric\ncurrents produce magnetic fields This discovery, which was made in the\nearly part of the nineteenth century is credited to Oersted, Ampere, Biot\nand Savart, among others In the present chapter, we take a look at magnetism as a subject in its\nown right Some of the commonly known ideas regarding magnetism are:\n(i)\nThe earth behaves as a magnet with the magnetic field pointing\napproximately from the geographic south to the north"}, {"Chapter": "1", "sentence_range": "4248-4251", "Text": "This discovery, which was made in the\nearly part of the nineteenth century is credited to Oersted, Ampere, Biot\nand Savart, among others In the present chapter, we take a look at magnetism as a subject in its\nown right Some of the commonly known ideas regarding magnetism are:\n(i)\nThe earth behaves as a magnet with the magnetic field pointing\napproximately from the geographic south to the north (ii) When a bar magnet is freely suspended, it points in  the north-south\ndirection"}, {"Chapter": "1", "sentence_range": "4249-4252", "Text": "In the present chapter, we take a look at magnetism as a subject in its\nown right Some of the commonly known ideas regarding magnetism are:\n(i)\nThe earth behaves as a magnet with the magnetic field pointing\napproximately from the geographic south to the north (ii) When a bar magnet is freely suspended, it points in  the north-south\ndirection The tip which points to the geographic north is called the\nnorth pole and the tip which points to the geographic south is called\nthe south pole of the magnet"}, {"Chapter": "1", "sentence_range": "4250-4253", "Text": "Some of the commonly known ideas regarding magnetism are:\n(i)\nThe earth behaves as a magnet with the magnetic field pointing\napproximately from the geographic south to the north (ii) When a bar magnet is freely suspended, it points in  the north-south\ndirection The tip which points to the geographic north is called the\nnorth pole and the tip which points to the geographic south is called\nthe south pole of the magnet Chapter Five\nMAGNETISM AND\nMATTER\nRationalised 2023-24\n137\nMagnetism and\nMatter\n(iii) There is a repulsive force when north poles ( or south poles ) of two\nmagnets are brought close together"}, {"Chapter": "1", "sentence_range": "4251-4254", "Text": "(ii) When a bar magnet is freely suspended, it points in  the north-south\ndirection The tip which points to the geographic north is called the\nnorth pole and the tip which points to the geographic south is called\nthe south pole of the magnet Chapter Five\nMAGNETISM AND\nMATTER\nRationalised 2023-24\n137\nMagnetism and\nMatter\n(iii) There is a repulsive force when north poles ( or south poles ) of two\nmagnets are brought close together Conversely, there is an attractive\nforce between the north pole of one magnet and the south pole of\nthe other"}, {"Chapter": "1", "sentence_range": "4252-4255", "Text": "The tip which points to the geographic north is called the\nnorth pole and the tip which points to the geographic south is called\nthe south pole of the magnet Chapter Five\nMAGNETISM AND\nMATTER\nRationalised 2023-24\n137\nMagnetism and\nMatter\n(iii) There is a repulsive force when north poles ( or south poles ) of two\nmagnets are brought close together Conversely, there is an attractive\nforce between the north pole of one magnet and the south pole of\nthe other (iv) We cannot isolate the north, or south pole of a magnet"}, {"Chapter": "1", "sentence_range": "4253-4256", "Text": "Chapter Five\nMAGNETISM AND\nMATTER\nRationalised 2023-24\n137\nMagnetism and\nMatter\n(iii) There is a repulsive force when north poles ( or south poles ) of two\nmagnets are brought close together Conversely, there is an attractive\nforce between the north pole of one magnet and the south pole of\nthe other (iv) We cannot isolate the north, or south pole of a magnet If a bar magnet\nis broken into two halves, we get two similar bar magnets with\nsomewhat weaker properties"}, {"Chapter": "1", "sentence_range": "4254-4257", "Text": "Conversely, there is an attractive\nforce between the north pole of one magnet and the south pole of\nthe other (iv) We cannot isolate the north, or south pole of a magnet If a bar magnet\nis broken into two halves, we get two similar bar magnets with\nsomewhat weaker properties Unlike electric charges, isolated magnetic\nnorth and south poles known as magnetic monopoles do not exist"}, {"Chapter": "1", "sentence_range": "4255-4258", "Text": "(iv) We cannot isolate the north, or south pole of a magnet If a bar magnet\nis broken into two halves, we get two similar bar magnets with\nsomewhat weaker properties Unlike electric charges, isolated magnetic\nnorth and south poles known as magnetic monopoles do not exist (v) It is possible to make magnets out of iron and its alloys"}, {"Chapter": "1", "sentence_range": "4256-4259", "Text": "If a bar magnet\nis broken into two halves, we get two similar bar magnets with\nsomewhat weaker properties Unlike electric charges, isolated magnetic\nnorth and south poles known as magnetic monopoles do not exist (v) It is possible to make magnets out of iron and its alloys We begin with a description of a bar magnet and its behaviour in an\nexternal magnetic field"}, {"Chapter": "1", "sentence_range": "4257-4260", "Text": "Unlike electric charges, isolated magnetic\nnorth and south poles known as magnetic monopoles do not exist (v) It is possible to make magnets out of iron and its alloys We begin with a description of a bar magnet and its behaviour in an\nexternal magnetic field We describe Gauss\u2019s law of magnetism"}, {"Chapter": "1", "sentence_range": "4258-4261", "Text": "(v) It is possible to make magnets out of iron and its alloys We begin with a description of a bar magnet and its behaviour in an\nexternal magnetic field We describe Gauss\u2019s law of magnetism We then\nfollow it up with an account of the earth\u2019s magnetic field"}, {"Chapter": "1", "sentence_range": "4259-4262", "Text": "We begin with a description of a bar magnet and its behaviour in an\nexternal magnetic field We describe Gauss\u2019s law of magnetism We then\nfollow it up with an account of the earth\u2019s magnetic field We next describe\nhow materials can be classified on the basis of their magnetic properties"}, {"Chapter": "1", "sentence_range": "4260-4263", "Text": "We describe Gauss\u2019s law of magnetism We then\nfollow it up with an account of the earth\u2019s magnetic field We next describe\nhow materials can be classified on the basis of their magnetic properties We describe para-, dia-, and ferromagnetism"}, {"Chapter": "1", "sentence_range": "4261-4264", "Text": "We then\nfollow it up with an account of the earth\u2019s magnetic field We next describe\nhow materials can be classified on the basis of their magnetic properties We describe para-, dia-, and ferromagnetism We conclude with a section\non electromagnets and permanent magnets"}, {"Chapter": "1", "sentence_range": "4262-4265", "Text": "We next describe\nhow materials can be classified on the basis of their magnetic properties We describe para-, dia-, and ferromagnetism We conclude with a section\non electromagnets and permanent magnets 5"}, {"Chapter": "1", "sentence_range": "4263-4266", "Text": "We describe para-, dia-, and ferromagnetism We conclude with a section\non electromagnets and permanent magnets 5 2  THE BAR MAGNET\nOne of the earliest childhood memories of the famous physicist Albert\nEinstein was that of a magnet gifted to him by a relative"}, {"Chapter": "1", "sentence_range": "4264-4267", "Text": "We conclude with a section\non electromagnets and permanent magnets 5 2  THE BAR MAGNET\nOne of the earliest childhood memories of the famous physicist Albert\nEinstein was that of a magnet gifted to him by a relative Einstein was\nfascinated, and played endlessly with it"}, {"Chapter": "1", "sentence_range": "4265-4268", "Text": "5 2  THE BAR MAGNET\nOne of the earliest childhood memories of the famous physicist Albert\nEinstein was that of a magnet gifted to him by a relative Einstein was\nfascinated, and played endlessly with it He wondered how the magnet\ncould affect objects such as nails or pins placed away from it and not in\nany way connected to it by a spring or string"}, {"Chapter": "1", "sentence_range": "4266-4269", "Text": "2  THE BAR MAGNET\nOne of the earliest childhood memories of the famous physicist Albert\nEinstein was that of a magnet gifted to him by a relative Einstein was\nfascinated, and played endlessly with it He wondered how the magnet\ncould affect objects such as nails or pins placed away from it and not in\nany way connected to it by a spring or string We begin our study by examining iron filings sprinkled on a sheet of\nglass placed over a short bar magnet"}, {"Chapter": "1", "sentence_range": "4267-4270", "Text": "Einstein was\nfascinated, and played endlessly with it He wondered how the magnet\ncould affect objects such as nails or pins placed away from it and not in\nany way connected to it by a spring or string We begin our study by examining iron filings sprinkled on a sheet of\nglass placed over a short bar magnet The arrangement of iron filings is\nshown in Fig"}, {"Chapter": "1", "sentence_range": "4268-4271", "Text": "He wondered how the magnet\ncould affect objects such as nails or pins placed away from it and not in\nany way connected to it by a spring or string We begin our study by examining iron filings sprinkled on a sheet of\nglass placed over a short bar magnet The arrangement of iron filings is\nshown in Fig 5"}, {"Chapter": "1", "sentence_range": "4269-4272", "Text": "We begin our study by examining iron filings sprinkled on a sheet of\nglass placed over a short bar magnet The arrangement of iron filings is\nshown in Fig 5 1"}, {"Chapter": "1", "sentence_range": "4270-4273", "Text": "The arrangement of iron filings is\nshown in Fig 5 1 The pattern of iron filings suggests that the magnet has two poles\nsimilar to the positive and negative charge of an electric dipole"}, {"Chapter": "1", "sentence_range": "4271-4274", "Text": "5 1 The pattern of iron filings suggests that the magnet has two poles\nsimilar to the positive and negative charge of an electric dipole As\nmentioned in the introductory section, one pole is designated the North\npole and the other, the South pole"}, {"Chapter": "1", "sentence_range": "4272-4275", "Text": "1 The pattern of iron filings suggests that the magnet has two poles\nsimilar to the positive and negative charge of an electric dipole As\nmentioned in the introductory section, one pole is designated the North\npole and the other, the South pole When suspended freely, these poles\npoint approximately towards the geographic north and south poles,\nrespectively"}, {"Chapter": "1", "sentence_range": "4273-4276", "Text": "The pattern of iron filings suggests that the magnet has two poles\nsimilar to the positive and negative charge of an electric dipole As\nmentioned in the introductory section, one pole is designated the North\npole and the other, the South pole When suspended freely, these poles\npoint approximately towards the geographic north and south poles,\nrespectively A similar pattern of iron filings is observed around a current\ncarrying solenoid"}, {"Chapter": "1", "sentence_range": "4274-4277", "Text": "As\nmentioned in the introductory section, one pole is designated the North\npole and the other, the South pole When suspended freely, these poles\npoint approximately towards the geographic north and south poles,\nrespectively A similar pattern of iron filings is observed around a current\ncarrying solenoid 5"}, {"Chapter": "1", "sentence_range": "4275-4278", "Text": "When suspended freely, these poles\npoint approximately towards the geographic north and south poles,\nrespectively A similar pattern of iron filings is observed around a current\ncarrying solenoid 5 2"}, {"Chapter": "1", "sentence_range": "4276-4279", "Text": "A similar pattern of iron filings is observed around a current\ncarrying solenoid 5 2 1  The magnetic field lines\nThe pattern of iron filings permits us to plot the magnetic field lines*"}, {"Chapter": "1", "sentence_range": "4277-4280", "Text": "5 2 1  The magnetic field lines\nThe pattern of iron filings permits us to plot the magnetic field lines* This is\nshown both for the bar-magnet and the current-carrying solenoid in\nFig"}, {"Chapter": "1", "sentence_range": "4278-4281", "Text": "2 1  The magnetic field lines\nThe pattern of iron filings permits us to plot the magnetic field lines* This is\nshown both for the bar-magnet and the current-carrying solenoid in\nFig 5"}, {"Chapter": "1", "sentence_range": "4279-4282", "Text": "1  The magnetic field lines\nThe pattern of iron filings permits us to plot the magnetic field lines* This is\nshown both for the bar-magnet and the current-carrying solenoid in\nFig 5 2"}, {"Chapter": "1", "sentence_range": "4280-4283", "Text": "This is\nshown both for the bar-magnet and the current-carrying solenoid in\nFig 5 2 For comparison refer to the Chapter 1, Figure 1"}, {"Chapter": "1", "sentence_range": "4281-4284", "Text": "5 2 For comparison refer to the Chapter 1, Figure 1 17(d)"}, {"Chapter": "1", "sentence_range": "4282-4285", "Text": "2 For comparison refer to the Chapter 1, Figure 1 17(d) Electric field\nlines of an electric dipole are also displayed in Fig"}, {"Chapter": "1", "sentence_range": "4283-4286", "Text": "For comparison refer to the Chapter 1, Figure 1 17(d) Electric field\nlines of an electric dipole are also displayed in Fig 5"}, {"Chapter": "1", "sentence_range": "4284-4287", "Text": "17(d) Electric field\nlines of an electric dipole are also displayed in Fig 5 2(c)"}, {"Chapter": "1", "sentence_range": "4285-4288", "Text": "Electric field\nlines of an electric dipole are also displayed in Fig 5 2(c) The magnetic field\nlines are a visual and intuitive realisation of the magnetic field"}, {"Chapter": "1", "sentence_range": "4286-4289", "Text": "5 2(c) The magnetic field\nlines are a visual and intuitive realisation of the magnetic field Their\nproperties are:\n(i)\nThe magnetic field lines of a magnet (or a solenoid) form continuous\nclosed loops"}, {"Chapter": "1", "sentence_range": "4287-4290", "Text": "2(c) The magnetic field\nlines are a visual and intuitive realisation of the magnetic field Their\nproperties are:\n(i)\nThe magnetic field lines of a magnet (or a solenoid) form continuous\nclosed loops This is unlike the electric dipole where these field lines\nbegin from a positive charge and end on the negative charge or escape\nto infinity"}, {"Chapter": "1", "sentence_range": "4288-4291", "Text": "The magnetic field\nlines are a visual and intuitive realisation of the magnetic field Their\nproperties are:\n(i)\nThe magnetic field lines of a magnet (or a solenoid) form continuous\nclosed loops This is unlike the electric dipole where these field lines\nbegin from a positive charge and end on the negative charge or escape\nto infinity FIGURE 5"}, {"Chapter": "1", "sentence_range": "4289-4292", "Text": "Their\nproperties are:\n(i)\nThe magnetic field lines of a magnet (or a solenoid) form continuous\nclosed loops This is unlike the electric dipole where these field lines\nbegin from a positive charge and end on the negative charge or escape\nto infinity FIGURE 5 1 The\narrangement of iron\nfilings surrounding a\nbar magnet"}, {"Chapter": "1", "sentence_range": "4290-4293", "Text": "This is unlike the electric dipole where these field lines\nbegin from a positive charge and end on the negative charge or escape\nto infinity FIGURE 5 1 The\narrangement of iron\nfilings surrounding a\nbar magnet The\npattern mimics\nmagnetic field lines"}, {"Chapter": "1", "sentence_range": "4291-4294", "Text": "FIGURE 5 1 The\narrangement of iron\nfilings surrounding a\nbar magnet The\npattern mimics\nmagnetic field lines The pattern suggests\nthat the bar magnet\nis a magnetic dipole"}, {"Chapter": "1", "sentence_range": "4292-4295", "Text": "1 The\narrangement of iron\nfilings surrounding a\nbar magnet The\npattern mimics\nmagnetic field lines The pattern suggests\nthat the bar magnet\nis a magnetic dipole *\nIn some textbooks the magnetic field lines are called magnetic lines of force"}, {"Chapter": "1", "sentence_range": "4293-4296", "Text": "The\npattern mimics\nmagnetic field lines The pattern suggests\nthat the bar magnet\nis a magnetic dipole *\nIn some textbooks the magnetic field lines are called magnetic lines of force This nomenclature is avoided since it can be confusing"}, {"Chapter": "1", "sentence_range": "4294-4297", "Text": "The pattern suggests\nthat the bar magnet\nis a magnetic dipole *\nIn some textbooks the magnetic field lines are called magnetic lines of force This nomenclature is avoided since it can be confusing Unlike electrostatics\nthe field lines in magnetism do not indicate the direction of the force on a\n(moving) charge"}, {"Chapter": "1", "sentence_range": "4295-4298", "Text": "*\nIn some textbooks the magnetic field lines are called magnetic lines of force This nomenclature is avoided since it can be confusing Unlike electrostatics\nthe field lines in magnetism do not indicate the direction of the force on a\n(moving) charge Rationalised 2023-24\nPhysics\n138\nFIGURE 5"}, {"Chapter": "1", "sentence_range": "4296-4299", "Text": "This nomenclature is avoided since it can be confusing Unlike electrostatics\nthe field lines in magnetism do not indicate the direction of the force on a\n(moving) charge Rationalised 2023-24\nPhysics\n138\nFIGURE 5 3 Calculation of (a) The axial field of a\nfinite solenoid in order to demonstrate its similarity\nto that of a bar magnet"}, {"Chapter": "1", "sentence_range": "4297-4300", "Text": "Unlike electrostatics\nthe field lines in magnetism do not indicate the direction of the force on a\n(moving) charge Rationalised 2023-24\nPhysics\n138\nFIGURE 5 3 Calculation of (a) The axial field of a\nfinite solenoid in order to demonstrate its similarity\nto that of a bar magnet (b) A magnetic needle\nin a uniform magnetic field B"}, {"Chapter": "1", "sentence_range": "4298-4301", "Text": "Rationalised 2023-24\nPhysics\n138\nFIGURE 5 3 Calculation of (a) The axial field of a\nfinite solenoid in order to demonstrate its similarity\nto that of a bar magnet (b) A magnetic needle\nin a uniform magnetic field B The\narrangement may be used to\ndetermine either B or the magnetic\nmoment m of the needle"}, {"Chapter": "1", "sentence_range": "4299-4302", "Text": "3 Calculation of (a) The axial field of a\nfinite solenoid in order to demonstrate its similarity\nto that of a bar magnet (b) A magnetic needle\nin a uniform magnetic field B The\narrangement may be used to\ndetermine either B or the magnetic\nmoment m of the needle (ii) The tangent to the field line at a given\npoint represents the direction of the net\nmagnetic field B at that point"}, {"Chapter": "1", "sentence_range": "4300-4303", "Text": "(b) A magnetic needle\nin a uniform magnetic field B The\narrangement may be used to\ndetermine either B or the magnetic\nmoment m of the needle (ii) The tangent to the field line at a given\npoint represents the direction of the net\nmagnetic field B at that point (iii) The larger the number of field lines\ncrossing per unit area, the stronger is\nthe magnitude of the magnetic field B"}, {"Chapter": "1", "sentence_range": "4301-4304", "Text": "The\narrangement may be used to\ndetermine either B or the magnetic\nmoment m of the needle (ii) The tangent to the field line at a given\npoint represents the direction of the net\nmagnetic field B at that point (iii) The larger the number of field lines\ncrossing per unit area, the stronger is\nthe magnitude of the magnetic field B In Fig"}, {"Chapter": "1", "sentence_range": "4302-4305", "Text": "(ii) The tangent to the field line at a given\npoint represents the direction of the net\nmagnetic field B at that point (iii) The larger the number of field lines\ncrossing per unit area, the stronger is\nthe magnitude of the magnetic field B In Fig 5"}, {"Chapter": "1", "sentence_range": "4303-4306", "Text": "(iii) The larger the number of field lines\ncrossing per unit area, the stronger is\nthe magnitude of the magnetic field B In Fig 5 2(a), B is larger around region\nii  than in region  i"}, {"Chapter": "1", "sentence_range": "4304-4307", "Text": "In Fig 5 2(a), B is larger around region\nii  than in region  i (iv) The magnetic field lines do not\nintersect, for if they did, the direction\nof the magnetic field would not be\nunique at the point of intersection"}, {"Chapter": "1", "sentence_range": "4305-4308", "Text": "5 2(a), B is larger around region\nii  than in region  i (iv) The magnetic field lines do not\nintersect, for if they did, the direction\nof the magnetic field would not be\nunique at the point of intersection One can plot the magnetic field lines\nin a variety of ways"}, {"Chapter": "1", "sentence_range": "4306-4309", "Text": "2(a), B is larger around region\nii  than in region  i (iv) The magnetic field lines do not\nintersect, for if they did, the direction\nof the magnetic field would not be\nunique at the point of intersection One can plot the magnetic field lines\nin a variety of ways One way is to place a\nsmall magnetic compass needle at various\npositions and note its orientation"}, {"Chapter": "1", "sentence_range": "4307-4310", "Text": "(iv) The magnetic field lines do not\nintersect, for if they did, the direction\nof the magnetic field would not be\nunique at the point of intersection One can plot the magnetic field lines\nin a variety of ways One way is to place a\nsmall magnetic compass needle at various\npositions and note its orientation This\ngives us an idea of the magnetic field\ndirection at various points in space"}, {"Chapter": "1", "sentence_range": "4308-4311", "Text": "One can plot the magnetic field lines\nin a variety of ways One way is to place a\nsmall magnetic compass needle at various\npositions and note its orientation This\ngives us an idea of the magnetic field\ndirection at various points in space 5"}, {"Chapter": "1", "sentence_range": "4309-4312", "Text": "One way is to place a\nsmall magnetic compass needle at various\npositions and note its orientation This\ngives us an idea of the magnetic field\ndirection at various points in space 5 2"}, {"Chapter": "1", "sentence_range": "4310-4313", "Text": "This\ngives us an idea of the magnetic field\ndirection at various points in space 5 2 2  Bar magnet as an equivalent\nsolenoid\nIn the previous chapter, we have explained\nhow a current loop acts as a magnetic\ndipole (Section 4"}, {"Chapter": "1", "sentence_range": "4311-4314", "Text": "5 2 2  Bar magnet as an equivalent\nsolenoid\nIn the previous chapter, we have explained\nhow a current loop acts as a magnetic\ndipole (Section 4 10)"}, {"Chapter": "1", "sentence_range": "4312-4315", "Text": "2 2  Bar magnet as an equivalent\nsolenoid\nIn the previous chapter, we have explained\nhow a current loop acts as a magnetic\ndipole (Section 4 10) We mentioned\nAmpere\u2019s hypothesis that all magnetic\nphenomena can be explained in terms of\ncirculating currents"}, {"Chapter": "1", "sentence_range": "4313-4316", "Text": "2  Bar magnet as an equivalent\nsolenoid\nIn the previous chapter, we have explained\nhow a current loop acts as a magnetic\ndipole (Section 4 10) We mentioned\nAmpere\u2019s hypothesis that all magnetic\nphenomena can be explained in terms of\ncirculating currents FIGURE 5"}, {"Chapter": "1", "sentence_range": "4314-4317", "Text": "10) We mentioned\nAmpere\u2019s hypothesis that all magnetic\nphenomena can be explained in terms of\ncirculating currents FIGURE 5 2 The field lines of (a) a bar magnet, (b) a current-carrying finite solenoid and\n(c) electric dipole"}, {"Chapter": "1", "sentence_range": "4315-4318", "Text": "We mentioned\nAmpere\u2019s hypothesis that all magnetic\nphenomena can be explained in terms of\ncirculating currents FIGURE 5 2 The field lines of (a) a bar magnet, (b) a current-carrying finite solenoid and\n(c) electric dipole At large distances, the field lines are very similar"}, {"Chapter": "1", "sentence_range": "4316-4319", "Text": "FIGURE 5 2 The field lines of (a) a bar magnet, (b) a current-carrying finite solenoid and\n(c) electric dipole At large distances, the field lines are very similar The curves\nlabelled  i  and ii are closed Gaussian surfaces"}, {"Chapter": "1", "sentence_range": "4317-4320", "Text": "2 The field lines of (a) a bar magnet, (b) a current-carrying finite solenoid and\n(c) electric dipole At large distances, the field lines are very similar The curves\nlabelled  i  and ii are closed Gaussian surfaces Rationalised 2023-24\n139\nMagnetism and\nMatter\nThe resemblance of magnetic field lines for a bar magnet and a solenoid\nsuggest that a bar magnet may be thought of as a large number of\ncirculating currents in analogy with a solenoid"}, {"Chapter": "1", "sentence_range": "4318-4321", "Text": "At large distances, the field lines are very similar The curves\nlabelled  i  and ii are closed Gaussian surfaces Rationalised 2023-24\n139\nMagnetism and\nMatter\nThe resemblance of magnetic field lines for a bar magnet and a solenoid\nsuggest that a bar magnet may be thought of as a large number of\ncirculating currents in analogy with a solenoid Cutting a bar magnet in\nhalf is like cutting a solenoid"}, {"Chapter": "1", "sentence_range": "4319-4322", "Text": "The curves\nlabelled  i  and ii are closed Gaussian surfaces Rationalised 2023-24\n139\nMagnetism and\nMatter\nThe resemblance of magnetic field lines for a bar magnet and a solenoid\nsuggest that a bar magnet may be thought of as a large number of\ncirculating currents in analogy with a solenoid Cutting a bar magnet in\nhalf is like cutting a solenoid We get two smaller solenoids with weaker\nmagnetic properties"}, {"Chapter": "1", "sentence_range": "4320-4323", "Text": "Rationalised 2023-24\n139\nMagnetism and\nMatter\nThe resemblance of magnetic field lines for a bar magnet and a solenoid\nsuggest that a bar magnet may be thought of as a large number of\ncirculating currents in analogy with a solenoid Cutting a bar magnet in\nhalf is like cutting a solenoid We get two smaller solenoids with weaker\nmagnetic properties The field lines remain continuous, emerging from\none face of the solenoid and entering into the other face"}, {"Chapter": "1", "sentence_range": "4321-4324", "Text": "Cutting a bar magnet in\nhalf is like cutting a solenoid We get two smaller solenoids with weaker\nmagnetic properties The field lines remain continuous, emerging from\none face of the solenoid and entering into the other face One can test this\nanalogy by moving a small compass needle in the neighbourhood of a\nbar magnet and a current-carrying finite solenoid and noting that the\ndeflections of the needle are similar in both cases"}, {"Chapter": "1", "sentence_range": "4322-4325", "Text": "We get two smaller solenoids with weaker\nmagnetic properties The field lines remain continuous, emerging from\none face of the solenoid and entering into the other face One can test this\nanalogy by moving a small compass needle in the neighbourhood of a\nbar magnet and a current-carrying finite solenoid and noting that the\ndeflections of the needle are similar in both cases To make this analogy more firm we calculate the axial field of a finite\nsolenoid depicted in Fig"}, {"Chapter": "1", "sentence_range": "4323-4326", "Text": "The field lines remain continuous, emerging from\none face of the solenoid and entering into the other face One can test this\nanalogy by moving a small compass needle in the neighbourhood of a\nbar magnet and a current-carrying finite solenoid and noting that the\ndeflections of the needle are similar in both cases To make this analogy more firm we calculate the axial field of a finite\nsolenoid depicted in Fig 5"}, {"Chapter": "1", "sentence_range": "4324-4327", "Text": "One can test this\nanalogy by moving a small compass needle in the neighbourhood of a\nbar magnet and a current-carrying finite solenoid and noting that the\ndeflections of the needle are similar in both cases To make this analogy more firm we calculate the axial field of a finite\nsolenoid depicted in Fig 5 3 (a)"}, {"Chapter": "1", "sentence_range": "4325-4328", "Text": "To make this analogy more firm we calculate the axial field of a finite\nsolenoid depicted in Fig 5 3 (a) We shall demonstrate that at large\ndistances this axial field resembles that of a bar magnet"}, {"Chapter": "1", "sentence_range": "4326-4329", "Text": "5 3 (a) We shall demonstrate that at large\ndistances this axial field resembles that of a bar magnet 0\n3\n2\n4\nm\nB\nr\n\u00b5\n\u03c0\n=\n(5"}, {"Chapter": "1", "sentence_range": "4327-4330", "Text": "3 (a) We shall demonstrate that at large\ndistances this axial field resembles that of a bar magnet 0\n3\n2\n4\nm\nB\nr\n\u00b5\n\u03c0\n=\n(5 1)\nThis is also the far axial magnetic field of a bar magnet which one may\nobtain experimentally"}, {"Chapter": "1", "sentence_range": "4328-4331", "Text": "We shall demonstrate that at large\ndistances this axial field resembles that of a bar magnet 0\n3\n2\n4\nm\nB\nr\n\u00b5\n\u03c0\n=\n(5 1)\nThis is also the far axial magnetic field of a bar magnet which one may\nobtain experimentally Thus, a bar magnet and a solenoid produce similar\nmagnetic fields"}, {"Chapter": "1", "sentence_range": "4329-4332", "Text": "0\n3\n2\n4\nm\nB\nr\n\u00b5\n\u03c0\n=\n(5 1)\nThis is also the far axial magnetic field of a bar magnet which one may\nobtain experimentally Thus, a bar magnet and a solenoid produce similar\nmagnetic fields The magnetic moment of a bar magnet is thus equal to\nthe magnetic moment of an equivalent solenoid that produces the same\nmagnetic field"}, {"Chapter": "1", "sentence_range": "4330-4333", "Text": "1)\nThis is also the far axial magnetic field of a bar magnet which one may\nobtain experimentally Thus, a bar magnet and a solenoid produce similar\nmagnetic fields The magnetic moment of a bar magnet is thus equal to\nthe magnetic moment of an equivalent solenoid that produces the same\nmagnetic field 5"}, {"Chapter": "1", "sentence_range": "4331-4334", "Text": "Thus, a bar magnet and a solenoid produce similar\nmagnetic fields The magnetic moment of a bar magnet is thus equal to\nthe magnetic moment of an equivalent solenoid that produces the same\nmagnetic field 5 2"}, {"Chapter": "1", "sentence_range": "4332-4335", "Text": "The magnetic moment of a bar magnet is thus equal to\nthe magnetic moment of an equivalent solenoid that produces the same\nmagnetic field 5 2 3  The dipole in a uniform magnetic field\nLet\u2019s place a small compass needle of known magnetic moment m and\nallowing it to oscillate in the magnetic field"}, {"Chapter": "1", "sentence_range": "4333-4336", "Text": "5 2 3  The dipole in a uniform magnetic field\nLet\u2019s place a small compass needle of known magnetic moment m and\nallowing it to oscillate in the magnetic field This arrangement is shown in\nFig"}, {"Chapter": "1", "sentence_range": "4334-4337", "Text": "2 3  The dipole in a uniform magnetic field\nLet\u2019s place a small compass needle of known magnetic moment m and\nallowing it to oscillate in the magnetic field This arrangement is shown in\nFig 5"}, {"Chapter": "1", "sentence_range": "4335-4338", "Text": "3  The dipole in a uniform magnetic field\nLet\u2019s place a small compass needle of known magnetic moment m and\nallowing it to oscillate in the magnetic field This arrangement is shown in\nFig 5 3(b)"}, {"Chapter": "1", "sentence_range": "4336-4339", "Text": "This arrangement is shown in\nFig 5 3(b) The torque on the needle is [see Eq"}, {"Chapter": "1", "sentence_range": "4337-4340", "Text": "5 3(b) The torque on the needle is [see Eq (4"}, {"Chapter": "1", "sentence_range": "4338-4341", "Text": "3(b) The torque on the needle is [see Eq (4 23)],\nttttt = m \u00d7 B\n(5"}, {"Chapter": "1", "sentence_range": "4339-4342", "Text": "The torque on the needle is [see Eq (4 23)],\nttttt = m \u00d7 B\n(5 2)\nIn magnitude t = mB sinq\nHere ttttt  is restoring torque and q is the angle between m and B"}, {"Chapter": "1", "sentence_range": "4340-4343", "Text": "(4 23)],\nttttt = m \u00d7 B\n(5 2)\nIn magnitude t = mB sinq\nHere ttttt  is restoring torque and q is the angle between m and B An expression for magnetic potential energy can also be obtained on\nlines similar to  electrostatic potential energy"}, {"Chapter": "1", "sentence_range": "4341-4344", "Text": "23)],\nttttt = m \u00d7 B\n(5 2)\nIn magnitude t = mB sinq\nHere ttttt  is restoring torque and q is the angle between m and B An expression for magnetic potential energy can also be obtained on\nlines similar to  electrostatic potential energy The magnetic potential energy Um is given by\nU\nm = \u222b\u03c4 \u03b8d\n\u03b8\n( )\n      =\n= \u2212\n\u222bmB\nd\nmB\nsin\ncos\n\u03b8\n\u03b8\n\u03b8\n      = \u2013m"}, {"Chapter": "1", "sentence_range": "4342-4345", "Text": "2)\nIn magnitude t = mB sinq\nHere ttttt  is restoring torque and q is the angle between m and B An expression for magnetic potential energy can also be obtained on\nlines similar to  electrostatic potential energy The magnetic potential energy Um is given by\nU\nm = \u222b\u03c4 \u03b8d\n\u03b8\n( )\n      =\n= \u2212\n\u222bmB\nd\nmB\nsin\ncos\n\u03b8\n\u03b8\n\u03b8\n      = \u2013m B\n(5"}, {"Chapter": "1", "sentence_range": "4343-4346", "Text": "An expression for magnetic potential energy can also be obtained on\nlines similar to  electrostatic potential energy The magnetic potential energy Um is given by\nU\nm = \u222b\u03c4 \u03b8d\n\u03b8\n( )\n      =\n= \u2212\n\u222bmB\nd\nmB\nsin\ncos\n\u03b8\n\u03b8\n\u03b8\n      = \u2013m B\n(5 3)\nWe have emphasised in Chapter 2 that the zero of potential energy\ncan be fixed at one\u2019s convenience"}, {"Chapter": "1", "sentence_range": "4344-4347", "Text": "The magnetic potential energy Um is given by\nU\nm = \u222b\u03c4 \u03b8d\n\u03b8\n( )\n      =\n= \u2212\n\u222bmB\nd\nmB\nsin\ncos\n\u03b8\n\u03b8\n\u03b8\n      = \u2013m B\n(5 3)\nWe have emphasised in Chapter 2 that the zero of potential energy\ncan be fixed at one\u2019s convenience Taking the constant of integration to be\nzero means fixing the zero of potential energy at q = 90\u00b0, i"}, {"Chapter": "1", "sentence_range": "4345-4348", "Text": "B\n(5 3)\nWe have emphasised in Chapter 2 that the zero of potential energy\ncan be fixed at one\u2019s convenience Taking the constant of integration to be\nzero means fixing the zero of potential energy at q = 90\u00b0, i e"}, {"Chapter": "1", "sentence_range": "4346-4349", "Text": "3)\nWe have emphasised in Chapter 2 that the zero of potential energy\ncan be fixed at one\u2019s convenience Taking the constant of integration to be\nzero means fixing the zero of potential energy at q = 90\u00b0, i e , when the\nneedle is perpendicular to the field"}, {"Chapter": "1", "sentence_range": "4347-4350", "Text": "Taking the constant of integration to be\nzero means fixing the zero of potential energy at q = 90\u00b0, i e , when the\nneedle is perpendicular to the field Equation (5"}, {"Chapter": "1", "sentence_range": "4348-4351", "Text": "e , when the\nneedle is perpendicular to the field Equation (5 6) shows that potential\nenergy is minimum (= \u2013mB) at q = 0\u00b0 (most stable position) and maximum\n(= +mB) at q = 180\u00b0 (most unstable position)"}, {"Chapter": "1", "sentence_range": "4349-4352", "Text": ", when the\nneedle is perpendicular to the field Equation (5 6) shows that potential\nenergy is minimum (= \u2013mB) at q = 0\u00b0 (most stable position) and maximum\n(= +mB) at q = 180\u00b0 (most unstable position) Example 5"}, {"Chapter": "1", "sentence_range": "4350-4353", "Text": "Equation (5 6) shows that potential\nenergy is minimum (= \u2013mB) at q = 0\u00b0 (most stable position) and maximum\n(= +mB) at q = 180\u00b0 (most unstable position) Example 5 1\n(a) What happens if a bar magnet is cut into two pieces: (i) transverse\nto its length, (ii) along its length"}, {"Chapter": "1", "sentence_range": "4351-4354", "Text": "6) shows that potential\nenergy is minimum (= \u2013mB) at q = 0\u00b0 (most stable position) and maximum\n(= +mB) at q = 180\u00b0 (most unstable position) Example 5 1\n(a) What happens if a bar magnet is cut into two pieces: (i) transverse\nto its length, (ii) along its length (b) A magnetised needle in a uniform magnetic field experiences a\ntorque but no net force"}, {"Chapter": "1", "sentence_range": "4352-4355", "Text": "Example 5 1\n(a) What happens if a bar magnet is cut into two pieces: (i) transverse\nto its length, (ii) along its length (b) A magnetised needle in a uniform magnetic field experiences a\ntorque but no net force An iron nail near a bar magnet, however,\nexperiences a force of attraction in addition to a torque"}, {"Chapter": "1", "sentence_range": "4353-4356", "Text": "1\n(a) What happens if a bar magnet is cut into two pieces: (i) transverse\nto its length, (ii) along its length (b) A magnetised needle in a uniform magnetic field experiences a\ntorque but no net force An iron nail near a bar magnet, however,\nexperiences a force of attraction in addition to a torque Why"}, {"Chapter": "1", "sentence_range": "4354-4357", "Text": "(b) A magnetised needle in a uniform magnetic field experiences a\ntorque but no net force An iron nail near a bar magnet, however,\nexperiences a force of attraction in addition to a torque Why EXAMPLE 5"}, {"Chapter": "1", "sentence_range": "4355-4358", "Text": "An iron nail near a bar magnet, however,\nexperiences a force of attraction in addition to a torque Why EXAMPLE 5 1\nRationalised 2023-24\nPhysics\n140\n EXAMPLE 5"}, {"Chapter": "1", "sentence_range": "4356-4359", "Text": "Why EXAMPLE 5 1\nRationalised 2023-24\nPhysics\n140\n EXAMPLE 5 1\n(c) Must every magnetic configuration have a north pole and a south\npole"}, {"Chapter": "1", "sentence_range": "4357-4360", "Text": "EXAMPLE 5 1\nRationalised 2023-24\nPhysics\n140\n EXAMPLE 5 1\n(c) Must every magnetic configuration have a north pole and a south\npole What about the field due to a toroid"}, {"Chapter": "1", "sentence_range": "4358-4361", "Text": "1\nRationalised 2023-24\nPhysics\n140\n EXAMPLE 5 1\n(c) Must every magnetic configuration have a north pole and a south\npole What about the field due to a toroid (d) Two identical looking iron bars A and B are given, one of which\nis definitely known to be magnetised"}, {"Chapter": "1", "sentence_range": "4359-4362", "Text": "1\n(c) Must every magnetic configuration have a north pole and a south\npole What about the field due to a toroid (d) Two identical looking iron bars A and B are given, one of which\nis definitely known to be magnetised (We do not know which\none"}, {"Chapter": "1", "sentence_range": "4360-4363", "Text": "What about the field due to a toroid (d) Two identical looking iron bars A and B are given, one of which\nis definitely known to be magnetised (We do not know which\none ) How would one ascertain whether or not both are\nmagnetised"}, {"Chapter": "1", "sentence_range": "4361-4364", "Text": "(d) Two identical looking iron bars A and B are given, one of which\nis definitely known to be magnetised (We do not know which\none ) How would one ascertain whether or not both are\nmagnetised If only one is magnetised, how does one ascertain\nwhich one"}, {"Chapter": "1", "sentence_range": "4362-4365", "Text": "(We do not know which\none ) How would one ascertain whether or not both are\nmagnetised If only one is magnetised, how does one ascertain\nwhich one [Use nothing else but the bars A and B"}, {"Chapter": "1", "sentence_range": "4363-4366", "Text": ") How would one ascertain whether or not both are\nmagnetised If only one is magnetised, how does one ascertain\nwhich one [Use nothing else but the bars A and B ]\nSolution\n(a) In either case, one gets two magnets, each with a north and\nsouth pole"}, {"Chapter": "1", "sentence_range": "4364-4367", "Text": "If only one is magnetised, how does one ascertain\nwhich one [Use nothing else but the bars A and B ]\nSolution\n(a) In either case, one gets two magnets, each with a north and\nsouth pole (b) No force if the field is uniform"}, {"Chapter": "1", "sentence_range": "4365-4368", "Text": "[Use nothing else but the bars A and B ]\nSolution\n(a) In either case, one gets two magnets, each with a north and\nsouth pole (b) No force if the field is uniform The iron nail experiences a non-\nuniform field due to the bar magnet"}, {"Chapter": "1", "sentence_range": "4366-4369", "Text": "]\nSolution\n(a) In either case, one gets two magnets, each with a north and\nsouth pole (b) No force if the field is uniform The iron nail experiences a non-\nuniform field due to the bar magnet There is induced magnetic\nmoment in the nail, therefore, it experiences both force and\ntorque"}, {"Chapter": "1", "sentence_range": "4367-4370", "Text": "(b) No force if the field is uniform The iron nail experiences a non-\nuniform field due to the bar magnet There is induced magnetic\nmoment in the nail, therefore, it experiences both force and\ntorque The net force is attractive because the induced south\npole (say) in the nail is closer to the north pole of magnet than\ninduced north pole"}, {"Chapter": "1", "sentence_range": "4368-4371", "Text": "The iron nail experiences a non-\nuniform field due to the bar magnet There is induced magnetic\nmoment in the nail, therefore, it experiences both force and\ntorque The net force is attractive because the induced south\npole (say) in the nail is closer to the north pole of magnet than\ninduced north pole (c) Not necessarily"}, {"Chapter": "1", "sentence_range": "4369-4372", "Text": "There is induced magnetic\nmoment in the nail, therefore, it experiences both force and\ntorque The net force is attractive because the induced south\npole (say) in the nail is closer to the north pole of magnet than\ninduced north pole (c) Not necessarily True only if the source of the field has a net\nnon-zero magnetic moment"}, {"Chapter": "1", "sentence_range": "4370-4373", "Text": "The net force is attractive because the induced south\npole (say) in the nail is closer to the north pole of magnet than\ninduced north pole (c) Not necessarily True only if the source of the field has a net\nnon-zero magnetic moment This is not so for a toroid or even for\na straight infinite conductor"}, {"Chapter": "1", "sentence_range": "4371-4374", "Text": "(c) Not necessarily True only if the source of the field has a net\nnon-zero magnetic moment This is not so for a toroid or even for\na straight infinite conductor (d) Try to bring different ends of the bars closer"}, {"Chapter": "1", "sentence_range": "4372-4375", "Text": "True only if the source of the field has a net\nnon-zero magnetic moment This is not so for a toroid or even for\na straight infinite conductor (d) Try to bring different ends of the bars closer A repulsive force in\nsome situation establishes that both are magnetised"}, {"Chapter": "1", "sentence_range": "4373-4376", "Text": "This is not so for a toroid or even for\na straight infinite conductor (d) Try to bring different ends of the bars closer A repulsive force in\nsome situation establishes that both are magnetised If it is\nalways attractive, then one of them is not magnetised"}, {"Chapter": "1", "sentence_range": "4374-4377", "Text": "(d) Try to bring different ends of the bars closer A repulsive force in\nsome situation establishes that both are magnetised If it is\nalways attractive, then one of them is not magnetised In a bar\nmagnet the intensity of the magnetic field is the strongest at the\ntwo ends (poles) and weakest at the central region"}, {"Chapter": "1", "sentence_range": "4375-4378", "Text": "A repulsive force in\nsome situation establishes that both are magnetised If it is\nalways attractive, then one of them is not magnetised In a bar\nmagnet the intensity of the magnetic field is the strongest at the\ntwo ends (poles) and weakest at the central region This fact\nmay be used to  determine whether A or B is the magnet"}, {"Chapter": "1", "sentence_range": "4376-4379", "Text": "If it is\nalways attractive, then one of them is not magnetised In a bar\nmagnet the intensity of the magnetic field is the strongest at the\ntwo ends (poles) and weakest at the central region This fact\nmay be used to  determine whether A or B is the magnet In this\ncase, to see which one of the two bars is a magnet, pick up one,\n(say, A) and lower one of its ends; first on one of the ends of the\nother (say, B), and then on the middle of B"}, {"Chapter": "1", "sentence_range": "4377-4380", "Text": "In a bar\nmagnet the intensity of the magnetic field is the strongest at the\ntwo ends (poles) and weakest at the central region This fact\nmay be used to  determine whether A or B is the magnet In this\ncase, to see which one of the two bars is a magnet, pick up one,\n(say, A) and lower one of its ends; first on one of the ends of the\nother (say, B), and then on the middle of B If you notice that in\nthe middle of B, A experiences no force, then B is magnetised"}, {"Chapter": "1", "sentence_range": "4378-4381", "Text": "This fact\nmay be used to  determine whether A or B is the magnet In this\ncase, to see which one of the two bars is a magnet, pick up one,\n(say, A) and lower one of its ends; first on one of the ends of the\nother (say, B), and then on the middle of B If you notice that in\nthe middle of B, A experiences no force, then B is magnetised If\nyou do not notice any change from the end to the middle of B,\nthen A is magnetised"}, {"Chapter": "1", "sentence_range": "4379-4382", "Text": "In this\ncase, to see which one of the two bars is a magnet, pick up one,\n(say, A) and lower one of its ends; first on one of the ends of the\nother (say, B), and then on the middle of B If you notice that in\nthe middle of B, A experiences no force, then B is magnetised If\nyou do not notice any change from the end to the middle of B,\nthen A is magnetised 5"}, {"Chapter": "1", "sentence_range": "4380-4383", "Text": "If you notice that in\nthe middle of B, A experiences no force, then B is magnetised If\nyou do not notice any change from the end to the middle of B,\nthen A is magnetised 5 2"}, {"Chapter": "1", "sentence_range": "4381-4384", "Text": "If\nyou do not notice any change from the end to the middle of B,\nthen A is magnetised 5 2 4  The electrostatic analog\nComparison of Eqs"}, {"Chapter": "1", "sentence_range": "4382-4385", "Text": "5 2 4  The electrostatic analog\nComparison of Eqs (5"}, {"Chapter": "1", "sentence_range": "4383-4386", "Text": "2 4  The electrostatic analog\nComparison of Eqs (5 2), (5"}, {"Chapter": "1", "sentence_range": "4384-4387", "Text": "4  The electrostatic analog\nComparison of Eqs (5 2), (5 3) and (5"}, {"Chapter": "1", "sentence_range": "4385-4388", "Text": "(5 2), (5 3) and (5 6) with the corresponding equations\nfor electric dipole (Chapter 1), suggests that magnetic field at large\ndistances due to a bar magnet of magnetic moment m can be obtained\nfrom the equation for electric field due to an electric dipole of dipole moment\np, by making the following replacements:\nE\u2192\nB , p\nm\n\u2192\n, \n0\n0\n1\n4\n4\n\u00b5\n\u03b5\n\u2192\n\u03c0\n\u03c0\nIn particular, we can write down the equatorial field (BE) of a bar magnet\nat a distance r, for r >> l, where l is the size of the magnet:\n0\n3\n4\nE\nr\n= \u2212\u00b5\n\u03c0\nm\nB\n(5"}, {"Chapter": "1", "sentence_range": "4386-4389", "Text": "2), (5 3) and (5 6) with the corresponding equations\nfor electric dipole (Chapter 1), suggests that magnetic field at large\ndistances due to a bar magnet of magnetic moment m can be obtained\nfrom the equation for electric field due to an electric dipole of dipole moment\np, by making the following replacements:\nE\u2192\nB , p\nm\n\u2192\n, \n0\n0\n1\n4\n4\n\u00b5\n\u03b5\n\u2192\n\u03c0\n\u03c0\nIn particular, we can write down the equatorial field (BE) of a bar magnet\nat a distance r, for r >> l, where l is the size of the magnet:\n0\n3\n4\nE\nr\n= \u2212\u00b5\n\u03c0\nm\nB\n(5 4)\nLikewise, the axial field (BA) of a bar magnet for r >> l is:\n0\n3\n2\n4\nA\nr\n=\u00b5\n\u03c0\nm\nB\n(5"}, {"Chapter": "1", "sentence_range": "4387-4390", "Text": "3) and (5 6) with the corresponding equations\nfor electric dipole (Chapter 1), suggests that magnetic field at large\ndistances due to a bar magnet of magnetic moment m can be obtained\nfrom the equation for electric field due to an electric dipole of dipole moment\np, by making the following replacements:\nE\u2192\nB , p\nm\n\u2192\n, \n0\n0\n1\n4\n4\n\u00b5\n\u03b5\n\u2192\n\u03c0\n\u03c0\nIn particular, we can write down the equatorial field (BE) of a bar magnet\nat a distance r, for r >> l, where l is the size of the magnet:\n0\n3\n4\nE\nr\n= \u2212\u00b5\n\u03c0\nm\nB\n(5 4)\nLikewise, the axial field (BA) of a bar magnet for r >> l is:\n0\n3\n2\n4\nA\nr\n=\u00b5\n\u03c0\nm\nB\n(5 5)\nRationalised 2023-24\n141\nMagnetism and\nMatter\nEquation (5"}, {"Chapter": "1", "sentence_range": "4388-4391", "Text": "6) with the corresponding equations\nfor electric dipole (Chapter 1), suggests that magnetic field at large\ndistances due to a bar magnet of magnetic moment m can be obtained\nfrom the equation for electric field due to an electric dipole of dipole moment\np, by making the following replacements:\nE\u2192\nB , p\nm\n\u2192\n, \n0\n0\n1\n4\n4\n\u00b5\n\u03b5\n\u2192\n\u03c0\n\u03c0\nIn particular, we can write down the equatorial field (BE) of a bar magnet\nat a distance r, for r >> l, where l is the size of the magnet:\n0\n3\n4\nE\nr\n= \u2212\u00b5\n\u03c0\nm\nB\n(5 4)\nLikewise, the axial field (BA) of a bar magnet for r >> l is:\n0\n3\n2\n4\nA\nr\n=\u00b5\n\u03c0\nm\nB\n(5 5)\nRationalised 2023-24\n141\nMagnetism and\nMatter\nEquation (5 8) is just Eq"}, {"Chapter": "1", "sentence_range": "4389-4392", "Text": "4)\nLikewise, the axial field (BA) of a bar magnet for r >> l is:\n0\n3\n2\n4\nA\nr\n=\u00b5\n\u03c0\nm\nB\n(5 5)\nRationalised 2023-24\n141\nMagnetism and\nMatter\nEquation (5 8) is just Eq (5"}, {"Chapter": "1", "sentence_range": "4390-4393", "Text": "5)\nRationalised 2023-24\n141\nMagnetism and\nMatter\nEquation (5 8) is just Eq (5 2) in the vector form"}, {"Chapter": "1", "sentence_range": "4391-4394", "Text": "8) is just Eq (5 2) in the vector form Table 5"}, {"Chapter": "1", "sentence_range": "4392-4395", "Text": "(5 2) in the vector form Table 5 1 summarises\nthe analogy between electric and magnetic dipoles"}, {"Chapter": "1", "sentence_range": "4393-4396", "Text": "2) in the vector form Table 5 1 summarises\nthe analogy between electric and magnetic dipoles Electrostatics\nMagnetism\n1/e0\nm0\nDipole moment\np\nm\nEquatorial Field for a short dipole\n\u2013p/4pe0r 3\n\u2013 m0 m / 4p r 3\nAxial Field for a short dipole\n2p/4pe0r 3\nm0 2m / 4p r 3\nExternal Field: torque\np \u00d7 E\nm \u00d7 B\nExternal Field: Energy\n\u2013p"}, {"Chapter": "1", "sentence_range": "4394-4397", "Text": "Table 5 1 summarises\nthe analogy between electric and magnetic dipoles Electrostatics\nMagnetism\n1/e0\nm0\nDipole moment\np\nm\nEquatorial Field for a short dipole\n\u2013p/4pe0r 3\n\u2013 m0 m / 4p r 3\nAxial Field for a short dipole\n2p/4pe0r 3\nm0 2m / 4p r 3\nExternal Field: torque\np \u00d7 E\nm \u00d7 B\nExternal Field: Energy\n\u2013p E\n\u2013m"}, {"Chapter": "1", "sentence_range": "4395-4398", "Text": "1 summarises\nthe analogy between electric and magnetic dipoles Electrostatics\nMagnetism\n1/e0\nm0\nDipole moment\np\nm\nEquatorial Field for a short dipole\n\u2013p/4pe0r 3\n\u2013 m0 m / 4p r 3\nAxial Field for a short dipole\n2p/4pe0r 3\nm0 2m / 4p r 3\nExternal Field: torque\np \u00d7 E\nm \u00d7 B\nExternal Field: Energy\n\u2013p E\n\u2013m B\nTABLE 5"}, {"Chapter": "1", "sentence_range": "4396-4399", "Text": "Electrostatics\nMagnetism\n1/e0\nm0\nDipole moment\np\nm\nEquatorial Field for a short dipole\n\u2013p/4pe0r 3\n\u2013 m0 m / 4p r 3\nAxial Field for a short dipole\n2p/4pe0r 3\nm0 2m / 4p r 3\nExternal Field: torque\np \u00d7 E\nm \u00d7 B\nExternal Field: Energy\n\u2013p E\n\u2013m B\nTABLE 5 1 THE DIPOLE ANALOGY\n EXAMPLE 5"}, {"Chapter": "1", "sentence_range": "4397-4400", "Text": "E\n\u2013m B\nTABLE 5 1 THE DIPOLE ANALOGY\n EXAMPLE 5 2\nExample 5"}, {"Chapter": "1", "sentence_range": "4398-4401", "Text": "B\nTABLE 5 1 THE DIPOLE ANALOGY\n EXAMPLE 5 2\nExample 5 2  Figure 5"}, {"Chapter": "1", "sentence_range": "4399-4402", "Text": "1 THE DIPOLE ANALOGY\n EXAMPLE 5 2\nExample 5 2  Figure 5 4 shows a small magnetised needle P placed at\na point O"}, {"Chapter": "1", "sentence_range": "4400-4403", "Text": "2\nExample 5 2  Figure 5 4 shows a small magnetised needle P placed at\na point O The arrow shows the direction of its magnetic moment"}, {"Chapter": "1", "sentence_range": "4401-4404", "Text": "2  Figure 5 4 shows a small magnetised needle P placed at\na point O The arrow shows the direction of its magnetic moment The\nother arrows show different positions (and orientations of the magnetic\nmoment) of another identical magnetised needle Q"}, {"Chapter": "1", "sentence_range": "4402-4405", "Text": "4 shows a small magnetised needle P placed at\na point O The arrow shows the direction of its magnetic moment The\nother arrows show different positions (and orientations of the magnetic\nmoment) of another identical magnetised needle Q (a) In which configuration the system is not in equilibrium"}, {"Chapter": "1", "sentence_range": "4403-4406", "Text": "The arrow shows the direction of its magnetic moment The\nother arrows show different positions (and orientations of the magnetic\nmoment) of another identical magnetised needle Q (a) In which configuration the system is not in equilibrium (b) In which configuration is the system in (i) stable, and (ii) unstable\nequilibrium"}, {"Chapter": "1", "sentence_range": "4404-4407", "Text": "The\nother arrows show different positions (and orientations of the magnetic\nmoment) of another identical magnetised needle Q (a) In which configuration the system is not in equilibrium (b) In which configuration is the system in (i) stable, and (ii) unstable\nequilibrium (c) Which configuration corresponds to the lowest potential energy\namong all the configurations shown"}, {"Chapter": "1", "sentence_range": "4405-4408", "Text": "(a) In which configuration the system is not in equilibrium (b) In which configuration is the system in (i) stable, and (ii) unstable\nequilibrium (c) Which configuration corresponds to the lowest potential energy\namong all the configurations shown FIGURE 5"}, {"Chapter": "1", "sentence_range": "4406-4409", "Text": "(b) In which configuration is the system in (i) stable, and (ii) unstable\nequilibrium (c) Which configuration corresponds to the lowest potential energy\namong all the configurations shown FIGURE 5 4\nSolution\nPotential energy of the configuration arises due to the potential energy of\none dipole (say, Q) in the magnetic field due to other (P)"}, {"Chapter": "1", "sentence_range": "4407-4410", "Text": "(c) Which configuration corresponds to the lowest potential energy\namong all the configurations shown FIGURE 5 4\nSolution\nPotential energy of the configuration arises due to the potential energy of\none dipole (say, Q) in the magnetic field due to other (P) Use the result\nthat the field due to P is given by the expression [Eqs"}, {"Chapter": "1", "sentence_range": "4408-4411", "Text": "FIGURE 5 4\nSolution\nPotential energy of the configuration arises due to the potential energy of\none dipole (say, Q) in the magnetic field due to other (P) Use the result\nthat the field due to P is given by the expression [Eqs (5"}, {"Chapter": "1", "sentence_range": "4409-4412", "Text": "4\nSolution\nPotential energy of the configuration arises due to the potential energy of\none dipole (say, Q) in the magnetic field due to other (P) Use the result\nthat the field due to P is given by the expression [Eqs (5 7) and (5"}, {"Chapter": "1", "sentence_range": "4410-4413", "Text": "Use the result\nthat the field due to P is given by the expression [Eqs (5 7) and (5 8)]:\n0\nP\nP\n3\n4\nr\n\u00b5\n\u03c0\n= \u2212\nm\nB\n     (on the normal bisector)\n0\nP\nP\n3\n42\nr\n\u00b5\n\u03c0\n=\nm\nB\n     (on the axis)\nwhere mP is the magnetic moment of the dipole P"}, {"Chapter": "1", "sentence_range": "4411-4414", "Text": "(5 7) and (5 8)]:\n0\nP\nP\n3\n4\nr\n\u00b5\n\u03c0\n= \u2212\nm\nB\n     (on the normal bisector)\n0\nP\nP\n3\n42\nr\n\u00b5\n\u03c0\n=\nm\nB\n     (on the axis)\nwhere mP is the magnetic moment of the dipole P Equilibrium is stable when mQ is parallel to BP, and unstable when it\nis anti-parallel to BP"}, {"Chapter": "1", "sentence_range": "4412-4415", "Text": "7) and (5 8)]:\n0\nP\nP\n3\n4\nr\n\u00b5\n\u03c0\n= \u2212\nm\nB\n     (on the normal bisector)\n0\nP\nP\n3\n42\nr\n\u00b5\n\u03c0\n=\nm\nB\n     (on the axis)\nwhere mP is the magnetic moment of the dipole P Equilibrium is stable when mQ is parallel to BP, and unstable when it\nis anti-parallel to BP Rationalised 2023-24\nPhysics\n142\nKARL FRIEDRICH GAUSS (1777 \u2013 1855)\nKarl Friedrich Gauss\n(1777 \u2013 1855) He was a\nchild prodigy and was gifted\nin mathematics, physics,\nengineering, astronomy\nand even land surveying"}, {"Chapter": "1", "sentence_range": "4413-4416", "Text": "8)]:\n0\nP\nP\n3\n4\nr\n\u00b5\n\u03c0\n= \u2212\nm\nB\n     (on the normal bisector)\n0\nP\nP\n3\n42\nr\n\u00b5\n\u03c0\n=\nm\nB\n     (on the axis)\nwhere mP is the magnetic moment of the dipole P Equilibrium is stable when mQ is parallel to BP, and unstable when it\nis anti-parallel to BP Rationalised 2023-24\nPhysics\n142\nKARL FRIEDRICH GAUSS (1777 \u2013 1855)\nKarl Friedrich Gauss\n(1777 \u2013 1855) He was a\nchild prodigy and was gifted\nin mathematics, physics,\nengineering, astronomy\nand even land surveying The properties of numbers\nfascinated him, and in his\nwork he anticipated major\nmathematical development\nof later times"}, {"Chapter": "1", "sentence_range": "4414-4417", "Text": "Equilibrium is stable when mQ is parallel to BP, and unstable when it\nis anti-parallel to BP Rationalised 2023-24\nPhysics\n142\nKARL FRIEDRICH GAUSS (1777 \u2013 1855)\nKarl Friedrich Gauss\n(1777 \u2013 1855) He was a\nchild prodigy and was gifted\nin mathematics, physics,\nengineering, astronomy\nand even land surveying The properties of numbers\nfascinated him, and in his\nwork he anticipated major\nmathematical development\nof later times Along with\nWilhelm Welser, he built the\nfirst electric telegraph in\n1833"}, {"Chapter": "1", "sentence_range": "4415-4418", "Text": "Rationalised 2023-24\nPhysics\n142\nKARL FRIEDRICH GAUSS (1777 \u2013 1855)\nKarl Friedrich Gauss\n(1777 \u2013 1855) He was a\nchild prodigy and was gifted\nin mathematics, physics,\nengineering, astronomy\nand even land surveying The properties of numbers\nfascinated him, and in his\nwork he anticipated major\nmathematical development\nof later times Along with\nWilhelm Welser, he built the\nfirst electric telegraph in\n1833 His mathematical\ntheory of curved surface\nlaid the foundation for the\nlater work of Riemann"}, {"Chapter": "1", "sentence_range": "4416-4419", "Text": "The properties of numbers\nfascinated him, and in his\nwork he anticipated major\nmathematical development\nof later times Along with\nWilhelm Welser, he built the\nfirst electric telegraph in\n1833 His mathematical\ntheory of curved surface\nlaid the foundation for the\nlater work of Riemann For instance for the configuration Q3 for which Q is along the\nperpendicular bisector of the dipole P, the magnetic moment of Q is\nparallel to the magnetic field at the position 3"}, {"Chapter": "1", "sentence_range": "4417-4420", "Text": "Along with\nWilhelm Welser, he built the\nfirst electric telegraph in\n1833 His mathematical\ntheory of curved surface\nlaid the foundation for the\nlater work of Riemann For instance for the configuration Q3 for which Q is along the\nperpendicular bisector of the dipole P, the magnetic moment of Q is\nparallel to the magnetic field at the position 3 Hence Q3 is stable"}, {"Chapter": "1", "sentence_range": "4418-4421", "Text": "His mathematical\ntheory of curved surface\nlaid the foundation for the\nlater work of Riemann For instance for the configuration Q3 for which Q is along the\nperpendicular bisector of the dipole P, the magnetic moment of Q is\nparallel to the magnetic field at the position 3 Hence Q3 is stable Thus,\n(a) PQ1 and PQ2\n(b) (i) PQ3, PQ6 (stable); (ii) PQ5, PQ4 (unstable)\n(c) PQ6\n5"}, {"Chapter": "1", "sentence_range": "4419-4422", "Text": "For instance for the configuration Q3 for which Q is along the\nperpendicular bisector of the dipole P, the magnetic moment of Q is\nparallel to the magnetic field at the position 3 Hence Q3 is stable Thus,\n(a) PQ1 and PQ2\n(b) (i) PQ3, PQ6 (stable); (ii) PQ5, PQ4 (unstable)\n(c) PQ6\n5 3  MAGNETISM AND GAUSS\u2019S LAW\nIn Chapter 1, we studied Gauss\u2019s law for electrostatics"}, {"Chapter": "1", "sentence_range": "4420-4423", "Text": "Hence Q3 is stable Thus,\n(a) PQ1 and PQ2\n(b) (i) PQ3, PQ6 (stable); (ii) PQ5, PQ4 (unstable)\n(c) PQ6\n5 3  MAGNETISM AND GAUSS\u2019S LAW\nIn Chapter 1, we studied Gauss\u2019s law for electrostatics In Fig 5"}, {"Chapter": "1", "sentence_range": "4421-4424", "Text": "Thus,\n(a) PQ1 and PQ2\n(b) (i) PQ3, PQ6 (stable); (ii) PQ5, PQ4 (unstable)\n(c) PQ6\n5 3  MAGNETISM AND GAUSS\u2019S LAW\nIn Chapter 1, we studied Gauss\u2019s law for electrostatics In Fig 5 3(c), we see that for a closed surface represented\nby  i , the number of lines leaving the surface is equal to\nthe number of lines entering it"}, {"Chapter": "1", "sentence_range": "4422-4425", "Text": "3  MAGNETISM AND GAUSS\u2019S LAW\nIn Chapter 1, we studied Gauss\u2019s law for electrostatics In Fig 5 3(c), we see that for a closed surface represented\nby  i , the number of lines leaving the surface is equal to\nthe number of lines entering it This is consistent with the\nfact that no net charge is enclosed by the surface"}, {"Chapter": "1", "sentence_range": "4423-4426", "Text": "In Fig 5 3(c), we see that for a closed surface represented\nby  i , the number of lines leaving the surface is equal to\nthe number of lines entering it This is consistent with the\nfact that no net charge is enclosed by the surface However,\nin the same figure, for the closed surface ii , there is a net\noutward flux, since it does include a net (positive) charge"}, {"Chapter": "1", "sentence_range": "4424-4427", "Text": "3(c), we see that for a closed surface represented\nby  i , the number of lines leaving the surface is equal to\nthe number of lines entering it This is consistent with the\nfact that no net charge is enclosed by the surface However,\nin the same figure, for the closed surface ii , there is a net\noutward flux, since it does include a net (positive) charge The situation is radically different for magnetic fields\nwhich are continuous and form closed loops"}, {"Chapter": "1", "sentence_range": "4425-4428", "Text": "This is consistent with the\nfact that no net charge is enclosed by the surface However,\nin the same figure, for the closed surface ii , there is a net\noutward flux, since it does include a net (positive) charge The situation is radically different for magnetic fields\nwhich are continuous and form closed loops Examine the\nGaussian surfaces represented by  i  or  ii  in Fig 5"}, {"Chapter": "1", "sentence_range": "4426-4429", "Text": "However,\nin the same figure, for the closed surface ii , there is a net\noutward flux, since it does include a net (positive) charge The situation is radically different for magnetic fields\nwhich are continuous and form closed loops Examine the\nGaussian surfaces represented by  i  or  ii  in Fig 5 3(a) or\nFig"}, {"Chapter": "1", "sentence_range": "4427-4430", "Text": "The situation is radically different for magnetic fields\nwhich are continuous and form closed loops Examine the\nGaussian surfaces represented by  i  or  ii  in Fig 5 3(a) or\nFig 5"}, {"Chapter": "1", "sentence_range": "4428-4431", "Text": "Examine the\nGaussian surfaces represented by  i  or  ii  in Fig 5 3(a) or\nFig 5 3(b)"}, {"Chapter": "1", "sentence_range": "4429-4432", "Text": "3(a) or\nFig 5 3(b) Both cases visually demonstrate that the\nnumber of magnetic field lines leaving the surface is\nbalanced by the number of lines entering it"}, {"Chapter": "1", "sentence_range": "4430-4433", "Text": "5 3(b) Both cases visually demonstrate that the\nnumber of magnetic field lines leaving the surface is\nbalanced by the number of lines entering it The net\nmagnetic flux is zero for both the surfaces"}, {"Chapter": "1", "sentence_range": "4431-4434", "Text": "3(b) Both cases visually demonstrate that the\nnumber of magnetic field lines leaving the surface is\nbalanced by the number of lines entering it The net\nmagnetic flux is zero for both the surfaces This is true\nfor any closed surface"}, {"Chapter": "1", "sentence_range": "4432-4435", "Text": "Both cases visually demonstrate that the\nnumber of magnetic field lines leaving the surface is\nbalanced by the number of lines entering it The net\nmagnetic flux is zero for both the surfaces This is true\nfor any closed surface FIGURE 5"}, {"Chapter": "1", "sentence_range": "4433-4436", "Text": "The net\nmagnetic flux is zero for both the surfaces This is true\nfor any closed surface FIGURE 5 5\nConsider a small vector area element DS of a closed surface S as in\nFig"}, {"Chapter": "1", "sentence_range": "4434-4437", "Text": "This is true\nfor any closed surface FIGURE 5 5\nConsider a small vector area element DS of a closed surface S as in\nFig 5"}, {"Chapter": "1", "sentence_range": "4435-4438", "Text": "FIGURE 5 5\nConsider a small vector area element DS of a closed surface S as in\nFig 5 5"}, {"Chapter": "1", "sentence_range": "4436-4439", "Text": "5\nConsider a small vector area element DS of a closed surface S as in\nFig 5 5 The magnetic flux through \u00c4S is defined as DfB = B"}, {"Chapter": "1", "sentence_range": "4437-4440", "Text": "5 5 The magnetic flux through \u00c4S is defined as DfB = B DS, where B\nis the field  at DS"}, {"Chapter": "1", "sentence_range": "4438-4441", "Text": "5 The magnetic flux through \u00c4S is defined as DfB = B DS, where B\nis the field  at DS We divide S into many small area elements and calculate\nthe individual flux through each"}, {"Chapter": "1", "sentence_range": "4439-4442", "Text": "The magnetic flux through \u00c4S is defined as DfB = B DS, where B\nis the field  at DS We divide S into many small area elements and calculate\nthe individual flux through each Then, the net flux fB is,\n\u03c6\n\u03c6\nB\nB\nall\nall\n=\n=\n=\n\u2211\n\u2211\n\u2206\n\u2206\n\u2019\n\u2019\n\u2019\n\u2019\nB"}, {"Chapter": "1", "sentence_range": "4440-4443", "Text": "DS, where B\nis the field  at DS We divide S into many small area elements and calculate\nthe individual flux through each Then, the net flux fB is,\n\u03c6\n\u03c6\nB\nB\nall\nall\n=\n=\n=\n\u2211\n\u2211\n\u2206\n\u2206\n\u2019\n\u2019\n\u2019\n\u2019\nB S\n0\n(5"}, {"Chapter": "1", "sentence_range": "4441-4444", "Text": "We divide S into many small area elements and calculate\nthe individual flux through each Then, the net flux fB is,\n\u03c6\n\u03c6\nB\nB\nall\nall\n=\n=\n=\n\u2211\n\u2211\n\u2206\n\u2206\n\u2019\n\u2019\n\u2019\n\u2019\nB S\n0\n(5 6)\nwhere \u2018all\u2019 stands for \u2018all area elements DS\u00a2"}, {"Chapter": "1", "sentence_range": "4442-4445", "Text": "Then, the net flux fB is,\n\u03c6\n\u03c6\nB\nB\nall\nall\n=\n=\n=\n\u2211\n\u2211\n\u2206\n\u2206\n\u2019\n\u2019\n\u2019\n\u2019\nB S\n0\n(5 6)\nwhere \u2018all\u2019 stands for \u2018all area elements DS\u00a2 Compare this with the Gauss\u2019s\nlaw of electrostatics"}, {"Chapter": "1", "sentence_range": "4443-4446", "Text": "S\n0\n(5 6)\nwhere \u2018all\u2019 stands for \u2018all area elements DS\u00a2 Compare this with the Gauss\u2019s\nlaw of electrostatics The flux through a closed surface in that case is\ngiven by\nE"}, {"Chapter": "1", "sentence_range": "4444-4447", "Text": "6)\nwhere \u2018all\u2019 stands for \u2018all area elements DS\u00a2 Compare this with the Gauss\u2019s\nlaw of electrostatics The flux through a closed surface in that case is\ngiven by\nE \u2206S\n=\n\u2211\nq\n\u03b50\n EXAMPLE 5"}, {"Chapter": "1", "sentence_range": "4445-4448", "Text": "Compare this with the Gauss\u2019s\nlaw of electrostatics The flux through a closed surface in that case is\ngiven by\nE \u2206S\n=\n\u2211\nq\n\u03b50\n EXAMPLE 5 2\nRationalised 2023-24\n143\nMagnetism and\nMatter\nwhere q is the electric charge enclosed by the surface"}, {"Chapter": "1", "sentence_range": "4446-4449", "Text": "The flux through a closed surface in that case is\ngiven by\nE \u2206S\n=\n\u2211\nq\n\u03b50\n EXAMPLE 5 2\nRationalised 2023-24\n143\nMagnetism and\nMatter\nwhere q is the electric charge enclosed by the surface The difference between the Gauss\u2019s law of magnetism and that for\nelectrostatics is a reflection of the fact that isolated magnetic poles (also\ncalled monopoles) are not known to exist"}, {"Chapter": "1", "sentence_range": "4447-4450", "Text": "\u2206S\n=\n\u2211\nq\n\u03b50\n EXAMPLE 5 2\nRationalised 2023-24\n143\nMagnetism and\nMatter\nwhere q is the electric charge enclosed by the surface The difference between the Gauss\u2019s law of magnetism and that for\nelectrostatics is a reflection of the fact that isolated magnetic poles (also\ncalled monopoles) are not known to exist There are no sources or sinks\nof B; the simplest magnetic element is a dipole or a current loop"}, {"Chapter": "1", "sentence_range": "4448-4451", "Text": "2\nRationalised 2023-24\n143\nMagnetism and\nMatter\nwhere q is the electric charge enclosed by the surface The difference between the Gauss\u2019s law of magnetism and that for\nelectrostatics is a reflection of the fact that isolated magnetic poles (also\ncalled monopoles) are not known to exist There are no sources or sinks\nof B; the simplest magnetic element is a dipole or a current loop All\nmagnetic phenomena can be explained in terms of an arrangement of\ndipoles and/or current loops"}, {"Chapter": "1", "sentence_range": "4449-4452", "Text": "The difference between the Gauss\u2019s law of magnetism and that for\nelectrostatics is a reflection of the fact that isolated magnetic poles (also\ncalled monopoles) are not known to exist There are no sources or sinks\nof B; the simplest magnetic element is a dipole or a current loop All\nmagnetic phenomena can be explained in terms of an arrangement of\ndipoles and/or current loops Thus, Gauss\u2019s law for magnetism is:\nThe net magnetic flux through any closed surface is zero"}, {"Chapter": "1", "sentence_range": "4450-4453", "Text": "There are no sources or sinks\nof B; the simplest magnetic element is a dipole or a current loop All\nmagnetic phenomena can be explained in terms of an arrangement of\ndipoles and/or current loops Thus, Gauss\u2019s law for magnetism is:\nThe net magnetic flux through any closed surface is zero Example 5"}, {"Chapter": "1", "sentence_range": "4451-4454", "Text": "All\nmagnetic phenomena can be explained in terms of an arrangement of\ndipoles and/or current loops Thus, Gauss\u2019s law for magnetism is:\nThe net magnetic flux through any closed surface is zero Example 5 3  Many of the diagrams given in Fig"}, {"Chapter": "1", "sentence_range": "4452-4455", "Text": "Thus, Gauss\u2019s law for magnetism is:\nThe net magnetic flux through any closed surface is zero Example 5 3  Many of the diagrams given in Fig 5"}, {"Chapter": "1", "sentence_range": "4453-4456", "Text": "Example 5 3  Many of the diagrams given in Fig 5 7 show magnetic\nfield lines (thick lines in the figure) wrongly"}, {"Chapter": "1", "sentence_range": "4454-4457", "Text": "3  Many of the diagrams given in Fig 5 7 show magnetic\nfield lines (thick lines in the figure) wrongly Point out what is wrong\nwith them"}, {"Chapter": "1", "sentence_range": "4455-4458", "Text": "5 7 show magnetic\nfield lines (thick lines in the figure) wrongly Point out what is wrong\nwith them Some of them may describe electrostatic field lines correctly"}, {"Chapter": "1", "sentence_range": "4456-4459", "Text": "7 show magnetic\nfield lines (thick lines in the figure) wrongly Point out what is wrong\nwith them Some of them may describe electrostatic field lines correctly Point out which ones"}, {"Chapter": "1", "sentence_range": "4457-4460", "Text": "Point out what is wrong\nwith them Some of them may describe electrostatic field lines correctly Point out which ones FIGURE 5"}, {"Chapter": "1", "sentence_range": "4458-4461", "Text": "Some of them may describe electrostatic field lines correctly Point out which ones FIGURE 5 6\n EXAMPLE 5"}, {"Chapter": "1", "sentence_range": "4459-4462", "Text": "Point out which ones FIGURE 5 6\n EXAMPLE 5 3\nRationalised 2023-24\nPhysics\n144\n EXAMPLE 5"}, {"Chapter": "1", "sentence_range": "4460-4463", "Text": "FIGURE 5 6\n EXAMPLE 5 3\nRationalised 2023-24\nPhysics\n144\n EXAMPLE 5 4\n EXAMPLE 5"}, {"Chapter": "1", "sentence_range": "4461-4464", "Text": "6\n EXAMPLE 5 3\nRationalised 2023-24\nPhysics\n144\n EXAMPLE 5 4\n EXAMPLE 5 3\nSolution\n(a) Wrong"}, {"Chapter": "1", "sentence_range": "4462-4465", "Text": "3\nRationalised 2023-24\nPhysics\n144\n EXAMPLE 5 4\n EXAMPLE 5 3\nSolution\n(a) Wrong Magnetic field lines can never emanate from a point, as\nshown in figure"}, {"Chapter": "1", "sentence_range": "4463-4466", "Text": "4\n EXAMPLE 5 3\nSolution\n(a) Wrong Magnetic field lines can never emanate from a point, as\nshown in figure Over any closed surface, the net flux of B must\nalways be zero, i"}, {"Chapter": "1", "sentence_range": "4464-4467", "Text": "3\nSolution\n(a) Wrong Magnetic field lines can never emanate from a point, as\nshown in figure Over any closed surface, the net flux of B must\nalways be zero, i e"}, {"Chapter": "1", "sentence_range": "4465-4468", "Text": "Magnetic field lines can never emanate from a point, as\nshown in figure Over any closed surface, the net flux of B must\nalways be zero, i e , pictorially as many field lines should seem to\nenter the surface as the number of lines leaving it"}, {"Chapter": "1", "sentence_range": "4466-4469", "Text": "Over any closed surface, the net flux of B must\nalways be zero, i e , pictorially as many field lines should seem to\nenter the surface as the number of lines leaving it The field lines\nshown, in fact, represent electric field of a long positively charged\nwire"}, {"Chapter": "1", "sentence_range": "4467-4470", "Text": "e , pictorially as many field lines should seem to\nenter the surface as the number of lines leaving it The field lines\nshown, in fact, represent electric field of a long positively charged\nwire The correct magnetic field lines are circling the straight\nconductor, as described in Chapter 4"}, {"Chapter": "1", "sentence_range": "4468-4471", "Text": ", pictorially as many field lines should seem to\nenter the surface as the number of lines leaving it The field lines\nshown, in fact, represent electric field of a long positively charged\nwire The correct magnetic field lines are circling the straight\nconductor, as described in Chapter 4 (b) Wrong"}, {"Chapter": "1", "sentence_range": "4469-4472", "Text": "The field lines\nshown, in fact, represent electric field of a long positively charged\nwire The correct magnetic field lines are circling the straight\nconductor, as described in Chapter 4 (b) Wrong Magnetic field lines (like electric field lines) can never cross\neach other, because otherwise the direction of field at the point of\nintersection is ambiguous"}, {"Chapter": "1", "sentence_range": "4470-4473", "Text": "The correct magnetic field lines are circling the straight\nconductor, as described in Chapter 4 (b) Wrong Magnetic field lines (like electric field lines) can never cross\neach other, because otherwise the direction of field at the point of\nintersection is ambiguous There is further error in the figure"}, {"Chapter": "1", "sentence_range": "4471-4474", "Text": "(b) Wrong Magnetic field lines (like electric field lines) can never cross\neach other, because otherwise the direction of field at the point of\nintersection is ambiguous There is further error in the figure Magnetostatic field lines can never form closed loops around empty\nspace"}, {"Chapter": "1", "sentence_range": "4472-4475", "Text": "Magnetic field lines (like electric field lines) can never cross\neach other, because otherwise the direction of field at the point of\nintersection is ambiguous There is further error in the figure Magnetostatic field lines can never form closed loops around empty\nspace A closed loop of static magnetic field line must enclose a\nregion across which a current is passing"}, {"Chapter": "1", "sentence_range": "4473-4476", "Text": "There is further error in the figure Magnetostatic field lines can never form closed loops around empty\nspace A closed loop of static magnetic field line must enclose a\nregion across which a current is passing By contrast, electrostatic\nfield lines can never form closed loops, neither in empty space,\nnor when the loop encloses charges"}, {"Chapter": "1", "sentence_range": "4474-4477", "Text": "Magnetostatic field lines can never form closed loops around empty\nspace A closed loop of static magnetic field line must enclose a\nregion across which a current is passing By contrast, electrostatic\nfield lines can never form closed loops, neither in empty space,\nnor when the loop encloses charges (c) Right"}, {"Chapter": "1", "sentence_range": "4475-4478", "Text": "A closed loop of static magnetic field line must enclose a\nregion across which a current is passing By contrast, electrostatic\nfield lines can never form closed loops, neither in empty space,\nnor when the loop encloses charges (c) Right Magnetic lines are completely confined within a toroid"}, {"Chapter": "1", "sentence_range": "4476-4479", "Text": "By contrast, electrostatic\nfield lines can never form closed loops, neither in empty space,\nnor when the loop encloses charges (c) Right Magnetic lines are completely confined within a toroid Nothing wrong here in field lines forming closed loops, since each\nloop encloses a region across which a current passes"}, {"Chapter": "1", "sentence_range": "4477-4480", "Text": "(c) Right Magnetic lines are completely confined within a toroid Nothing wrong here in field lines forming closed loops, since each\nloop encloses a region across which a current passes Note, for\nclarity of figure, only a few field lines within the toroid have been\nshown"}, {"Chapter": "1", "sentence_range": "4478-4481", "Text": "Magnetic lines are completely confined within a toroid Nothing wrong here in field lines forming closed loops, since each\nloop encloses a region across which a current passes Note, for\nclarity of figure, only a few field lines within the toroid have been\nshown Actually, the entire region enclosed by the windings\ncontains magnetic field"}, {"Chapter": "1", "sentence_range": "4479-4482", "Text": "Nothing wrong here in field lines forming closed loops, since each\nloop encloses a region across which a current passes Note, for\nclarity of figure, only a few field lines within the toroid have been\nshown Actually, the entire region enclosed by the windings\ncontains magnetic field (d) Wrong"}, {"Chapter": "1", "sentence_range": "4480-4483", "Text": "Note, for\nclarity of figure, only a few field lines within the toroid have been\nshown Actually, the entire region enclosed by the windings\ncontains magnetic field (d) Wrong Field lines due to a solenoid at its ends and outside cannot\nbe so completely straight and confined; such a thing violates\nAmpere\u2019s law"}, {"Chapter": "1", "sentence_range": "4481-4484", "Text": "Actually, the entire region enclosed by the windings\ncontains magnetic field (d) Wrong Field lines due to a solenoid at its ends and outside cannot\nbe so completely straight and confined; such a thing violates\nAmpere\u2019s law The lines should curve out at both ends, and meet\neventually to form closed loops"}, {"Chapter": "1", "sentence_range": "4482-4485", "Text": "(d) Wrong Field lines due to a solenoid at its ends and outside cannot\nbe so completely straight and confined; such a thing violates\nAmpere\u2019s law The lines should curve out at both ends, and meet\neventually to form closed loops (e) Right"}, {"Chapter": "1", "sentence_range": "4483-4486", "Text": "Field lines due to a solenoid at its ends and outside cannot\nbe so completely straight and confined; such a thing violates\nAmpere\u2019s law The lines should curve out at both ends, and meet\neventually to form closed loops (e) Right These are field lines outside and inside a bar magnet"}, {"Chapter": "1", "sentence_range": "4484-4487", "Text": "The lines should curve out at both ends, and meet\neventually to form closed loops (e) Right These are field lines outside and inside a bar magnet Note\ncarefully the direction of field lines inside"}, {"Chapter": "1", "sentence_range": "4485-4488", "Text": "(e) Right These are field lines outside and inside a bar magnet Note\ncarefully the direction of field lines inside Not all field lines emanate\nout of a north pole (or converge into a south pole)"}, {"Chapter": "1", "sentence_range": "4486-4489", "Text": "These are field lines outside and inside a bar magnet Note\ncarefully the direction of field lines inside Not all field lines emanate\nout of a north pole (or converge into a south pole) Around both\nthe N-pole, and the S-pole, the net flux of the field is zero"}, {"Chapter": "1", "sentence_range": "4487-4490", "Text": "Note\ncarefully the direction of field lines inside Not all field lines emanate\nout of a north pole (or converge into a south pole) Around both\nthe N-pole, and the S-pole, the net flux of the field is zero (f ) Wrong"}, {"Chapter": "1", "sentence_range": "4488-4491", "Text": "Not all field lines emanate\nout of a north pole (or converge into a south pole) Around both\nthe N-pole, and the S-pole, the net flux of the field is zero (f ) Wrong These field lines cannot possibly represent a magnetic field"}, {"Chapter": "1", "sentence_range": "4489-4492", "Text": "Around both\nthe N-pole, and the S-pole, the net flux of the field is zero (f ) Wrong These field lines cannot possibly represent a magnetic field Look at the upper region"}, {"Chapter": "1", "sentence_range": "4490-4493", "Text": "(f ) Wrong These field lines cannot possibly represent a magnetic field Look at the upper region All the field lines seem to emanate out of\nthe shaded plate"}, {"Chapter": "1", "sentence_range": "4491-4494", "Text": "These field lines cannot possibly represent a magnetic field Look at the upper region All the field lines seem to emanate out of\nthe shaded plate The net flux through a surface surrounding the\nshaded plate is not zero"}, {"Chapter": "1", "sentence_range": "4492-4495", "Text": "Look at the upper region All the field lines seem to emanate out of\nthe shaded plate The net flux through a surface surrounding the\nshaded plate is not zero This is impossible for a magnetic field"}, {"Chapter": "1", "sentence_range": "4493-4496", "Text": "All the field lines seem to emanate out of\nthe shaded plate The net flux through a surface surrounding the\nshaded plate is not zero This is impossible for a magnetic field The given field lines, in fact, show the electrostatic field lines\naround a positively charged upper plate and a negatively charged\nlower plate"}, {"Chapter": "1", "sentence_range": "4494-4497", "Text": "The net flux through a surface surrounding the\nshaded plate is not zero This is impossible for a magnetic field The given field lines, in fact, show the electrostatic field lines\naround a positively charged upper plate and a negatively charged\nlower plate The difference between Fig"}, {"Chapter": "1", "sentence_range": "4495-4498", "Text": "This is impossible for a magnetic field The given field lines, in fact, show the electrostatic field lines\naround a positively charged upper plate and a negatively charged\nlower plate The difference between Fig [5"}, {"Chapter": "1", "sentence_range": "4496-4499", "Text": "The given field lines, in fact, show the electrostatic field lines\naround a positively charged upper plate and a negatively charged\nlower plate The difference between Fig [5 7(e) and (f)]  should be\ncarefully grasped"}, {"Chapter": "1", "sentence_range": "4497-4500", "Text": "The difference between Fig [5 7(e) and (f)]  should be\ncarefully grasped (g) Wrong"}, {"Chapter": "1", "sentence_range": "4498-4501", "Text": "[5 7(e) and (f)]  should be\ncarefully grasped (g) Wrong Magnetic field lines between two pole pieces cannot be\nprecisely straight at the ends"}, {"Chapter": "1", "sentence_range": "4499-4502", "Text": "7(e) and (f)]  should be\ncarefully grasped (g) Wrong Magnetic field lines between two pole pieces cannot be\nprecisely straight at the ends Some fringing of lines is inevitable"}, {"Chapter": "1", "sentence_range": "4500-4503", "Text": "(g) Wrong Magnetic field lines between two pole pieces cannot be\nprecisely straight at the ends Some fringing of lines is inevitable Otherwise, Ampere\u2019s law is violated"}, {"Chapter": "1", "sentence_range": "4501-4504", "Text": "Magnetic field lines between two pole pieces cannot be\nprecisely straight at the ends Some fringing of lines is inevitable Otherwise, Ampere\u2019s law is violated This is also true for electric\nfield lines"}, {"Chapter": "1", "sentence_range": "4502-4505", "Text": "Some fringing of lines is inevitable Otherwise, Ampere\u2019s law is violated This is also true for electric\nfield lines Example 5"}, {"Chapter": "1", "sentence_range": "4503-4506", "Text": "Otherwise, Ampere\u2019s law is violated This is also true for electric\nfield lines Example 5 4\n(a) Magnetic field lines show the direction (at every point) along which\na small magnetised needle aligns (at the point)"}, {"Chapter": "1", "sentence_range": "4504-4507", "Text": "This is also true for electric\nfield lines Example 5 4\n(a) Magnetic field lines show the direction (at every point) along which\na small magnetised needle aligns (at the point) Do the magnetic\nfield lines also represent the lines of force on a moving charged\nparticle at every point"}, {"Chapter": "1", "sentence_range": "4505-4508", "Text": "Example 5 4\n(a) Magnetic field lines show the direction (at every point) along which\na small magnetised needle aligns (at the point) Do the magnetic\nfield lines also represent the lines of force on a moving charged\nparticle at every point (b) If magnetic monopoles existed, how would the Gauss\u2019s law of\nmagnetism be modified"}, {"Chapter": "1", "sentence_range": "4506-4509", "Text": "4\n(a) Magnetic field lines show the direction (at every point) along which\na small magnetised needle aligns (at the point) Do the magnetic\nfield lines also represent the lines of force on a moving charged\nparticle at every point (b) If magnetic monopoles existed, how would the Gauss\u2019s law of\nmagnetism be modified (c) Does a bar magnet exert a torque on itself due to its own field"}, {"Chapter": "1", "sentence_range": "4507-4510", "Text": "Do the magnetic\nfield lines also represent the lines of force on a moving charged\nparticle at every point (b) If magnetic monopoles existed, how would the Gauss\u2019s law of\nmagnetism be modified (c) Does a bar magnet exert a torque on itself due to its own field Does one element of a current-carrying wire exert a force on another\nelement of the same wire"}, {"Chapter": "1", "sentence_range": "4508-4511", "Text": "(b) If magnetic monopoles existed, how would the Gauss\u2019s law of\nmagnetism be modified (c) Does a bar magnet exert a torque on itself due to its own field Does one element of a current-carrying wire exert a force on another\nelement of the same wire Rationalised 2023-24\n145\nMagnetism and\nMatter\n EXAMPLE 5"}, {"Chapter": "1", "sentence_range": "4509-4512", "Text": "(c) Does a bar magnet exert a torque on itself due to its own field Does one element of a current-carrying wire exert a force on another\nelement of the same wire Rationalised 2023-24\n145\nMagnetism and\nMatter\n EXAMPLE 5 4\n(d) Magnetic field arises due to charges in motion"}, {"Chapter": "1", "sentence_range": "4510-4513", "Text": "Does one element of a current-carrying wire exert a force on another\nelement of the same wire Rationalised 2023-24\n145\nMagnetism and\nMatter\n EXAMPLE 5 4\n(d) Magnetic field arises due to charges in motion Can a system\nhave magnetic moments even though its net charge is zero"}, {"Chapter": "1", "sentence_range": "4511-4514", "Text": "Rationalised 2023-24\n145\nMagnetism and\nMatter\n EXAMPLE 5 4\n(d) Magnetic field arises due to charges in motion Can a system\nhave magnetic moments even though its net charge is zero Solution\n(a) No"}, {"Chapter": "1", "sentence_range": "4512-4515", "Text": "4\n(d) Magnetic field arises due to charges in motion Can a system\nhave magnetic moments even though its net charge is zero Solution\n(a) No The magnetic force is always normal to B (remember magnetic\nforce = qv \u00d7 B)"}, {"Chapter": "1", "sentence_range": "4513-4516", "Text": "Can a system\nhave magnetic moments even though its net charge is zero Solution\n(a) No The magnetic force is always normal to B (remember magnetic\nforce = qv \u00d7 B) It is misleading to call magnetic field lines as lines\nof force"}, {"Chapter": "1", "sentence_range": "4514-4517", "Text": "Solution\n(a) No The magnetic force is always normal to B (remember magnetic\nforce = qv \u00d7 B) It is misleading to call magnetic field lines as lines\nof force (b) Gauss\u2019s law of magnetism states that the flux of B through any\nclosed surface is always zero \nB"}, {"Chapter": "1", "sentence_range": "4515-4518", "Text": "The magnetic force is always normal to B (remember magnetic\nforce = qv \u00d7 B) It is misleading to call magnetic field lines as lines\nof force (b) Gauss\u2019s law of magnetism states that the flux of B through any\nclosed surface is always zero \nB \u2206s\n=\n\u222b\n0\ns"}, {"Chapter": "1", "sentence_range": "4516-4519", "Text": "It is misleading to call magnetic field lines as lines\nof force (b) Gauss\u2019s law of magnetism states that the flux of B through any\nclosed surface is always zero \nB \u2206s\n=\n\u222b\n0\ns If monopoles existed, the right hand side would be equal to the\nmonopole (magnetic charge) qm enclosed by S"}, {"Chapter": "1", "sentence_range": "4517-4520", "Text": "(b) Gauss\u2019s law of magnetism states that the flux of B through any\nclosed surface is always zero \nB \u2206s\n=\n\u222b\n0\ns If monopoles existed, the right hand side would be equal to the\nmonopole (magnetic charge) qm enclosed by S [Analogous to\nGauss\u2019s law of electrostatics, \nB"}, {"Chapter": "1", "sentence_range": "4518-4521", "Text": "\u2206s\n=\n\u222b\n0\ns If monopoles existed, the right hand side would be equal to the\nmonopole (magnetic charge) qm enclosed by S [Analogous to\nGauss\u2019s law of electrostatics, \nB \u2206s\n=\n\u222b\n\u00b50qm\nS\n where qm is the\n(monopole) magnetic charge enclosed by S"}, {"Chapter": "1", "sentence_range": "4519-4522", "Text": "If monopoles existed, the right hand side would be equal to the\nmonopole (magnetic charge) qm enclosed by S [Analogous to\nGauss\u2019s law of electrostatics, \nB \u2206s\n=\n\u222b\n\u00b50qm\nS\n where qm is the\n(monopole) magnetic charge enclosed by S ]\n(c) No"}, {"Chapter": "1", "sentence_range": "4520-4523", "Text": "[Analogous to\nGauss\u2019s law of electrostatics, \nB \u2206s\n=\n\u222b\n\u00b50qm\nS\n where qm is the\n(monopole) magnetic charge enclosed by S ]\n(c) No There is no force or torque on an element due to the field\nproduced by that element itself"}, {"Chapter": "1", "sentence_range": "4521-4524", "Text": "\u2206s\n=\n\u222b\n\u00b50qm\nS\n where qm is the\n(monopole) magnetic charge enclosed by S ]\n(c) No There is no force or torque on an element due to the field\nproduced by that element itself But there is a force (or torque)\non an element of the same wire"}, {"Chapter": "1", "sentence_range": "4522-4525", "Text": "]\n(c) No There is no force or torque on an element due to the field\nproduced by that element itself But there is a force (or torque)\non an element of the same wire (For the special case of a straight\nwire, this force is zero"}, {"Chapter": "1", "sentence_range": "4523-4526", "Text": "There is no force or torque on an element due to the field\nproduced by that element itself But there is a force (or torque)\non an element of the same wire (For the special case of a straight\nwire, this force is zero )\n(d) Yes"}, {"Chapter": "1", "sentence_range": "4524-4527", "Text": "But there is a force (or torque)\non an element of the same wire (For the special case of a straight\nwire, this force is zero )\n(d) Yes The average of the charge in the system may be zero"}, {"Chapter": "1", "sentence_range": "4525-4528", "Text": "(For the special case of a straight\nwire, this force is zero )\n(d) Yes The average of the charge in the system may be zero Yet,\nthe mean of the magnetic moments due to various current loops\nmay not be zero"}, {"Chapter": "1", "sentence_range": "4526-4529", "Text": ")\n(d) Yes The average of the charge in the system may be zero Yet,\nthe mean of the magnetic moments due to various current loops\nmay not be zero We will come across such examples in connection\nwith paramagnetic material where atoms have net dipole moment\nthrough their net charge is zero"}, {"Chapter": "1", "sentence_range": "4527-4530", "Text": "The average of the charge in the system may be zero Yet,\nthe mean of the magnetic moments due to various current loops\nmay not be zero We will come across such examples in connection\nwith paramagnetic material where atoms have net dipole moment\nthrough their net charge is zero 5"}, {"Chapter": "1", "sentence_range": "4528-4531", "Text": "Yet,\nthe mean of the magnetic moments due to various current loops\nmay not be zero We will come across such examples in connection\nwith paramagnetic material where atoms have net dipole moment\nthrough their net charge is zero 5 4  MAGNETISATION AND MAGNETIC INTENSITY\nThe earth abounds with a bewildering variety of elements and compounds"}, {"Chapter": "1", "sentence_range": "4529-4532", "Text": "We will come across such examples in connection\nwith paramagnetic material where atoms have net dipole moment\nthrough their net charge is zero 5 4  MAGNETISATION AND MAGNETIC INTENSITY\nThe earth abounds with a bewildering variety of elements and compounds In addition, we have been synthesising new alloys, compounds and even\nelements"}, {"Chapter": "1", "sentence_range": "4530-4533", "Text": "5 4  MAGNETISATION AND MAGNETIC INTENSITY\nThe earth abounds with a bewildering variety of elements and compounds In addition, we have been synthesising new alloys, compounds and even\nelements One would like to classify the magnetic properties of these\nsubstances"}, {"Chapter": "1", "sentence_range": "4531-4534", "Text": "4  MAGNETISATION AND MAGNETIC INTENSITY\nThe earth abounds with a bewildering variety of elements and compounds In addition, we have been synthesising new alloys, compounds and even\nelements One would like to classify the magnetic properties of these\nsubstances In the present section, we define and explain certain terms\nwhich will help us to carry out this exercise"}, {"Chapter": "1", "sentence_range": "4532-4535", "Text": "In addition, we have been synthesising new alloys, compounds and even\nelements One would like to classify the magnetic properties of these\nsubstances In the present section, we define and explain certain terms\nwhich will help us to carry out this exercise We have seen that a circulating electron in an atom has a magnetic\nmoment"}, {"Chapter": "1", "sentence_range": "4533-4536", "Text": "One would like to classify the magnetic properties of these\nsubstances In the present section, we define and explain certain terms\nwhich will help us to carry out this exercise We have seen that a circulating electron in an atom has a magnetic\nmoment In a bulk material, these moments add up vectorially and they\ncan give a net magnetic moment which is non-zero"}, {"Chapter": "1", "sentence_range": "4534-4537", "Text": "In the present section, we define and explain certain terms\nwhich will help us to carry out this exercise We have seen that a circulating electron in an atom has a magnetic\nmoment In a bulk material, these moments add up vectorially and they\ncan give a net magnetic moment which is non-zero We define\nmagnetisation M of a sample to be equal to its net magnetic moment per\nunit volume:\nM= mVnet\n(5"}, {"Chapter": "1", "sentence_range": "4535-4538", "Text": "We have seen that a circulating electron in an atom has a magnetic\nmoment In a bulk material, these moments add up vectorially and they\ncan give a net magnetic moment which is non-zero We define\nmagnetisation M of a sample to be equal to its net magnetic moment per\nunit volume:\nM= mVnet\n(5 7)\nM is a vector with dimensions L\u20131 A and is measured in a units of A m\u20131"}, {"Chapter": "1", "sentence_range": "4536-4539", "Text": "In a bulk material, these moments add up vectorially and they\ncan give a net magnetic moment which is non-zero We define\nmagnetisation M of a sample to be equal to its net magnetic moment per\nunit volume:\nM= mVnet\n(5 7)\nM is a vector with dimensions L\u20131 A and is measured in a units of A m\u20131 Consider a long solenoid of n turns per unit length and carrying a\ncurrent I"}, {"Chapter": "1", "sentence_range": "4537-4540", "Text": "We define\nmagnetisation M of a sample to be equal to its net magnetic moment per\nunit volume:\nM= mVnet\n(5 7)\nM is a vector with dimensions L\u20131 A and is measured in a units of A m\u20131 Consider a long solenoid of n turns per unit length and carrying a\ncurrent I The magnetic field in the interior of the solenoid was shown to\nbe given by\nB0 = m0 nI\n(5"}, {"Chapter": "1", "sentence_range": "4538-4541", "Text": "7)\nM is a vector with dimensions L\u20131 A and is measured in a units of A m\u20131 Consider a long solenoid of n turns per unit length and carrying a\ncurrent I The magnetic field in the interior of the solenoid was shown to\nbe given by\nB0 = m0 nI\n(5 8)\nIf the interior of the solenoid is filled with a material with non-zero\nmagnetisation, the field inside the solenoid will be greater than B0"}, {"Chapter": "1", "sentence_range": "4539-4542", "Text": "Consider a long solenoid of n turns per unit length and carrying a\ncurrent I The magnetic field in the interior of the solenoid was shown to\nbe given by\nB0 = m0 nI\n(5 8)\nIf the interior of the solenoid is filled with a material with non-zero\nmagnetisation, the field inside the solenoid will be greater than B0 The\nnet B field in the interior of the solenoid may be expressed as\nB = B0 + Bm\n(5"}, {"Chapter": "1", "sentence_range": "4540-4543", "Text": "The magnetic field in the interior of the solenoid was shown to\nbe given by\nB0 = m0 nI\n(5 8)\nIf the interior of the solenoid is filled with a material with non-zero\nmagnetisation, the field inside the solenoid will be greater than B0 The\nnet B field in the interior of the solenoid may be expressed as\nB = B0 + Bm\n(5 9)\nRationalised 2023-24\nPhysics\n146\n EXAMPLE 5"}, {"Chapter": "1", "sentence_range": "4541-4544", "Text": "8)\nIf the interior of the solenoid is filled with a material with non-zero\nmagnetisation, the field inside the solenoid will be greater than B0 The\nnet B field in the interior of the solenoid may be expressed as\nB = B0 + Bm\n(5 9)\nRationalised 2023-24\nPhysics\n146\n EXAMPLE 5 5\nwhere Bm is the field contributed by the material core"}, {"Chapter": "1", "sentence_range": "4542-4545", "Text": "The\nnet B field in the interior of the solenoid may be expressed as\nB = B0 + Bm\n(5 9)\nRationalised 2023-24\nPhysics\n146\n EXAMPLE 5 5\nwhere Bm is the field contributed by the material core It turns out that\nthis additional field Bm is proportional to the magnetisation M of the\nmaterial and is expressed as\nBm = m0M\n(5"}, {"Chapter": "1", "sentence_range": "4543-4546", "Text": "9)\nRationalised 2023-24\nPhysics\n146\n EXAMPLE 5 5\nwhere Bm is the field contributed by the material core It turns out that\nthis additional field Bm is proportional to the magnetisation M of the\nmaterial and is expressed as\nBm = m0M\n(5 10)\nwhere m0 is the same constant (permittivity of vacuum) that appears in\nBiot-Savart\u2019s law"}, {"Chapter": "1", "sentence_range": "4544-4547", "Text": "5\nwhere Bm is the field contributed by the material core It turns out that\nthis additional field Bm is proportional to the magnetisation M of the\nmaterial and is expressed as\nBm = m0M\n(5 10)\nwhere m0 is the same constant (permittivity of vacuum) that appears in\nBiot-Savart\u2019s law It is convenient to introduce another vector field H, called the magnetic\nintensity, which is defined by\n0\n\u2013\n\u00b5\nH= B\nM\n(5"}, {"Chapter": "1", "sentence_range": "4545-4548", "Text": "It turns out that\nthis additional field Bm is proportional to the magnetisation M of the\nmaterial and is expressed as\nBm = m0M\n(5 10)\nwhere m0 is the same constant (permittivity of vacuum) that appears in\nBiot-Savart\u2019s law It is convenient to introduce another vector field H, called the magnetic\nintensity, which is defined by\n0\n\u2013\n\u00b5\nH= B\nM\n(5 11)\nwhere H has the same dimensions as M and is measured in units of A m\u20131"}, {"Chapter": "1", "sentence_range": "4546-4549", "Text": "10)\nwhere m0 is the same constant (permittivity of vacuum) that appears in\nBiot-Savart\u2019s law It is convenient to introduce another vector field H, called the magnetic\nintensity, which is defined by\n0\n\u2013\n\u00b5\nH= B\nM\n(5 11)\nwhere H has the same dimensions as M and is measured in units of A m\u20131 Thus, the total magnetic field B is written as\nB = m0 (H + M)\n(5"}, {"Chapter": "1", "sentence_range": "4547-4550", "Text": "It is convenient to introduce another vector field H, called the magnetic\nintensity, which is defined by\n0\n\u2013\n\u00b5\nH= B\nM\n(5 11)\nwhere H has the same dimensions as M and is measured in units of A m\u20131 Thus, the total magnetic field B is written as\nB = m0 (H + M)\n(5 12)\nWe repeat our defining procedure"}, {"Chapter": "1", "sentence_range": "4548-4551", "Text": "11)\nwhere H has the same dimensions as M and is measured in units of A m\u20131 Thus, the total magnetic field B is written as\nB = m0 (H + M)\n(5 12)\nWe repeat our defining procedure We have partitioned the contribution\nto the total magnetic field inside the sample into two parts: one, due to\nexternal factors such as the current in the solenoid"}, {"Chapter": "1", "sentence_range": "4549-4552", "Text": "Thus, the total magnetic field B is written as\nB = m0 (H + M)\n(5 12)\nWe repeat our defining procedure We have partitioned the contribution\nto the total magnetic field inside the sample into two parts: one, due to\nexternal factors such as the current in the solenoid This is represented\nby H"}, {"Chapter": "1", "sentence_range": "4550-4553", "Text": "12)\nWe repeat our defining procedure We have partitioned the contribution\nto the total magnetic field inside the sample into two parts: one, due to\nexternal factors such as the current in the solenoid This is represented\nby H The other is due to the specific nature of the magnetic material,\nnamely M"}, {"Chapter": "1", "sentence_range": "4551-4554", "Text": "We have partitioned the contribution\nto the total magnetic field inside the sample into two parts: one, due to\nexternal factors such as the current in the solenoid This is represented\nby H The other is due to the specific nature of the magnetic material,\nnamely M The latter quantity can be influenced by external factors"}, {"Chapter": "1", "sentence_range": "4552-4555", "Text": "This is represented\nby H The other is due to the specific nature of the magnetic material,\nnamely M The latter quantity can be influenced by external factors This\ninfluence is mathematically expressed as\nM=\u03c7\nH\n(5"}, {"Chapter": "1", "sentence_range": "4553-4556", "Text": "The other is due to the specific nature of the magnetic material,\nnamely M The latter quantity can be influenced by external factors This\ninfluence is mathematically expressed as\nM=\u03c7\nH\n(5 13)\nwhere c , a dimensionless quantity, is appropriately called the magnetic\nsusceptibility"}, {"Chapter": "1", "sentence_range": "4554-4557", "Text": "The latter quantity can be influenced by external factors This\ninfluence is mathematically expressed as\nM=\u03c7\nH\n(5 13)\nwhere c , a dimensionless quantity, is appropriately called the magnetic\nsusceptibility It is a measure of how a magnetic material responds to an\nexternal field"}, {"Chapter": "1", "sentence_range": "4555-4558", "Text": "This\ninfluence is mathematically expressed as\nM=\u03c7\nH\n(5 13)\nwhere c , a dimensionless quantity, is appropriately called the magnetic\nsusceptibility It is a measure of how a magnetic material responds to an\nexternal field c is small and positive for materials, which are called\nparamagnetic"}, {"Chapter": "1", "sentence_range": "4556-4559", "Text": "13)\nwhere c , a dimensionless quantity, is appropriately called the magnetic\nsusceptibility It is a measure of how a magnetic material responds to an\nexternal field c is small and positive for materials, which are called\nparamagnetic It is small and negative for materials, which are termed\ndiamagnetic"}, {"Chapter": "1", "sentence_range": "4557-4560", "Text": "It is a measure of how a magnetic material responds to an\nexternal field c is small and positive for materials, which are called\nparamagnetic It is small and negative for materials, which are termed\ndiamagnetic In the latter case M and H are opposite in direction"}, {"Chapter": "1", "sentence_range": "4558-4561", "Text": "c is small and positive for materials, which are called\nparamagnetic It is small and negative for materials, which are termed\ndiamagnetic In the latter case M and H are opposite in direction From\nEqs"}, {"Chapter": "1", "sentence_range": "4559-4562", "Text": "It is small and negative for materials, which are termed\ndiamagnetic In the latter case M and H are opposite in direction From\nEqs (5"}, {"Chapter": "1", "sentence_range": "4560-4563", "Text": "In the latter case M and H are opposite in direction From\nEqs (5 12) and (5"}, {"Chapter": "1", "sentence_range": "4561-4564", "Text": "From\nEqs (5 12) and (5 13) we obtain,\n0(1\n)\n\u00b5\n\u03c7\n=\n+\nB\nH\n(5"}, {"Chapter": "1", "sentence_range": "4562-4565", "Text": "(5 12) and (5 13) we obtain,\n0(1\n)\n\u00b5\n\u03c7\n=\n+\nB\nH\n(5 14)\n= m0 mr H\n=  m H\n(5"}, {"Chapter": "1", "sentence_range": "4563-4566", "Text": "12) and (5 13) we obtain,\n0(1\n)\n\u00b5\n\u03c7\n=\n+\nB\nH\n(5 14)\n= m0 mr H\n=  m H\n(5 15)\nwhere mr= 1 + c, is a dimensionless quantity called the relative magnetic\npermeability of the substance"}, {"Chapter": "1", "sentence_range": "4564-4567", "Text": "13) we obtain,\n0(1\n)\n\u00b5\n\u03c7\n=\n+\nB\nH\n(5 14)\n= m0 mr H\n=  m H\n(5 15)\nwhere mr= 1 + c, is a dimensionless quantity called the relative magnetic\npermeability of the substance It is the analog of the dielectric constant in\nelectrostatics"}, {"Chapter": "1", "sentence_range": "4565-4568", "Text": "14)\n= m0 mr H\n=  m H\n(5 15)\nwhere mr= 1 + c, is a dimensionless quantity called the relative magnetic\npermeability of the substance It is the analog of the dielectric constant in\nelectrostatics The magnetic permeability of the substance is m and it has\nthe same dimensions and units as m0;\nm = m0mr = m0 (1+c)"}, {"Chapter": "1", "sentence_range": "4566-4569", "Text": "15)\nwhere mr= 1 + c, is a dimensionless quantity called the relative magnetic\npermeability of the substance It is the analog of the dielectric constant in\nelectrostatics The magnetic permeability of the substance is m and it has\nthe same dimensions and units as m0;\nm = m0mr = m0 (1+c) The three quantities c, mr and m are interrelated and only one of\nthem is independent"}, {"Chapter": "1", "sentence_range": "4567-4570", "Text": "It is the analog of the dielectric constant in\nelectrostatics The magnetic permeability of the substance is m and it has\nthe same dimensions and units as m0;\nm = m0mr = m0 (1+c) The three quantities c, mr and m are interrelated and only one of\nthem is independent Given one, the other two may be easily determined"}, {"Chapter": "1", "sentence_range": "4568-4571", "Text": "The magnetic permeability of the substance is m and it has\nthe same dimensions and units as m0;\nm = m0mr = m0 (1+c) The three quantities c, mr and m are interrelated and only one of\nthem is independent Given one, the other two may be easily determined Example 5"}, {"Chapter": "1", "sentence_range": "4569-4572", "Text": "The three quantities c, mr and m are interrelated and only one of\nthem is independent Given one, the other two may be easily determined Example 5 5 A solenoid has a core of a material with relative\npermeability 400"}, {"Chapter": "1", "sentence_range": "4570-4573", "Text": "Given one, the other two may be easily determined Example 5 5 A solenoid has a core of a material with relative\npermeability 400 The windings of the solenoid are insulated from the\ncore and carry a current of 2A"}, {"Chapter": "1", "sentence_range": "4571-4574", "Text": "Example 5 5 A solenoid has a core of a material with relative\npermeability 400 The windings of the solenoid are insulated from the\ncore and carry a current of 2A If the number of turns is 1000 per\nmetre, calculate (a) H, (b) M, (c) B and (d) the magnetising current Im"}, {"Chapter": "1", "sentence_range": "4572-4575", "Text": "5 A solenoid has a core of a material with relative\npermeability 400 The windings of the solenoid are insulated from the\ncore and carry a current of 2A If the number of turns is 1000 per\nmetre, calculate (a) H, (b) M, (c) B and (d) the magnetising current Im Rationalised 2023-24\n147\nMagnetism and\nMatter\n EXAMPLE 5"}, {"Chapter": "1", "sentence_range": "4573-4576", "Text": "The windings of the solenoid are insulated from the\ncore and carry a current of 2A If the number of turns is 1000 per\nmetre, calculate (a) H, (b) M, (c) B and (d) the magnetising current Im Rationalised 2023-24\n147\nMagnetism and\nMatter\n EXAMPLE 5 5\nSolution\n(a) The field H is dependent of the material of the core, and is\nH = nI = 1000 \u00d7 2"}, {"Chapter": "1", "sentence_range": "4574-4577", "Text": "If the number of turns is 1000 per\nmetre, calculate (a) H, (b) M, (c) B and (d) the magnetising current Im Rationalised 2023-24\n147\nMagnetism and\nMatter\n EXAMPLE 5 5\nSolution\n(a) The field H is dependent of the material of the core, and is\nH = nI = 1000 \u00d7 2 0 = 2 \u00d7103 A/m"}, {"Chapter": "1", "sentence_range": "4575-4578", "Text": "Rationalised 2023-24\n147\nMagnetism and\nMatter\n EXAMPLE 5 5\nSolution\n(a) The field H is dependent of the material of the core, and is\nH = nI = 1000 \u00d7 2 0 = 2 \u00d7103 A/m (b) The magnetic field B is given by\nB = mr m0 H\n   = 400 \u00d7 4p \u00d710\u20137 (N/A2) \u00d7 2 \u00d7 103 (A/m)\n   = 1"}, {"Chapter": "1", "sentence_range": "4576-4579", "Text": "5\nSolution\n(a) The field H is dependent of the material of the core, and is\nH = nI = 1000 \u00d7 2 0 = 2 \u00d7103 A/m (b) The magnetic field B is given by\nB = mr m0 H\n   = 400 \u00d7 4p \u00d710\u20137 (N/A2) \u00d7 2 \u00d7 103 (A/m)\n   = 1 0 T\n(c) Magnetisation is given by\nM = (B\u2013 m0 H)/ m0\n    = (mr m0 H\u2013m0 H)/m0 = (mr \u2013 1)H = 399 \u00d7 H\n     @ 8 \u00d7 105 A/m\n(d) The magnetising current IM is the additional current that needs to\nbe passed through the windings of the solenoid in the absence of\nthe core which would give a B value as in the presence of the core"}, {"Chapter": "1", "sentence_range": "4577-4580", "Text": "0 = 2 \u00d7103 A/m (b) The magnetic field B is given by\nB = mr m0 H\n   = 400 \u00d7 4p \u00d710\u20137 (N/A2) \u00d7 2 \u00d7 103 (A/m)\n   = 1 0 T\n(c) Magnetisation is given by\nM = (B\u2013 m0 H)/ m0\n    = (mr m0 H\u2013m0 H)/m0 = (mr \u2013 1)H = 399 \u00d7 H\n     @ 8 \u00d7 105 A/m\n(d) The magnetising current IM is the additional current that needs to\nbe passed through the windings of the solenoid in the absence of\nthe core which would give a B value as in the presence of the core Thus B = mr n (I + IM)"}, {"Chapter": "1", "sentence_range": "4578-4581", "Text": "(b) The magnetic field B is given by\nB = mr m0 H\n   = 400 \u00d7 4p \u00d710\u20137 (N/A2) \u00d7 2 \u00d7 103 (A/m)\n   = 1 0 T\n(c) Magnetisation is given by\nM = (B\u2013 m0 H)/ m0\n    = (mr m0 H\u2013m0 H)/m0 = (mr \u2013 1)H = 399 \u00d7 H\n     @ 8 \u00d7 105 A/m\n(d) The magnetising current IM is the additional current that needs to\nbe passed through the windings of the solenoid in the absence of\nthe core which would give a B value as in the presence of the core Thus B = mr n (I + IM) Using I = 2A, B = 1 T, we get IM = 794 A"}, {"Chapter": "1", "sentence_range": "4579-4582", "Text": "0 T\n(c) Magnetisation is given by\nM = (B\u2013 m0 H)/ m0\n    = (mr m0 H\u2013m0 H)/m0 = (mr \u2013 1)H = 399 \u00d7 H\n     @ 8 \u00d7 105 A/m\n(d) The magnetising current IM is the additional current that needs to\nbe passed through the windings of the solenoid in the absence of\nthe core which would give a B value as in the presence of the core Thus B = mr n (I + IM) Using I = 2A, B = 1 T, we get IM = 794 A 5"}, {"Chapter": "1", "sentence_range": "4580-4583", "Text": "Thus B = mr n (I + IM) Using I = 2A, B = 1 T, we get IM = 794 A 5 5 MAGNETIC PROPERTIES OF MATERIALS\nThe discussion in the previous section helps us to classify materials as\ndiamagnetic, paramagnetic or ferromagnetic"}, {"Chapter": "1", "sentence_range": "4581-4584", "Text": "Using I = 2A, B = 1 T, we get IM = 794 A 5 5 MAGNETIC PROPERTIES OF MATERIALS\nThe discussion in the previous section helps us to classify materials as\ndiamagnetic, paramagnetic or ferromagnetic In terms of the susceptibility\nc, a material is diamagnetic if c is negative, para- if c is positive and small,\nand ferro- if c is large and positive"}, {"Chapter": "1", "sentence_range": "4582-4585", "Text": "5 5 MAGNETIC PROPERTIES OF MATERIALS\nThe discussion in the previous section helps us to classify materials as\ndiamagnetic, paramagnetic or ferromagnetic In terms of the susceptibility\nc, a material is diamagnetic if c is negative, para- if c is positive and small,\nand ferro- if c is large and positive A glance at Table 5"}, {"Chapter": "1", "sentence_range": "4583-4586", "Text": "5 MAGNETIC PROPERTIES OF MATERIALS\nThe discussion in the previous section helps us to classify materials as\ndiamagnetic, paramagnetic or ferromagnetic In terms of the susceptibility\nc, a material is diamagnetic if c is negative, para- if c is positive and small,\nand ferro- if c is large and positive A glance at Table 5 3 gives one a better feeling for these materials"}, {"Chapter": "1", "sentence_range": "4584-4587", "Text": "In terms of the susceptibility\nc, a material is diamagnetic if c is negative, para- if c is positive and small,\nand ferro- if c is large and positive A glance at Table 5 3 gives one a better feeling for these materials Here e is a small positive number introduced to quantify paramagnetic\nmaterials"}, {"Chapter": "1", "sentence_range": "4585-4588", "Text": "A glance at Table 5 3 gives one a better feeling for these materials Here e is a small positive number introduced to quantify paramagnetic\nmaterials Next, we describe these materials in some detail"}, {"Chapter": "1", "sentence_range": "4586-4589", "Text": "3 gives one a better feeling for these materials Here e is a small positive number introduced to quantify paramagnetic\nmaterials Next, we describe these materials in some detail TABLE 5"}, {"Chapter": "1", "sentence_range": "4587-4590", "Text": "Here e is a small positive number introduced to quantify paramagnetic\nmaterials Next, we describe these materials in some detail TABLE 5 3\nDiamagnetic\nParamagnetic\nFerromagnetic\n\u20131 \u00a3 c < 0\n0 < c <  e\nc >> 1\n0 \u00a3 mr < 1\n1< mr < 1+ e\nmr >> 1\nm < m0\nm > m0\nm >> m0\n5"}, {"Chapter": "1", "sentence_range": "4588-4591", "Text": "Next, we describe these materials in some detail TABLE 5 3\nDiamagnetic\nParamagnetic\nFerromagnetic\n\u20131 \u00a3 c < 0\n0 < c <  e\nc >> 1\n0 \u00a3 mr < 1\n1< mr < 1+ e\nmr >> 1\nm < m0\nm > m0\nm >> m0\n5 5"}, {"Chapter": "1", "sentence_range": "4589-4592", "Text": "TABLE 5 3\nDiamagnetic\nParamagnetic\nFerromagnetic\n\u20131 \u00a3 c < 0\n0 < c <  e\nc >> 1\n0 \u00a3 mr < 1\n1< mr < 1+ e\nmr >> 1\nm < m0\nm > m0\nm >> m0\n5 5 1  Diamagnetism\nDiamagnetic substances are those which have tendency to move from\nstronger to the weaker part of the external magnetic field"}, {"Chapter": "1", "sentence_range": "4590-4593", "Text": "3\nDiamagnetic\nParamagnetic\nFerromagnetic\n\u20131 \u00a3 c < 0\n0 < c <  e\nc >> 1\n0 \u00a3 mr < 1\n1< mr < 1+ e\nmr >> 1\nm < m0\nm > m0\nm >> m0\n5 5 1  Diamagnetism\nDiamagnetic substances are those which have tendency to move from\nstronger to the weaker part of the external magnetic field In other words,\nunlike the way a magnet attracts metals like iron, it would repel a\ndiamagnetic substance"}, {"Chapter": "1", "sentence_range": "4591-4594", "Text": "5 1  Diamagnetism\nDiamagnetic substances are those which have tendency to move from\nstronger to the weaker part of the external magnetic field In other words,\nunlike the way a magnet attracts metals like iron, it would repel a\ndiamagnetic substance Figure 5"}, {"Chapter": "1", "sentence_range": "4592-4595", "Text": "1  Diamagnetism\nDiamagnetic substances are those which have tendency to move from\nstronger to the weaker part of the external magnetic field In other words,\nunlike the way a magnet attracts metals like iron, it would repel a\ndiamagnetic substance Figure 5 7(a) shows a bar of diamagnetic material placed in an external\nmagnetic field"}, {"Chapter": "1", "sentence_range": "4593-4596", "Text": "In other words,\nunlike the way a magnet attracts metals like iron, it would repel a\ndiamagnetic substance Figure 5 7(a) shows a bar of diamagnetic material placed in an external\nmagnetic field The field lines are repelled or expelled and the field inside\nthe material is reduced"}, {"Chapter": "1", "sentence_range": "4594-4597", "Text": "Figure 5 7(a) shows a bar of diamagnetic material placed in an external\nmagnetic field The field lines are repelled or expelled and the field inside\nthe material is reduced In most cases, this reduction is slight, being one\npart in 105"}, {"Chapter": "1", "sentence_range": "4595-4598", "Text": "7(a) shows a bar of diamagnetic material placed in an external\nmagnetic field The field lines are repelled or expelled and the field inside\nthe material is reduced In most cases, this reduction is slight, being one\npart in 105 When placed in a non-uniform magnetic field, the bar will tend\nto move from high to low field"}, {"Chapter": "1", "sentence_range": "4596-4599", "Text": "The field lines are repelled or expelled and the field inside\nthe material is reduced In most cases, this reduction is slight, being one\npart in 105 When placed in a non-uniform magnetic field, the bar will tend\nto move from high to low field FIGURE 5"}, {"Chapter": "1", "sentence_range": "4597-4600", "Text": "In most cases, this reduction is slight, being one\npart in 105 When placed in a non-uniform magnetic field, the bar will tend\nto move from high to low field FIGURE 5 7\nBehaviour of\nmagnetic field lines\nnear a\n(a) diamagnetic,\n(b) paramagnetic\nsubstance"}, {"Chapter": "1", "sentence_range": "4598-4601", "Text": "When placed in a non-uniform magnetic field, the bar will tend\nto move from high to low field FIGURE 5 7\nBehaviour of\nmagnetic field lines\nnear a\n(a) diamagnetic,\n(b) paramagnetic\nsubstance Rationalised 2023-24\nPhysics\n148\nThe simplest explanation for diamagnetism is as follows"}, {"Chapter": "1", "sentence_range": "4599-4602", "Text": "FIGURE 5 7\nBehaviour of\nmagnetic field lines\nnear a\n(a) diamagnetic,\n(b) paramagnetic\nsubstance Rationalised 2023-24\nPhysics\n148\nThe simplest explanation for diamagnetism is as follows Electrons in\nan atom orbiting around nucleus possess orbital angular momentum"}, {"Chapter": "1", "sentence_range": "4600-4603", "Text": "7\nBehaviour of\nmagnetic field lines\nnear a\n(a) diamagnetic,\n(b) paramagnetic\nsubstance Rationalised 2023-24\nPhysics\n148\nThe simplest explanation for diamagnetism is as follows Electrons in\nan atom orbiting around nucleus possess orbital angular momentum These orbiting electrons are equivalent to current-carrying loop and thus\npossess orbital magnetic moment"}, {"Chapter": "1", "sentence_range": "4601-4604", "Text": "Rationalised 2023-24\nPhysics\n148\nThe simplest explanation for diamagnetism is as follows Electrons in\nan atom orbiting around nucleus possess orbital angular momentum These orbiting electrons are equivalent to current-carrying loop and thus\npossess orbital magnetic moment Diamagnetic substances are the ones\nin which resultant magnetic moment in an atom is zero"}, {"Chapter": "1", "sentence_range": "4602-4605", "Text": "Electrons in\nan atom orbiting around nucleus possess orbital angular momentum These orbiting electrons are equivalent to current-carrying loop and thus\npossess orbital magnetic moment Diamagnetic substances are the ones\nin which resultant magnetic moment in an atom is zero When magnetic\nfield is applied, those electrons having orbital magnetic moment in the\nsame direction slow down and those in the opposite direction speed up"}, {"Chapter": "1", "sentence_range": "4603-4606", "Text": "These orbiting electrons are equivalent to current-carrying loop and thus\npossess orbital magnetic moment Diamagnetic substances are the ones\nin which resultant magnetic moment in an atom is zero When magnetic\nfield is applied, those electrons having orbital magnetic moment in the\nsame direction slow down and those in the opposite direction speed up This happens due to induced current in accordance with Lenz\u2019s law which\nyou will study in Chapter 6"}, {"Chapter": "1", "sentence_range": "4604-4607", "Text": "Diamagnetic substances are the ones\nin which resultant magnetic moment in an atom is zero When magnetic\nfield is applied, those electrons having orbital magnetic moment in the\nsame direction slow down and those in the opposite direction speed up This happens due to induced current in accordance with Lenz\u2019s law which\nyou will study in Chapter 6 Thus, the substance develops a net magnetic\nmoment in direction opposite to that of the applied field and hence repulsion"}, {"Chapter": "1", "sentence_range": "4605-4608", "Text": "When magnetic\nfield is applied, those electrons having orbital magnetic moment in the\nsame direction slow down and those in the opposite direction speed up This happens due to induced current in accordance with Lenz\u2019s law which\nyou will study in Chapter 6 Thus, the substance develops a net magnetic\nmoment in direction opposite to that of the applied field and hence repulsion Some diamagnetic materials are bismuth, copper, lead, silicon,\nnitrogen (at STP), water and sodium chloride"}, {"Chapter": "1", "sentence_range": "4606-4609", "Text": "This happens due to induced current in accordance with Lenz\u2019s law which\nyou will study in Chapter 6 Thus, the substance develops a net magnetic\nmoment in direction opposite to that of the applied field and hence repulsion Some diamagnetic materials are bismuth, copper, lead, silicon,\nnitrogen (at STP), water and sodium chloride Diamagnetism is present\nin all the substances"}, {"Chapter": "1", "sentence_range": "4607-4610", "Text": "Thus, the substance develops a net magnetic\nmoment in direction opposite to that of the applied field and hence repulsion Some diamagnetic materials are bismuth, copper, lead, silicon,\nnitrogen (at STP), water and sodium chloride Diamagnetism is present\nin all the substances However, the effect is so weak in most cases that it\ngets shifted by other effects like paramagnetism, ferromagnetism, etc"}, {"Chapter": "1", "sentence_range": "4608-4611", "Text": "Some diamagnetic materials are bismuth, copper, lead, silicon,\nnitrogen (at STP), water and sodium chloride Diamagnetism is present\nin all the substances However, the effect is so weak in most cases that it\ngets shifted by other effects like paramagnetism, ferromagnetism, etc The most exotic diamagnetic materials are superconductors"}, {"Chapter": "1", "sentence_range": "4609-4612", "Text": "Diamagnetism is present\nin all the substances However, the effect is so weak in most cases that it\ngets shifted by other effects like paramagnetism, ferromagnetism, etc The most exotic diamagnetic materials are superconductors These\nare metals, cooled to very low temperatures which exhibits both perfect\nconductivity and perfect diamagnetism"}, {"Chapter": "1", "sentence_range": "4610-4613", "Text": "However, the effect is so weak in most cases that it\ngets shifted by other effects like paramagnetism, ferromagnetism, etc The most exotic diamagnetic materials are superconductors These\nare metals, cooled to very low temperatures which exhibits both perfect\nconductivity and perfect diamagnetism Here the field lines are completely\nexpelled"}, {"Chapter": "1", "sentence_range": "4611-4614", "Text": "The most exotic diamagnetic materials are superconductors These\nare metals, cooled to very low temperatures which exhibits both perfect\nconductivity and perfect diamagnetism Here the field lines are completely\nexpelled c = \u20131 and mr = 0"}, {"Chapter": "1", "sentence_range": "4612-4615", "Text": "These\nare metals, cooled to very low temperatures which exhibits both perfect\nconductivity and perfect diamagnetism Here the field lines are completely\nexpelled c = \u20131 and mr = 0 A superconductor repels a magnet and (by\nNewton\u2019s third law) is repelled by the magnet"}, {"Chapter": "1", "sentence_range": "4613-4616", "Text": "Here the field lines are completely\nexpelled c = \u20131 and mr = 0 A superconductor repels a magnet and (by\nNewton\u2019s third law) is repelled by the magnet The phenomenon of perfect\ndiamagnetism in superconductors is called the Meissner effect, after the\nname of its discoverer"}, {"Chapter": "1", "sentence_range": "4614-4617", "Text": "c = \u20131 and mr = 0 A superconductor repels a magnet and (by\nNewton\u2019s third law) is repelled by the magnet The phenomenon of perfect\ndiamagnetism in superconductors is called the Meissner effect, after the\nname of its discoverer Superconducting magnets can be gainfully\nexploited in variety of situations, for example, for running magnetically\nlevitated superfast trains"}, {"Chapter": "1", "sentence_range": "4615-4618", "Text": "A superconductor repels a magnet and (by\nNewton\u2019s third law) is repelled by the magnet The phenomenon of perfect\ndiamagnetism in superconductors is called the Meissner effect, after the\nname of its discoverer Superconducting magnets can be gainfully\nexploited in variety of situations, for example, for running magnetically\nlevitated superfast trains 5"}, {"Chapter": "1", "sentence_range": "4616-4619", "Text": "The phenomenon of perfect\ndiamagnetism in superconductors is called the Meissner effect, after the\nname of its discoverer Superconducting magnets can be gainfully\nexploited in variety of situations, for example, for running magnetically\nlevitated superfast trains 5 5"}, {"Chapter": "1", "sentence_range": "4617-4620", "Text": "Superconducting magnets can be gainfully\nexploited in variety of situations, for example, for running magnetically\nlevitated superfast trains 5 5 2  Paramagnetism\nParamagnetic substances are those which get weakly magnetised when\nplaced in an external magnetic field"}, {"Chapter": "1", "sentence_range": "4618-4621", "Text": "5 5 2  Paramagnetism\nParamagnetic substances are those which get weakly magnetised when\nplaced in an external magnetic field They have tendency to move from a\nregion of weak magnetic field to strong magnetic field, i"}, {"Chapter": "1", "sentence_range": "4619-4622", "Text": "5 2  Paramagnetism\nParamagnetic substances are those which get weakly magnetised when\nplaced in an external magnetic field They have tendency to move from a\nregion of weak magnetic field to strong magnetic field, i e"}, {"Chapter": "1", "sentence_range": "4620-4623", "Text": "2  Paramagnetism\nParamagnetic substances are those which get weakly magnetised when\nplaced in an external magnetic field They have tendency to move from a\nregion of weak magnetic field to strong magnetic field, i e , they get weakly\nattracted to a magnet"}, {"Chapter": "1", "sentence_range": "4621-4624", "Text": "They have tendency to move from a\nregion of weak magnetic field to strong magnetic field, i e , they get weakly\nattracted to a magnet The individual atoms (or ions or molecules) of a paramagnetic material\npossess a permanent magnetic dipole moment of their own"}, {"Chapter": "1", "sentence_range": "4622-4625", "Text": "e , they get weakly\nattracted to a magnet The individual atoms (or ions or molecules) of a paramagnetic material\npossess a permanent magnetic dipole moment of their own On account\nof the ceaseless random thermal motion of the atoms, no net magnetisation\nis seen"}, {"Chapter": "1", "sentence_range": "4623-4626", "Text": ", they get weakly\nattracted to a magnet The individual atoms (or ions or molecules) of a paramagnetic material\npossess a permanent magnetic dipole moment of their own On account\nof the ceaseless random thermal motion of the atoms, no net magnetisation\nis seen In the presence of an external field B0, which is strong enough,\nand at low temperatures, the individual atomic dipole moment can be\nmade to align and point in the same direction as B0"}, {"Chapter": "1", "sentence_range": "4624-4627", "Text": "The individual atoms (or ions or molecules) of a paramagnetic material\npossess a permanent magnetic dipole moment of their own On account\nof the ceaseless random thermal motion of the atoms, no net magnetisation\nis seen In the presence of an external field B0, which is strong enough,\nand at low temperatures, the individual atomic dipole moment can be\nmade to align and point in the same direction as B0 Figure 5"}, {"Chapter": "1", "sentence_range": "4625-4628", "Text": "On account\nof the ceaseless random thermal motion of the atoms, no net magnetisation\nis seen In the presence of an external field B0, which is strong enough,\nand at low temperatures, the individual atomic dipole moment can be\nmade to align and point in the same direction as B0 Figure 5 7(b) shows\na bar of paramagnetic material placed in an external field"}, {"Chapter": "1", "sentence_range": "4626-4629", "Text": "In the presence of an external field B0, which is strong enough,\nand at low temperatures, the individual atomic dipole moment can be\nmade to align and point in the same direction as B0 Figure 5 7(b) shows\na bar of paramagnetic material placed in an external field The field lines\ngets concentrated inside the material, and the field inside is enhanced"}, {"Chapter": "1", "sentence_range": "4627-4630", "Text": "Figure 5 7(b) shows\na bar of paramagnetic material placed in an external field The field lines\ngets concentrated inside the material, and the field inside is enhanced In\nmost cases, this enhancement is slight, being one part in 105"}, {"Chapter": "1", "sentence_range": "4628-4631", "Text": "7(b) shows\na bar of paramagnetic material placed in an external field The field lines\ngets concentrated inside the material, and the field inside is enhanced In\nmost cases, this enhancement is slight, being one part in 105 When placed\nin a non-uniform magnetic field, the bar will tend to move from weak field\nto strong"}, {"Chapter": "1", "sentence_range": "4629-4632", "Text": "The field lines\ngets concentrated inside the material, and the field inside is enhanced In\nmost cases, this enhancement is slight, being one part in 105 When placed\nin a non-uniform magnetic field, the bar will tend to move from weak field\nto strong Some paramagnetic materials are aluminium, sodium, calcium,\noxygen (at STP) and copper chloride"}, {"Chapter": "1", "sentence_range": "4630-4633", "Text": "In\nmost cases, this enhancement is slight, being one part in 105 When placed\nin a non-uniform magnetic field, the bar will tend to move from weak field\nto strong Some paramagnetic materials are aluminium, sodium, calcium,\noxygen (at STP) and copper chloride For a paramagnetic material both c\nand mr depend not only on the material, but also (in a simple fashion) on\nthe sample temperature"}, {"Chapter": "1", "sentence_range": "4631-4634", "Text": "When placed\nin a non-uniform magnetic field, the bar will tend to move from weak field\nto strong Some paramagnetic materials are aluminium, sodium, calcium,\noxygen (at STP) and copper chloride For a paramagnetic material both c\nand mr depend not only on the material, but also (in a simple fashion) on\nthe sample temperature As the field is increased or the temperature is\nlowered, the magnetisation increases until it reaches the saturation value\nat which point all the dipoles are perfectly aligned with the field"}, {"Chapter": "1", "sentence_range": "4632-4635", "Text": "Some paramagnetic materials are aluminium, sodium, calcium,\noxygen (at STP) and copper chloride For a paramagnetic material both c\nand mr depend not only on the material, but also (in a simple fashion) on\nthe sample temperature As the field is increased or the temperature is\nlowered, the magnetisation increases until it reaches the saturation value\nat which point all the dipoles are perfectly aligned with the field 5"}, {"Chapter": "1", "sentence_range": "4633-4636", "Text": "For a paramagnetic material both c\nand mr depend not only on the material, but also (in a simple fashion) on\nthe sample temperature As the field is increased or the temperature is\nlowered, the magnetisation increases until it reaches the saturation value\nat which point all the dipoles are perfectly aligned with the field 5 5"}, {"Chapter": "1", "sentence_range": "4634-4637", "Text": "As the field is increased or the temperature is\nlowered, the magnetisation increases until it reaches the saturation value\nat which point all the dipoles are perfectly aligned with the field 5 5 3  Ferromagnetism\nFerromagnetic substances are those which gets strongly magnetised when\nplaced in an external magnetic field"}, {"Chapter": "1", "sentence_range": "4635-4638", "Text": "5 5 3  Ferromagnetism\nFerromagnetic substances are those which gets strongly magnetised when\nplaced in an external magnetic field They have strong tendency to move\nRationalised 2023-24\n149\nMagnetism and\nMatter\nfrom a region of weak magnetic field to strong magnetic field, i"}, {"Chapter": "1", "sentence_range": "4636-4639", "Text": "5 3  Ferromagnetism\nFerromagnetic substances are those which gets strongly magnetised when\nplaced in an external magnetic field They have strong tendency to move\nRationalised 2023-24\n149\nMagnetism and\nMatter\nfrom a region of weak magnetic field to strong magnetic field, i e"}, {"Chapter": "1", "sentence_range": "4637-4640", "Text": "3  Ferromagnetism\nFerromagnetic substances are those which gets strongly magnetised when\nplaced in an external magnetic field They have strong tendency to move\nRationalised 2023-24\n149\nMagnetism and\nMatter\nfrom a region of weak magnetic field to strong magnetic field, i e , they get\nstrongly attracted to a magnet"}, {"Chapter": "1", "sentence_range": "4638-4641", "Text": "They have strong tendency to move\nRationalised 2023-24\n149\nMagnetism and\nMatter\nfrom a region of weak magnetic field to strong magnetic field, i e , they get\nstrongly attracted to a magnet The individual atoms (or ions or molecules) in a ferromagnetic material\npossess a dipole moment as in a paramagnetic material"}, {"Chapter": "1", "sentence_range": "4639-4642", "Text": "e , they get\nstrongly attracted to a magnet The individual atoms (or ions or molecules) in a ferromagnetic material\npossess a dipole moment as in a paramagnetic material However, they\ninteract with one another in such a way that they spontaneously align\nthemselves in a common direction over a macroscopic volume called\ndomain"}, {"Chapter": "1", "sentence_range": "4640-4643", "Text": ", they get\nstrongly attracted to a magnet The individual atoms (or ions or molecules) in a ferromagnetic material\npossess a dipole moment as in a paramagnetic material However, they\ninteract with one another in such a way that they spontaneously align\nthemselves in a common direction over a macroscopic volume called\ndomain The explanation of this cooperative effect requires quantum\nmechanics and is beyond the scope of this textbook"}, {"Chapter": "1", "sentence_range": "4641-4644", "Text": "The individual atoms (or ions or molecules) in a ferromagnetic material\npossess a dipole moment as in a paramagnetic material However, they\ninteract with one another in such a way that they spontaneously align\nthemselves in a common direction over a macroscopic volume called\ndomain The explanation of this cooperative effect requires quantum\nmechanics and is beyond the scope of this textbook Each domain has a\nnet magnetisation"}, {"Chapter": "1", "sentence_range": "4642-4645", "Text": "However, they\ninteract with one another in such a way that they spontaneously align\nthemselves in a common direction over a macroscopic volume called\ndomain The explanation of this cooperative effect requires quantum\nmechanics and is beyond the scope of this textbook Each domain has a\nnet magnetisation Typical domain size is 1mm and the domain contains\nabout 1011 atoms"}, {"Chapter": "1", "sentence_range": "4643-4646", "Text": "The explanation of this cooperative effect requires quantum\nmechanics and is beyond the scope of this textbook Each domain has a\nnet magnetisation Typical domain size is 1mm and the domain contains\nabout 1011 atoms In the first instant, the magnetisation varies randomly\nfrom domain to domain and there is no bulk magnetisation"}, {"Chapter": "1", "sentence_range": "4644-4647", "Text": "Each domain has a\nnet magnetisation Typical domain size is 1mm and the domain contains\nabout 1011 atoms In the first instant, the magnetisation varies randomly\nfrom domain to domain and there is no bulk magnetisation This is shown\nin Fig"}, {"Chapter": "1", "sentence_range": "4645-4648", "Text": "Typical domain size is 1mm and the domain contains\nabout 1011 atoms In the first instant, the magnetisation varies randomly\nfrom domain to domain and there is no bulk magnetisation This is shown\nin Fig 5"}, {"Chapter": "1", "sentence_range": "4646-4649", "Text": "In the first instant, the magnetisation varies randomly\nfrom domain to domain and there is no bulk magnetisation This is shown\nin Fig 5 8(a)"}, {"Chapter": "1", "sentence_range": "4647-4650", "Text": "This is shown\nin Fig 5 8(a) When we apply an external magnetic field B0, the domains\norient themselves in the direction of B0 and simultaneously the domain\noriented in the direction of B0 grow in size"}, {"Chapter": "1", "sentence_range": "4648-4651", "Text": "5 8(a) When we apply an external magnetic field B0, the domains\norient themselves in the direction of B0 and simultaneously the domain\noriented in the direction of B0 grow in size This existence of domains and\ntheir motion in B0 are not speculations"}, {"Chapter": "1", "sentence_range": "4649-4652", "Text": "8(a) When we apply an external magnetic field B0, the domains\norient themselves in the direction of B0 and simultaneously the domain\noriented in the direction of B0 grow in size This existence of domains and\ntheir motion in B0 are not speculations One may observe this under a\nmicroscope after sprinkling a liquid suspension of powdered\nferromagnetic substance of samples"}, {"Chapter": "1", "sentence_range": "4650-4653", "Text": "When we apply an external magnetic field B0, the domains\norient themselves in the direction of B0 and simultaneously the domain\noriented in the direction of B0 grow in size This existence of domains and\ntheir motion in B0 are not speculations One may observe this under a\nmicroscope after sprinkling a liquid suspension of powdered\nferromagnetic substance of samples This motion of suspension can be\nobserved"}, {"Chapter": "1", "sentence_range": "4651-4654", "Text": "This existence of domains and\ntheir motion in B0 are not speculations One may observe this under a\nmicroscope after sprinkling a liquid suspension of powdered\nferromagnetic substance of samples This motion of suspension can be\nobserved Fig"}, {"Chapter": "1", "sentence_range": "4652-4655", "Text": "One may observe this under a\nmicroscope after sprinkling a liquid suspension of powdered\nferromagnetic substance of samples This motion of suspension can be\nobserved Fig 5"}, {"Chapter": "1", "sentence_range": "4653-4656", "Text": "This motion of suspension can be\nobserved Fig 5 8(b) shows the situation when the domains have aligned\nand amalgamated to form a single \u2018giant\u2019 domain"}, {"Chapter": "1", "sentence_range": "4654-4657", "Text": "Fig 5 8(b) shows the situation when the domains have aligned\nand amalgamated to form a single \u2018giant\u2019 domain Thus, in a ferromagnetic material the field lines are highly\nconcentrated"}, {"Chapter": "1", "sentence_range": "4655-4658", "Text": "5 8(b) shows the situation when the domains have aligned\nand amalgamated to form a single \u2018giant\u2019 domain Thus, in a ferromagnetic material the field lines are highly\nconcentrated In non-uniform magnetic field, the sample tends to move\ntowards the region of high field"}, {"Chapter": "1", "sentence_range": "4656-4659", "Text": "8(b) shows the situation when the domains have aligned\nand amalgamated to form a single \u2018giant\u2019 domain Thus, in a ferromagnetic material the field lines are highly\nconcentrated In non-uniform magnetic field, the sample tends to move\ntowards the region of high field We may wonder as to what happens\nwhen the external field is removed"}, {"Chapter": "1", "sentence_range": "4657-4660", "Text": "Thus, in a ferromagnetic material the field lines are highly\nconcentrated In non-uniform magnetic field, the sample tends to move\ntowards the region of high field We may wonder as to what happens\nwhen the external field is removed In some ferromagnetic materials the\nmagnetisation persists"}, {"Chapter": "1", "sentence_range": "4658-4661", "Text": "In non-uniform magnetic field, the sample tends to move\ntowards the region of high field We may wonder as to what happens\nwhen the external field is removed In some ferromagnetic materials the\nmagnetisation persists Such materials are called hard magnetic materials\nor hard ferromagnets"}, {"Chapter": "1", "sentence_range": "4659-4662", "Text": "We may wonder as to what happens\nwhen the external field is removed In some ferromagnetic materials the\nmagnetisation persists Such materials are called hard magnetic materials\nor hard ferromagnets Alnico, an alloy of iron, aluminium, nickel, cobalt\nand copper, is one such material"}, {"Chapter": "1", "sentence_range": "4660-4663", "Text": "In some ferromagnetic materials the\nmagnetisation persists Such materials are called hard magnetic materials\nor hard ferromagnets Alnico, an alloy of iron, aluminium, nickel, cobalt\nand copper, is one such material The naturally occurring lodestone is\nanother"}, {"Chapter": "1", "sentence_range": "4661-4664", "Text": "Such materials are called hard magnetic materials\nor hard ferromagnets Alnico, an alloy of iron, aluminium, nickel, cobalt\nand copper, is one such material The naturally occurring lodestone is\nanother Such materials form permanent magnets to be used among other\nthings as a compass needle"}, {"Chapter": "1", "sentence_range": "4662-4665", "Text": "Alnico, an alloy of iron, aluminium, nickel, cobalt\nand copper, is one such material The naturally occurring lodestone is\nanother Such materials form permanent magnets to be used among other\nthings as a compass needle On the other hand, there is a class of\nferromagnetic materials in which the magnetisation disappears on removal\nof the external field"}, {"Chapter": "1", "sentence_range": "4663-4666", "Text": "The naturally occurring lodestone is\nanother Such materials form permanent magnets to be used among other\nthings as a compass needle On the other hand, there is a class of\nferromagnetic materials in which the magnetisation disappears on removal\nof the external field Soft iron is one such material"}, {"Chapter": "1", "sentence_range": "4664-4667", "Text": "Such materials form permanent magnets to be used among other\nthings as a compass needle On the other hand, there is a class of\nferromagnetic materials in which the magnetisation disappears on removal\nof the external field Soft iron is one such material Appropriately enough,\nsuch materials are called soft ferromagnetic materials"}, {"Chapter": "1", "sentence_range": "4665-4668", "Text": "On the other hand, there is a class of\nferromagnetic materials in which the magnetisation disappears on removal\nof the external field Soft iron is one such material Appropriately enough,\nsuch materials are called soft ferromagnetic materials There are a number\nof elements, which are ferromagnetic: iron, cobalt, nickel, gadolinium,\netc"}, {"Chapter": "1", "sentence_range": "4666-4669", "Text": "Soft iron is one such material Appropriately enough,\nsuch materials are called soft ferromagnetic materials There are a number\nof elements, which are ferromagnetic: iron, cobalt, nickel, gadolinium,\netc The relative magnetic permeability is >1000"}, {"Chapter": "1", "sentence_range": "4667-4670", "Text": "Appropriately enough,\nsuch materials are called soft ferromagnetic materials There are a number\nof elements, which are ferromagnetic: iron, cobalt, nickel, gadolinium,\netc The relative magnetic permeability is >1000 The ferromagnetic property depends on temperature"}, {"Chapter": "1", "sentence_range": "4668-4671", "Text": "There are a number\nof elements, which are ferromagnetic: iron, cobalt, nickel, gadolinium,\netc The relative magnetic permeability is >1000 The ferromagnetic property depends on temperature At high enough\ntemperature, a ferromagnet becomes a paramagnet"}, {"Chapter": "1", "sentence_range": "4669-4672", "Text": "The relative magnetic permeability is >1000 The ferromagnetic property depends on temperature At high enough\ntemperature, a ferromagnet becomes a paramagnet The domain structure\ndisintegrates with temperature"}, {"Chapter": "1", "sentence_range": "4670-4673", "Text": "The ferromagnetic property depends on temperature At high enough\ntemperature, a ferromagnet becomes a paramagnet The domain structure\ndisintegrates with temperature This disappearance of magnetisation with\ntemperature is gradual"}, {"Chapter": "1", "sentence_range": "4671-4674", "Text": "At high enough\ntemperature, a ferromagnet becomes a paramagnet The domain structure\ndisintegrates with temperature This disappearance of magnetisation with\ntemperature is gradual FIGURE 5"}, {"Chapter": "1", "sentence_range": "4672-4675", "Text": "The domain structure\ndisintegrates with temperature This disappearance of magnetisation with\ntemperature is gradual FIGURE 5 8\n(a) Randomly\noriented domains,\n(b) Aligned domains"}, {"Chapter": "1", "sentence_range": "4673-4676", "Text": "This disappearance of magnetisation with\ntemperature is gradual FIGURE 5 8\n(a) Randomly\noriented domains,\n(b) Aligned domains SUMMARY\n1"}, {"Chapter": "1", "sentence_range": "4674-4677", "Text": "FIGURE 5 8\n(a) Randomly\noriented domains,\n(b) Aligned domains SUMMARY\n1 The science of magnetism is old"}, {"Chapter": "1", "sentence_range": "4675-4678", "Text": "8\n(a) Randomly\noriented domains,\n(b) Aligned domains SUMMARY\n1 The science of magnetism is old It has been known since ancient times\nthat magnetic materials tend to point in the north-south direction; like\nmagnetic poles repel and unlike ones attract; and cutting a bar magnet\nin two leads to two smaller magnets"}, {"Chapter": "1", "sentence_range": "4676-4679", "Text": "SUMMARY\n1 The science of magnetism is old It has been known since ancient times\nthat magnetic materials tend to point in the north-south direction; like\nmagnetic poles repel and unlike ones attract; and cutting a bar magnet\nin two leads to two smaller magnets Magnetic poles cannot be isolated"}, {"Chapter": "1", "sentence_range": "4677-4680", "Text": "The science of magnetism is old It has been known since ancient times\nthat magnetic materials tend to point in the north-south direction; like\nmagnetic poles repel and unlike ones attract; and cutting a bar magnet\nin two leads to two smaller magnets Magnetic poles cannot be isolated 2"}, {"Chapter": "1", "sentence_range": "4678-4681", "Text": "It has been known since ancient times\nthat magnetic materials tend to point in the north-south direction; like\nmagnetic poles repel and unlike ones attract; and cutting a bar magnet\nin two leads to two smaller magnets Magnetic poles cannot be isolated 2 When a bar magnet of dipole moment m is placed in a uniform magnetic\nfield B,\nRationalised 2023-24\nPhysics\n150\n     (a)\nthe force on it is zero,\n     (b)\nthe torque on it is m \u00d7 B,\n     (c)\nits potential energy is \u2013m"}, {"Chapter": "1", "sentence_range": "4679-4682", "Text": "Magnetic poles cannot be isolated 2 When a bar magnet of dipole moment m is placed in a uniform magnetic\nfield B,\nRationalised 2023-24\nPhysics\n150\n     (a)\nthe force on it is zero,\n     (b)\nthe torque on it is m \u00d7 B,\n     (c)\nits potential energy is \u2013m B, where we choose the zero of energy at\nthe orientation when m is perpendicular to B"}, {"Chapter": "1", "sentence_range": "4680-4683", "Text": "2 When a bar magnet of dipole moment m is placed in a uniform magnetic\nfield B,\nRationalised 2023-24\nPhysics\n150\n     (a)\nthe force on it is zero,\n     (b)\nthe torque on it is m \u00d7 B,\n     (c)\nits potential energy is \u2013m B, where we choose the zero of energy at\nthe orientation when m is perpendicular to B 3"}, {"Chapter": "1", "sentence_range": "4681-4684", "Text": "When a bar magnet of dipole moment m is placed in a uniform magnetic\nfield B,\nRationalised 2023-24\nPhysics\n150\n     (a)\nthe force on it is zero,\n     (b)\nthe torque on it is m \u00d7 B,\n     (c)\nits potential energy is \u2013m B, where we choose the zero of energy at\nthe orientation when m is perpendicular to B 3 Consider a bar magnet of size l and magnetic moment m, at a distance\nr from its mid-point, where r >>l, the magnetic field B due to this bar\nis,\n0\n3\n2\nr\n=\u00b5\n\u03c0\nm\nB\n    (along axis)\n   =\n0\n3\n\u2013 4\nr\n\u00b5\n\u03c0\nm     (along equator)\n4"}, {"Chapter": "1", "sentence_range": "4682-4685", "Text": "B, where we choose the zero of energy at\nthe orientation when m is perpendicular to B 3 Consider a bar magnet of size l and magnetic moment m, at a distance\nr from its mid-point, where r >>l, the magnetic field B due to this bar\nis,\n0\n3\n2\nr\n=\u00b5\n\u03c0\nm\nB\n    (along axis)\n   =\n0\n3\n\u2013 4\nr\n\u00b5\n\u03c0\nm     (along equator)\n4 Gauss\u2019s law for magnetism states that the net magnetic flux through\nany closed surface is zero\n0\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\nS\nB\niS\nB\nall area\nelements\n5"}, {"Chapter": "1", "sentence_range": "4683-4686", "Text": "3 Consider a bar magnet of size l and magnetic moment m, at a distance\nr from its mid-point, where r >>l, the magnetic field B due to this bar\nis,\n0\n3\n2\nr\n=\u00b5\n\u03c0\nm\nB\n    (along axis)\n   =\n0\n3\n\u2013 4\nr\n\u00b5\n\u03c0\nm     (along equator)\n4 Gauss\u2019s law for magnetism states that the net magnetic flux through\nany closed surface is zero\n0\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\nS\nB\niS\nB\nall area\nelements\n5 Consider a material placed in an external magnetic field B0"}, {"Chapter": "1", "sentence_range": "4684-4687", "Text": "Consider a bar magnet of size l and magnetic moment m, at a distance\nr from its mid-point, where r >>l, the magnetic field B due to this bar\nis,\n0\n3\n2\nr\n=\u00b5\n\u03c0\nm\nB\n    (along axis)\n   =\n0\n3\n\u2013 4\nr\n\u00b5\n\u03c0\nm     (along equator)\n4 Gauss\u2019s law for magnetism states that the net magnetic flux through\nany closed surface is zero\n0\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\nS\nB\niS\nB\nall area\nelements\n5 Consider a material placed in an external magnetic field B0 The\nmagnetic intensity is defined as,\n0\n= B\u00b50\nH\nThe magnetisation M of the material is its dipole moment per unit volume"}, {"Chapter": "1", "sentence_range": "4685-4688", "Text": "Gauss\u2019s law for magnetism states that the net magnetic flux through\nany closed surface is zero\n0\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\nS\nB\niS\nB\nall area\nelements\n5 Consider a material placed in an external magnetic field B0 The\nmagnetic intensity is defined as,\n0\n= B\u00b50\nH\nThe magnetisation M of the material is its dipole moment per unit volume The magnetic field B in the material is,\n      B = m0 (H + M)\n6"}, {"Chapter": "1", "sentence_range": "4686-4689", "Text": "Consider a material placed in an external magnetic field B0 The\nmagnetic intensity is defined as,\n0\n= B\u00b50\nH\nThe magnetisation M of the material is its dipole moment per unit volume The magnetic field B in the material is,\n      B = m0 (H + M)\n6 For a linear material M = c H"}, {"Chapter": "1", "sentence_range": "4687-4690", "Text": "The\nmagnetic intensity is defined as,\n0\n= B\u00b50\nH\nThe magnetisation M of the material is its dipole moment per unit volume The magnetic field B in the material is,\n      B = m0 (H + M)\n6 For a linear material M = c H So that B = m H and c is called the\nmagnetic susceptibility of the material"}, {"Chapter": "1", "sentence_range": "4688-4691", "Text": "The magnetic field B in the material is,\n      B = m0 (H + M)\n6 For a linear material M = c H So that B = m H and c is called the\nmagnetic susceptibility of the material The three quantities, c, the\nrelative magnetic permeability mr, and the magnetic permeability m  are\nrelated as follows:\nm = m0 mr\n      mr = 1+ c\n7"}, {"Chapter": "1", "sentence_range": "4689-4692", "Text": "For a linear material M = c H So that B = m H and c is called the\nmagnetic susceptibility of the material The three quantities, c, the\nrelative magnetic permeability mr, and the magnetic permeability m  are\nrelated as follows:\nm = m0 mr\n      mr = 1+ c\n7 Magnetic materials are broadly classified as: diamagnetic, paramagnetic,\nand ferromagnetic"}, {"Chapter": "1", "sentence_range": "4690-4693", "Text": "So that B = m H and c is called the\nmagnetic susceptibility of the material The three quantities, c, the\nrelative magnetic permeability mr, and the magnetic permeability m  are\nrelated as follows:\nm = m0 mr\n      mr = 1+ c\n7 Magnetic materials are broadly classified as: diamagnetic, paramagnetic,\nand ferromagnetic For diamagnetic materials c is negative and small\nand for paramagnetic materials it is positive and small"}, {"Chapter": "1", "sentence_range": "4691-4694", "Text": "The three quantities, c, the\nrelative magnetic permeability mr, and the magnetic permeability m  are\nrelated as follows:\nm = m0 mr\n      mr = 1+ c\n7 Magnetic materials are broadly classified as: diamagnetic, paramagnetic,\nand ferromagnetic For diamagnetic materials c is negative and small\nand for paramagnetic materials it is positive and small Ferromagnetic\nmaterials have large c and are characterised by non-linear relation\nbetween B and H"}, {"Chapter": "1", "sentence_range": "4692-4695", "Text": "Magnetic materials are broadly classified as: diamagnetic, paramagnetic,\nand ferromagnetic For diamagnetic materials c is negative and small\nand for paramagnetic materials it is positive and small Ferromagnetic\nmaterials have large c and are characterised by non-linear relation\nbetween B and H 8"}, {"Chapter": "1", "sentence_range": "4693-4696", "Text": "For diamagnetic materials c is negative and small\nand for paramagnetic materials it is positive and small Ferromagnetic\nmaterials have large c and are characterised by non-linear relation\nbetween B and H 8 Substances, which at room temperature, retain their ferromagnetic\nproperty for a long period of time are called permanent magnets"}, {"Chapter": "1", "sentence_range": "4694-4697", "Text": "Ferromagnetic\nmaterials have large c and are characterised by non-linear relation\nbetween B and H 8 Substances, which at room temperature, retain their ferromagnetic\nproperty for a long period of time are called permanent magnets Physical quantity\nSymbol\nNature\nDimensions\nUnits\nRemarks\nPermeability of\nm0\nScalar\n[MLT\u20132 A\u20132]\nT m A\u20131\nm0/4p = 10\u20137\nfree space\nMagnetic field,\nB\nVector\n[MT\u20132 A\u20131]\nT (tesla)\n104 G (gauss) = 1 T\nMagnetic induction,\nMagnetic flux density\nMagnetic moment\nm\nVector\n[L\u20132 A]\nA m2\nRationalised 2023-24\n151\nMagnetism and\nMatter\nPOINTS TO PONDER\n1"}, {"Chapter": "1", "sentence_range": "4695-4698", "Text": "8 Substances, which at room temperature, retain their ferromagnetic\nproperty for a long period of time are called permanent magnets Physical quantity\nSymbol\nNature\nDimensions\nUnits\nRemarks\nPermeability of\nm0\nScalar\n[MLT\u20132 A\u20132]\nT m A\u20131\nm0/4p = 10\u20137\nfree space\nMagnetic field,\nB\nVector\n[MT\u20132 A\u20131]\nT (tesla)\n104 G (gauss) = 1 T\nMagnetic induction,\nMagnetic flux density\nMagnetic moment\nm\nVector\n[L\u20132 A]\nA m2\nRationalised 2023-24\n151\nMagnetism and\nMatter\nPOINTS TO PONDER\n1 A satisfactory understanding of magnetic phenomenon in terms of moving\ncharges/currents was arrived at after 1800 AD"}, {"Chapter": "1", "sentence_range": "4696-4699", "Text": "Substances, which at room temperature, retain their ferromagnetic\nproperty for a long period of time are called permanent magnets Physical quantity\nSymbol\nNature\nDimensions\nUnits\nRemarks\nPermeability of\nm0\nScalar\n[MLT\u20132 A\u20132]\nT m A\u20131\nm0/4p = 10\u20137\nfree space\nMagnetic field,\nB\nVector\n[MT\u20132 A\u20131]\nT (tesla)\n104 G (gauss) = 1 T\nMagnetic induction,\nMagnetic flux density\nMagnetic moment\nm\nVector\n[L\u20132 A]\nA m2\nRationalised 2023-24\n151\nMagnetism and\nMatter\nPOINTS TO PONDER\n1 A satisfactory understanding of magnetic phenomenon in terms of moving\ncharges/currents was arrived at after 1800 AD But technological\nexploitation of the directional properties of magnets predates this scientific\nunderstanding by two thousand years"}, {"Chapter": "1", "sentence_range": "4697-4700", "Text": "Physical quantity\nSymbol\nNature\nDimensions\nUnits\nRemarks\nPermeability of\nm0\nScalar\n[MLT\u20132 A\u20132]\nT m A\u20131\nm0/4p = 10\u20137\nfree space\nMagnetic field,\nB\nVector\n[MT\u20132 A\u20131]\nT (tesla)\n104 G (gauss) = 1 T\nMagnetic induction,\nMagnetic flux density\nMagnetic moment\nm\nVector\n[L\u20132 A]\nA m2\nRationalised 2023-24\n151\nMagnetism and\nMatter\nPOINTS TO PONDER\n1 A satisfactory understanding of magnetic phenomenon in terms of moving\ncharges/currents was arrived at after 1800 AD But technological\nexploitation of the directional properties of magnets predates this scientific\nunderstanding by two thousand years Thus, scientific understanding is\nnot a necessary condition for engineering applications"}, {"Chapter": "1", "sentence_range": "4698-4701", "Text": "A satisfactory understanding of magnetic phenomenon in terms of moving\ncharges/currents was arrived at after 1800 AD But technological\nexploitation of the directional properties of magnets predates this scientific\nunderstanding by two thousand years Thus, scientific understanding is\nnot a necessary condition for engineering applications Ideally, science\nand engineering go hand-in-hand, one leading and assisting the other in\ntandem"}, {"Chapter": "1", "sentence_range": "4699-4702", "Text": "But technological\nexploitation of the directional properties of magnets predates this scientific\nunderstanding by two thousand years Thus, scientific understanding is\nnot a necessary condition for engineering applications Ideally, science\nand engineering go hand-in-hand, one leading and assisting the other in\ntandem 2"}, {"Chapter": "1", "sentence_range": "4700-4703", "Text": "Thus, scientific understanding is\nnot a necessary condition for engineering applications Ideally, science\nand engineering go hand-in-hand, one leading and assisting the other in\ntandem 2 Magnetic monopoles do not exist"}, {"Chapter": "1", "sentence_range": "4701-4704", "Text": "Ideally, science\nand engineering go hand-in-hand, one leading and assisting the other in\ntandem 2 Magnetic monopoles do not exist If you slice a magnet in half, you get\ntwo smaller magnets"}, {"Chapter": "1", "sentence_range": "4702-4705", "Text": "2 Magnetic monopoles do not exist If you slice a magnet in half, you get\ntwo smaller magnets On the other hand, isolated positive and negative\ncharges exist"}, {"Chapter": "1", "sentence_range": "4703-4706", "Text": "Magnetic monopoles do not exist If you slice a magnet in half, you get\ntwo smaller magnets On the other hand, isolated positive and negative\ncharges exist There exists a smallest unit of charge, for example, the\nelectronic charge with value |e| = 1"}, {"Chapter": "1", "sentence_range": "4704-4707", "Text": "If you slice a magnet in half, you get\ntwo smaller magnets On the other hand, isolated positive and negative\ncharges exist There exists a smallest unit of charge, for example, the\nelectronic charge with value |e| = 1 6 \u00d710\u201319 C"}, {"Chapter": "1", "sentence_range": "4705-4708", "Text": "On the other hand, isolated positive and negative\ncharges exist There exists a smallest unit of charge, for example, the\nelectronic charge with value |e| = 1 6 \u00d710\u201319 C All other charges are\nintegral multiples of this smallest unit charge"}, {"Chapter": "1", "sentence_range": "4706-4709", "Text": "There exists a smallest unit of charge, for example, the\nelectronic charge with value |e| = 1 6 \u00d710\u201319 C All other charges are\nintegral multiples of this smallest unit charge In other words, charge is\nquantised"}, {"Chapter": "1", "sentence_range": "4707-4710", "Text": "6 \u00d710\u201319 C All other charges are\nintegral multiples of this smallest unit charge In other words, charge is\nquantised We do not know why magnetic monopoles do not exist or why\nelectric charge is quantised"}, {"Chapter": "1", "sentence_range": "4708-4711", "Text": "All other charges are\nintegral multiples of this smallest unit charge In other words, charge is\nquantised We do not know why magnetic monopoles do not exist or why\nelectric charge is quantised 3"}, {"Chapter": "1", "sentence_range": "4709-4712", "Text": "In other words, charge is\nquantised We do not know why magnetic monopoles do not exist or why\nelectric charge is quantised 3 A consequence of the fact that magnetic monopoles do not exist is that\nthe magnetic field lines are continuous and form closed loops"}, {"Chapter": "1", "sentence_range": "4710-4713", "Text": "We do not know why magnetic monopoles do not exist or why\nelectric charge is quantised 3 A consequence of the fact that magnetic monopoles do not exist is that\nthe magnetic field lines are continuous and form closed loops In contrast,\nthe electrostatic lines of force begin on a positive charge and terminate\non the negative charge (or fade out at infinity)"}, {"Chapter": "1", "sentence_range": "4711-4714", "Text": "3 A consequence of the fact that magnetic monopoles do not exist is that\nthe magnetic field lines are continuous and form closed loops In contrast,\nthe electrostatic lines of force begin on a positive charge and terminate\non the negative charge (or fade out at infinity) 4"}, {"Chapter": "1", "sentence_range": "4712-4715", "Text": "A consequence of the fact that magnetic monopoles do not exist is that\nthe magnetic field lines are continuous and form closed loops In contrast,\nthe electrostatic lines of force begin on a positive charge and terminate\non the negative charge (or fade out at infinity) 4 A miniscule difference in the value of c, the magnetic susceptibility, yields\nradically different behaviour: diamagnetic versus paramagnetic"}, {"Chapter": "1", "sentence_range": "4713-4716", "Text": "In contrast,\nthe electrostatic lines of force begin on a positive charge and terminate\non the negative charge (or fade out at infinity) 4 A miniscule difference in the value of c, the magnetic susceptibility, yields\nradically different behaviour: diamagnetic versus paramagnetic For\ndiamagnetic materials c = \u201310\u20135 whereas c = +10\u20135 for paramagnetic\nmaterials"}, {"Chapter": "1", "sentence_range": "4714-4717", "Text": "4 A miniscule difference in the value of c, the magnetic susceptibility, yields\nradically different behaviour: diamagnetic versus paramagnetic For\ndiamagnetic materials c = \u201310\u20135 whereas c = +10\u20135 for paramagnetic\nmaterials 5"}, {"Chapter": "1", "sentence_range": "4715-4718", "Text": "A miniscule difference in the value of c, the magnetic susceptibility, yields\nradically different behaviour: diamagnetic versus paramagnetic For\ndiamagnetic materials c = \u201310\u20135 whereas c = +10\u20135 for paramagnetic\nmaterials 5 There exists a perfect diamagnet, namely, a superconductor"}, {"Chapter": "1", "sentence_range": "4716-4719", "Text": "For\ndiamagnetic materials c = \u201310\u20135 whereas c = +10\u20135 for paramagnetic\nmaterials 5 There exists a perfect diamagnet, namely, a superconductor This is a\nmetal at very low temperatures"}, {"Chapter": "1", "sentence_range": "4717-4720", "Text": "5 There exists a perfect diamagnet, namely, a superconductor This is a\nmetal at very low temperatures In this case c = \u20131, mr = 0, m = 0"}, {"Chapter": "1", "sentence_range": "4718-4721", "Text": "There exists a perfect diamagnet, namely, a superconductor This is a\nmetal at very low temperatures In this case c = \u20131, mr = 0, m = 0 The\nexternal magnetic field is totally expelled"}, {"Chapter": "1", "sentence_range": "4719-4722", "Text": "This is a\nmetal at very low temperatures In this case c = \u20131, mr = 0, m = 0 The\nexternal magnetic field is totally expelled Interestingly, this material is\nalso a perfect conductor"}, {"Chapter": "1", "sentence_range": "4720-4723", "Text": "In this case c = \u20131, mr = 0, m = 0 The\nexternal magnetic field is totally expelled Interestingly, this material is\nalso a perfect conductor However, there exists no classical theory which\nties these two properties together"}, {"Chapter": "1", "sentence_range": "4721-4724", "Text": "The\nexternal magnetic field is totally expelled Interestingly, this material is\nalso a perfect conductor However, there exists no classical theory which\nties these two properties together A quantum-mechanical theory by\nBardeen, Cooper, and Schrieffer (BCS theory) explains these effects"}, {"Chapter": "1", "sentence_range": "4722-4725", "Text": "Interestingly, this material is\nalso a perfect conductor However, there exists no classical theory which\nties these two properties together A quantum-mechanical theory by\nBardeen, Cooper, and Schrieffer (BCS theory) explains these effects The\nBCS theory was proposed in1957 and was eventually recognised by a Nobel\nPrize in physics in 1970"}, {"Chapter": "1", "sentence_range": "4723-4726", "Text": "However, there exists no classical theory which\nties these two properties together A quantum-mechanical theory by\nBardeen, Cooper, and Schrieffer (BCS theory) explains these effects The\nBCS theory was proposed in1957 and was eventually recognised by a Nobel\nPrize in physics in 1970 Magnetic flux\nfB\nScalar\n[ML2T\u20132 A\u20131]\nW (weber)\nW = T m2\nMagnetisation\nM\nVector\n[L\u20131 A]\nA m\u20131\nMagnetic moment\nVolume\nMagnetic intensity\nH\nVector\n[L\u20131 A]\nA m\u20131\nB = m0 (H + M)\nMagnetic field\nstrength\nMagnetic\nc\nScalar\n-\n-\nM = cH\nsusceptibility\nRelative magnetic\nmr\nScalar\n-\n-\nB = m0 mr H\npermeability\nMagnetic permeability\nm\nScalar\n[MLT\u20132 A\u20132]\nT m A\u20131\nm = m0 mr\nN A\u20132\nB = m H\nRationalised 2023-24\nPhysics\n152\n6"}, {"Chapter": "1", "sentence_range": "4724-4727", "Text": "A quantum-mechanical theory by\nBardeen, Cooper, and Schrieffer (BCS theory) explains these effects The\nBCS theory was proposed in1957 and was eventually recognised by a Nobel\nPrize in physics in 1970 Magnetic flux\nfB\nScalar\n[ML2T\u20132 A\u20131]\nW (weber)\nW = T m2\nMagnetisation\nM\nVector\n[L\u20131 A]\nA m\u20131\nMagnetic moment\nVolume\nMagnetic intensity\nH\nVector\n[L\u20131 A]\nA m\u20131\nB = m0 (H + M)\nMagnetic field\nstrength\nMagnetic\nc\nScalar\n-\n-\nM = cH\nsusceptibility\nRelative magnetic\nmr\nScalar\n-\n-\nB = m0 mr H\npermeability\nMagnetic permeability\nm\nScalar\n[MLT\u20132 A\u20132]\nT m A\u20131\nm = m0 mr\nN A\u20132\nB = m H\nRationalised 2023-24\nPhysics\n152\n6 Diamagnetism is universal"}, {"Chapter": "1", "sentence_range": "4725-4728", "Text": "The\nBCS theory was proposed in1957 and was eventually recognised by a Nobel\nPrize in physics in 1970 Magnetic flux\nfB\nScalar\n[ML2T\u20132 A\u20131]\nW (weber)\nW = T m2\nMagnetisation\nM\nVector\n[L\u20131 A]\nA m\u20131\nMagnetic moment\nVolume\nMagnetic intensity\nH\nVector\n[L\u20131 A]\nA m\u20131\nB = m0 (H + M)\nMagnetic field\nstrength\nMagnetic\nc\nScalar\n-\n-\nM = cH\nsusceptibility\nRelative magnetic\nmr\nScalar\n-\n-\nB = m0 mr H\npermeability\nMagnetic permeability\nm\nScalar\n[MLT\u20132 A\u20132]\nT m A\u20131\nm = m0 mr\nN A\u20132\nB = m H\nRationalised 2023-24\nPhysics\n152\n6 Diamagnetism is universal It is present in all materials"}, {"Chapter": "1", "sentence_range": "4726-4729", "Text": "Magnetic flux\nfB\nScalar\n[ML2T\u20132 A\u20131]\nW (weber)\nW = T m2\nMagnetisation\nM\nVector\n[L\u20131 A]\nA m\u20131\nMagnetic moment\nVolume\nMagnetic intensity\nH\nVector\n[L\u20131 A]\nA m\u20131\nB = m0 (H + M)\nMagnetic field\nstrength\nMagnetic\nc\nScalar\n-\n-\nM = cH\nsusceptibility\nRelative magnetic\nmr\nScalar\n-\n-\nB = m0 mr H\npermeability\nMagnetic permeability\nm\nScalar\n[MLT\u20132 A\u20132]\nT m A\u20131\nm = m0 mr\nN A\u20132\nB = m H\nRationalised 2023-24\nPhysics\n152\n6 Diamagnetism is universal It is present in all materials But it\nis weak and hard to detect if the substance is para- or ferromagnetic"}, {"Chapter": "1", "sentence_range": "4727-4730", "Text": "Diamagnetism is universal It is present in all materials But it\nis weak and hard to detect if the substance is para- or ferromagnetic 7"}, {"Chapter": "1", "sentence_range": "4728-4731", "Text": "It is present in all materials But it\nis weak and hard to detect if the substance is para- or ferromagnetic 7 We have classified materials as diamagnetic, paramagnetic, and\nferromagnetic"}, {"Chapter": "1", "sentence_range": "4729-4732", "Text": "But it\nis weak and hard to detect if the substance is para- or ferromagnetic 7 We have classified materials as diamagnetic, paramagnetic, and\nferromagnetic However, there exist additional types of magnetic material\nsuch as ferrimagnetic, anti-ferromagnetic, spin glass, etc"}, {"Chapter": "1", "sentence_range": "4730-4733", "Text": "7 We have classified materials as diamagnetic, paramagnetic, and\nferromagnetic However, there exist additional types of magnetic material\nsuch as ferrimagnetic, anti-ferromagnetic, spin glass, etc with properties\nwhich are exotic and mysterious"}, {"Chapter": "1", "sentence_range": "4731-4734", "Text": "We have classified materials as diamagnetic, paramagnetic, and\nferromagnetic However, there exist additional types of magnetic material\nsuch as ferrimagnetic, anti-ferromagnetic, spin glass, etc with properties\nwhich are exotic and mysterious EXERCISES\n5"}, {"Chapter": "1", "sentence_range": "4732-4735", "Text": "However, there exist additional types of magnetic material\nsuch as ferrimagnetic, anti-ferromagnetic, spin glass, etc with properties\nwhich are exotic and mysterious EXERCISES\n5 1\nA short bar magnet placed with its axis at 30\u00b0 with a uniform external\nmagnetic field of 0"}, {"Chapter": "1", "sentence_range": "4733-4736", "Text": "with properties\nwhich are exotic and mysterious EXERCISES\n5 1\nA short bar magnet placed with its axis at 30\u00b0 with a uniform external\nmagnetic field of 0 25 T experiences a torque of magnitude equal to\n4"}, {"Chapter": "1", "sentence_range": "4734-4737", "Text": "EXERCISES\n5 1\nA short bar magnet placed with its axis at 30\u00b0 with a uniform external\nmagnetic field of 0 25 T experiences a torque of magnitude equal to\n4 5 \u00d7 10\u20132 J"}, {"Chapter": "1", "sentence_range": "4735-4738", "Text": "1\nA short bar magnet placed with its axis at 30\u00b0 with a uniform external\nmagnetic field of 0 25 T experiences a torque of magnitude equal to\n4 5 \u00d7 10\u20132 J What is the magnitude of magnetic moment of the magnet"}, {"Chapter": "1", "sentence_range": "4736-4739", "Text": "25 T experiences a torque of magnitude equal to\n4 5 \u00d7 10\u20132 J What is the magnitude of magnetic moment of the magnet 5"}, {"Chapter": "1", "sentence_range": "4737-4740", "Text": "5 \u00d7 10\u20132 J What is the magnitude of magnetic moment of the magnet 5 2\nA short bar magnet of magnetic moment m = 0"}, {"Chapter": "1", "sentence_range": "4738-4741", "Text": "What is the magnitude of magnetic moment of the magnet 5 2\nA short bar magnet of magnetic moment m = 0 32 JT \u20131 is placed in a\nuniform magnetic field of 0"}, {"Chapter": "1", "sentence_range": "4739-4742", "Text": "5 2\nA short bar magnet of magnetic moment m = 0 32 JT \u20131 is placed in a\nuniform magnetic field of 0 15 T"}, {"Chapter": "1", "sentence_range": "4740-4743", "Text": "2\nA short bar magnet of magnetic moment m = 0 32 JT \u20131 is placed in a\nuniform magnetic field of 0 15 T If the bar is free to rotate in the\nplane of the field, which orientation would correspond to its (a) stable,\nand (b) unstable equilibrium"}, {"Chapter": "1", "sentence_range": "4741-4744", "Text": "32 JT \u20131 is placed in a\nuniform magnetic field of 0 15 T If the bar is free to rotate in the\nplane of the field, which orientation would correspond to its (a) stable,\nand (b) unstable equilibrium What is the potential energy of the\nmagnet in each case"}, {"Chapter": "1", "sentence_range": "4742-4745", "Text": "15 T If the bar is free to rotate in the\nplane of the field, which orientation would correspond to its (a) stable,\nand (b) unstable equilibrium What is the potential energy of the\nmagnet in each case 5"}, {"Chapter": "1", "sentence_range": "4743-4746", "Text": "If the bar is free to rotate in the\nplane of the field, which orientation would correspond to its (a) stable,\nand (b) unstable equilibrium What is the potential energy of the\nmagnet in each case 5 3\nA closely wound solenoid of 800 turns and area of cross section\n2"}, {"Chapter": "1", "sentence_range": "4744-4747", "Text": "What is the potential energy of the\nmagnet in each case 5 3\nA closely wound solenoid of 800 turns and area of cross section\n2 5 \u00d7 10\u20134 m2 carries a current of 3"}, {"Chapter": "1", "sentence_range": "4745-4748", "Text": "5 3\nA closely wound solenoid of 800 turns and area of cross section\n2 5 \u00d7 10\u20134 m2 carries a current of 3 0 A"}, {"Chapter": "1", "sentence_range": "4746-4749", "Text": "3\nA closely wound solenoid of 800 turns and area of cross section\n2 5 \u00d7 10\u20134 m2 carries a current of 3 0 A Explain the sense in which\nthe solenoid acts like a bar magnet"}, {"Chapter": "1", "sentence_range": "4747-4750", "Text": "5 \u00d7 10\u20134 m2 carries a current of 3 0 A Explain the sense in which\nthe solenoid acts like a bar magnet What is its associated magnetic\nmoment"}, {"Chapter": "1", "sentence_range": "4748-4751", "Text": "0 A Explain the sense in which\nthe solenoid acts like a bar magnet What is its associated magnetic\nmoment 5"}, {"Chapter": "1", "sentence_range": "4749-4752", "Text": "Explain the sense in which\nthe solenoid acts like a bar magnet What is its associated magnetic\nmoment 5 4\nIf the solenoid in Exercise 5"}, {"Chapter": "1", "sentence_range": "4750-4753", "Text": "What is its associated magnetic\nmoment 5 4\nIf the solenoid in Exercise 5 5 is free to turn about the vertical\ndirection and a uniform horizontal magnetic field of 0"}, {"Chapter": "1", "sentence_range": "4751-4754", "Text": "5 4\nIf the solenoid in Exercise 5 5 is free to turn about the vertical\ndirection and a uniform horizontal magnetic field of 0 25 T is applied,\nwhat is the magnitude of torque on the solenoid when its axis makes\nan angle of 30\u00b0 with the direction of applied field"}, {"Chapter": "1", "sentence_range": "4752-4755", "Text": "4\nIf the solenoid in Exercise 5 5 is free to turn about the vertical\ndirection and a uniform horizontal magnetic field of 0 25 T is applied,\nwhat is the magnitude of torque on the solenoid when its axis makes\nan angle of 30\u00b0 with the direction of applied field 5"}, {"Chapter": "1", "sentence_range": "4753-4756", "Text": "5 is free to turn about the vertical\ndirection and a uniform horizontal magnetic field of 0 25 T is applied,\nwhat is the magnitude of torque on the solenoid when its axis makes\nan angle of 30\u00b0 with the direction of applied field 5 5\nA bar magnet of magnetic moment 1"}, {"Chapter": "1", "sentence_range": "4754-4757", "Text": "25 T is applied,\nwhat is the magnitude of torque on the solenoid when its axis makes\nan angle of 30\u00b0 with the direction of applied field 5 5\nA bar magnet of magnetic moment 1 5 J T \u20131 lies aligned with the\ndirection of a uniform magnetic field of 0"}, {"Chapter": "1", "sentence_range": "4755-4758", "Text": "5 5\nA bar magnet of magnetic moment 1 5 J T \u20131 lies aligned with the\ndirection of a uniform magnetic field of 0 22 T"}, {"Chapter": "1", "sentence_range": "4756-4759", "Text": "5\nA bar magnet of magnetic moment 1 5 J T \u20131 lies aligned with the\ndirection of a uniform magnetic field of 0 22 T (a) What is the amount of work required by an external torque to\nturn the magnet so as to align its magnetic moment: (i) normal\nto the field direction, (ii) opposite to the field direction"}, {"Chapter": "1", "sentence_range": "4757-4760", "Text": "5 J T \u20131 lies aligned with the\ndirection of a uniform magnetic field of 0 22 T (a) What is the amount of work required by an external torque to\nturn the magnet so as to align its magnetic moment: (i) normal\nto the field direction, (ii) opposite to the field direction (b) What is the torque on the magnet in cases (i) and (ii)"}, {"Chapter": "1", "sentence_range": "4758-4761", "Text": "22 T (a) What is the amount of work required by an external torque to\nturn the magnet so as to align its magnetic moment: (i) normal\nto the field direction, (ii) opposite to the field direction (b) What is the torque on the magnet in cases (i) and (ii) 5"}, {"Chapter": "1", "sentence_range": "4759-4762", "Text": "(a) What is the amount of work required by an external torque to\nturn the magnet so as to align its magnetic moment: (i) normal\nto the field direction, (ii) opposite to the field direction (b) What is the torque on the magnet in cases (i) and (ii) 5 6\nA closely wound solenoid of 2000 turns and area of cross-section\n1"}, {"Chapter": "1", "sentence_range": "4760-4763", "Text": "(b) What is the torque on the magnet in cases (i) and (ii) 5 6\nA closely wound solenoid of 2000 turns and area of cross-section\n1 6 \u00d7 10 \u20134 m2,  carrying a current of 4"}, {"Chapter": "1", "sentence_range": "4761-4764", "Text": "5 6\nA closely wound solenoid of 2000 turns and area of cross-section\n1 6 \u00d7 10 \u20134 m2,  carrying a current of 4 0 A, is suspended through its\ncentre allowing it to turn in a horizontal plane"}, {"Chapter": "1", "sentence_range": "4762-4765", "Text": "6\nA closely wound solenoid of 2000 turns and area of cross-section\n1 6 \u00d7 10 \u20134 m2,  carrying a current of 4 0 A, is suspended through its\ncentre allowing it to turn in a horizontal plane (a) What is the magnetic moment associated with the solenoid"}, {"Chapter": "1", "sentence_range": "4763-4766", "Text": "6 \u00d7 10 \u20134 m2,  carrying a current of 4 0 A, is suspended through its\ncentre allowing it to turn in a horizontal plane (a) What is the magnetic moment associated with the solenoid (b) What is the force and torque on the solenoid if a uniform\nhorizontal magnetic field of 7"}, {"Chapter": "1", "sentence_range": "4764-4767", "Text": "0 A, is suspended through its\ncentre allowing it to turn in a horizontal plane (a) What is the magnetic moment associated with the solenoid (b) What is the force and torque on the solenoid if a uniform\nhorizontal magnetic field of 7 5 \u00d7 10\u20132 T is set up at an angle of\n30\u00b0 with the axis of the solenoid"}, {"Chapter": "1", "sentence_range": "4765-4768", "Text": "(a) What is the magnetic moment associated with the solenoid (b) What is the force and torque on the solenoid if a uniform\nhorizontal magnetic field of 7 5 \u00d7 10\u20132 T is set up at an angle of\n30\u00b0 with the axis of the solenoid 5"}, {"Chapter": "1", "sentence_range": "4766-4769", "Text": "(b) What is the force and torque on the solenoid if a uniform\nhorizontal magnetic field of 7 5 \u00d7 10\u20132 T is set up at an angle of\n30\u00b0 with the axis of the solenoid 5 7\nA short bar magnet has a magnetic moment of 0"}, {"Chapter": "1", "sentence_range": "4767-4770", "Text": "5 \u00d7 10\u20132 T is set up at an angle of\n30\u00b0 with the axis of the solenoid 5 7\nA short bar magnet has a magnetic moment of 0 48 J T \u20131"}, {"Chapter": "1", "sentence_range": "4768-4771", "Text": "5 7\nA short bar magnet has a magnetic moment of 0 48 J T \u20131 Give the\ndirection and magnitude of the magnetic field produced by the magnet\nat a distance of 10 cm from the centre of the magnet on (a) the axis,\n(b) the equatorial lines (normal bisector) of the magnet"}, {"Chapter": "1", "sentence_range": "4769-4772", "Text": "7\nA short bar magnet has a magnetic moment of 0 48 J T \u20131 Give the\ndirection and magnitude of the magnetic field produced by the magnet\nat a distance of 10 cm from the centre of the magnet on (a) the axis,\n(b) the equatorial lines (normal bisector) of the magnet Rationalised 2023-24\n153\nMagnetism and\nMatter\n5"}, {"Chapter": "1", "sentence_range": "4770-4773", "Text": "48 J T \u20131 Give the\ndirection and magnitude of the magnetic field produced by the magnet\nat a distance of 10 cm from the centre of the magnet on (a) the axis,\n(b) the equatorial lines (normal bisector) of the magnet Rationalised 2023-24\n153\nMagnetism and\nMatter\n5 8\nA short bar magnet placed in a horizontal plane has its axis aligned\nalong the magnetic north-south direction"}, {"Chapter": "1", "sentence_range": "4771-4774", "Text": "Give the\ndirection and magnitude of the magnetic field produced by the magnet\nat a distance of 10 cm from the centre of the magnet on (a) the axis,\n(b) the equatorial lines (normal bisector) of the magnet Rationalised 2023-24\n153\nMagnetism and\nMatter\n5 8\nA short bar magnet placed in a horizontal plane has its axis aligned\nalong the magnetic north-south direction Null points are found on\nthe axis of the magnet at 14 cm from the centre of the magnet"}, {"Chapter": "1", "sentence_range": "4772-4775", "Text": "Rationalised 2023-24\n153\nMagnetism and\nMatter\n5 8\nA short bar magnet placed in a horizontal plane has its axis aligned\nalong the magnetic north-south direction Null points are found on\nthe axis of the magnet at 14 cm from the centre of the magnet The\nearth\u2019s magnetic field at the place is 0"}, {"Chapter": "1", "sentence_range": "4773-4776", "Text": "8\nA short bar magnet placed in a horizontal plane has its axis aligned\nalong the magnetic north-south direction Null points are found on\nthe axis of the magnet at 14 cm from the centre of the magnet The\nearth\u2019s magnetic field at the place is 0 36 G and the angle of dip is\nzero"}, {"Chapter": "1", "sentence_range": "4774-4777", "Text": "Null points are found on\nthe axis of the magnet at 14 cm from the centre of the magnet The\nearth\u2019s magnetic field at the place is 0 36 G and the angle of dip is\nzero What is the total magnetic field on the normal bisector of the\nmagnet at the same distance as the null\u2013point (i"}, {"Chapter": "1", "sentence_range": "4775-4778", "Text": "The\nearth\u2019s magnetic field at the place is 0 36 G and the angle of dip is\nzero What is the total magnetic field on the normal bisector of the\nmagnet at the same distance as the null\u2013point (i e"}, {"Chapter": "1", "sentence_range": "4776-4779", "Text": "36 G and the angle of dip is\nzero What is the total magnetic field on the normal bisector of the\nmagnet at the same distance as the null\u2013point (i e , 14 cm) from the\ncentre of the magnet"}, {"Chapter": "1", "sentence_range": "4777-4780", "Text": "What is the total magnetic field on the normal bisector of the\nmagnet at the same distance as the null\u2013point (i e , 14 cm) from the\ncentre of the magnet (At null points, field due to a magnet is equal\nand opposite to the horizontal component of earth\u2019s magnetic field"}, {"Chapter": "1", "sentence_range": "4778-4781", "Text": "e , 14 cm) from the\ncentre of the magnet (At null points, field due to a magnet is equal\nand opposite to the horizontal component of earth\u2019s magnetic field )\n5"}, {"Chapter": "1", "sentence_range": "4779-4782", "Text": ", 14 cm) from the\ncentre of the magnet (At null points, field due to a magnet is equal\nand opposite to the horizontal component of earth\u2019s magnetic field )\n5 9\nIf the bar magnet in exercise 5"}, {"Chapter": "1", "sentence_range": "4780-4783", "Text": "(At null points, field due to a magnet is equal\nand opposite to the horizontal component of earth\u2019s magnetic field )\n5 9\nIf the bar magnet in exercise 5 13 is turned around by 180\u00b0, where\nwill the new null points be located"}, {"Chapter": "1", "sentence_range": "4781-4784", "Text": ")\n5 9\nIf the bar magnet in exercise 5 13 is turned around by 180\u00b0, where\nwill the new null points be located Rationalised 2023-24\nPhysics\n154\n6"}, {"Chapter": "1", "sentence_range": "4782-4785", "Text": "9\nIf the bar magnet in exercise 5 13 is turned around by 180\u00b0, where\nwill the new null points be located Rationalised 2023-24\nPhysics\n154\n6 1  INTRODUCTION\nElectricity and magnetism were considered separate and unrelated\nphenomena for a long time"}, {"Chapter": "1", "sentence_range": "4783-4786", "Text": "13 is turned around by 180\u00b0, where\nwill the new null points be located Rationalised 2023-24\nPhysics\n154\n6 1  INTRODUCTION\nElectricity and magnetism were considered separate and unrelated\nphenomena for a long time In the early decades of the nineteenth century,\nexperiments on electric current by Oersted, Ampere and a few others\nestablished the fact that electricity and magnetism are inter-related"}, {"Chapter": "1", "sentence_range": "4784-4787", "Text": "Rationalised 2023-24\nPhysics\n154\n6 1  INTRODUCTION\nElectricity and magnetism were considered separate and unrelated\nphenomena for a long time In the early decades of the nineteenth century,\nexperiments on electric current by Oersted, Ampere and a few others\nestablished the fact that electricity and magnetism are inter-related They\nfound that moving electric charges produce magnetic fields"}, {"Chapter": "1", "sentence_range": "4785-4788", "Text": "1  INTRODUCTION\nElectricity and magnetism were considered separate and unrelated\nphenomena for a long time In the early decades of the nineteenth century,\nexperiments on electric current by Oersted, Ampere and a few others\nestablished the fact that electricity and magnetism are inter-related They\nfound that moving electric charges produce magnetic fields For example,\nan electric current deflects a magnetic compass needle placed in its vicinity"}, {"Chapter": "1", "sentence_range": "4786-4789", "Text": "In the early decades of the nineteenth century,\nexperiments on electric current by Oersted, Ampere and a few others\nestablished the fact that electricity and magnetism are inter-related They\nfound that moving electric charges produce magnetic fields For example,\nan electric current deflects a magnetic compass needle placed in its vicinity This naturally raises the questions like: Is the converse effect possible"}, {"Chapter": "1", "sentence_range": "4787-4790", "Text": "They\nfound that moving electric charges produce magnetic fields For example,\nan electric current deflects a magnetic compass needle placed in its vicinity This naturally raises the questions like: Is the converse effect possible Can moving magnets produce electric currents"}, {"Chapter": "1", "sentence_range": "4788-4791", "Text": "For example,\nan electric current deflects a magnetic compass needle placed in its vicinity This naturally raises the questions like: Is the converse effect possible Can moving magnets produce electric currents Does the nature permit\nsuch a relation between electricity and magnetism"}, {"Chapter": "1", "sentence_range": "4789-4792", "Text": "This naturally raises the questions like: Is the converse effect possible Can moving magnets produce electric currents Does the nature permit\nsuch a relation between electricity and magnetism The answer is\nresounding yes"}, {"Chapter": "1", "sentence_range": "4790-4793", "Text": "Can moving magnets produce electric currents Does the nature permit\nsuch a relation between electricity and magnetism The answer is\nresounding yes The experiments of Michael Faraday in England and\nJoseph Henry in USA, conducted around 1830, demonstrated\nconclusively that electric currents were induced in closed coils when\nsubjected to changing magnetic fields"}, {"Chapter": "1", "sentence_range": "4791-4794", "Text": "Does the nature permit\nsuch a relation between electricity and magnetism The answer is\nresounding yes The experiments of Michael Faraday in England and\nJoseph Henry in USA, conducted around 1830, demonstrated\nconclusively that electric currents were induced in closed coils when\nsubjected to changing magnetic fields In this chapter, we will study the\nphenomena associated with changing magnetic fields and understand\nthe underlying principles"}, {"Chapter": "1", "sentence_range": "4792-4795", "Text": "The answer is\nresounding yes The experiments of Michael Faraday in England and\nJoseph Henry in USA, conducted around 1830, demonstrated\nconclusively that electric currents were induced in closed coils when\nsubjected to changing magnetic fields In this chapter, we will study the\nphenomena associated with changing magnetic fields and understand\nthe underlying principles The phenomenon in which electric current is\ngenerated by varying magnetic fields is appropriately called\nelectromagnetic induction"}, {"Chapter": "1", "sentence_range": "4793-4796", "Text": "The experiments of Michael Faraday in England and\nJoseph Henry in USA, conducted around 1830, demonstrated\nconclusively that electric currents were induced in closed coils when\nsubjected to changing magnetic fields In this chapter, we will study the\nphenomena associated with changing magnetic fields and understand\nthe underlying principles The phenomenon in which electric current is\ngenerated by varying magnetic fields is appropriately called\nelectromagnetic induction When Faraday first made public his discovery that relative motion\nbetween a bar magnet and a wire loop produced a small current in the\nlatter, he was asked, \u201cWhat is the use of it"}, {"Chapter": "1", "sentence_range": "4794-4797", "Text": "In this chapter, we will study the\nphenomena associated with changing magnetic fields and understand\nthe underlying principles The phenomenon in which electric current is\ngenerated by varying magnetic fields is appropriately called\nelectromagnetic induction When Faraday first made public his discovery that relative motion\nbetween a bar magnet and a wire loop produced a small current in the\nlatter, he was asked, \u201cWhat is the use of it \u201d His reply was: \u201cWhat is the\nuse of a new born baby"}, {"Chapter": "1", "sentence_range": "4795-4798", "Text": "The phenomenon in which electric current is\ngenerated by varying magnetic fields is appropriately called\nelectromagnetic induction When Faraday first made public his discovery that relative motion\nbetween a bar magnet and a wire loop produced a small current in the\nlatter, he was asked, \u201cWhat is the use of it \u201d His reply was: \u201cWhat is the\nuse of a new born baby \u201d The phenomenon of electromagnetic induction\nChapter Six\nELECTROMAGNETIC\nINDUCTION\nRationalised 2023-24\nElectromagnetic\nInduction\n155\nis not merely of theoretical or academic interest but also\nof practical utility"}, {"Chapter": "1", "sentence_range": "4796-4799", "Text": "When Faraday first made public his discovery that relative motion\nbetween a bar magnet and a wire loop produced a small current in the\nlatter, he was asked, \u201cWhat is the use of it \u201d His reply was: \u201cWhat is the\nuse of a new born baby \u201d The phenomenon of electromagnetic induction\nChapter Six\nELECTROMAGNETIC\nINDUCTION\nRationalised 2023-24\nElectromagnetic\nInduction\n155\nis not merely of theoretical or academic interest but also\nof practical utility Imagine a world where there is no\nelectricity \u2013 no electric lights, no trains, no telephones and\nno personal computers"}, {"Chapter": "1", "sentence_range": "4797-4800", "Text": "\u201d His reply was: \u201cWhat is the\nuse of a new born baby \u201d The phenomenon of electromagnetic induction\nChapter Six\nELECTROMAGNETIC\nINDUCTION\nRationalised 2023-24\nElectromagnetic\nInduction\n155\nis not merely of theoretical or academic interest but also\nof practical utility Imagine a world where there is no\nelectricity \u2013 no electric lights, no trains, no telephones and\nno personal computers The pioneering experiments of\nFaraday and Henry have led directly to the development\nof modern day generators and transformers"}, {"Chapter": "1", "sentence_range": "4798-4801", "Text": "\u201d The phenomenon of electromagnetic induction\nChapter Six\nELECTROMAGNETIC\nINDUCTION\nRationalised 2023-24\nElectromagnetic\nInduction\n155\nis not merely of theoretical or academic interest but also\nof practical utility Imagine a world where there is no\nelectricity \u2013 no electric lights, no trains, no telephones and\nno personal computers The pioneering experiments of\nFaraday and Henry have led directly to the development\nof modern day generators and transformers Today\u2019s\ncivilisation owes its progress to a great extent to the\ndiscovery of electromagnetic induction"}, {"Chapter": "1", "sentence_range": "4799-4802", "Text": "Imagine a world where there is no\nelectricity \u2013 no electric lights, no trains, no telephones and\nno personal computers The pioneering experiments of\nFaraday and Henry have led directly to the development\nof modern day generators and transformers Today\u2019s\ncivilisation owes its progress to a great extent to the\ndiscovery of electromagnetic induction 6"}, {"Chapter": "1", "sentence_range": "4800-4803", "Text": "The pioneering experiments of\nFaraday and Henry have led directly to the development\nof modern day generators and transformers Today\u2019s\ncivilisation owes its progress to a great extent to the\ndiscovery of electromagnetic induction 6 2 THE EXPERIMENTS OF FARADAY AND\nHENRY\nThe discovery and understanding of electromagnetic\ninduction are based on a long series of experiments carried\nout by Faraday and Henry"}, {"Chapter": "1", "sentence_range": "4801-4804", "Text": "Today\u2019s\ncivilisation owes its progress to a great extent to the\ndiscovery of electromagnetic induction 6 2 THE EXPERIMENTS OF FARADAY AND\nHENRY\nThe discovery and understanding of electromagnetic\ninduction are based on a long series of experiments carried\nout by Faraday and Henry We shall now describe some\nof these experiments"}, {"Chapter": "1", "sentence_range": "4802-4805", "Text": "6 2 THE EXPERIMENTS OF FARADAY AND\nHENRY\nThe discovery and understanding of electromagnetic\ninduction are based on a long series of experiments carried\nout by Faraday and Henry We shall now describe some\nof these experiments Experiment 6"}, {"Chapter": "1", "sentence_range": "4803-4806", "Text": "2 THE EXPERIMENTS OF FARADAY AND\nHENRY\nThe discovery and understanding of electromagnetic\ninduction are based on a long series of experiments carried\nout by Faraday and Henry We shall now describe some\nof these experiments Experiment 6 1\nFigure 6"}, {"Chapter": "1", "sentence_range": "4804-4807", "Text": "We shall now describe some\nof these experiments Experiment 6 1\nFigure 6 1 shows a coil C1* connected to a galvanometer\nG"}, {"Chapter": "1", "sentence_range": "4805-4808", "Text": "Experiment 6 1\nFigure 6 1 shows a coil C1* connected to a galvanometer\nG When the North-pole of a bar magnet is pushed\ntowards the coil, the pointer in the galvanometer deflects,\nindicating  the presence of electric  current in the coil"}, {"Chapter": "1", "sentence_range": "4806-4809", "Text": "1\nFigure 6 1 shows a coil C1* connected to a galvanometer\nG When the North-pole of a bar magnet is pushed\ntowards the coil, the pointer in the galvanometer deflects,\nindicating  the presence of electric  current in the coil The\ndeflection lasts as long as the bar magnet is in motion"}, {"Chapter": "1", "sentence_range": "4807-4810", "Text": "1 shows a coil C1* connected to a galvanometer\nG When the North-pole of a bar magnet is pushed\ntowards the coil, the pointer in the galvanometer deflects,\nindicating  the presence of electric  current in the coil The\ndeflection lasts as long as the bar magnet is in motion The galvanometer does not show any deflection when the\nmagnet is held stationary"}, {"Chapter": "1", "sentence_range": "4808-4811", "Text": "When the North-pole of a bar magnet is pushed\ntowards the coil, the pointer in the galvanometer deflects,\nindicating  the presence of electric  current in the coil The\ndeflection lasts as long as the bar magnet is in motion The galvanometer does not show any deflection when the\nmagnet is held stationary When the magnet is pulled\naway from the coil, the galvanometer shows deflection in\nthe opposite direction, which indicates reversal of the\ncurrent\u2019s direction"}, {"Chapter": "1", "sentence_range": "4809-4812", "Text": "The\ndeflection lasts as long as the bar magnet is in motion The galvanometer does not show any deflection when the\nmagnet is held stationary When the magnet is pulled\naway from the coil, the galvanometer shows deflection in\nthe opposite direction, which indicates reversal of the\ncurrent\u2019s direction Moreover, when the South-pole of\nthe bar magnet is moved towards or away from the\ncoil, the deflections in the galvanometer are opposite\nto that observed with the North-pole for similar\nmovements"}, {"Chapter": "1", "sentence_range": "4810-4813", "Text": "The galvanometer does not show any deflection when the\nmagnet is held stationary When the magnet is pulled\naway from the coil, the galvanometer shows deflection in\nthe opposite direction, which indicates reversal of the\ncurrent\u2019s direction Moreover, when the South-pole of\nthe bar magnet is moved towards or away from the\ncoil, the deflections in the galvanometer are opposite\nto that observed with the North-pole for similar\nmovements Further, the deflection (and hence current)\nis found to be larger when the magnet is pushed\ntowards or pulled away from the coil faster"}, {"Chapter": "1", "sentence_range": "4811-4814", "Text": "When the magnet is pulled\naway from the coil, the galvanometer shows deflection in\nthe opposite direction, which indicates reversal of the\ncurrent\u2019s direction Moreover, when the South-pole of\nthe bar magnet is moved towards or away from the\ncoil, the deflections in the galvanometer are opposite\nto that observed with the North-pole for similar\nmovements Further, the deflection (and hence current)\nis found to be larger when the magnet is pushed\ntowards or pulled away from the coil faster Instead,\nwhen the bar magnet is held fixed and the coil C1 is\nmoved towards or away from the magnet, the same\neffects are observed"}, {"Chapter": "1", "sentence_range": "4812-4815", "Text": "Moreover, when the South-pole of\nthe bar magnet is moved towards or away from the\ncoil, the deflections in the galvanometer are opposite\nto that observed with the North-pole for similar\nmovements Further, the deflection (and hence current)\nis found to be larger when the magnet is pushed\ntowards or pulled away from the coil faster Instead,\nwhen the bar magnet is held fixed and the coil C1 is\nmoved towards or away from the magnet, the same\neffects are observed It shows that it is the relative\nmotion between the magnet and the coil that is\nresponsible for generation (induction) of electric\ncurrent in the coil"}, {"Chapter": "1", "sentence_range": "4813-4816", "Text": "Further, the deflection (and hence current)\nis found to be larger when the magnet is pushed\ntowards or pulled away from the coil faster Instead,\nwhen the bar magnet is held fixed and the coil C1 is\nmoved towards or away from the magnet, the same\neffects are observed It shows that it is the relative\nmotion between the magnet and the coil that is\nresponsible for generation (induction) of electric\ncurrent in the coil Experiment 6"}, {"Chapter": "1", "sentence_range": "4814-4817", "Text": "Instead,\nwhen the bar magnet is held fixed and the coil C1 is\nmoved towards or away from the magnet, the same\neffects are observed It shows that it is the relative\nmotion between the magnet and the coil that is\nresponsible for generation (induction) of electric\ncurrent in the coil Experiment 6 2\nIn Fig"}, {"Chapter": "1", "sentence_range": "4815-4818", "Text": "It shows that it is the relative\nmotion between the magnet and the coil that is\nresponsible for generation (induction) of electric\ncurrent in the coil Experiment 6 2\nIn Fig 6"}, {"Chapter": "1", "sentence_range": "4816-4819", "Text": "Experiment 6 2\nIn Fig 6 2 the bar magnet is replaced by a second coil\nC2 connected to a battery"}, {"Chapter": "1", "sentence_range": "4817-4820", "Text": "2\nIn Fig 6 2 the bar magnet is replaced by a second coil\nC2 connected to a battery The steady current in the\ncoil C2 produces a steady magnetic field"}, {"Chapter": "1", "sentence_range": "4818-4821", "Text": "6 2 the bar magnet is replaced by a second coil\nC2 connected to a battery The steady current in the\ncoil C2 produces a steady magnetic field As coil C2 is\n*\nWherever the term \u2018coil\u2019 or \u2018loop\u2019 is used, it is assumed that they are made up of\nconducting material and are prepared using wires which are coated with insulating\nmaterial"}, {"Chapter": "1", "sentence_range": "4819-4822", "Text": "2 the bar magnet is replaced by a second coil\nC2 connected to a battery The steady current in the\ncoil C2 produces a steady magnetic field As coil C2 is\n*\nWherever the term \u2018coil\u2019 or \u2018loop\u2019 is used, it is assumed that they are made up of\nconducting material and are prepared using wires which are coated with insulating\nmaterial FIGURE 6"}, {"Chapter": "1", "sentence_range": "4820-4823", "Text": "The steady current in the\ncoil C2 produces a steady magnetic field As coil C2 is\n*\nWherever the term \u2018coil\u2019 or \u2018loop\u2019 is used, it is assumed that they are made up of\nconducting material and are prepared using wires which are coated with insulating\nmaterial FIGURE 6 1 When the bar magnet is\npushed towards the coil, the pointer in\nthe galvanometer G deflects"}, {"Chapter": "1", "sentence_range": "4821-4824", "Text": "As coil C2 is\n*\nWherever the term \u2018coil\u2019 or \u2018loop\u2019 is used, it is assumed that they are made up of\nconducting material and are prepared using wires which are coated with insulating\nmaterial FIGURE 6 1 When the bar magnet is\npushed towards the coil, the pointer in\nthe galvanometer G deflects Josheph Henry [1797 \u2013\n1878] American experimental\nphysicist, \nprofessor \nat\nPrinceton University and first\ndirector of the Smithsonian\nInstitution"}, {"Chapter": "1", "sentence_range": "4822-4825", "Text": "FIGURE 6 1 When the bar magnet is\npushed towards the coil, the pointer in\nthe galvanometer G deflects Josheph Henry [1797 \u2013\n1878] American experimental\nphysicist, \nprofessor \nat\nPrinceton University and first\ndirector of the Smithsonian\nInstitution He made important\nimprovements in electro-\nmagnets by winding coils of\ninsulated wire around iron\npole pieces and  invented an\nelectromagnetic motor and a\nnew, efficient telegraph"}, {"Chapter": "1", "sentence_range": "4823-4826", "Text": "1 When the bar magnet is\npushed towards the coil, the pointer in\nthe galvanometer G deflects Josheph Henry [1797 \u2013\n1878] American experimental\nphysicist, \nprofessor \nat\nPrinceton University and first\ndirector of the Smithsonian\nInstitution He made important\nimprovements in electro-\nmagnets by winding coils of\ninsulated wire around iron\npole pieces and  invented an\nelectromagnetic motor and a\nnew, efficient telegraph He\ndiscoverd self-induction and\ninvestigated how currents in\none circuit induce currents in\nanother"}, {"Chapter": "1", "sentence_range": "4824-4827", "Text": "Josheph Henry [1797 \u2013\n1878] American experimental\nphysicist, \nprofessor \nat\nPrinceton University and first\ndirector of the Smithsonian\nInstitution He made important\nimprovements in electro-\nmagnets by winding coils of\ninsulated wire around iron\npole pieces and  invented an\nelectromagnetic motor and a\nnew, efficient telegraph He\ndiscoverd self-induction and\ninvestigated how currents in\none circuit induce currents in\nanother JOSEPH HENRY (1797 \u2013 1878)\nRationalised 2023-24\nPhysics\n156\nmoved towards the coil C1, the galvanometer shows a\ndeflection"}, {"Chapter": "1", "sentence_range": "4825-4828", "Text": "He made important\nimprovements in electro-\nmagnets by winding coils of\ninsulated wire around iron\npole pieces and  invented an\nelectromagnetic motor and a\nnew, efficient telegraph He\ndiscoverd self-induction and\ninvestigated how currents in\none circuit induce currents in\nanother JOSEPH HENRY (1797 \u2013 1878)\nRationalised 2023-24\nPhysics\n156\nmoved towards the coil C1, the galvanometer shows a\ndeflection This indicates that electric current is induced in\ncoil C1"}, {"Chapter": "1", "sentence_range": "4826-4829", "Text": "He\ndiscoverd self-induction and\ninvestigated how currents in\none circuit induce currents in\nanother JOSEPH HENRY (1797 \u2013 1878)\nRationalised 2023-24\nPhysics\n156\nmoved towards the coil C1, the galvanometer shows a\ndeflection This indicates that electric current is induced in\ncoil C1 When C2 is moved away, the galvanometer shows a\ndeflection again, but this time in the opposite direction"}, {"Chapter": "1", "sentence_range": "4827-4830", "Text": "JOSEPH HENRY (1797 \u2013 1878)\nRationalised 2023-24\nPhysics\n156\nmoved towards the coil C1, the galvanometer shows a\ndeflection This indicates that electric current is induced in\ncoil C1 When C2 is moved away, the galvanometer shows a\ndeflection again, but this time in the opposite direction The\ndeflection lasts as long as coil C2 is in motion"}, {"Chapter": "1", "sentence_range": "4828-4831", "Text": "This indicates that electric current is induced in\ncoil C1 When C2 is moved away, the galvanometer shows a\ndeflection again, but this time in the opposite direction The\ndeflection lasts as long as coil C2 is in motion When the coil\nC2 is held fixed and C1 is moved, the same effects are observed"}, {"Chapter": "1", "sentence_range": "4829-4832", "Text": "When C2 is moved away, the galvanometer shows a\ndeflection again, but this time in the opposite direction The\ndeflection lasts as long as coil C2 is in motion When the coil\nC2 is held fixed and C1 is moved, the same effects are observed Again, it is the relative motion between the coils that induces\nthe electric current"}, {"Chapter": "1", "sentence_range": "4830-4833", "Text": "The\ndeflection lasts as long as coil C2 is in motion When the coil\nC2 is held fixed and C1 is moved, the same effects are observed Again, it is the relative motion between the coils that induces\nthe electric current Experiment 6"}, {"Chapter": "1", "sentence_range": "4831-4834", "Text": "When the coil\nC2 is held fixed and C1 is moved, the same effects are observed Again, it is the relative motion between the coils that induces\nthe electric current Experiment 6 3\nThe above two experiments involved relative motion between\na magnet and a coil and between two coils, respectively"}, {"Chapter": "1", "sentence_range": "4832-4835", "Text": "Again, it is the relative motion between the coils that induces\nthe electric current Experiment 6 3\nThe above two experiments involved relative motion between\na magnet and a coil and between two coils, respectively Through another experiment, Faraday showed that this\nrelative motion is not an absolute requirement"}, {"Chapter": "1", "sentence_range": "4833-4836", "Text": "Experiment 6 3\nThe above two experiments involved relative motion between\na magnet and a coil and between two coils, respectively Through another experiment, Faraday showed that this\nrelative motion is not an absolute requirement Figure 6"}, {"Chapter": "1", "sentence_range": "4834-4837", "Text": "3\nThe above two experiments involved relative motion between\na magnet and a coil and between two coils, respectively Through another experiment, Faraday showed that this\nrelative motion is not an absolute requirement Figure 6 3\nshows two coils C1 and C2 held stationary"}, {"Chapter": "1", "sentence_range": "4835-4838", "Text": "Through another experiment, Faraday showed that this\nrelative motion is not an absolute requirement Figure 6 3\nshows two coils C1 and C2 held stationary Coil C1 is connected\nto galvanometer G while the second coil C2 is connected to a\nbattery through a tapping key K"}, {"Chapter": "1", "sentence_range": "4836-4839", "Text": "Figure 6 3\nshows two coils C1 and C2 held stationary Coil C1 is connected\nto galvanometer G while the second coil C2 is connected to a\nbattery through a tapping key K FIGURE 6"}, {"Chapter": "1", "sentence_range": "4837-4840", "Text": "3\nshows two coils C1 and C2 held stationary Coil C1 is connected\nto galvanometer G while the second coil C2 is connected to a\nbattery through a tapping key K FIGURE 6 2  Current is\ninduced in coil C1 due to motion\nof the current carrying coil C2"}, {"Chapter": "1", "sentence_range": "4838-4841", "Text": "Coil C1 is connected\nto galvanometer G while the second coil C2 is connected to a\nbattery through a tapping key K FIGURE 6 2  Current is\ninduced in coil C1 due to motion\nof the current carrying coil C2 FIGURE 6"}, {"Chapter": "1", "sentence_range": "4839-4842", "Text": "FIGURE 6 2  Current is\ninduced in coil C1 due to motion\nof the current carrying coil C2 FIGURE 6 3 Experimental set-up for Experiment 6"}, {"Chapter": "1", "sentence_range": "4840-4843", "Text": "2  Current is\ninduced in coil C1 due to motion\nof the current carrying coil C2 FIGURE 6 3 Experimental set-up for Experiment 6 3"}, {"Chapter": "1", "sentence_range": "4841-4844", "Text": "FIGURE 6 3 Experimental set-up for Experiment 6 3 It is observed that the galvanometer shows a momentary deflection\nwhen the tapping key K is pressed"}, {"Chapter": "1", "sentence_range": "4842-4845", "Text": "3 Experimental set-up for Experiment 6 3 It is observed that the galvanometer shows a momentary deflection\nwhen the tapping key K is pressed The pointer in the galvanometer returns\nto zero immediately"}, {"Chapter": "1", "sentence_range": "4843-4846", "Text": "3 It is observed that the galvanometer shows a momentary deflection\nwhen the tapping key K is pressed The pointer in the galvanometer returns\nto zero immediately If the key is held pressed continuously, there is no\ndeflection in the galvanometer"}, {"Chapter": "1", "sentence_range": "4844-4847", "Text": "It is observed that the galvanometer shows a momentary deflection\nwhen the tapping key K is pressed The pointer in the galvanometer returns\nto zero immediately If the key is held pressed continuously, there is no\ndeflection in the galvanometer When the key is released, a momentory\ndeflection is observed again, but in the opposite direction"}, {"Chapter": "1", "sentence_range": "4845-4848", "Text": "The pointer in the galvanometer returns\nto zero immediately If the key is held pressed continuously, there is no\ndeflection in the galvanometer When the key is released, a momentory\ndeflection is observed again, but in the opposite direction It is also observed\nthat the deflection increases dramatically when an iron rod is inserted\ninto the coils along their axis"}, {"Chapter": "1", "sentence_range": "4846-4849", "Text": "If the key is held pressed continuously, there is no\ndeflection in the galvanometer When the key is released, a momentory\ndeflection is observed again, but in the opposite direction It is also observed\nthat the deflection increases dramatically when an iron rod is inserted\ninto the coils along their axis 6"}, {"Chapter": "1", "sentence_range": "4847-4850", "Text": "When the key is released, a momentory\ndeflection is observed again, but in the opposite direction It is also observed\nthat the deflection increases dramatically when an iron rod is inserted\ninto the coils along their axis 6 3  MAGNETIC FLUX\nFaraday\u2019s great insight lay in discovering a simple mathematical relation\nto explain the series of experiments he carried out on electromagnetic\ninduction"}, {"Chapter": "1", "sentence_range": "4848-4851", "Text": "It is also observed\nthat the deflection increases dramatically when an iron rod is inserted\ninto the coils along their axis 6 3  MAGNETIC FLUX\nFaraday\u2019s great insight lay in discovering a simple mathematical relation\nto explain the series of experiments he carried out on electromagnetic\ninduction However, before we state and appreciate his laws, we must get\nfamiliar with the notion of magnetic flux, F B"}, {"Chapter": "1", "sentence_range": "4849-4852", "Text": "6 3  MAGNETIC FLUX\nFaraday\u2019s great insight lay in discovering a simple mathematical relation\nto explain the series of experiments he carried out on electromagnetic\ninduction However, before we state and appreciate his laws, we must get\nfamiliar with the notion of magnetic flux, F B Magnetic flux is defined in\nthe same way as electric flux is defined in Chapter 1"}, {"Chapter": "1", "sentence_range": "4850-4853", "Text": "3  MAGNETIC FLUX\nFaraday\u2019s great insight lay in discovering a simple mathematical relation\nto explain the series of experiments he carried out on electromagnetic\ninduction However, before we state and appreciate his laws, we must get\nfamiliar with the notion of magnetic flux, F B Magnetic flux is defined in\nthe same way as electric flux is defined in Chapter 1 Magnetic flux through\nRationalised 2023-24\nElectromagnetic\nInduction\n157\na plane of area A placed in a uniform magnetic field B (Fig"}, {"Chapter": "1", "sentence_range": "4851-4854", "Text": "However, before we state and appreciate his laws, we must get\nfamiliar with the notion of magnetic flux, F B Magnetic flux is defined in\nthe same way as electric flux is defined in Chapter 1 Magnetic flux through\nRationalised 2023-24\nElectromagnetic\nInduction\n157\na plane of area A placed in a uniform magnetic field B (Fig 6"}, {"Chapter": "1", "sentence_range": "4852-4855", "Text": "Magnetic flux is defined in\nthe same way as electric flux is defined in Chapter 1 Magnetic flux through\nRationalised 2023-24\nElectromagnetic\nInduction\n157\na plane of area A placed in a uniform magnetic field B (Fig 6 4) can\nbe written as\nF B  = B"}, {"Chapter": "1", "sentence_range": "4853-4856", "Text": "Magnetic flux through\nRationalised 2023-24\nElectromagnetic\nInduction\n157\na plane of area A placed in a uniform magnetic field B (Fig 6 4) can\nbe written as\nF B  = B A = BA cos q\n(6"}, {"Chapter": "1", "sentence_range": "4854-4857", "Text": "6 4) can\nbe written as\nF B  = B A = BA cos q\n(6 1)\nwhere q   is angle between B and A"}, {"Chapter": "1", "sentence_range": "4855-4858", "Text": "4) can\nbe written as\nF B  = B A = BA cos q\n(6 1)\nwhere q   is angle between B and A The notion of the area as a vector\nhas been discussed earlier in Chapter 1"}, {"Chapter": "1", "sentence_range": "4856-4859", "Text": "A = BA cos q\n(6 1)\nwhere q   is angle between B and A The notion of the area as a vector\nhas been discussed earlier in Chapter 1 Equation (6"}, {"Chapter": "1", "sentence_range": "4857-4860", "Text": "1)\nwhere q   is angle between B and A The notion of the area as a vector\nhas been discussed earlier in Chapter 1 Equation (6 1) can be\nextended to curved surfaces and nonuniform fields"}, {"Chapter": "1", "sentence_range": "4858-4861", "Text": "The notion of the area as a vector\nhas been discussed earlier in Chapter 1 Equation (6 1) can be\nextended to curved surfaces and nonuniform fields If the magnetic field has different magnitudes and directions at\nvarious parts of a surface as shown in Fig"}, {"Chapter": "1", "sentence_range": "4859-4862", "Text": "Equation (6 1) can be\nextended to curved surfaces and nonuniform fields If the magnetic field has different magnitudes and directions at\nvarious parts of a surface as shown in Fig 6"}, {"Chapter": "1", "sentence_range": "4860-4863", "Text": "1) can be\nextended to curved surfaces and nonuniform fields If the magnetic field has different magnitudes and directions at\nvarious parts of a surface as shown in Fig 6 5, then the magnetic\nflux through the surface is given by\n1\n1\n2\n2\nd\nd\n\u03a6 =\n+\n+\nB\nA\nB\nA"}, {"Chapter": "1", "sentence_range": "4861-4864", "Text": "If the magnetic field has different magnitudes and directions at\nvarious parts of a surface as shown in Fig 6 5, then the magnetic\nflux through the surface is given by\n1\n1\n2\n2\nd\nd\n\u03a6 =\n+\n+\nB\nA\nB\nA B"}, {"Chapter": "1", "sentence_range": "4862-4865", "Text": "6 5, then the magnetic\nflux through the surface is given by\n1\n1\n2\n2\nd\nd\n\u03a6 =\n+\n+\nB\nA\nB\nA B =\nB\ni"}, {"Chapter": "1", "sentence_range": "4863-4866", "Text": "5, then the magnetic\nflux through the surface is given by\n1\n1\n2\n2\nd\nd\n\u03a6 =\n+\n+\nB\nA\nB\nA B =\nB\ni A\ni\nd\nall\u2211\n(6"}, {"Chapter": "1", "sentence_range": "4864-4867", "Text": "B =\nB\ni A\ni\nd\nall\u2211\n(6 2)\nwhere \u2018all\u2019 stands for summation over all the area elements dAi\ncomprising the surface and Bi is the magnetic field at the area element\ndAi"}, {"Chapter": "1", "sentence_range": "4865-4868", "Text": "=\nB\ni A\ni\nd\nall\u2211\n(6 2)\nwhere \u2018all\u2019 stands for summation over all the area elements dAi\ncomprising the surface and Bi is the magnetic field at the area element\ndAi The SI unit of magnetic flux is weber (Wb) or tesla meter\nsquared (T m2)"}, {"Chapter": "1", "sentence_range": "4866-4869", "Text": "A\ni\nd\nall\u2211\n(6 2)\nwhere \u2018all\u2019 stands for summation over all the area elements dAi\ncomprising the surface and Bi is the magnetic field at the area element\ndAi The SI unit of magnetic flux is weber (Wb) or tesla meter\nsquared (T m2) Magnetic flux is a scalar quantity"}, {"Chapter": "1", "sentence_range": "4867-4870", "Text": "2)\nwhere \u2018all\u2019 stands for summation over all the area elements dAi\ncomprising the surface and Bi is the magnetic field at the area element\ndAi The SI unit of magnetic flux is weber (Wb) or tesla meter\nsquared (T m2) Magnetic flux is a scalar quantity 6"}, {"Chapter": "1", "sentence_range": "4868-4871", "Text": "The SI unit of magnetic flux is weber (Wb) or tesla meter\nsquared (T m2) Magnetic flux is a scalar quantity 6 4  FARADAY\u2019S LAW OF INDUCTION\nFrom the experimental observations, Faraday arrived at a\nconclusion that an emf is induced in a coil when magnetic flux\nthrough the coil changes with time"}, {"Chapter": "1", "sentence_range": "4869-4872", "Text": "Magnetic flux is a scalar quantity 6 4  FARADAY\u2019S LAW OF INDUCTION\nFrom the experimental observations, Faraday arrived at a\nconclusion that an emf is induced in a coil when magnetic flux\nthrough the coil changes with time Experimental observations\ndiscussed in Section 6"}, {"Chapter": "1", "sentence_range": "4870-4873", "Text": "6 4  FARADAY\u2019S LAW OF INDUCTION\nFrom the experimental observations, Faraday arrived at a\nconclusion that an emf is induced in a coil when magnetic flux\nthrough the coil changes with time Experimental observations\ndiscussed in Section 6 2 can be explained using this concept"}, {"Chapter": "1", "sentence_range": "4871-4874", "Text": "4  FARADAY\u2019S LAW OF INDUCTION\nFrom the experimental observations, Faraday arrived at a\nconclusion that an emf is induced in a coil when magnetic flux\nthrough the coil changes with time Experimental observations\ndiscussed in Section 6 2 can be explained using this concept The motion of a magnet towards or away from coil C1 in\nExperiment 6"}, {"Chapter": "1", "sentence_range": "4872-4875", "Text": "Experimental observations\ndiscussed in Section 6 2 can be explained using this concept The motion of a magnet towards or away from coil C1 in\nExperiment 6 1 and moving a current-carrying coil C2 towards\nor away from coil C1 in Experiment 6"}, {"Chapter": "1", "sentence_range": "4873-4876", "Text": "2 can be explained using this concept The motion of a magnet towards or away from coil C1 in\nExperiment 6 1 and moving a current-carrying coil C2 towards\nor away from coil C1 in Experiment 6 2, change the magnetic\nflux associated with coil C1"}, {"Chapter": "1", "sentence_range": "4874-4877", "Text": "The motion of a magnet towards or away from coil C1 in\nExperiment 6 1 and moving a current-carrying coil C2 towards\nor away from coil C1 in Experiment 6 2, change the magnetic\nflux associated with coil C1 The change in magnetic flux induces\nemf in coil C1"}, {"Chapter": "1", "sentence_range": "4875-4878", "Text": "1 and moving a current-carrying coil C2 towards\nor away from coil C1 in Experiment 6 2, change the magnetic\nflux associated with coil C1 The change in magnetic flux induces\nemf in coil C1 It was this induced emf which caused electric\ncurrent to flow in coil C1 and through the galvanometer"}, {"Chapter": "1", "sentence_range": "4876-4879", "Text": "2, change the magnetic\nflux associated with coil C1 The change in magnetic flux induces\nemf in coil C1 It was this induced emf which caused electric\ncurrent to flow in coil C1 and through the galvanometer A\nplausible explanation for the observations of Experiment 6"}, {"Chapter": "1", "sentence_range": "4877-4880", "Text": "The change in magnetic flux induces\nemf in coil C1 It was this induced emf which caused electric\ncurrent to flow in coil C1 and through the galvanometer A\nplausible explanation for the observations of Experiment 6 3 is\nas follows: When the tapping key K is pressed, the current in\ncoil C2 (and the resulting magnetic field) rises from zero to a\nmaximum value in a short time"}, {"Chapter": "1", "sentence_range": "4878-4881", "Text": "It was this induced emf which caused electric\ncurrent to flow in coil C1 and through the galvanometer A\nplausible explanation for the observations of Experiment 6 3 is\nas follows: When the tapping key K is pressed, the current in\ncoil C2 (and the resulting magnetic field) rises from zero to a\nmaximum value in a short time Consequently, the magnetic\nflux through the neighbouring coil C1 also increases"}, {"Chapter": "1", "sentence_range": "4879-4882", "Text": "A\nplausible explanation for the observations of Experiment 6 3 is\nas follows: When the tapping key K is pressed, the current in\ncoil C2 (and the resulting magnetic field) rises from zero to a\nmaximum value in a short time Consequently, the magnetic\nflux through the neighbouring coil C1 also increases It is the change in\nmagnetic flux through coil C1 that produces an induced emf in coil C1"}, {"Chapter": "1", "sentence_range": "4880-4883", "Text": "3 is\nas follows: When the tapping key K is pressed, the current in\ncoil C2 (and the resulting magnetic field) rises from zero to a\nmaximum value in a short time Consequently, the magnetic\nflux through the neighbouring coil C1 also increases It is the change in\nmagnetic flux through coil C1 that produces an induced emf in coil C1 When the key is held pressed, current in coil C2 is constant"}, {"Chapter": "1", "sentence_range": "4881-4884", "Text": "Consequently, the magnetic\nflux through the neighbouring coil C1 also increases It is the change in\nmagnetic flux through coil C1 that produces an induced emf in coil C1 When the key is held pressed, current in coil C2 is constant Therefore,\nthere is no change in the magnetic flux through coil C1 and the current in\ncoil C1 drops to zero"}, {"Chapter": "1", "sentence_range": "4882-4885", "Text": "It is the change in\nmagnetic flux through coil C1 that produces an induced emf in coil C1 When the key is held pressed, current in coil C2 is constant Therefore,\nthere is no change in the magnetic flux through coil C1 and the current in\ncoil C1 drops to zero When the key is released, the current in C2 and the\nresulting magnetic field decreases from the maximum value to zero in a\nshort time"}, {"Chapter": "1", "sentence_range": "4883-4886", "Text": "When the key is held pressed, current in coil C2 is constant Therefore,\nthere is no change in the magnetic flux through coil C1 and the current in\ncoil C1 drops to zero When the key is released, the current in C2 and the\nresulting magnetic field decreases from the maximum value to zero in a\nshort time This results in a decrease in magnetic flux through coil C1\nand hence again induces an electric current in coil C1*"}, {"Chapter": "1", "sentence_range": "4884-4887", "Text": "Therefore,\nthere is no change in the magnetic flux through coil C1 and the current in\ncoil C1 drops to zero When the key is released, the current in C2 and the\nresulting magnetic field decreases from the maximum value to zero in a\nshort time This results in a decrease in magnetic flux through coil C1\nand hence again induces an electric current in coil C1* The common\npoint in all these observations is that the time rate of change of magnetic\nflux through a circuit induces emf in it"}, {"Chapter": "1", "sentence_range": "4885-4888", "Text": "When the key is released, the current in C2 and the\nresulting magnetic field decreases from the maximum value to zero in a\nshort time This results in a decrease in magnetic flux through coil C1\nand hence again induces an electric current in coil C1* The common\npoint in all these observations is that the time rate of change of magnetic\nflux through a circuit induces emf in it Faraday stated experimental\nobservations in the form of a law called Faraday\u2019s law of electromagnetic\ninduction"}, {"Chapter": "1", "sentence_range": "4886-4889", "Text": "This results in a decrease in magnetic flux through coil C1\nand hence again induces an electric current in coil C1* The common\npoint in all these observations is that the time rate of change of magnetic\nflux through a circuit induces emf in it Faraday stated experimental\nobservations in the form of a law called Faraday\u2019s law of electromagnetic\ninduction The law is stated below"}, {"Chapter": "1", "sentence_range": "4887-4890", "Text": "The common\npoint in all these observations is that the time rate of change of magnetic\nflux through a circuit induces emf in it Faraday stated experimental\nobservations in the form of a law called Faraday\u2019s law of electromagnetic\ninduction The law is stated below FIGURE 6"}, {"Chapter": "1", "sentence_range": "4888-4891", "Text": "Faraday stated experimental\nobservations in the form of a law called Faraday\u2019s law of electromagnetic\ninduction The law is stated below FIGURE 6 4 A plane of\nsurface area A placed in a\nuniform magnetic field B"}, {"Chapter": "1", "sentence_range": "4889-4892", "Text": "The law is stated below FIGURE 6 4 A plane of\nsurface area A placed in a\nuniform magnetic field B FIGURE 6"}, {"Chapter": "1", "sentence_range": "4890-4893", "Text": "FIGURE 6 4 A plane of\nsurface area A placed in a\nuniform magnetic field B FIGURE 6 5 Magnetic field Bi\nat the ith area element"}, {"Chapter": "1", "sentence_range": "4891-4894", "Text": "4 A plane of\nsurface area A placed in a\nuniform magnetic field B FIGURE 6 5 Magnetic field Bi\nat the ith area element dAi\nrepresents area vector of the\nith area element"}, {"Chapter": "1", "sentence_range": "4892-4895", "Text": "FIGURE 6 5 Magnetic field Bi\nat the ith area element dAi\nrepresents area vector of the\nith area element *\nNote that sensitive electrical instruments in the vicinity of an electromagnet\ncan be damaged due to the induced emfs (and the resulting currents)  when the\nelectromagnet is turned on or off"}, {"Chapter": "1", "sentence_range": "4893-4896", "Text": "5 Magnetic field Bi\nat the ith area element dAi\nrepresents area vector of the\nith area element *\nNote that sensitive electrical instruments in the vicinity of an electromagnet\ncan be damaged due to the induced emfs (and the resulting currents)  when the\nelectromagnet is turned on or off Rationalised 2023-24\nPhysics\n158\n EXAMPLE 6"}, {"Chapter": "1", "sentence_range": "4894-4897", "Text": "dAi\nrepresents area vector of the\nith area element *\nNote that sensitive electrical instruments in the vicinity of an electromagnet\ncan be damaged due to the induced emfs (and the resulting currents)  when the\nelectromagnet is turned on or off Rationalised 2023-24\nPhysics\n158\n EXAMPLE 6 1\nThe magnitude of the induced emf in a circuit is equal\nto the time rate of change of magnetic flux through the\ncircuit"}, {"Chapter": "1", "sentence_range": "4895-4898", "Text": "*\nNote that sensitive electrical instruments in the vicinity of an electromagnet\ncan be damaged due to the induced emfs (and the resulting currents)  when the\nelectromagnet is turned on or off Rationalised 2023-24\nPhysics\n158\n EXAMPLE 6 1\nThe magnitude of the induced emf in a circuit is equal\nto the time rate of change of magnetic flux through the\ncircuit Mathematically, the induced emf is given by\nd\n\u2013 d\n\u03a6tB\n\u03b5 =\n(6"}, {"Chapter": "1", "sentence_range": "4896-4899", "Text": "Rationalised 2023-24\nPhysics\n158\n EXAMPLE 6 1\nThe magnitude of the induced emf in a circuit is equal\nto the time rate of change of magnetic flux through the\ncircuit Mathematically, the induced emf is given by\nd\n\u2013 d\n\u03a6tB\n\u03b5 =\n(6 3)\nThe negative sign indicates the direction of e  and hence\nthe direction of current in a closed loop"}, {"Chapter": "1", "sentence_range": "4897-4900", "Text": "1\nThe magnitude of the induced emf in a circuit is equal\nto the time rate of change of magnetic flux through the\ncircuit Mathematically, the induced emf is given by\nd\n\u2013 d\n\u03a6tB\n\u03b5 =\n(6 3)\nThe negative sign indicates the direction of e  and hence\nthe direction of current in a closed loop This will be\ndiscussed in detail in the next section"}, {"Chapter": "1", "sentence_range": "4898-4901", "Text": "Mathematically, the induced emf is given by\nd\n\u2013 d\n\u03a6tB\n\u03b5 =\n(6 3)\nThe negative sign indicates the direction of e  and hence\nthe direction of current in a closed loop This will be\ndiscussed in detail in the next section In the case of a closely wound coil of N turns, change\nof flux associated with each turn, is the same"}, {"Chapter": "1", "sentence_range": "4899-4902", "Text": "3)\nThe negative sign indicates the direction of e  and hence\nthe direction of current in a closed loop This will be\ndiscussed in detail in the next section In the case of a closely wound coil of N turns, change\nof flux associated with each turn, is the same Therefore,\nthe expression for the total induced emf is given by\nd\n\u2013\nd\nB\nN\n\u03a6t\n\u03b5 =\n(6"}, {"Chapter": "1", "sentence_range": "4900-4903", "Text": "This will be\ndiscussed in detail in the next section In the case of a closely wound coil of N turns, change\nof flux associated with each turn, is the same Therefore,\nthe expression for the total induced emf is given by\nd\n\u2013\nd\nB\nN\n\u03a6t\n\u03b5 =\n(6 4)\nThe induced emf can be increased by increasing the\nnumber of turns N of a closed coil"}, {"Chapter": "1", "sentence_range": "4901-4904", "Text": "In the case of a closely wound coil of N turns, change\nof flux associated with each turn, is the same Therefore,\nthe expression for the total induced emf is given by\nd\n\u2013\nd\nB\nN\n\u03a6t\n\u03b5 =\n(6 4)\nThe induced emf can be increased by increasing the\nnumber of turns N of a closed coil From Eqs"}, {"Chapter": "1", "sentence_range": "4902-4905", "Text": "Therefore,\nthe expression for the total induced emf is given by\nd\n\u2013\nd\nB\nN\n\u03a6t\n\u03b5 =\n(6 4)\nThe induced emf can be increased by increasing the\nnumber of turns N of a closed coil From Eqs (6"}, {"Chapter": "1", "sentence_range": "4903-4906", "Text": "4)\nThe induced emf can be increased by increasing the\nnumber of turns N of a closed coil From Eqs (6 1) and (6"}, {"Chapter": "1", "sentence_range": "4904-4907", "Text": "From Eqs (6 1) and (6 2), we see that the flux can be\nvaried by changing any one or more of the terms B, A and\nq"}, {"Chapter": "1", "sentence_range": "4905-4908", "Text": "(6 1) and (6 2), we see that the flux can be\nvaried by changing any one or more of the terms B, A and\nq In Experiments 6"}, {"Chapter": "1", "sentence_range": "4906-4909", "Text": "1) and (6 2), we see that the flux can be\nvaried by changing any one or more of the terms B, A and\nq In Experiments 6 1 and 6"}, {"Chapter": "1", "sentence_range": "4907-4910", "Text": "2), we see that the flux can be\nvaried by changing any one or more of the terms B, A and\nq In Experiments 6 1 and 6 2 in Section 6"}, {"Chapter": "1", "sentence_range": "4908-4911", "Text": "In Experiments 6 1 and 6 2 in Section 6 2, the flux is\nchanged by varying B"}, {"Chapter": "1", "sentence_range": "4909-4912", "Text": "1 and 6 2 in Section 6 2, the flux is\nchanged by varying B The flux can also be altered by\nchanging the shape of a coil (that is, by shrinking it or\nstretching it) in a magnetic field, or rotating a coil in a\nmagnetic field such that the angle q  between B and A\nchanges"}, {"Chapter": "1", "sentence_range": "4910-4913", "Text": "2 in Section 6 2, the flux is\nchanged by varying B The flux can also be altered by\nchanging the shape of a coil (that is, by shrinking it or\nstretching it) in a magnetic field, or rotating a coil in a\nmagnetic field such that the angle q  between B and A\nchanges In these cases too, an emf is induced in the\nrespective coils"}, {"Chapter": "1", "sentence_range": "4911-4914", "Text": "2, the flux is\nchanged by varying B The flux can also be altered by\nchanging the shape of a coil (that is, by shrinking it or\nstretching it) in a magnetic field, or rotating a coil in a\nmagnetic field such that the angle q  between B and A\nchanges In these cases too, an emf is induced in the\nrespective coils Example 6"}, {"Chapter": "1", "sentence_range": "4912-4915", "Text": "The flux can also be altered by\nchanging the shape of a coil (that is, by shrinking it or\nstretching it) in a magnetic field, or rotating a coil in a\nmagnetic field such that the angle q  between B and A\nchanges In these cases too, an emf is induced in the\nrespective coils Example 6 1  Consider Experiment 6"}, {"Chapter": "1", "sentence_range": "4913-4916", "Text": "In these cases too, an emf is induced in the\nrespective coils Example 6 1  Consider Experiment 6 2"}, {"Chapter": "1", "sentence_range": "4914-4917", "Text": "Example 6 1  Consider Experiment 6 2 (a) What would you do to obtain\na large deflection of the galvanometer"}, {"Chapter": "1", "sentence_range": "4915-4918", "Text": "1  Consider Experiment 6 2 (a) What would you do to obtain\na large deflection of the galvanometer (b) How would you demonstrate\nthe presence of an induced current in the absence of a galvanometer"}, {"Chapter": "1", "sentence_range": "4916-4919", "Text": "2 (a) What would you do to obtain\na large deflection of the galvanometer (b) How would you demonstrate\nthe presence of an induced current in the absence of a galvanometer Solution\n(a) To obtain a large deflection, one or more of the following steps can\nbe taken:  (i) Use a rod made of soft iron inside the coil C2, (ii) Connect\nthe coil to a powerful battery, and (iii) Move the arrangement rapidly\ntowards the test coil C1"}, {"Chapter": "1", "sentence_range": "4917-4920", "Text": "(a) What would you do to obtain\na large deflection of the galvanometer (b) How would you demonstrate\nthe presence of an induced current in the absence of a galvanometer Solution\n(a) To obtain a large deflection, one or more of the following steps can\nbe taken:  (i) Use a rod made of soft iron inside the coil C2, (ii) Connect\nthe coil to a powerful battery, and (iii) Move the arrangement rapidly\ntowards the test coil C1 (b) Replace the galvanometer by a small bulb, the kind one finds in a\nsmall torch light"}, {"Chapter": "1", "sentence_range": "4918-4921", "Text": "(b) How would you demonstrate\nthe presence of an induced current in the absence of a galvanometer Solution\n(a) To obtain a large deflection, one or more of the following steps can\nbe taken:  (i) Use a rod made of soft iron inside the coil C2, (ii) Connect\nthe coil to a powerful battery, and (iii) Move the arrangement rapidly\ntowards the test coil C1 (b) Replace the galvanometer by a small bulb, the kind one finds in a\nsmall torch light The relative motion between the two coils will cause\nthe bulb to glow and thus demonstrate the presence of an induced\ncurrent"}, {"Chapter": "1", "sentence_range": "4919-4922", "Text": "Solution\n(a) To obtain a large deflection, one or more of the following steps can\nbe taken:  (i) Use a rod made of soft iron inside the coil C2, (ii) Connect\nthe coil to a powerful battery, and (iii) Move the arrangement rapidly\ntowards the test coil C1 (b) Replace the galvanometer by a small bulb, the kind one finds in a\nsmall torch light The relative motion between the two coils will cause\nthe bulb to glow and thus demonstrate the presence of an induced\ncurrent In experimental physics one must learn to innovate"}, {"Chapter": "1", "sentence_range": "4920-4923", "Text": "(b) Replace the galvanometer by a small bulb, the kind one finds in a\nsmall torch light The relative motion between the two coils will cause\nthe bulb to glow and thus demonstrate the presence of an induced\ncurrent In experimental physics one must learn to innovate Michael Faraday\nwho is ranked as one of the best experimentalists ever, was legendary\nfor his innovative skills"}, {"Chapter": "1", "sentence_range": "4921-4924", "Text": "The relative motion between the two coils will cause\nthe bulb to glow and thus demonstrate the presence of an induced\ncurrent In experimental physics one must learn to innovate Michael Faraday\nwho is ranked as one of the best experimentalists ever, was legendary\nfor his innovative skills Example 6"}, {"Chapter": "1", "sentence_range": "4922-4925", "Text": "In experimental physics one must learn to innovate Michael Faraday\nwho is ranked as one of the best experimentalists ever, was legendary\nfor his innovative skills Example 6 2 A square loop of side 10 cm and resistance 0"}, {"Chapter": "1", "sentence_range": "4923-4926", "Text": "Michael Faraday\nwho is ranked as one of the best experimentalists ever, was legendary\nfor his innovative skills Example 6 2 A square loop of side 10 cm and resistance 0 5 W is\nplaced vertically in the east-west  plane"}, {"Chapter": "1", "sentence_range": "4924-4927", "Text": "Example 6 2 A square loop of side 10 cm and resistance 0 5 W is\nplaced vertically in the east-west  plane A uniform magnetic field of\n0"}, {"Chapter": "1", "sentence_range": "4925-4928", "Text": "2 A square loop of side 10 cm and resistance 0 5 W is\nplaced vertically in the east-west  plane A uniform magnetic field of\n0 10 T is set up across the plane in the north-east direction"}, {"Chapter": "1", "sentence_range": "4926-4929", "Text": "5 W is\nplaced vertically in the east-west  plane A uniform magnetic field of\n0 10 T is set up across the plane in the north-east direction The\nmagnetic field is decreased to zero in 0"}, {"Chapter": "1", "sentence_range": "4927-4930", "Text": "A uniform magnetic field of\n0 10 T is set up across the plane in the north-east direction The\nmagnetic field is decreased to zero in 0 70 s at a steady rate"}, {"Chapter": "1", "sentence_range": "4928-4931", "Text": "10 T is set up across the plane in the north-east direction The\nmagnetic field is decreased to zero in 0 70 s at a steady rate Determine\nthe magnitudes of induced emf and current during this time-interval"}, {"Chapter": "1", "sentence_range": "4929-4932", "Text": "The\nmagnetic field is decreased to zero in 0 70 s at a steady rate Determine\nthe magnitudes of induced emf and current during this time-interval Michael Faraday  [1791\u2013\n1867] \nFaraday \nmade\nnumerous contributions to\nscience, viz"}, {"Chapter": "1", "sentence_range": "4930-4933", "Text": "70 s at a steady rate Determine\nthe magnitudes of induced emf and current during this time-interval Michael Faraday  [1791\u2013\n1867] \nFaraday \nmade\nnumerous contributions to\nscience, viz , the discovery\nof \nelectromagnetic\ninduction, the laws of\nelectrolysis, benzene, and\nthe fact that the plane of\npolarisation is rotated in an\nelectric field"}, {"Chapter": "1", "sentence_range": "4931-4934", "Text": "Determine\nthe magnitudes of induced emf and current during this time-interval Michael Faraday  [1791\u2013\n1867] \nFaraday \nmade\nnumerous contributions to\nscience, viz , the discovery\nof \nelectromagnetic\ninduction, the laws of\nelectrolysis, benzene, and\nthe fact that the plane of\npolarisation is rotated in an\nelectric field He is also\ncredited with the invention\nof the electric motor, the\nelectric generator and the\ntransformer"}, {"Chapter": "1", "sentence_range": "4932-4935", "Text": "Michael Faraday  [1791\u2013\n1867] \nFaraday \nmade\nnumerous contributions to\nscience, viz , the discovery\nof \nelectromagnetic\ninduction, the laws of\nelectrolysis, benzene, and\nthe fact that the plane of\npolarisation is rotated in an\nelectric field He is also\ncredited with the invention\nof the electric motor, the\nelectric generator and the\ntransformer He is widely\nregarded as the greatest\nexperimental scientist of\nthe nineteenth century"}, {"Chapter": "1", "sentence_range": "4933-4936", "Text": ", the discovery\nof \nelectromagnetic\ninduction, the laws of\nelectrolysis, benzene, and\nthe fact that the plane of\npolarisation is rotated in an\nelectric field He is also\ncredited with the invention\nof the electric motor, the\nelectric generator and the\ntransformer He is widely\nregarded as the greatest\nexperimental scientist of\nthe nineteenth century MICHAEL FARADAY (1791\u20131867)\n EXAMPLE 6"}, {"Chapter": "1", "sentence_range": "4934-4937", "Text": "He is also\ncredited with the invention\nof the electric motor, the\nelectric generator and the\ntransformer He is widely\nregarded as the greatest\nexperimental scientist of\nthe nineteenth century MICHAEL FARADAY (1791\u20131867)\n EXAMPLE 6 2\nRationalised 2023-24\nElectromagnetic\nInduction\n159\n EXAMPLE 6"}, {"Chapter": "1", "sentence_range": "4935-4938", "Text": "He is widely\nregarded as the greatest\nexperimental scientist of\nthe nineteenth century MICHAEL FARADAY (1791\u20131867)\n EXAMPLE 6 2\nRationalised 2023-24\nElectromagnetic\nInduction\n159\n EXAMPLE 6 2\nSolution The angle q made by the area vector of the coil with the\nmagnetic field is 45\u00b0"}, {"Chapter": "1", "sentence_range": "4936-4939", "Text": "MICHAEL FARADAY (1791\u20131867)\n EXAMPLE 6 2\nRationalised 2023-24\nElectromagnetic\nInduction\n159\n EXAMPLE 6 2\nSolution The angle q made by the area vector of the coil with the\nmagnetic field is 45\u00b0 From Eq"}, {"Chapter": "1", "sentence_range": "4937-4940", "Text": "2\nRationalised 2023-24\nElectromagnetic\nInduction\n159\n EXAMPLE 6 2\nSolution The angle q made by the area vector of the coil with the\nmagnetic field is 45\u00b0 From Eq (6"}, {"Chapter": "1", "sentence_range": "4938-4941", "Text": "2\nSolution The angle q made by the area vector of the coil with the\nmagnetic field is 45\u00b0 From Eq (6 1), the initial magnetic flux is\nF = BA cos q\n\u20132\n0"}, {"Chapter": "1", "sentence_range": "4939-4942", "Text": "From Eq (6 1), the initial magnetic flux is\nF = BA cos q\n\u20132\n0 1\n10\nWb\n2\n\u00d7\n=\nFinal flux, Fmin = 0\nThe change in flux is brought about in 0"}, {"Chapter": "1", "sentence_range": "4940-4943", "Text": "(6 1), the initial magnetic flux is\nF = BA cos q\n\u20132\n0 1\n10\nWb\n2\n\u00d7\n=\nFinal flux, Fmin = 0\nThe change in flux is brought about in 0 70 s"}, {"Chapter": "1", "sentence_range": "4941-4944", "Text": "1), the initial magnetic flux is\nF = BA cos q\n\u20132\n0 1\n10\nWb\n2\n\u00d7\n=\nFinal flux, Fmin = 0\nThe change in flux is brought about in 0 70 s From Eq"}, {"Chapter": "1", "sentence_range": "4942-4945", "Text": "1\n10\nWb\n2\n\u00d7\n=\nFinal flux, Fmin = 0\nThe change in flux is brought about in 0 70 s From Eq (6"}, {"Chapter": "1", "sentence_range": "4943-4946", "Text": "70 s From Eq (6 3), the\nmagnitude of the induced emf is given by\n(\n\u2013 0)\ntB\nt\n\u03a6\n\u03a6\n\u03b5\n=\u2206\n=\n\u2206\n\u2206\n \n10\u20133\n=\n1"}, {"Chapter": "1", "sentence_range": "4944-4947", "Text": "From Eq (6 3), the\nmagnitude of the induced emf is given by\n(\n\u2013 0)\ntB\nt\n\u03a6\n\u03a6\n\u03b5\n=\u2206\n=\n\u2206\n\u2206\n \n10\u20133\n=\n1 0 mV\n2\n0"}, {"Chapter": "1", "sentence_range": "4945-4948", "Text": "(6 3), the\nmagnitude of the induced emf is given by\n(\n\u2013 0)\ntB\nt\n\u03a6\n\u03a6\n\u03b5\n=\u2206\n=\n\u2206\n\u2206\n \n10\u20133\n=\n1 0 mV\n2\n0 7\n=\n\u00d7\nAnd the magnitude of the current is\n10\u20133\nV\n2 mA\n0"}, {"Chapter": "1", "sentence_range": "4946-4949", "Text": "3), the\nmagnitude of the induced emf is given by\n(\n\u2013 0)\ntB\nt\n\u03a6\n\u03a6\n\u03b5\n=\u2206\n=\n\u2206\n\u2206\n \n10\u20133\n=\n1 0 mV\n2\n0 7\n=\n\u00d7\nAnd the magnitude of the current is\n10\u20133\nV\n2 mA\n0 5\nI\n=\u03b5R\n=\n=\n\u2126\nNote that the earth\u2019s magnetic field also produces a flux through the\nloop"}, {"Chapter": "1", "sentence_range": "4947-4950", "Text": "0 mV\n2\n0 7\n=\n\u00d7\nAnd the magnitude of the current is\n10\u20133\nV\n2 mA\n0 5\nI\n=\u03b5R\n=\n=\n\u2126\nNote that the earth\u2019s magnetic field also produces a flux through the\nloop But it is a steady field (which does not change within the time\nspan of the experiment) and hence does not induce any emf"}, {"Chapter": "1", "sentence_range": "4948-4951", "Text": "7\n=\n\u00d7\nAnd the magnitude of the current is\n10\u20133\nV\n2 mA\n0 5\nI\n=\u03b5R\n=\n=\n\u2126\nNote that the earth\u2019s magnetic field also produces a flux through the\nloop But it is a steady field (which does not change within the time\nspan of the experiment) and hence does not induce any emf Example 6"}, {"Chapter": "1", "sentence_range": "4949-4952", "Text": "5\nI\n=\u03b5R\n=\n=\n\u2126\nNote that the earth\u2019s magnetic field also produces a flux through the\nloop But it is a steady field (which does not change within the time\nspan of the experiment) and hence does not induce any emf Example 6 3\nA circular coil of radius 10 cm, 500 turns and resistance 2 W is placed\nwith its plane perpendicular to the horizontal component of the earth\u2019s\nmagnetic field"}, {"Chapter": "1", "sentence_range": "4950-4953", "Text": "But it is a steady field (which does not change within the time\nspan of the experiment) and hence does not induce any emf Example 6 3\nA circular coil of radius 10 cm, 500 turns and resistance 2 W is placed\nwith its plane perpendicular to the horizontal component of the earth\u2019s\nmagnetic field It is rotated about its vertical diameter through 180\u00b0\nin 0"}, {"Chapter": "1", "sentence_range": "4951-4954", "Text": "Example 6 3\nA circular coil of radius 10 cm, 500 turns and resistance 2 W is placed\nwith its plane perpendicular to the horizontal component of the earth\u2019s\nmagnetic field It is rotated about its vertical diameter through 180\u00b0\nin 0 25 s"}, {"Chapter": "1", "sentence_range": "4952-4955", "Text": "3\nA circular coil of radius 10 cm, 500 turns and resistance 2 W is placed\nwith its plane perpendicular to the horizontal component of the earth\u2019s\nmagnetic field It is rotated about its vertical diameter through 180\u00b0\nin 0 25 s Estimate the magnitudes of the emf and current induced in\nthe coil"}, {"Chapter": "1", "sentence_range": "4953-4956", "Text": "It is rotated about its vertical diameter through 180\u00b0\nin 0 25 s Estimate the magnitudes of the emf and current induced in\nthe coil Horizontal component of the earth\u2019s magnetic field at the\nplace is 3"}, {"Chapter": "1", "sentence_range": "4954-4957", "Text": "25 s Estimate the magnitudes of the emf and current induced in\nthe coil Horizontal component of the earth\u2019s magnetic field at the\nplace is 3 0 \u00d7 10\u20135 T"}, {"Chapter": "1", "sentence_range": "4955-4958", "Text": "Estimate the magnitudes of the emf and current induced in\nthe coil Horizontal component of the earth\u2019s magnetic field at the\nplace is 3 0 \u00d7 10\u20135 T Solution\nInitial flux through the coil,\nFB (initial) = BA cos q\n= 3"}, {"Chapter": "1", "sentence_range": "4956-4959", "Text": "Horizontal component of the earth\u2019s magnetic field at the\nplace is 3 0 \u00d7 10\u20135 T Solution\nInitial flux through the coil,\nFB (initial) = BA cos q\n= 3 0 \u00d7 10\u20135 \u00d7 (p \u00d710\u20132) \u00d7 cos 0\u00b0\n= 3p \u00d7 10\u20137 Wb\nFinal flux after the rotation,\nFB (final)    = 3"}, {"Chapter": "1", "sentence_range": "4957-4960", "Text": "0 \u00d7 10\u20135 T Solution\nInitial flux through the coil,\nFB (initial) = BA cos q\n= 3 0 \u00d7 10\u20135 \u00d7 (p \u00d710\u20132) \u00d7 cos 0\u00b0\n= 3p \u00d7 10\u20137 Wb\nFinal flux after the rotation,\nFB (final)    = 3 0 \u00d7 10\u20135 \u00d7 (p \u00d710\u20132) \u00d7 cos 180\u00b0\n= \u20133p \u00d7 10\u20137 Wb\nTherefore, estimated value of the induced emf is,\nN\n\u03a6t\n\u03b5\n\u2206\n=\n\u2206\n   = 500 \u00d7 (6p \u00d7 10\u20137)/0"}, {"Chapter": "1", "sentence_range": "4958-4961", "Text": "Solution\nInitial flux through the coil,\nFB (initial) = BA cos q\n= 3 0 \u00d7 10\u20135 \u00d7 (p \u00d710\u20132) \u00d7 cos 0\u00b0\n= 3p \u00d7 10\u20137 Wb\nFinal flux after the rotation,\nFB (final)    = 3 0 \u00d7 10\u20135 \u00d7 (p \u00d710\u20132) \u00d7 cos 180\u00b0\n= \u20133p \u00d7 10\u20137 Wb\nTherefore, estimated value of the induced emf is,\nN\n\u03a6t\n\u03b5\n\u2206\n=\n\u2206\n   = 500 \u00d7 (6p \u00d7 10\u20137)/0 25\n   = 3"}, {"Chapter": "1", "sentence_range": "4959-4962", "Text": "0 \u00d7 10\u20135 \u00d7 (p \u00d710\u20132) \u00d7 cos 0\u00b0\n= 3p \u00d7 10\u20137 Wb\nFinal flux after the rotation,\nFB (final)    = 3 0 \u00d7 10\u20135 \u00d7 (p \u00d710\u20132) \u00d7 cos 180\u00b0\n= \u20133p \u00d7 10\u20137 Wb\nTherefore, estimated value of the induced emf is,\nN\n\u03a6t\n\u03b5\n\u2206\n=\n\u2206\n   = 500 \u00d7 (6p \u00d7 10\u20137)/0 25\n   = 3 8  \u00d7 10\u20133  V\nI = e/R = 1"}, {"Chapter": "1", "sentence_range": "4960-4963", "Text": "0 \u00d7 10\u20135 \u00d7 (p \u00d710\u20132) \u00d7 cos 180\u00b0\n= \u20133p \u00d7 10\u20137 Wb\nTherefore, estimated value of the induced emf is,\nN\n\u03a6t\n\u03b5\n\u2206\n=\n\u2206\n   = 500 \u00d7 (6p \u00d7 10\u20137)/0 25\n   = 3 8  \u00d7 10\u20133  V\nI = e/R = 1 9 \u00d7 10\u20133 A\nNote that the magnitudes of e and I are the estimated values"}, {"Chapter": "1", "sentence_range": "4961-4964", "Text": "25\n   = 3 8  \u00d7 10\u20133  V\nI = e/R = 1 9 \u00d7 10\u20133 A\nNote that the magnitudes of e and I are the estimated values Their\ninstantaneous values are different and depend upon the speed of\nrotation at the particular instant"}, {"Chapter": "1", "sentence_range": "4962-4965", "Text": "8  \u00d7 10\u20133  V\nI = e/R = 1 9 \u00d7 10\u20133 A\nNote that the magnitudes of e and I are the estimated values Their\ninstantaneous values are different and depend upon the speed of\nrotation at the particular instant EXAMPLE 6"}, {"Chapter": "1", "sentence_range": "4963-4966", "Text": "9 \u00d7 10\u20133 A\nNote that the magnitudes of e and I are the estimated values Their\ninstantaneous values are different and depend upon the speed of\nrotation at the particular instant EXAMPLE 6 3\nRationalised 2023-24\nPhysics\n160\n6"}, {"Chapter": "1", "sentence_range": "4964-4967", "Text": "Their\ninstantaneous values are different and depend upon the speed of\nrotation at the particular instant EXAMPLE 6 3\nRationalised 2023-24\nPhysics\n160\n6 5  LENZ\u2019S LAW AND CONSERVATION OF ENERGY\nIn 1834, German physicist Heinrich Friedrich Lenz (1804-1865) deduced\na rule, known as Lenz\u2019s law which gives the polarity of the induced emf\nin a clear and concise fashion"}, {"Chapter": "1", "sentence_range": "4965-4968", "Text": "EXAMPLE 6 3\nRationalised 2023-24\nPhysics\n160\n6 5  LENZ\u2019S LAW AND CONSERVATION OF ENERGY\nIn 1834, German physicist Heinrich Friedrich Lenz (1804-1865) deduced\na rule, known as Lenz\u2019s law which gives the polarity of the induced emf\nin a clear and concise fashion The statement of the law is:\nThe polarity of induced emf is such that it tends to produce a current\nwhich opposes the change in magnetic flux that produced it"}, {"Chapter": "1", "sentence_range": "4966-4969", "Text": "3\nRationalised 2023-24\nPhysics\n160\n6 5  LENZ\u2019S LAW AND CONSERVATION OF ENERGY\nIn 1834, German physicist Heinrich Friedrich Lenz (1804-1865) deduced\na rule, known as Lenz\u2019s law which gives the polarity of the induced emf\nin a clear and concise fashion The statement of the law is:\nThe polarity of induced emf is such that it tends to produce a current\nwhich opposes the change in magnetic flux that produced it The negative sign shown in Eq"}, {"Chapter": "1", "sentence_range": "4967-4970", "Text": "5  LENZ\u2019S LAW AND CONSERVATION OF ENERGY\nIn 1834, German physicist Heinrich Friedrich Lenz (1804-1865) deduced\na rule, known as Lenz\u2019s law which gives the polarity of the induced emf\nin a clear and concise fashion The statement of the law is:\nThe polarity of induced emf is such that it tends to produce a current\nwhich opposes the change in magnetic flux that produced it The negative sign shown in Eq (6"}, {"Chapter": "1", "sentence_range": "4968-4971", "Text": "The statement of the law is:\nThe polarity of induced emf is such that it tends to produce a current\nwhich opposes the change in magnetic flux that produced it The negative sign shown in Eq (6 3) represents this effect"}, {"Chapter": "1", "sentence_range": "4969-4972", "Text": "The negative sign shown in Eq (6 3) represents this effect We can\nunderstand Lenz\u2019s law by examining Experiment 6"}, {"Chapter": "1", "sentence_range": "4970-4973", "Text": "(6 3) represents this effect We can\nunderstand Lenz\u2019s law by examining Experiment 6 1 in Section 6"}, {"Chapter": "1", "sentence_range": "4971-4974", "Text": "3) represents this effect We can\nunderstand Lenz\u2019s law by examining Experiment 6 1 in Section 6 2"}, {"Chapter": "1", "sentence_range": "4972-4975", "Text": "We can\nunderstand Lenz\u2019s law by examining Experiment 6 1 in Section 6 2 1"}, {"Chapter": "1", "sentence_range": "4973-4976", "Text": "1 in Section 6 2 1 In\nFig"}, {"Chapter": "1", "sentence_range": "4974-4977", "Text": "2 1 In\nFig 6"}, {"Chapter": "1", "sentence_range": "4975-4978", "Text": "1 In\nFig 6 1, we see that the North-pole of a bar magnet is being pushed\ntowards the closed coil"}, {"Chapter": "1", "sentence_range": "4976-4979", "Text": "In\nFig 6 1, we see that the North-pole of a bar magnet is being pushed\ntowards the closed coil As the North-pole of the bar magnet moves towards\nthe coil, the magnetic flux through the coil increases"}, {"Chapter": "1", "sentence_range": "4977-4980", "Text": "6 1, we see that the North-pole of a bar magnet is being pushed\ntowards the closed coil As the North-pole of the bar magnet moves towards\nthe coil, the magnetic flux through the coil increases Hence current is\ninduced in the coil in such a direction that it opposes the increase in flux"}, {"Chapter": "1", "sentence_range": "4978-4981", "Text": "1, we see that the North-pole of a bar magnet is being pushed\ntowards the closed coil As the North-pole of the bar magnet moves towards\nthe coil, the magnetic flux through the coil increases Hence current is\ninduced in the coil in such a direction that it opposes the increase in flux This is possible only if the current in the coil is in a counter-clockwise\ndirection with respect to an observer situated on the side of the magnet"}, {"Chapter": "1", "sentence_range": "4979-4982", "Text": "As the North-pole of the bar magnet moves towards\nthe coil, the magnetic flux through the coil increases Hence current is\ninduced in the coil in such a direction that it opposes the increase in flux This is possible only if the current in the coil is in a counter-clockwise\ndirection with respect to an observer situated on the side of the magnet Note that magnetic moment associated with this current has North polarity\ntowards the North-pole of the approaching magnet"}, {"Chapter": "1", "sentence_range": "4980-4983", "Text": "Hence current is\ninduced in the coil in such a direction that it opposes the increase in flux This is possible only if the current in the coil is in a counter-clockwise\ndirection with respect to an observer situated on the side of the magnet Note that magnetic moment associated with this current has North polarity\ntowards the North-pole of the approaching magnet Similarly, if the North-\npole of the magnet is being withdrawn from the coil, the magnetic flux\nthrough the coil will decrease"}, {"Chapter": "1", "sentence_range": "4981-4984", "Text": "This is possible only if the current in the coil is in a counter-clockwise\ndirection with respect to an observer situated on the side of the magnet Note that magnetic moment associated with this current has North polarity\ntowards the North-pole of the approaching magnet Similarly, if the North-\npole of the magnet is being withdrawn from the coil, the magnetic flux\nthrough the coil will decrease To counter this decrease in magnetic flux,\nthe induced current in the coil flows in clockwise direction and its South-\npole faces the receding North-pole of the bar magnet"}, {"Chapter": "1", "sentence_range": "4982-4985", "Text": "Note that magnetic moment associated with this current has North polarity\ntowards the North-pole of the approaching magnet Similarly, if the North-\npole of the magnet is being withdrawn from the coil, the magnetic flux\nthrough the coil will decrease To counter this decrease in magnetic flux,\nthe induced current in the coil flows in clockwise direction and its South-\npole faces the receding North-pole of the bar magnet This would result in\nan attractive force which opposes the motion of the magnet and the\ncorresponding decrease in flux"}, {"Chapter": "1", "sentence_range": "4983-4986", "Text": "Similarly, if the North-\npole of the magnet is being withdrawn from the coil, the magnetic flux\nthrough the coil will decrease To counter this decrease in magnetic flux,\nthe induced current in the coil flows in clockwise direction and its South-\npole faces the receding North-pole of the bar magnet This would result in\nan attractive force which opposes the motion of the magnet and the\ncorresponding decrease in flux What will happen if an open circuit is used in place of the closed loop\nin the above example"}, {"Chapter": "1", "sentence_range": "4984-4987", "Text": "To counter this decrease in magnetic flux,\nthe induced current in the coil flows in clockwise direction and its South-\npole faces the receding North-pole of the bar magnet This would result in\nan attractive force which opposes the motion of the magnet and the\ncorresponding decrease in flux What will happen if an open circuit is used in place of the closed loop\nin the above example In this case too, an emf is induced across the open\nends of the circuit"}, {"Chapter": "1", "sentence_range": "4985-4988", "Text": "This would result in\nan attractive force which opposes the motion of the magnet and the\ncorresponding decrease in flux What will happen if an open circuit is used in place of the closed loop\nin the above example In this case too, an emf is induced across the open\nends of the circuit The direction of the induced emf can be found\nusing Lenz\u2019s law"}, {"Chapter": "1", "sentence_range": "4986-4989", "Text": "What will happen if an open circuit is used in place of the closed loop\nin the above example In this case too, an emf is induced across the open\nends of the circuit The direction of the induced emf can be found\nusing Lenz\u2019s law Consider Figs"}, {"Chapter": "1", "sentence_range": "4987-4990", "Text": "In this case too, an emf is induced across the open\nends of the circuit The direction of the induced emf can be found\nusing Lenz\u2019s law Consider Figs 6"}, {"Chapter": "1", "sentence_range": "4988-4991", "Text": "The direction of the induced emf can be found\nusing Lenz\u2019s law Consider Figs 6 6 (a) and (b)"}, {"Chapter": "1", "sentence_range": "4989-4992", "Text": "Consider Figs 6 6 (a) and (b) They provide an easier\nway to understand the direction of induced currents"}, {"Chapter": "1", "sentence_range": "4990-4993", "Text": "6 6 (a) and (b) They provide an easier\nway to understand the direction of induced currents Note that the\ndirection shown by \n and \n indicate the directions of the induced\ncurrents"}, {"Chapter": "1", "sentence_range": "4991-4994", "Text": "6 (a) and (b) They provide an easier\nway to understand the direction of induced currents Note that the\ndirection shown by \n and \n indicate the directions of the induced\ncurrents A little reflection on this matter should convince us on the\ncorrectness of Lenz\u2019s law"}, {"Chapter": "1", "sentence_range": "4992-4995", "Text": "They provide an easier\nway to understand the direction of induced currents Note that the\ndirection shown by \n and \n indicate the directions of the induced\ncurrents A little reflection on this matter should convince us on the\ncorrectness of Lenz\u2019s law Suppose that the induced current was in\nthe direction opposite to the one depicted in Fig"}, {"Chapter": "1", "sentence_range": "4993-4996", "Text": "Note that the\ndirection shown by \n and \n indicate the directions of the induced\ncurrents A little reflection on this matter should convince us on the\ncorrectness of Lenz\u2019s law Suppose that the induced current was in\nthe direction opposite to the one depicted in Fig 6"}, {"Chapter": "1", "sentence_range": "4994-4997", "Text": "A little reflection on this matter should convince us on the\ncorrectness of Lenz\u2019s law Suppose that the induced current was in\nthe direction opposite to the one depicted in Fig 6 6(a)"}, {"Chapter": "1", "sentence_range": "4995-4998", "Text": "Suppose that the induced current was in\nthe direction opposite to the one depicted in Fig 6 6(a) In that case,\nthe South-pole due to the induced current will face the approaching\nNorth-pole of the magnet"}, {"Chapter": "1", "sentence_range": "4996-4999", "Text": "6 6(a) In that case,\nthe South-pole due to the induced current will face the approaching\nNorth-pole of the magnet The bar magnet will then be attracted\ntowards the coil at an ever increasing acceleration"}, {"Chapter": "1", "sentence_range": "4997-5000", "Text": "6(a) In that case,\nthe South-pole due to the induced current will face the approaching\nNorth-pole of the magnet The bar magnet will then be attracted\ntowards the coil at an ever increasing acceleration A gentle push on\nthe magnet will initiate the process and its velocity and kinetic energy\nwill continuously increase without expending any energy"}, {"Chapter": "1", "sentence_range": "4998-5001", "Text": "In that case,\nthe South-pole due to the induced current will face the approaching\nNorth-pole of the magnet The bar magnet will then be attracted\ntowards the coil at an ever increasing acceleration A gentle push on\nthe magnet will initiate the process and its velocity and kinetic energy\nwill continuously increase without expending any energy If this can\nhappen, one could construct a perpetual-motion machine by a\nsuitable arrangement"}, {"Chapter": "1", "sentence_range": "4999-5002", "Text": "The bar magnet will then be attracted\ntowards the coil at an ever increasing acceleration A gentle push on\nthe magnet will initiate the process and its velocity and kinetic energy\nwill continuously increase without expending any energy If this can\nhappen, one could construct a perpetual-motion machine by a\nsuitable arrangement This violates the law of conservation of energy\nand hence can not happen"}, {"Chapter": "1", "sentence_range": "5000-5003", "Text": "A gentle push on\nthe magnet will initiate the process and its velocity and kinetic energy\nwill continuously increase without expending any energy If this can\nhappen, one could construct a perpetual-motion machine by a\nsuitable arrangement This violates the law of conservation of energy\nand hence can not happen Now consider the correct case shown in Fig"}, {"Chapter": "1", "sentence_range": "5001-5004", "Text": "If this can\nhappen, one could construct a perpetual-motion machine by a\nsuitable arrangement This violates the law of conservation of energy\nand hence can not happen Now consider the correct case shown in Fig 6"}, {"Chapter": "1", "sentence_range": "5002-5005", "Text": "This violates the law of conservation of energy\nand hence can not happen Now consider the correct case shown in Fig 6 6(a)"}, {"Chapter": "1", "sentence_range": "5003-5006", "Text": "Now consider the correct case shown in Fig 6 6(a) In this situation,\nthe bar magnet experiences a repulsive force due to the induced\ncurrent"}, {"Chapter": "1", "sentence_range": "5004-5007", "Text": "6 6(a) In this situation,\nthe bar magnet experiences a repulsive force due to the induced\ncurrent Therefore, a person has to do work in moving the magnet"}, {"Chapter": "1", "sentence_range": "5005-5008", "Text": "6(a) In this situation,\nthe bar magnet experiences a repulsive force due to the induced\ncurrent Therefore, a person has to do work in moving the magnet Where does the energy spent by the person go"}, {"Chapter": "1", "sentence_range": "5006-5009", "Text": "In this situation,\nthe bar magnet experiences a repulsive force due to the induced\ncurrent Therefore, a person has to do work in moving the magnet Where does the energy spent by the person go This energy is\ndissipated by Joule heating produced by the induced current"}, {"Chapter": "1", "sentence_range": "5007-5010", "Text": "Therefore, a person has to do work in moving the magnet Where does the energy spent by the person go This energy is\ndissipated by Joule heating produced by the induced current FIGURE 6"}, {"Chapter": "1", "sentence_range": "5008-5011", "Text": "Where does the energy spent by the person go This energy is\ndissipated by Joule heating produced by the induced current FIGURE 6 6\nIllustration of\nLenz\u2019s law"}, {"Chapter": "1", "sentence_range": "5009-5012", "Text": "This energy is\ndissipated by Joule heating produced by the induced current FIGURE 6 6\nIllustration of\nLenz\u2019s law Rationalised 2023-24\nElectromagnetic\nInduction\n161\n EXAMPLE 6"}, {"Chapter": "1", "sentence_range": "5010-5013", "Text": "FIGURE 6 6\nIllustration of\nLenz\u2019s law Rationalised 2023-24\nElectromagnetic\nInduction\n161\n EXAMPLE 6 4\nExample 6"}, {"Chapter": "1", "sentence_range": "5011-5014", "Text": "6\nIllustration of\nLenz\u2019s law Rationalised 2023-24\nElectromagnetic\nInduction\n161\n EXAMPLE 6 4\nExample 6 4\nFigure 6"}, {"Chapter": "1", "sentence_range": "5012-5015", "Text": "Rationalised 2023-24\nElectromagnetic\nInduction\n161\n EXAMPLE 6 4\nExample 6 4\nFigure 6 7 shows planar loops of different shapes moving out of or\ninto a region of a magnetic field which is directed normal to the plane\nof the loop away from the reader"}, {"Chapter": "1", "sentence_range": "5013-5016", "Text": "4\nExample 6 4\nFigure 6 7 shows planar loops of different shapes moving out of or\ninto a region of a magnetic field which is directed normal to the plane\nof the loop away from the reader Determine the direction of induced\ncurrent in each loop using Lenz\u2019s law"}, {"Chapter": "1", "sentence_range": "5014-5017", "Text": "4\nFigure 6 7 shows planar loops of different shapes moving out of or\ninto a region of a magnetic field which is directed normal to the plane\nof the loop away from the reader Determine the direction of induced\ncurrent in each loop using Lenz\u2019s law FIGURE 6"}, {"Chapter": "1", "sentence_range": "5015-5018", "Text": "7 shows planar loops of different shapes moving out of or\ninto a region of a magnetic field which is directed normal to the plane\nof the loop away from the reader Determine the direction of induced\ncurrent in each loop using Lenz\u2019s law FIGURE 6 7\nSolution\n(i)\nThe magnetic flux through the rectangular loop abcd increases,\ndue to the motion of the loop into the region of magnetic field, The\ninduced current must flow along the path bcdab so that it opposes\nthe increasing flux"}, {"Chapter": "1", "sentence_range": "5016-5019", "Text": "Determine the direction of induced\ncurrent in each loop using Lenz\u2019s law FIGURE 6 7\nSolution\n(i)\nThe magnetic flux through the rectangular loop abcd increases,\ndue to the motion of the loop into the region of magnetic field, The\ninduced current must flow along the path bcdab so that it opposes\nthe increasing flux (ii) Due to the outward motion, magnetic flux through the triangular\nloop abc decreases due to which the induced current flows along\nbacb, so as to oppose the change in flux"}, {"Chapter": "1", "sentence_range": "5017-5020", "Text": "FIGURE 6 7\nSolution\n(i)\nThe magnetic flux through the rectangular loop abcd increases,\ndue to the motion of the loop into the region of magnetic field, The\ninduced current must flow along the path bcdab so that it opposes\nthe increasing flux (ii) Due to the outward motion, magnetic flux through the triangular\nloop abc decreases due to which the induced current flows along\nbacb, so as to oppose the change in flux (iii) As the magnetic flux decreases due to motion of the irregular\nshaped loop abcd out of the region of magnetic field, the induced\ncurrent flows along cdabc, so as to oppose change in flux"}, {"Chapter": "1", "sentence_range": "5018-5021", "Text": "7\nSolution\n(i)\nThe magnetic flux through the rectangular loop abcd increases,\ndue to the motion of the loop into the region of magnetic field, The\ninduced current must flow along the path bcdab so that it opposes\nthe increasing flux (ii) Due to the outward motion, magnetic flux through the triangular\nloop abc decreases due to which the induced current flows along\nbacb, so as to oppose the change in flux (iii) As the magnetic flux decreases due to motion of the irregular\nshaped loop abcd out of the region of magnetic field, the induced\ncurrent flows along cdabc, so as to oppose change in flux Note that there are no induced current as long as the loops are\ncompletely inside or outside the region of the magnetic field"}, {"Chapter": "1", "sentence_range": "5019-5022", "Text": "(ii) Due to the outward motion, magnetic flux through the triangular\nloop abc decreases due to which the induced current flows along\nbacb, so as to oppose the change in flux (iii) As the magnetic flux decreases due to motion of the irregular\nshaped loop abcd out of the region of magnetic field, the induced\ncurrent flows along cdabc, so as to oppose change in flux Note that there are no induced current as long as the loops are\ncompletely inside or outside the region of the magnetic field Example 6"}, {"Chapter": "1", "sentence_range": "5020-5023", "Text": "(iii) As the magnetic flux decreases due to motion of the irregular\nshaped loop abcd out of the region of magnetic field, the induced\ncurrent flows along cdabc, so as to oppose change in flux Note that there are no induced current as long as the loops are\ncompletely inside or outside the region of the magnetic field Example 6 5\n(a) A closed loop is held stationary in the magnetic field between the\nnorth and south poles of two permanent magnets held fixed"}, {"Chapter": "1", "sentence_range": "5021-5024", "Text": "Note that there are no induced current as long as the loops are\ncompletely inside or outside the region of the magnetic field Example 6 5\n(a) A closed loop is held stationary in the magnetic field between the\nnorth and south poles of two permanent magnets held fixed Can\nwe hope to generate current in the loop by using very strong\nmagnets"}, {"Chapter": "1", "sentence_range": "5022-5025", "Text": "Example 6 5\n(a) A closed loop is held stationary in the magnetic field between the\nnorth and south poles of two permanent magnets held fixed Can\nwe hope to generate current in the loop by using very strong\nmagnets (b) A closed loop moves normal to the constant electric field between\nthe plates of a large capacitor"}, {"Chapter": "1", "sentence_range": "5023-5026", "Text": "5\n(a) A closed loop is held stationary in the magnetic field between the\nnorth and south poles of two permanent magnets held fixed Can\nwe hope to generate current in the loop by using very strong\nmagnets (b) A closed loop moves normal to the constant electric field between\nthe plates of a large capacitor Is a current induced in the loop\n(i) when it is wholly inside the region between the capacitor plates\n(ii) when it is partially outside the plates of the capacitor"}, {"Chapter": "1", "sentence_range": "5024-5027", "Text": "Can\nwe hope to generate current in the loop by using very strong\nmagnets (b) A closed loop moves normal to the constant electric field between\nthe plates of a large capacitor Is a current induced in the loop\n(i) when it is wholly inside the region between the capacitor plates\n(ii) when it is partially outside the plates of the capacitor The\nelectric field is normal to the plane of the loop"}, {"Chapter": "1", "sentence_range": "5025-5028", "Text": "(b) A closed loop moves normal to the constant electric field between\nthe plates of a large capacitor Is a current induced in the loop\n(i) when it is wholly inside the region between the capacitor plates\n(ii) when it is partially outside the plates of the capacitor The\nelectric field is normal to the plane of the loop (c) A rectangular loop and a circular loop are moving out of a uniform\nmagnetic field region (Fig"}, {"Chapter": "1", "sentence_range": "5026-5029", "Text": "Is a current induced in the loop\n(i) when it is wholly inside the region between the capacitor plates\n(ii) when it is partially outside the plates of the capacitor The\nelectric field is normal to the plane of the loop (c) A rectangular loop and a circular loop are moving out of a uniform\nmagnetic field region (Fig 6"}, {"Chapter": "1", "sentence_range": "5027-5030", "Text": "The\nelectric field is normal to the plane of the loop (c) A rectangular loop and a circular loop are moving out of a uniform\nmagnetic field region (Fig 6 8) to a field-free region with a constant\nvelocity v"}, {"Chapter": "1", "sentence_range": "5028-5031", "Text": "(c) A rectangular loop and a circular loop are moving out of a uniform\nmagnetic field region (Fig 6 8) to a field-free region with a constant\nvelocity v In which loop do you expect the induced emf to be\nconstant during the passage out of the field region"}, {"Chapter": "1", "sentence_range": "5029-5032", "Text": "6 8) to a field-free region with a constant\nvelocity v In which loop do you expect the induced emf to be\nconstant during the passage out of the field region The field is\nnormal to the loops"}, {"Chapter": "1", "sentence_range": "5030-5033", "Text": "8) to a field-free region with a constant\nvelocity v In which loop do you expect the induced emf to be\nconstant during the passage out of the field region The field is\nnormal to the loops EXAMPLE 6"}, {"Chapter": "1", "sentence_range": "5031-5034", "Text": "In which loop do you expect the induced emf to be\nconstant during the passage out of the field region The field is\nnormal to the loops EXAMPLE 6 5\nRationalised 2023-24\nPhysics\n162\n EXAMPLE 6"}, {"Chapter": "1", "sentence_range": "5032-5035", "Text": "The field is\nnormal to the loops EXAMPLE 6 5\nRationalised 2023-24\nPhysics\n162\n EXAMPLE 6 5\nFIGURE 6"}, {"Chapter": "1", "sentence_range": "5033-5036", "Text": "EXAMPLE 6 5\nRationalised 2023-24\nPhysics\n162\n EXAMPLE 6 5\nFIGURE 6 8\n(d) Predict the polarity of the capacitor in the situation described by\nFig"}, {"Chapter": "1", "sentence_range": "5034-5037", "Text": "5\nRationalised 2023-24\nPhysics\n162\n EXAMPLE 6 5\nFIGURE 6 8\n(d) Predict the polarity of the capacitor in the situation described by\nFig 6"}, {"Chapter": "1", "sentence_range": "5035-5038", "Text": "5\nFIGURE 6 8\n(d) Predict the polarity of the capacitor in the situation described by\nFig 6 9"}, {"Chapter": "1", "sentence_range": "5036-5039", "Text": "8\n(d) Predict the polarity of the capacitor in the situation described by\nFig 6 9 FIGURE 6"}, {"Chapter": "1", "sentence_range": "5037-5040", "Text": "6 9 FIGURE 6 9\nSolution\n(a) No"}, {"Chapter": "1", "sentence_range": "5038-5041", "Text": "9 FIGURE 6 9\nSolution\n(a) No However strong the magnet may be, current can be induced\nonly by changing the magnetic flux through the loop"}, {"Chapter": "1", "sentence_range": "5039-5042", "Text": "FIGURE 6 9\nSolution\n(a) No However strong the magnet may be, current can be induced\nonly by changing the magnetic flux through the loop (b) No current is induced in either case"}, {"Chapter": "1", "sentence_range": "5040-5043", "Text": "9\nSolution\n(a) No However strong the magnet may be, current can be induced\nonly by changing the magnetic flux through the loop (b) No current is induced in either case Current can not be induced\nby changing the electric flux"}, {"Chapter": "1", "sentence_range": "5041-5044", "Text": "However strong the magnet may be, current can be induced\nonly by changing the magnetic flux through the loop (b) No current is induced in either case Current can not be induced\nby changing the electric flux (c) The induced emf is expected to be constant only in the case of the\nrectangular loop"}, {"Chapter": "1", "sentence_range": "5042-5045", "Text": "(b) No current is induced in either case Current can not be induced\nby changing the electric flux (c) The induced emf is expected to be constant only in the case of the\nrectangular loop In the case of circular loop, the rate of change of\narea of the loop during its passage out of the field region is not\nconstant, hence induced emf will vary accordingly"}, {"Chapter": "1", "sentence_range": "5043-5046", "Text": "Current can not be induced\nby changing the electric flux (c) The induced emf is expected to be constant only in the case of the\nrectangular loop In the case of circular loop, the rate of change of\narea of the loop during its passage out of the field region is not\nconstant, hence induced emf will vary accordingly (d) The polarity of plate \u2018A\u2019 will be positive with respect to plate \u2018B\u2019 in\nthe capacitor"}, {"Chapter": "1", "sentence_range": "5044-5047", "Text": "(c) The induced emf is expected to be constant only in the case of the\nrectangular loop In the case of circular loop, the rate of change of\narea of the loop during its passage out of the field region is not\nconstant, hence induced emf will vary accordingly (d) The polarity of plate \u2018A\u2019 will be positive with respect to plate \u2018B\u2019 in\nthe capacitor 6"}, {"Chapter": "1", "sentence_range": "5045-5048", "Text": "In the case of circular loop, the rate of change of\narea of the loop during its passage out of the field region is not\nconstant, hence induced emf will vary accordingly (d) The polarity of plate \u2018A\u2019 will be positive with respect to plate \u2018B\u2019 in\nthe capacitor 6 6  MOTIONAL ELECTROMOTIVE FORCE\nLet us consider a straight conductor moving in a uniform and time-\nindependent magnetic field"}, {"Chapter": "1", "sentence_range": "5046-5049", "Text": "(d) The polarity of plate \u2018A\u2019 will be positive with respect to plate \u2018B\u2019 in\nthe capacitor 6 6  MOTIONAL ELECTROMOTIVE FORCE\nLet us consider a straight conductor moving in a uniform and time-\nindependent magnetic field Figure 6"}, {"Chapter": "1", "sentence_range": "5047-5050", "Text": "6 6  MOTIONAL ELECTROMOTIVE FORCE\nLet us consider a straight conductor moving in a uniform and time-\nindependent magnetic field Figure 6 10 shows a rectangular conductor\nPQRS in which the conductor PQ is free to move"}, {"Chapter": "1", "sentence_range": "5048-5051", "Text": "6  MOTIONAL ELECTROMOTIVE FORCE\nLet us consider a straight conductor moving in a uniform and time-\nindependent magnetic field Figure 6 10 shows a rectangular conductor\nPQRS in which the conductor PQ is free to move The rod PQ is moved\ntowards the left with a constant velocity v as\nshown in the figure"}, {"Chapter": "1", "sentence_range": "5049-5052", "Text": "Figure 6 10 shows a rectangular conductor\nPQRS in which the conductor PQ is free to move The rod PQ is moved\ntowards the left with a constant velocity v as\nshown in the figure Assume that there is no\nloss of energy due to friction"}, {"Chapter": "1", "sentence_range": "5050-5053", "Text": "10 shows a rectangular conductor\nPQRS in which the conductor PQ is free to move The rod PQ is moved\ntowards the left with a constant velocity v as\nshown in the figure Assume that there is no\nloss of energy due to friction PQRS forms a\nclosed circuit enclosing an area that changes\nas PQ moves"}, {"Chapter": "1", "sentence_range": "5051-5054", "Text": "The rod PQ is moved\ntowards the left with a constant velocity v as\nshown in the figure Assume that there is no\nloss of energy due to friction PQRS forms a\nclosed circuit enclosing an area that changes\nas PQ moves It is placed in a uniform magnetic\nfield B which is perpendicular to the plane of\nthis system"}, {"Chapter": "1", "sentence_range": "5052-5055", "Text": "Assume that there is no\nloss of energy due to friction PQRS forms a\nclosed circuit enclosing an area that changes\nas PQ moves It is placed in a uniform magnetic\nfield B which is perpendicular to the plane of\nthis system If the length RQ = x and RS = l, the\nmagnetic flux FB enclosed  by the loop PQRS\nwill be\nFB = Blx\nSince x is changing with time, the rate of change\nof flux FB will induce an emf given by:\n(\n)\n\u2013 d\n\u2013d\nd\nd\nB\nBlx\nt\nt\n\u03a6\n\u03b5 =\n=\n   = \nd\n\u2013\nd\nx\nBl\nt =Blv\n(6"}, {"Chapter": "1", "sentence_range": "5053-5056", "Text": "PQRS forms a\nclosed circuit enclosing an area that changes\nas PQ moves It is placed in a uniform magnetic\nfield B which is perpendicular to the plane of\nthis system If the length RQ = x and RS = l, the\nmagnetic flux FB enclosed  by the loop PQRS\nwill be\nFB = Blx\nSince x is changing with time, the rate of change\nof flux FB will induce an emf given by:\n(\n)\n\u2013 d\n\u2013d\nd\nd\nB\nBlx\nt\nt\n\u03a6\n\u03b5 =\n=\n   = \nd\n\u2013\nd\nx\nBl\nt =Blv\n(6 5)\nFIGURE 6"}, {"Chapter": "1", "sentence_range": "5054-5057", "Text": "It is placed in a uniform magnetic\nfield B which is perpendicular to the plane of\nthis system If the length RQ = x and RS = l, the\nmagnetic flux FB enclosed  by the loop PQRS\nwill be\nFB = Blx\nSince x is changing with time, the rate of change\nof flux FB will induce an emf given by:\n(\n)\n\u2013 d\n\u2013d\nd\nd\nB\nBlx\nt\nt\n\u03a6\n\u03b5 =\n=\n   = \nd\n\u2013\nd\nx\nBl\nt =Blv\n(6 5)\nFIGURE 6 10  The arm PQ is moved to the left\nside, thus decreasing the area of the\nrectangular loop"}, {"Chapter": "1", "sentence_range": "5055-5058", "Text": "If the length RQ = x and RS = l, the\nmagnetic flux FB enclosed  by the loop PQRS\nwill be\nFB = Blx\nSince x is changing with time, the rate of change\nof flux FB will induce an emf given by:\n(\n)\n\u2013 d\n\u2013d\nd\nd\nB\nBlx\nt\nt\n\u03a6\n\u03b5 =\n=\n   = \nd\n\u2013\nd\nx\nBl\nt =Blv\n(6 5)\nFIGURE 6 10  The arm PQ is moved to the left\nside, thus decreasing the area of the\nrectangular loop This movement\ninduces a current I as shown"}, {"Chapter": "1", "sentence_range": "5056-5059", "Text": "5)\nFIGURE 6 10  The arm PQ is moved to the left\nside, thus decreasing the area of the\nrectangular loop This movement\ninduces a current I as shown Rationalised 2023-24\nElectromagnetic\nInduction\n163\nwhere we have used dx/dt = \u2013v  which is the speed of the conductor PQ"}, {"Chapter": "1", "sentence_range": "5057-5060", "Text": "10  The arm PQ is moved to the left\nside, thus decreasing the area of the\nrectangular loop This movement\ninduces a current I as shown Rationalised 2023-24\nElectromagnetic\nInduction\n163\nwhere we have used dx/dt = \u2013v  which is the speed of the conductor PQ The induced emf Blv is called motional emf"}, {"Chapter": "1", "sentence_range": "5058-5061", "Text": "This movement\ninduces a current I as shown Rationalised 2023-24\nElectromagnetic\nInduction\n163\nwhere we have used dx/dt = \u2013v  which is the speed of the conductor PQ The induced emf Blv is called motional emf Thus, we are able to produce\ninduced emf by moving a conductor instead of varying the magnetic field,\nthat is, by changing the magnetic flux enclosed by the circuit"}, {"Chapter": "1", "sentence_range": "5059-5062", "Text": "Rationalised 2023-24\nElectromagnetic\nInduction\n163\nwhere we have used dx/dt = \u2013v  which is the speed of the conductor PQ The induced emf Blv is called motional emf Thus, we are able to produce\ninduced emf by moving a conductor instead of varying the magnetic field,\nthat is, by changing the magnetic flux enclosed by the circuit It is also possible to explain the motional emf expression in Eq"}, {"Chapter": "1", "sentence_range": "5060-5063", "Text": "The induced emf Blv is called motional emf Thus, we are able to produce\ninduced emf by moving a conductor instead of varying the magnetic field,\nthat is, by changing the magnetic flux enclosed by the circuit It is also possible to explain the motional emf expression in Eq (6"}, {"Chapter": "1", "sentence_range": "5061-5064", "Text": "Thus, we are able to produce\ninduced emf by moving a conductor instead of varying the magnetic field,\nthat is, by changing the magnetic flux enclosed by the circuit It is also possible to explain the motional emf expression in Eq (6 5)\nby invoking the Lorentz force acting on the free charge carriers of conductor\nPQ"}, {"Chapter": "1", "sentence_range": "5062-5065", "Text": "It is also possible to explain the motional emf expression in Eq (6 5)\nby invoking the Lorentz force acting on the free charge carriers of conductor\nPQ Consider any arbitrary charge q in the conductor PQ"}, {"Chapter": "1", "sentence_range": "5063-5066", "Text": "(6 5)\nby invoking the Lorentz force acting on the free charge carriers of conductor\nPQ Consider any arbitrary charge q in the conductor PQ When the rod\nmoves with speed v, the charge will also be moving with speed v in the\nmagnetic field B"}, {"Chapter": "1", "sentence_range": "5064-5067", "Text": "5)\nby invoking the Lorentz force acting on the free charge carriers of conductor\nPQ Consider any arbitrary charge q in the conductor PQ When the rod\nmoves with speed v, the charge will also be moving with speed v in the\nmagnetic field B The Lorentz force on this charge is qvB in magnitude,\nand its direction is towards Q"}, {"Chapter": "1", "sentence_range": "5065-5068", "Text": "Consider any arbitrary charge q in the conductor PQ When the rod\nmoves with speed v, the charge will also be moving with speed v in the\nmagnetic field B The Lorentz force on this charge is qvB in magnitude,\nand its direction is towards Q All charges experience the same force, in\nmagnitude and direction, irrespective of their position in the rod PQ"}, {"Chapter": "1", "sentence_range": "5066-5069", "Text": "When the rod\nmoves with speed v, the charge will also be moving with speed v in the\nmagnetic field B The Lorentz force on this charge is qvB in magnitude,\nand its direction is towards Q All charges experience the same force, in\nmagnitude and direction, irrespective of their position in the rod PQ The work done in moving the charge from P to Q is,\nW = qvBl\nSince emf is the work done per unit charge,\nW\nq\n\u03b5 =\n    = Blv\nThis equation gives emf induced across the rod PQ and is identical\nto Eq"}, {"Chapter": "1", "sentence_range": "5067-5070", "Text": "The Lorentz force on this charge is qvB in magnitude,\nand its direction is towards Q All charges experience the same force, in\nmagnitude and direction, irrespective of their position in the rod PQ The work done in moving the charge from P to Q is,\nW = qvBl\nSince emf is the work done per unit charge,\nW\nq\n\u03b5 =\n    = Blv\nThis equation gives emf induced across the rod PQ and is identical\nto Eq (6"}, {"Chapter": "1", "sentence_range": "5068-5071", "Text": "All charges experience the same force, in\nmagnitude and direction, irrespective of their position in the rod PQ The work done in moving the charge from P to Q is,\nW = qvBl\nSince emf is the work done per unit charge,\nW\nq\n\u03b5 =\n    = Blv\nThis equation gives emf induced across the rod PQ and is identical\nto Eq (6 5)"}, {"Chapter": "1", "sentence_range": "5069-5072", "Text": "The work done in moving the charge from P to Q is,\nW = qvBl\nSince emf is the work done per unit charge,\nW\nq\n\u03b5 =\n    = Blv\nThis equation gives emf induced across the rod PQ and is identical\nto Eq (6 5) We stress that our presentation is not wholly rigorous"}, {"Chapter": "1", "sentence_range": "5070-5073", "Text": "(6 5) We stress that our presentation is not wholly rigorous But\nit does help us to understand the basis of Faraday\u2019s law when\nthe conductor is moving in a uniform and time-independent\nmagnetic field"}, {"Chapter": "1", "sentence_range": "5071-5074", "Text": "5) We stress that our presentation is not wholly rigorous But\nit does help us to understand the basis of Faraday\u2019s law when\nthe conductor is moving in a uniform and time-independent\nmagnetic field On the other hand, it is not obvious how an emf is induced when a\nconductor is stationary and the magnetic field is changing \u2013 a fact which\nFaraday verified by numerous experiments"}, {"Chapter": "1", "sentence_range": "5072-5075", "Text": "We stress that our presentation is not wholly rigorous But\nit does help us to understand the basis of Faraday\u2019s law when\nthe conductor is moving in a uniform and time-independent\nmagnetic field On the other hand, it is not obvious how an emf is induced when a\nconductor is stationary and the magnetic field is changing \u2013 a fact which\nFaraday verified by numerous experiments In the case of a stationary\nconductor, the force on its charges is given by\nF = q (E + v \u00b4\u00b4\u00b4\u00b4\u00b4 B) = qE\n(6"}, {"Chapter": "1", "sentence_range": "5073-5076", "Text": "But\nit does help us to understand the basis of Faraday\u2019s law when\nthe conductor is moving in a uniform and time-independent\nmagnetic field On the other hand, it is not obvious how an emf is induced when a\nconductor is stationary and the magnetic field is changing \u2013 a fact which\nFaraday verified by numerous experiments In the case of a stationary\nconductor, the force on its charges is given by\nF = q (E + v \u00b4\u00b4\u00b4\u00b4\u00b4 B) = qE\n(6 6)\nsince v = 0"}, {"Chapter": "1", "sentence_range": "5074-5077", "Text": "On the other hand, it is not obvious how an emf is induced when a\nconductor is stationary and the magnetic field is changing \u2013 a fact which\nFaraday verified by numerous experiments In the case of a stationary\nconductor, the force on its charges is given by\nF = q (E + v \u00b4\u00b4\u00b4\u00b4\u00b4 B) = qE\n(6 6)\nsince v = 0 Thus, any force on the charge must arise from the electric\nfield term E alone"}, {"Chapter": "1", "sentence_range": "5075-5078", "Text": "In the case of a stationary\nconductor, the force on its charges is given by\nF = q (E + v \u00b4\u00b4\u00b4\u00b4\u00b4 B) = qE\n(6 6)\nsince v = 0 Thus, any force on the charge must arise from the electric\nfield term E alone Therefore, to explain the existence of induced emf or\ninduced current, we must assume that a time-varying magnetic field\ngenerates an electric field"}, {"Chapter": "1", "sentence_range": "5076-5079", "Text": "6)\nsince v = 0 Thus, any force on the charge must arise from the electric\nfield term E alone Therefore, to explain the existence of induced emf or\ninduced current, we must assume that a time-varying magnetic field\ngenerates an electric field However, we hasten to add that electric fields\nproduced by static electric charges have properties different from those\nproduced by time-varying magnetic fields"}, {"Chapter": "1", "sentence_range": "5077-5080", "Text": "Thus, any force on the charge must arise from the electric\nfield term E alone Therefore, to explain the existence of induced emf or\ninduced current, we must assume that a time-varying magnetic field\ngenerates an electric field However, we hasten to add that electric fields\nproduced by static electric charges have properties different from those\nproduced by time-varying magnetic fields In Chapter 4, we learnt that\ncharges in motion (current) can exert force/torque on a stationary magnet"}, {"Chapter": "1", "sentence_range": "5078-5081", "Text": "Therefore, to explain the existence of induced emf or\ninduced current, we must assume that a time-varying magnetic field\ngenerates an electric field However, we hasten to add that electric fields\nproduced by static electric charges have properties different from those\nproduced by time-varying magnetic fields In Chapter 4, we learnt that\ncharges in motion (current) can exert force/torque on a stationary magnet Conversely, a bar magnet in motion (or more generally, a changing\nmagnetic field) can exert a force on the stationary charge"}, {"Chapter": "1", "sentence_range": "5079-5082", "Text": "However, we hasten to add that electric fields\nproduced by static electric charges have properties different from those\nproduced by time-varying magnetic fields In Chapter 4, we learnt that\ncharges in motion (current) can exert force/torque on a stationary magnet Conversely, a bar magnet in motion (or more generally, a changing\nmagnetic field) can exert a force on the stationary charge This is the\nfundamental significance of the Faraday\u2019s discovery"}, {"Chapter": "1", "sentence_range": "5080-5083", "Text": "In Chapter 4, we learnt that\ncharges in motion (current) can exert force/torque on a stationary magnet Conversely, a bar magnet in motion (or more generally, a changing\nmagnetic field) can exert a force on the stationary charge This is the\nfundamental significance of the Faraday\u2019s discovery Electricity and\nmagnetism are related"}, {"Chapter": "1", "sentence_range": "5081-5084", "Text": "Conversely, a bar magnet in motion (or more generally, a changing\nmagnetic field) can exert a force on the stationary charge This is the\nfundamental significance of the Faraday\u2019s discovery Electricity and\nmagnetism are related Example 6"}, {"Chapter": "1", "sentence_range": "5082-5085", "Text": "This is the\nfundamental significance of the Faraday\u2019s discovery Electricity and\nmagnetism are related Example 6 6  A metallic rod of 1 m length is rotated with a frequency\nof 50 rev/s, with one end hinged at the centre and the other end at the\ncircumference of a circular metallic ring of radius 1 m, about an axis\npassing through the centre and perpendicular to the plane of the ring\n(Fig"}, {"Chapter": "1", "sentence_range": "5083-5086", "Text": "Electricity and\nmagnetism are related Example 6 6  A metallic rod of 1 m length is rotated with a frequency\nof 50 rev/s, with one end hinged at the centre and the other end at the\ncircumference of a circular metallic ring of radius 1 m, about an axis\npassing through the centre and perpendicular to the plane of the ring\n(Fig 6"}, {"Chapter": "1", "sentence_range": "5084-5087", "Text": "Example 6 6  A metallic rod of 1 m length is rotated with a frequency\nof 50 rev/s, with one end hinged at the centre and the other end at the\ncircumference of a circular metallic ring of radius 1 m, about an axis\npassing through the centre and perpendicular to the plane of the ring\n(Fig 6 11)"}, {"Chapter": "1", "sentence_range": "5085-5088", "Text": "6  A metallic rod of 1 m length is rotated with a frequency\nof 50 rev/s, with one end hinged at the centre and the other end at the\ncircumference of a circular metallic ring of radius 1 m, about an axis\npassing through the centre and perpendicular to the plane of the ring\n(Fig 6 11) A constant and uniform magnetic field of 1 T parallel to the\naxis is present everywhere"}, {"Chapter": "1", "sentence_range": "5086-5089", "Text": "6 11) A constant and uniform magnetic field of 1 T parallel to the\naxis is present everywhere What is the emf between the centre and\nthe metallic ring"}, {"Chapter": "1", "sentence_range": "5087-5090", "Text": "11) A constant and uniform magnetic field of 1 T parallel to the\naxis is present everywhere What is the emf between the centre and\nthe metallic ring EXAMPLE 6"}, {"Chapter": "1", "sentence_range": "5088-5091", "Text": "A constant and uniform magnetic field of 1 T parallel to the\naxis is present everywhere What is the emf between the centre and\nthe metallic ring EXAMPLE 6 6\nRationalised 2023-24\nPhysics\n164\n EXAMPLE 6"}, {"Chapter": "1", "sentence_range": "5089-5092", "Text": "What is the emf between the centre and\nthe metallic ring EXAMPLE 6 6\nRationalised 2023-24\nPhysics\n164\n EXAMPLE 6 6\nFIGURE 6"}, {"Chapter": "1", "sentence_range": "5090-5093", "Text": "EXAMPLE 6 6\nRationalised 2023-24\nPhysics\n164\n EXAMPLE 6 6\nFIGURE 6 11\nSolution\nMethod I\nAs the rod is rotated, free electrons in the rod move towards the outer\nend due to Lorentz force and get distributed over the ring"}, {"Chapter": "1", "sentence_range": "5091-5094", "Text": "6\nRationalised 2023-24\nPhysics\n164\n EXAMPLE 6 6\nFIGURE 6 11\nSolution\nMethod I\nAs the rod is rotated, free electrons in the rod move towards the outer\nend due to Lorentz force and get distributed over the ring Thus, the\nresulting separation of charges produces an emf across the ends of\nthe rod"}, {"Chapter": "1", "sentence_range": "5092-5095", "Text": "6\nFIGURE 6 11\nSolution\nMethod I\nAs the rod is rotated, free electrons in the rod move towards the outer\nend due to Lorentz force and get distributed over the ring Thus, the\nresulting separation of charges produces an emf across the ends of\nthe rod At a certain value of emf, there is no more flow of electrons\nand a steady state is reached"}, {"Chapter": "1", "sentence_range": "5093-5096", "Text": "11\nSolution\nMethod I\nAs the rod is rotated, free electrons in the rod move towards the outer\nend due to Lorentz force and get distributed over the ring Thus, the\nresulting separation of charges produces an emf across the ends of\nthe rod At a certain value of emf, there is no more flow of electrons\nand a steady state is reached Using Eq"}, {"Chapter": "1", "sentence_range": "5094-5097", "Text": "Thus, the\nresulting separation of charges produces an emf across the ends of\nthe rod At a certain value of emf, there is no more flow of electrons\nand a steady state is reached Using Eq (6"}, {"Chapter": "1", "sentence_range": "5095-5098", "Text": "At a certain value of emf, there is no more flow of electrons\nand a steady state is reached Using Eq (6 5), the magnitude of the\nemf generated across a length dr of the rod as it moves at right angles\nto the magnetic field is given by\nd\nBvd\nr\n\u03b5 ="}, {"Chapter": "1", "sentence_range": "5096-5099", "Text": "Using Eq (6 5), the magnitude of the\nemf generated across a length dr of the rod as it moves at right angles\nto the magnetic field is given by\nd\nBvd\nr\n\u03b5 = Hence,\n\u03b5\n\u03b5\n=\n= \u222b\n\u222b\nd\nBvd\nr\nR\n0\n =\n=\n\u222b B\nr\nr\nB\nR\nR\n\u03c9\n\u03c9\nd\n2\n0\n2\nNote that we have used v = w r"}, {"Chapter": "1", "sentence_range": "5097-5100", "Text": "(6 5), the magnitude of the\nemf generated across a length dr of the rod as it moves at right angles\nto the magnetic field is given by\nd\nBvd\nr\n\u03b5 = Hence,\n\u03b5\n\u03b5\n=\n= \u222b\n\u222b\nd\nBvd\nr\nR\n0\n =\n=\n\u222b B\nr\nr\nB\nR\nR\n\u03c9\n\u03c9\nd\n2\n0\n2\nNote that we have used v = w r This gives\ne \n2\n1\n1"}, {"Chapter": "1", "sentence_range": "5098-5101", "Text": "5), the magnitude of the\nemf generated across a length dr of the rod as it moves at right angles\nto the magnetic field is given by\nd\nBvd\nr\n\u03b5 = Hence,\n\u03b5\n\u03b5\n=\n= \u222b\n\u222b\nd\nBvd\nr\nR\n0\n =\n=\n\u222b B\nr\nr\nB\nR\nR\n\u03c9\n\u03c9\nd\n2\n0\n2\nNote that we have used v = w r This gives\ne \n2\n1\n1 0\n2\n50\n(1 )\n=2\n\u00d7\n\u00d7\n\u03c0 \u00d7\n\u00d7\n= 157 V\nMethod II\nTo calculate the emf, we can imagine a closed loop OPQ in which\npoint O and P are connected with a resistor R and OQ is the rotating\nrod"}, {"Chapter": "1", "sentence_range": "5099-5102", "Text": "Hence,\n\u03b5\n\u03b5\n=\n= \u222b\n\u222b\nd\nBvd\nr\nR\n0\n =\n=\n\u222b B\nr\nr\nB\nR\nR\n\u03c9\n\u03c9\nd\n2\n0\n2\nNote that we have used v = w r This gives\ne \n2\n1\n1 0\n2\n50\n(1 )\n=2\n\u00d7\n\u00d7\n\u03c0 \u00d7\n\u00d7\n= 157 V\nMethod II\nTo calculate the emf, we can imagine a closed loop OPQ in which\npoint O and P are connected with a resistor R and OQ is the rotating\nrod The potential difference across the resistor is then equal to the\ninduced emf and equals B \u00d7 (rate of change of area of loop)"}, {"Chapter": "1", "sentence_range": "5100-5103", "Text": "This gives\ne \n2\n1\n1 0\n2\n50\n(1 )\n=2\n\u00d7\n\u00d7\n\u03c0 \u00d7\n\u00d7\n= 157 V\nMethod II\nTo calculate the emf, we can imagine a closed loop OPQ in which\npoint O and P are connected with a resistor R and OQ is the rotating\nrod The potential difference across the resistor is then equal to the\ninduced emf and equals B \u00d7 (rate of change of area of loop) If q is the\nangle between the rod and the radius of the circle at P at time t, the\narea of the sector OPQ is given by\n2\n2\n1\n2\n2\nR\nR\n\u03b8\n\u03b8\n\u03c0\n\u00d7\n=\n\u03c0\nwhere R is the radius of the circle"}, {"Chapter": "1", "sentence_range": "5101-5104", "Text": "0\n2\n50\n(1 )\n=2\n\u00d7\n\u00d7\n\u03c0 \u00d7\n\u00d7\n= 157 V\nMethod II\nTo calculate the emf, we can imagine a closed loop OPQ in which\npoint O and P are connected with a resistor R and OQ is the rotating\nrod The potential difference across the resistor is then equal to the\ninduced emf and equals B \u00d7 (rate of change of area of loop) If q is the\nangle between the rod and the radius of the circle at P at time t, the\narea of the sector OPQ is given by\n2\n2\n1\n2\n2\nR\nR\n\u03b8\n\u03b8\n\u03c0\n\u00d7\n=\n\u03c0\nwhere R is the radius of the circle Hence, the induced emf is\ne = B\nt\nR\n\u00d7\n\uf8f0\uf8ef\uf8ee\n\uf8fb\uf8fa\uf8f9\ndd\n1\n2\n2\u03b8  = \n2\n2\n1\nd\n2\nd\n2\n\u03b8\n\u03c9\n= B R\nBR\nt\n[Note: \nd\n2\ndt\n\u03b8\n\u03c9\n\u03bd\n=\n=\n\u03c0\n]\nThis expression is identical to the expression obtained by Method I\nand we get the same value of e"}, {"Chapter": "1", "sentence_range": "5102-5105", "Text": "The potential difference across the resistor is then equal to the\ninduced emf and equals B \u00d7 (rate of change of area of loop) If q is the\nangle between the rod and the radius of the circle at P at time t, the\narea of the sector OPQ is given by\n2\n2\n1\n2\n2\nR\nR\n\u03b8\n\u03b8\n\u03c0\n\u00d7\n=\n\u03c0\nwhere R is the radius of the circle Hence, the induced emf is\ne = B\nt\nR\n\u00d7\n\uf8f0\uf8ef\uf8ee\n\uf8fb\uf8fa\uf8f9\ndd\n1\n2\n2\u03b8  = \n2\n2\n1\nd\n2\nd\n2\n\u03b8\n\u03c9\n= B R\nBR\nt\n[Note: \nd\n2\ndt\n\u03b8\n\u03c9\n\u03bd\n=\n=\n\u03c0\n]\nThis expression is identical to the expression obtained by Method I\nand we get the same value of e Rationalised 2023-24\nElectromagnetic\nInduction\n165\n EXAMPLE 6"}, {"Chapter": "1", "sentence_range": "5103-5106", "Text": "If q is the\nangle between the rod and the radius of the circle at P at time t, the\narea of the sector OPQ is given by\n2\n2\n1\n2\n2\nR\nR\n\u03b8\n\u03b8\n\u03c0\n\u00d7\n=\n\u03c0\nwhere R is the radius of the circle Hence, the induced emf is\ne = B\nt\nR\n\u00d7\n\uf8f0\uf8ef\uf8ee\n\uf8fb\uf8fa\uf8f9\ndd\n1\n2\n2\u03b8  = \n2\n2\n1\nd\n2\nd\n2\n\u03b8\n\u03c9\n= B R\nBR\nt\n[Note: \nd\n2\ndt\n\u03b8\n\u03c9\n\u03bd\n=\n=\n\u03c0\n]\nThis expression is identical to the expression obtained by Method I\nand we get the same value of e Rationalised 2023-24\nElectromagnetic\nInduction\n165\n EXAMPLE 6 7\nExample 6"}, {"Chapter": "1", "sentence_range": "5104-5107", "Text": "Hence, the induced emf is\ne = B\nt\nR\n\u00d7\n\uf8f0\uf8ef\uf8ee\n\uf8fb\uf8fa\uf8f9\ndd\n1\n2\n2\u03b8  = \n2\n2\n1\nd\n2\nd\n2\n\u03b8\n\u03c9\n= B R\nBR\nt\n[Note: \nd\n2\ndt\n\u03b8\n\u03c9\n\u03bd\n=\n=\n\u03c0\n]\nThis expression is identical to the expression obtained by Method I\nand we get the same value of e Rationalised 2023-24\nElectromagnetic\nInduction\n165\n EXAMPLE 6 7\nExample 6 7\nA wheel with 10 metallic spokes each 0"}, {"Chapter": "1", "sentence_range": "5105-5108", "Text": "Rationalised 2023-24\nElectromagnetic\nInduction\n165\n EXAMPLE 6 7\nExample 6 7\nA wheel with 10 metallic spokes each 0 5 m long is rotated with a\nspeed of 120 rev/min in a plane normal to the horizontal component\nof earth\u2019s magnetic field HE at a place"}, {"Chapter": "1", "sentence_range": "5106-5109", "Text": "7\nExample 6 7\nA wheel with 10 metallic spokes each 0 5 m long is rotated with a\nspeed of 120 rev/min in a plane normal to the horizontal component\nof earth\u2019s magnetic field HE at a place If HE = 0"}, {"Chapter": "1", "sentence_range": "5107-5110", "Text": "7\nA wheel with 10 metallic spokes each 0 5 m long is rotated with a\nspeed of 120 rev/min in a plane normal to the horizontal component\nof earth\u2019s magnetic field HE at a place If HE = 0 4 G at the place, what\nis the induced emf between the axle and the rim of the wheel"}, {"Chapter": "1", "sentence_range": "5108-5111", "Text": "5 m long is rotated with a\nspeed of 120 rev/min in a plane normal to the horizontal component\nof earth\u2019s magnetic field HE at a place If HE = 0 4 G at the place, what\nis the induced emf between the axle and the rim of the wheel Note\nthat 1 G = 10\u20134 T"}, {"Chapter": "1", "sentence_range": "5109-5112", "Text": "If HE = 0 4 G at the place, what\nis the induced emf between the axle and the rim of the wheel Note\nthat 1 G = 10\u20134 T Solution\nInduced emf = (1/2) w B R2\n= (1/2) \u00d7 4p \u00d7 0"}, {"Chapter": "1", "sentence_range": "5110-5113", "Text": "4 G at the place, what\nis the induced emf between the axle and the rim of the wheel Note\nthat 1 G = 10\u20134 T Solution\nInduced emf = (1/2) w B R2\n= (1/2) \u00d7 4p \u00d7 0 4 \u00d7 10\u20134 \u00d7 (0"}, {"Chapter": "1", "sentence_range": "5111-5114", "Text": "Note\nthat 1 G = 10\u20134 T Solution\nInduced emf = (1/2) w B R2\n= (1/2) \u00d7 4p \u00d7 0 4 \u00d7 10\u20134 \u00d7 (0 5)2\n= 6"}, {"Chapter": "1", "sentence_range": "5112-5115", "Text": "Solution\nInduced emf = (1/2) w B R2\n= (1/2) \u00d7 4p \u00d7 0 4 \u00d7 10\u20134 \u00d7 (0 5)2\n= 6 28 \u00d7 10\u20135 V\nThe number of spokes is immaterial because the emf\u2019s across the\nspokes are in parallel"}, {"Chapter": "1", "sentence_range": "5113-5116", "Text": "4 \u00d7 10\u20134 \u00d7 (0 5)2\n= 6 28 \u00d7 10\u20135 V\nThe number of spokes is immaterial because the emf\u2019s across the\nspokes are in parallel 6"}, {"Chapter": "1", "sentence_range": "5114-5117", "Text": "5)2\n= 6 28 \u00d7 10\u20135 V\nThe number of spokes is immaterial because the emf\u2019s across the\nspokes are in parallel 6 7  INDUCTANCE\nAn electric current can be induced in a coil by flux change produced by\nanother coil in its vicinity or flux change produced by the same coil"}, {"Chapter": "1", "sentence_range": "5115-5118", "Text": "28 \u00d7 10\u20135 V\nThe number of spokes is immaterial because the emf\u2019s across the\nspokes are in parallel 6 7  INDUCTANCE\nAn electric current can be induced in a coil by flux change produced by\nanother coil in its vicinity or flux change produced by the same coil These\ntwo situations are described separately in the next two sub-sections"}, {"Chapter": "1", "sentence_range": "5116-5119", "Text": "6 7  INDUCTANCE\nAn electric current can be induced in a coil by flux change produced by\nanother coil in its vicinity or flux change produced by the same coil These\ntwo situations are described separately in the next two sub-sections However, in both the cases, the flux through a coil is proportional to the\ncurrent"}, {"Chapter": "1", "sentence_range": "5117-5120", "Text": "7  INDUCTANCE\nAn electric current can be induced in a coil by flux change produced by\nanother coil in its vicinity or flux change produced by the same coil These\ntwo situations are described separately in the next two sub-sections However, in both the cases, the flux through a coil is proportional to the\ncurrent That is,  FB a I"}, {"Chapter": "1", "sentence_range": "5118-5121", "Text": "These\ntwo situations are described separately in the next two sub-sections However, in both the cases, the flux through a coil is proportional to the\ncurrent That is,  FB a I Further, if the geometry of the coil does not vary with time then,\nd\nd\nd\nd\nB\nI\nt\nt\n\u03a6\n\u221d\nFor a closely wound coil of N turns, the same magnetic flux is linked\nwith all the turns"}, {"Chapter": "1", "sentence_range": "5119-5122", "Text": "However, in both the cases, the flux through a coil is proportional to the\ncurrent That is,  FB a I Further, if the geometry of the coil does not vary with time then,\nd\nd\nd\nd\nB\nI\nt\nt\n\u03a6\n\u221d\nFor a closely wound coil of N turns, the same magnetic flux is linked\nwith all the turns When the flux FB through the coil changes, each turn\ncontributes to the induced emf"}, {"Chapter": "1", "sentence_range": "5120-5123", "Text": "That is,  FB a I Further, if the geometry of the coil does not vary with time then,\nd\nd\nd\nd\nB\nI\nt\nt\n\u03a6\n\u221d\nFor a closely wound coil of N turns, the same magnetic flux is linked\nwith all the turns When the flux FB through the coil changes, each turn\ncontributes to the induced emf Therefore, a term called flux linkage is\nused which is equal to NFB for a closely wound coil and in such a case\nNFB\u221d  I\nThe constant of proportionality, in this relation, is called inductance"}, {"Chapter": "1", "sentence_range": "5121-5124", "Text": "Further, if the geometry of the coil does not vary with time then,\nd\nd\nd\nd\nB\nI\nt\nt\n\u03a6\n\u221d\nFor a closely wound coil of N turns, the same magnetic flux is linked\nwith all the turns When the flux FB through the coil changes, each turn\ncontributes to the induced emf Therefore, a term called flux linkage is\nused which is equal to NFB for a closely wound coil and in such a case\nNFB\u221d  I\nThe constant of proportionality, in this relation, is called inductance We shall see that inductance depends only on the geometry of the coil\nand intrinsic material properties"}, {"Chapter": "1", "sentence_range": "5122-5125", "Text": "When the flux FB through the coil changes, each turn\ncontributes to the induced emf Therefore, a term called flux linkage is\nused which is equal to NFB for a closely wound coil and in such a case\nNFB\u221d  I\nThe constant of proportionality, in this relation, is called inductance We shall see that inductance depends only on the geometry of the coil\nand intrinsic material properties This aspect is akin to capacitance which\nfor a parallel plate capacitor depends on the plate area and plate separation\n(geometry) and the dielectric constant K of the intervening medium\n(intrinsic material property)"}, {"Chapter": "1", "sentence_range": "5123-5126", "Text": "Therefore, a term called flux linkage is\nused which is equal to NFB for a closely wound coil and in such a case\nNFB\u221d  I\nThe constant of proportionality, in this relation, is called inductance We shall see that inductance depends only on the geometry of the coil\nand intrinsic material properties This aspect is akin to capacitance which\nfor a parallel plate capacitor depends on the plate area and plate separation\n(geometry) and the dielectric constant K of the intervening medium\n(intrinsic material property) Inductance is a scalar quantity"}, {"Chapter": "1", "sentence_range": "5124-5127", "Text": "We shall see that inductance depends only on the geometry of the coil\nand intrinsic material properties This aspect is akin to capacitance which\nfor a parallel plate capacitor depends on the plate area and plate separation\n(geometry) and the dielectric constant K of the intervening medium\n(intrinsic material property) Inductance is a scalar quantity It has the dimensions of [M L2 T \u20132 A\u20132]\ngiven by the dimensions of flux divided by the dimensions of current"}, {"Chapter": "1", "sentence_range": "5125-5128", "Text": "This aspect is akin to capacitance which\nfor a parallel plate capacitor depends on the plate area and plate separation\n(geometry) and the dielectric constant K of the intervening medium\n(intrinsic material property) Inductance is a scalar quantity It has the dimensions of [M L2 T \u20132 A\u20132]\ngiven by the dimensions of flux divided by the dimensions of current The\nSI unit of inductance is henry and is denoted by H"}, {"Chapter": "1", "sentence_range": "5126-5129", "Text": "Inductance is a scalar quantity It has the dimensions of [M L2 T \u20132 A\u20132]\ngiven by the dimensions of flux divided by the dimensions of current The\nSI unit of inductance is henry and is denoted by H It is named in honour\nof Joseph Henry who discovered electromagnetic induction in USA,\nindependently of Faraday in England"}, {"Chapter": "1", "sentence_range": "5127-5130", "Text": "It has the dimensions of [M L2 T \u20132 A\u20132]\ngiven by the dimensions of flux divided by the dimensions of current The\nSI unit of inductance is henry and is denoted by H It is named in honour\nof Joseph Henry who discovered electromagnetic induction in USA,\nindependently of Faraday in England 6"}, {"Chapter": "1", "sentence_range": "5128-5131", "Text": "The\nSI unit of inductance is henry and is denoted by H It is named in honour\nof Joseph Henry who discovered electromagnetic induction in USA,\nindependently of Faraday in England 6 7"}, {"Chapter": "1", "sentence_range": "5129-5132", "Text": "It is named in honour\nof Joseph Henry who discovered electromagnetic induction in USA,\nindependently of Faraday in England 6 7 1  Mutual inductance\nConsider Fig"}, {"Chapter": "1", "sentence_range": "5130-5133", "Text": "6 7 1  Mutual inductance\nConsider Fig 6"}, {"Chapter": "1", "sentence_range": "5131-5134", "Text": "7 1  Mutual inductance\nConsider Fig 6 11 which shows two long co-axial solenoids each of length\nl"}, {"Chapter": "1", "sentence_range": "5132-5135", "Text": "1  Mutual inductance\nConsider Fig 6 11 which shows two long co-axial solenoids each of length\nl We denote the radius of the inner solenoid S1\n by r1 and the number of\nturns per unit length by n1"}, {"Chapter": "1", "sentence_range": "5133-5136", "Text": "6 11 which shows two long co-axial solenoids each of length\nl We denote the radius of the inner solenoid S1\n by r1 and the number of\nturns per unit length by n1 The corresponding quantities for the outer\nsolenoid S2 are r2 and n2, respectively"}, {"Chapter": "1", "sentence_range": "5134-5137", "Text": "11 which shows two long co-axial solenoids each of length\nl We denote the radius of the inner solenoid S1\n by r1 and the number of\nturns per unit length by n1 The corresponding quantities for the outer\nsolenoid S2 are r2 and n2, respectively Let N1 and N2 be the total number\nof turns of coils S1 and S2, respectively"}, {"Chapter": "1", "sentence_range": "5135-5138", "Text": "We denote the radius of the inner solenoid S1\n by r1 and the number of\nturns per unit length by n1 The corresponding quantities for the outer\nsolenoid S2 are r2 and n2, respectively Let N1 and N2 be the total number\nof turns of coils S1 and S2, respectively Rationalised 2023-24\nPhysics\n166\nWhen a current I2 is set up through S2, it in turn sets\nup a magnetic flux through S1"}, {"Chapter": "1", "sentence_range": "5136-5139", "Text": "The corresponding quantities for the outer\nsolenoid S2 are r2 and n2, respectively Let N1 and N2 be the total number\nof turns of coils S1 and S2, respectively Rationalised 2023-24\nPhysics\n166\nWhen a current I2 is set up through S2, it in turn sets\nup a magnetic flux through S1 Let us denote it by F1"}, {"Chapter": "1", "sentence_range": "5137-5140", "Text": "Let N1 and N2 be the total number\nof turns of coils S1 and S2, respectively Rationalised 2023-24\nPhysics\n166\nWhen a current I2 is set up through S2, it in turn sets\nup a magnetic flux through S1 Let us denote it by F1 The corresponding flux linkage with  solenoid S1 is\nN1\n1\nM12 2\nI\n\u03a6 =\n(6"}, {"Chapter": "1", "sentence_range": "5138-5141", "Text": "Rationalised 2023-24\nPhysics\n166\nWhen a current I2 is set up through S2, it in turn sets\nup a magnetic flux through S1 Let us denote it by F1 The corresponding flux linkage with  solenoid S1 is\nN1\n1\nM12 2\nI\n\u03a6 =\n(6 7)\nM12 is called the mutual inductance of solenoid S1 with\nrespect to solenoid S2"}, {"Chapter": "1", "sentence_range": "5139-5142", "Text": "Let us denote it by F1 The corresponding flux linkage with  solenoid S1 is\nN1\n1\nM12 2\nI\n\u03a6 =\n(6 7)\nM12 is called the mutual inductance of solenoid S1 with\nrespect to solenoid S2 It is also referred to as the\ncoefficient of mutual induction"}, {"Chapter": "1", "sentence_range": "5140-5143", "Text": "The corresponding flux linkage with  solenoid S1 is\nN1\n1\nM12 2\nI\n\u03a6 =\n(6 7)\nM12 is called the mutual inductance of solenoid S1 with\nrespect to solenoid S2 It is also referred to as the\ncoefficient of mutual induction For these simple co-axial solenoids it is possible to\ncalculate M12"}, {"Chapter": "1", "sentence_range": "5141-5144", "Text": "7)\nM12 is called the mutual inductance of solenoid S1 with\nrespect to solenoid S2 It is also referred to as the\ncoefficient of mutual induction For these simple co-axial solenoids it is possible to\ncalculate M12 The magnetic field due to the current I2 in\nS2 is m0n2I2"}, {"Chapter": "1", "sentence_range": "5142-5145", "Text": "It is also referred to as the\ncoefficient of mutual induction For these simple co-axial solenoids it is possible to\ncalculate M12 The magnetic field due to the current I2 in\nS2 is m0n2I2 The resulting flux linkage with coil S1 is,\n(\n) (\n) (\n)\n2\n1\n1\n1\n1\n0\n2 2\nN\nn l\nr\nn I\n\u03a6\n\u00b5\n=\n\u03c0\n         \n2\n0\n1\n2\n1\n2\nn n\nr l I\n=\u00b5\n\u03c0\n(6"}, {"Chapter": "1", "sentence_range": "5143-5146", "Text": "For these simple co-axial solenoids it is possible to\ncalculate M12 The magnetic field due to the current I2 in\nS2 is m0n2I2 The resulting flux linkage with coil S1 is,\n(\n) (\n) (\n)\n2\n1\n1\n1\n1\n0\n2 2\nN\nn l\nr\nn I\n\u03a6\n\u00b5\n=\n\u03c0\n         \n2\n0\n1\n2\n1\n2\nn n\nr l I\n=\u00b5\n\u03c0\n(6 8)\nwhere n1l is the total number of turns in solenoid S1"}, {"Chapter": "1", "sentence_range": "5144-5147", "Text": "The magnetic field due to the current I2 in\nS2 is m0n2I2 The resulting flux linkage with coil S1 is,\n(\n) (\n) (\n)\n2\n1\n1\n1\n1\n0\n2 2\nN\nn l\nr\nn I\n\u03a6\n\u00b5\n=\n\u03c0\n         \n2\n0\n1\n2\n1\n2\nn n\nr l I\n=\u00b5\n\u03c0\n(6 8)\nwhere n1l is the total number of turns in solenoid S1 Thus,\nfrom Eq"}, {"Chapter": "1", "sentence_range": "5145-5148", "Text": "The resulting flux linkage with coil S1 is,\n(\n) (\n) (\n)\n2\n1\n1\n1\n1\n0\n2 2\nN\nn l\nr\nn I\n\u03a6\n\u00b5\n=\n\u03c0\n         \n2\n0\n1\n2\n1\n2\nn n\nr l I\n=\u00b5\n\u03c0\n(6 8)\nwhere n1l is the total number of turns in solenoid S1 Thus,\nfrom Eq (6"}, {"Chapter": "1", "sentence_range": "5146-5149", "Text": "8)\nwhere n1l is the total number of turns in solenoid S1 Thus,\nfrom Eq (6 9) and Eq"}, {"Chapter": "1", "sentence_range": "5147-5150", "Text": "Thus,\nfrom Eq (6 9) and Eq (6"}, {"Chapter": "1", "sentence_range": "5148-5151", "Text": "(6 9) and Eq (6 10),\nM12 = m0n1n2pr 2\n1l\n(6"}, {"Chapter": "1", "sentence_range": "5149-5152", "Text": "9) and Eq (6 10),\nM12 = m0n1n2pr 2\n1l\n(6 9)\nNote that we neglected the edge effects and considered\nthe magnetic field m0n2I2 to be uniform throughout the\nlength and width of the solenoid S2"}, {"Chapter": "1", "sentence_range": "5150-5153", "Text": "(6 10),\nM12 = m0n1n2pr 2\n1l\n(6 9)\nNote that we neglected the edge effects and considered\nthe magnetic field m0n2I2 to be uniform throughout the\nlength and width of the solenoid S2 This is a good approximation keeping\nin mind that the solenoid is long, implying l  >> r2"}, {"Chapter": "1", "sentence_range": "5151-5154", "Text": "10),\nM12 = m0n1n2pr 2\n1l\n(6 9)\nNote that we neglected the edge effects and considered\nthe magnetic field m0n2I2 to be uniform throughout the\nlength and width of the solenoid S2 This is a good approximation keeping\nin mind that the solenoid is long, implying l  >> r2 We now consider the reverse case"}, {"Chapter": "1", "sentence_range": "5152-5155", "Text": "9)\nNote that we neglected the edge effects and considered\nthe magnetic field m0n2I2 to be uniform throughout the\nlength and width of the solenoid S2 This is a good approximation keeping\nin mind that the solenoid is long, implying l  >> r2 We now consider the reverse case A current I1 is passed through the\nsolenoid S1 and the flux linkage with coil S2 is,\nN2F2 = M21 I1\n(6"}, {"Chapter": "1", "sentence_range": "5153-5156", "Text": "This is a good approximation keeping\nin mind that the solenoid is long, implying l  >> r2 We now consider the reverse case A current I1 is passed through the\nsolenoid S1 and the flux linkage with coil S2 is,\nN2F2 = M21 I1\n(6 10)\nM21 is called the mutual inductance of solenoid S2 with respect to\nsolenoid S1"}, {"Chapter": "1", "sentence_range": "5154-5157", "Text": "We now consider the reverse case A current I1 is passed through the\nsolenoid S1 and the flux linkage with coil S2 is,\nN2F2 = M21 I1\n(6 10)\nM21 is called the mutual inductance of solenoid S2 with respect to\nsolenoid S1 The flux due to the current I1 in S1 can be assumed to be confined\nsolely inside S1 since the solenoids are very long"}, {"Chapter": "1", "sentence_range": "5155-5158", "Text": "A current I1 is passed through the\nsolenoid S1 and the flux linkage with coil S2 is,\nN2F2 = M21 I1\n(6 10)\nM21 is called the mutual inductance of solenoid S2 with respect to\nsolenoid S1 The flux due to the current I1 in S1 can be assumed to be confined\nsolely inside S1 since the solenoids are very long Thus, flux linkage with\nsolenoid S2 is\n(\n) (\n) (\n)\n2\n2\n2\n2\n1\n0\n1 1\nN\nn l\nr\nn I\n\u03a6\n\u00b5\n=\n\u03c0\nwhere n2l is the total number of turns of S2"}, {"Chapter": "1", "sentence_range": "5156-5159", "Text": "10)\nM21 is called the mutual inductance of solenoid S2 with respect to\nsolenoid S1 The flux due to the current I1 in S1 can be assumed to be confined\nsolely inside S1 since the solenoids are very long Thus, flux linkage with\nsolenoid S2 is\n(\n) (\n) (\n)\n2\n2\n2\n2\n1\n0\n1 1\nN\nn l\nr\nn I\n\u03a6\n\u00b5\n=\n\u03c0\nwhere n2l is the total number of turns of S2 From Eq"}, {"Chapter": "1", "sentence_range": "5157-5160", "Text": "The flux due to the current I1 in S1 can be assumed to be confined\nsolely inside S1 since the solenoids are very long Thus, flux linkage with\nsolenoid S2 is\n(\n) (\n) (\n)\n2\n2\n2\n2\n1\n0\n1 1\nN\nn l\nr\nn I\n\u03a6\n\u00b5\n=\n\u03c0\nwhere n2l is the total number of turns of S2 From Eq (6"}, {"Chapter": "1", "sentence_range": "5158-5161", "Text": "Thus, flux linkage with\nsolenoid S2 is\n(\n) (\n) (\n)\n2\n2\n2\n2\n1\n0\n1 1\nN\nn l\nr\nn I\n\u03a6\n\u00b5\n=\n\u03c0\nwhere n2l is the total number of turns of S2 From Eq (6 12),\nM21 = m0n1n2pr 2\n1l\n(6"}, {"Chapter": "1", "sentence_range": "5159-5162", "Text": "From Eq (6 12),\nM21 = m0n1n2pr 2\n1l\n(6 11)\nUsing Eq"}, {"Chapter": "1", "sentence_range": "5160-5163", "Text": "(6 12),\nM21 = m0n1n2pr 2\n1l\n(6 11)\nUsing Eq (6"}, {"Chapter": "1", "sentence_range": "5161-5164", "Text": "12),\nM21 = m0n1n2pr 2\n1l\n(6 11)\nUsing Eq (6 11) and Eq"}, {"Chapter": "1", "sentence_range": "5162-5165", "Text": "11)\nUsing Eq (6 11) and Eq (6"}, {"Chapter": "1", "sentence_range": "5163-5166", "Text": "(6 11) and Eq (6 12), we get\nM12 = M21= M (say)\n(6"}, {"Chapter": "1", "sentence_range": "5164-5167", "Text": "11) and Eq (6 12), we get\nM12 = M21= M (say)\n(6 12)\nWe have demonstrated this equality for long co-axial solenoids"}, {"Chapter": "1", "sentence_range": "5165-5168", "Text": "(6 12), we get\nM12 = M21= M (say)\n(6 12)\nWe have demonstrated this equality for long co-axial solenoids However, the relation is far more general"}, {"Chapter": "1", "sentence_range": "5166-5169", "Text": "12), we get\nM12 = M21= M (say)\n(6 12)\nWe have demonstrated this equality for long co-axial solenoids However, the relation is far more general Note that if the inner solenoid\nwas much shorter than (and placed well inside) the outer solenoid, then\nwe could still have calculated the flux linkage N1F1 because the inner\nsolenoid is effectively immersed in a uniform magnetic field due to the\nouter solenoid"}, {"Chapter": "1", "sentence_range": "5167-5170", "Text": "12)\nWe have demonstrated this equality for long co-axial solenoids However, the relation is far more general Note that if the inner solenoid\nwas much shorter than (and placed well inside) the outer solenoid, then\nwe could still have calculated the flux linkage N1F1 because the inner\nsolenoid is effectively immersed in a uniform magnetic field due to the\nouter solenoid In this case, the calculation of M12 would be easy"}, {"Chapter": "1", "sentence_range": "5168-5171", "Text": "However, the relation is far more general Note that if the inner solenoid\nwas much shorter than (and placed well inside) the outer solenoid, then\nwe could still have calculated the flux linkage N1F1 because the inner\nsolenoid is effectively immersed in a uniform magnetic field due to the\nouter solenoid In this case, the calculation of M12 would be easy However,\nit would be extremely difficult to calculate the flux linkage with the outer\nsolenoid as the magnetic field due to the inner solenoid would vary across\nthe length as well as cross section of the outer solenoid"}, {"Chapter": "1", "sentence_range": "5169-5172", "Text": "Note that if the inner solenoid\nwas much shorter than (and placed well inside) the outer solenoid, then\nwe could still have calculated the flux linkage N1F1 because the inner\nsolenoid is effectively immersed in a uniform magnetic field due to the\nouter solenoid In this case, the calculation of M12 would be easy However,\nit would be extremely difficult to calculate the flux linkage with the outer\nsolenoid as the magnetic field due to the inner solenoid would vary across\nthe length as well as cross section of the outer solenoid Therefore, the\ncalculation of M21 would also be extremely difficult in this case"}, {"Chapter": "1", "sentence_range": "5170-5173", "Text": "In this case, the calculation of M12 would be easy However,\nit would be extremely difficult to calculate the flux linkage with the outer\nsolenoid as the magnetic field due to the inner solenoid would vary across\nthe length as well as cross section of the outer solenoid Therefore, the\ncalculation of M21 would also be extremely difficult in this case The\nequality M12=M21 is very useful in such situations"}, {"Chapter": "1", "sentence_range": "5171-5174", "Text": "However,\nit would be extremely difficult to calculate the flux linkage with the outer\nsolenoid as the magnetic field due to the inner solenoid would vary across\nthe length as well as cross section of the outer solenoid Therefore, the\ncalculation of M21 would also be extremely difficult in this case The\nequality M12=M21 is very useful in such situations FIGURE 6"}, {"Chapter": "1", "sentence_range": "5172-5175", "Text": "Therefore, the\ncalculation of M21 would also be extremely difficult in this case The\nequality M12=M21 is very useful in such situations FIGURE 6 12 Two long co-axial\nsolenoids of same\nlength l"}, {"Chapter": "1", "sentence_range": "5173-5176", "Text": "The\nequality M12=M21 is very useful in such situations FIGURE 6 12 Two long co-axial\nsolenoids of same\nlength l Rationalised 2023-24\nElectromagnetic\nInduction\n167\n EXAMPLE 6"}, {"Chapter": "1", "sentence_range": "5174-5177", "Text": "FIGURE 6 12 Two long co-axial\nsolenoids of same\nlength l Rationalised 2023-24\nElectromagnetic\nInduction\n167\n EXAMPLE 6 8\nWe explained the above example with air as the medium within the\nsolenoids"}, {"Chapter": "1", "sentence_range": "5175-5178", "Text": "12 Two long co-axial\nsolenoids of same\nlength l Rationalised 2023-24\nElectromagnetic\nInduction\n167\n EXAMPLE 6 8\nWe explained the above example with air as the medium within the\nsolenoids Instead, if a medium of relative permeability mr had been present,\nthe mutual inductance would be\nM =mr m0 n1n2p r2\n1 l\nIt is also important to know that the mutual inductance of a pair of\ncoils, solenoids, etc"}, {"Chapter": "1", "sentence_range": "5176-5179", "Text": "Rationalised 2023-24\nElectromagnetic\nInduction\n167\n EXAMPLE 6 8\nWe explained the above example with air as the medium within the\nsolenoids Instead, if a medium of relative permeability mr had been present,\nthe mutual inductance would be\nM =mr m0 n1n2p r2\n1 l\nIt is also important to know that the mutual inductance of a pair of\ncoils, solenoids, etc , depends on their separation as well as their relative\norientation"}, {"Chapter": "1", "sentence_range": "5177-5180", "Text": "8\nWe explained the above example with air as the medium within the\nsolenoids Instead, if a medium of relative permeability mr had been present,\nthe mutual inductance would be\nM =mr m0 n1n2p r2\n1 l\nIt is also important to know that the mutual inductance of a pair of\ncoils, solenoids, etc , depends on their separation as well as their relative\norientation Example 6"}, {"Chapter": "1", "sentence_range": "5178-5181", "Text": "Instead, if a medium of relative permeability mr had been present,\nthe mutual inductance would be\nM =mr m0 n1n2p r2\n1 l\nIt is also important to know that the mutual inductance of a pair of\ncoils, solenoids, etc , depends on their separation as well as their relative\norientation Example 6 8 Two concentric circular coils, one of small radius r1 and\nthe other of large radius r2, such that r1 << r2,  are placed co-axially\nwith centres coinciding"}, {"Chapter": "1", "sentence_range": "5179-5182", "Text": ", depends on their separation as well as their relative\norientation Example 6 8 Two concentric circular coils, one of small radius r1 and\nthe other of large radius r2, such that r1 << r2,  are placed co-axially\nwith centres coinciding Obtain the mutual inductance of the\narrangement"}, {"Chapter": "1", "sentence_range": "5180-5183", "Text": "Example 6 8 Two concentric circular coils, one of small radius r1 and\nthe other of large radius r2, such that r1 << r2,  are placed co-axially\nwith centres coinciding Obtain the mutual inductance of the\narrangement Solution  Let a current I2 flow through the outer circular coil"}, {"Chapter": "1", "sentence_range": "5181-5184", "Text": "8 Two concentric circular coils, one of small radius r1 and\nthe other of large radius r2, such that r1 << r2,  are placed co-axially\nwith centres coinciding Obtain the mutual inductance of the\narrangement Solution  Let a current I2 flow through the outer circular coil The\nfield at the centre of the coil is B2 = m0I2 / 2r2"}, {"Chapter": "1", "sentence_range": "5182-5185", "Text": "Obtain the mutual inductance of the\narrangement Solution  Let a current I2 flow through the outer circular coil The\nfield at the centre of the coil is B2 = m0I2 / 2r2 Since the other\nco-axially placed coil has a very small radius, B2 may be considered\nconstant over its cross-sectional area"}, {"Chapter": "1", "sentence_range": "5183-5186", "Text": "Solution  Let a current I2 flow through the outer circular coil The\nfield at the centre of the coil is B2 = m0I2 / 2r2 Since the other\nco-axially placed coil has a very small radius, B2 may be considered\nconstant over its cross-sectional area Hence,\nF1 = pr 2\n1B2\n     \n2\n0\n1\n2\n22\nr\nI\n\u00b5 \u03c0r\n=\n     = M12 I2\nThus,\n2\n0\n1\n12\n2\n2\nr\nM\nr\n\u00b5 \u03c0\n=\nFrom Eq"}, {"Chapter": "1", "sentence_range": "5184-5187", "Text": "The\nfield at the centre of the coil is B2 = m0I2 / 2r2 Since the other\nco-axially placed coil has a very small radius, B2 may be considered\nconstant over its cross-sectional area Hence,\nF1 = pr 2\n1B2\n     \n2\n0\n1\n2\n22\nr\nI\n\u00b5 \u03c0r\n=\n     = M12 I2\nThus,\n2\n0\n1\n12\n2\n2\nr\nM\nr\n\u00b5 \u03c0\n=\nFrom Eq (6"}, {"Chapter": "1", "sentence_range": "5185-5188", "Text": "Since the other\nco-axially placed coil has a very small radius, B2 may be considered\nconstant over its cross-sectional area Hence,\nF1 = pr 2\n1B2\n     \n2\n0\n1\n2\n22\nr\nI\n\u00b5 \u03c0r\n=\n     = M12 I2\nThus,\n2\n0\n1\n12\n2\n2\nr\nM\nr\n\u00b5 \u03c0\n=\nFrom Eq (6 12)\n2\n0\n1\n12\n21\n2\n2\nr\nM\nM\nr\n\u00b5 \u03c0\n=\n=\nNote that we calculated M12 from an approximate value of F1, assuming\nthe magnetic field B2 to be uniform over the area p r1\n2"}, {"Chapter": "1", "sentence_range": "5186-5189", "Text": "Hence,\nF1 = pr 2\n1B2\n     \n2\n0\n1\n2\n22\nr\nI\n\u00b5 \u03c0r\n=\n     = M12 I2\nThus,\n2\n0\n1\n12\n2\n2\nr\nM\nr\n\u00b5 \u03c0\n=\nFrom Eq (6 12)\n2\n0\n1\n12\n21\n2\n2\nr\nM\nM\nr\n\u00b5 \u03c0\n=\n=\nNote that we calculated M12 from an approximate value of F1, assuming\nthe magnetic field B2 to be uniform over the area p r1\n2 However, we\ncan accept this value because r1 << r2"}, {"Chapter": "1", "sentence_range": "5187-5190", "Text": "(6 12)\n2\n0\n1\n12\n21\n2\n2\nr\nM\nM\nr\n\u00b5 \u03c0\n=\n=\nNote that we calculated M12 from an approximate value of F1, assuming\nthe magnetic field B2 to be uniform over the area p r1\n2 However, we\ncan accept this value because r1 << r2 Now, let us recollect Experiment 6"}, {"Chapter": "1", "sentence_range": "5188-5191", "Text": "12)\n2\n0\n1\n12\n21\n2\n2\nr\nM\nM\nr\n\u00b5 \u03c0\n=\n=\nNote that we calculated M12 from an approximate value of F1, assuming\nthe magnetic field B2 to be uniform over the area p r1\n2 However, we\ncan accept this value because r1 << r2 Now, let us recollect Experiment 6 3 in Section 6"}, {"Chapter": "1", "sentence_range": "5189-5192", "Text": "However, we\ncan accept this value because r1 << r2 Now, let us recollect Experiment 6 3 in Section 6 2"}, {"Chapter": "1", "sentence_range": "5190-5193", "Text": "Now, let us recollect Experiment 6 3 in Section 6 2 In that experiment,\nemf is induced in coil C1 wherever there was any change in current through\ncoil C2"}, {"Chapter": "1", "sentence_range": "5191-5194", "Text": "3 in Section 6 2 In that experiment,\nemf is induced in coil C1 wherever there was any change in current through\ncoil C2 Let F1 be the flux through coil C1 (say of N1 turns) when current in\ncoil C2 is I2"}, {"Chapter": "1", "sentence_range": "5192-5195", "Text": "2 In that experiment,\nemf is induced in coil C1 wherever there was any change in current through\ncoil C2 Let F1 be the flux through coil C1 (say of N1 turns) when current in\ncoil C2 is I2 Then, from Eq"}, {"Chapter": "1", "sentence_range": "5193-5196", "Text": "In that experiment,\nemf is induced in coil C1 wherever there was any change in current through\ncoil C2 Let F1 be the flux through coil C1 (say of N1 turns) when current in\ncoil C2 is I2 Then, from Eq (6"}, {"Chapter": "1", "sentence_range": "5194-5197", "Text": "Let F1 be the flux through coil C1 (say of N1 turns) when current in\ncoil C2 is I2 Then, from Eq (6 7), we have\nN1F1 = MI2\nFor currents varrying with time,\n(\n)\n(\n)\n1\n1\n2\nd\nd\nd\nd\nN\nMI\nt\nt\n\u03a6\n=\nSince induced emf in coil C1 is given by\n(\n)\n1\n1\n\u2013d\nd\nN\nt\n\u03a6\n\u03b51 =\nWe get,\nd2\n\u2013\nId\nM\nt\n\u03b51 =\nRationalised 2023-24\nPhysics\n168\nIt shows that varying current in a coil can induce emf in a neighbouring\ncoil"}, {"Chapter": "1", "sentence_range": "5195-5198", "Text": "Then, from Eq (6 7), we have\nN1F1 = MI2\nFor currents varrying with time,\n(\n)\n(\n)\n1\n1\n2\nd\nd\nd\nd\nN\nMI\nt\nt\n\u03a6\n=\nSince induced emf in coil C1 is given by\n(\n)\n1\n1\n\u2013d\nd\nN\nt\n\u03a6\n\u03b51 =\nWe get,\nd2\n\u2013\nId\nM\nt\n\u03b51 =\nRationalised 2023-24\nPhysics\n168\nIt shows that varying current in a coil can induce emf in a neighbouring\ncoil The magnitude of the induced emf depends upon the rate of change\nof current and mutual inductance of the two coils"}, {"Chapter": "1", "sentence_range": "5196-5199", "Text": "(6 7), we have\nN1F1 = MI2\nFor currents varrying with time,\n(\n)\n(\n)\n1\n1\n2\nd\nd\nd\nd\nN\nMI\nt\nt\n\u03a6\n=\nSince induced emf in coil C1 is given by\n(\n)\n1\n1\n\u2013d\nd\nN\nt\n\u03a6\n\u03b51 =\nWe get,\nd2\n\u2013\nId\nM\nt\n\u03b51 =\nRationalised 2023-24\nPhysics\n168\nIt shows that varying current in a coil can induce emf in a neighbouring\ncoil The magnitude of the induced emf depends upon the rate of change\nof current and mutual inductance of the two coils 6"}, {"Chapter": "1", "sentence_range": "5197-5200", "Text": "7), we have\nN1F1 = MI2\nFor currents varrying with time,\n(\n)\n(\n)\n1\n1\n2\nd\nd\nd\nd\nN\nMI\nt\nt\n\u03a6\n=\nSince induced emf in coil C1 is given by\n(\n)\n1\n1\n\u2013d\nd\nN\nt\n\u03a6\n\u03b51 =\nWe get,\nd2\n\u2013\nId\nM\nt\n\u03b51 =\nRationalised 2023-24\nPhysics\n168\nIt shows that varying current in a coil can induce emf in a neighbouring\ncoil The magnitude of the induced emf depends upon the rate of change\nof current and mutual inductance of the two coils 6 7"}, {"Chapter": "1", "sentence_range": "5198-5201", "Text": "The magnitude of the induced emf depends upon the rate of change\nof current and mutual inductance of the two coils 6 7 2  Self-inductance\nIn the previous sub-section, we considered the flux in one solenoid due\nto the current in the other"}, {"Chapter": "1", "sentence_range": "5199-5202", "Text": "6 7 2  Self-inductance\nIn the previous sub-section, we considered the flux in one solenoid due\nto the current in the other It is also possible that emf is induced in a\nsingle isolated coil due to change of flux through the coil by means of\nvarying the current through the same coil"}, {"Chapter": "1", "sentence_range": "5200-5203", "Text": "7 2  Self-inductance\nIn the previous sub-section, we considered the flux in one solenoid due\nto the current in the other It is also possible that emf is induced in a\nsingle isolated coil due to change of flux through the coil by means of\nvarying the current through the same coil This phenomenon is called\nself-induction"}, {"Chapter": "1", "sentence_range": "5201-5204", "Text": "2  Self-inductance\nIn the previous sub-section, we considered the flux in one solenoid due\nto the current in the other It is also possible that emf is induced in a\nsingle isolated coil due to change of flux through the coil by means of\nvarying the current through the same coil This phenomenon is called\nself-induction In this case, flux linkage through a coil of N turns is\nproportional to the current through the coil and is expressed as\nNB\nI\n\u03a6 \u221d\nB\nL\nN\nI\n\u03a6\n=\n(6"}, {"Chapter": "1", "sentence_range": "5202-5205", "Text": "It is also possible that emf is induced in a\nsingle isolated coil due to change of flux through the coil by means of\nvarying the current through the same coil This phenomenon is called\nself-induction In this case, flux linkage through a coil of N turns is\nproportional to the current through the coil and is expressed as\nNB\nI\n\u03a6 \u221d\nB\nL\nN\nI\n\u03a6\n=\n(6 13)\nwhere constant of proportionality L is called self-inductance of the coil"}, {"Chapter": "1", "sentence_range": "5203-5206", "Text": "This phenomenon is called\nself-induction In this case, flux linkage through a coil of N turns is\nproportional to the current through the coil and is expressed as\nNB\nI\n\u03a6 \u221d\nB\nL\nN\nI\n\u03a6\n=\n(6 13)\nwhere constant of proportionality L is called self-inductance of the coil It\nis also called the coefficient of self-induction of the coil"}, {"Chapter": "1", "sentence_range": "5204-5207", "Text": "In this case, flux linkage through a coil of N turns is\nproportional to the current through the coil and is expressed as\nNB\nI\n\u03a6 \u221d\nB\nL\nN\nI\n\u03a6\n=\n(6 13)\nwhere constant of proportionality L is called self-inductance of the coil It\nis also called the coefficient of self-induction of the coil When the current\nis varied, the flux linked with the coil also changes and an emf is induced\nin the coil"}, {"Chapter": "1", "sentence_range": "5205-5208", "Text": "13)\nwhere constant of proportionality L is called self-inductance of the coil It\nis also called the coefficient of self-induction of the coil When the current\nis varied, the flux linked with the coil also changes and an emf is induced\nin the coil Using Eq"}, {"Chapter": "1", "sentence_range": "5206-5209", "Text": "It\nis also called the coefficient of self-induction of the coil When the current\nis varied, the flux linked with the coil also changes and an emf is induced\nin the coil Using Eq (6"}, {"Chapter": "1", "sentence_range": "5207-5210", "Text": "When the current\nis varied, the flux linked with the coil also changes and an emf is induced\nin the coil Using Eq (6 13), the induced emf is given by\n(\nB)\n\u2013d\nd\nN\nt\n\u03a6\n\u03b5 =\nd\n\u2013\nd\nI\nL\nt\n\u03b5 =\n(6"}, {"Chapter": "1", "sentence_range": "5208-5211", "Text": "Using Eq (6 13), the induced emf is given by\n(\nB)\n\u2013d\nd\nN\nt\n\u03a6\n\u03b5 =\nd\n\u2013\nd\nI\nL\nt\n\u03b5 =\n(6 14)\nThus, the self-induced emf always opposes any change  (increase or\ndecrease) of current in the coil"}, {"Chapter": "1", "sentence_range": "5209-5212", "Text": "(6 13), the induced emf is given by\n(\nB)\n\u2013d\nd\nN\nt\n\u03a6\n\u03b5 =\nd\n\u2013\nd\nI\nL\nt\n\u03b5 =\n(6 14)\nThus, the self-induced emf always opposes any change  (increase or\ndecrease) of current in the coil It is possible to calculate the self-inductance for circuits with simple\ngeometries"}, {"Chapter": "1", "sentence_range": "5210-5213", "Text": "13), the induced emf is given by\n(\nB)\n\u2013d\nd\nN\nt\n\u03a6\n\u03b5 =\nd\n\u2013\nd\nI\nL\nt\n\u03b5 =\n(6 14)\nThus, the self-induced emf always opposes any change  (increase or\ndecrease) of current in the coil It is possible to calculate the self-inductance for circuits with simple\ngeometries Let us calculate the self-inductance of a long solenoid of cross-\nsectional area A and length l, having n turns per unit length"}, {"Chapter": "1", "sentence_range": "5211-5214", "Text": "14)\nThus, the self-induced emf always opposes any change  (increase or\ndecrease) of current in the coil It is possible to calculate the self-inductance for circuits with simple\ngeometries Let us calculate the self-inductance of a long solenoid of cross-\nsectional area A and length l, having n turns per unit length The magnetic\nfield due to a current I flowing in the solenoid is B = m0 n I  (neglecting edge\neffects, as before)"}, {"Chapter": "1", "sentence_range": "5212-5215", "Text": "It is possible to calculate the self-inductance for circuits with simple\ngeometries Let us calculate the self-inductance of a long solenoid of cross-\nsectional area A and length l, having n turns per unit length The magnetic\nfield due to a current I flowing in the solenoid is B = m0 n I  (neglecting edge\neffects, as before) The total flux linked with the solenoid is\n(\n)(\n)( )\n0\nNB\nnl\nn I\nA\n\u03a6\n\u00b5\n=\n\uf06d0n2AlI\n\uf03d\nwhere nl is the total number of turns"}, {"Chapter": "1", "sentence_range": "5213-5216", "Text": "Let us calculate the self-inductance of a long solenoid of cross-\nsectional area A and length l, having n turns per unit length The magnetic\nfield due to a current I flowing in the solenoid is B = m0 n I  (neglecting edge\neffects, as before) The total flux linked with the solenoid is\n(\n)(\n)( )\n0\nNB\nnl\nn I\nA\n\u03a6\n\u00b5\n=\n\uf06d0n2AlI\n\uf03d\nwhere nl is the total number of turns Thus, the self-inductance is,\nL\nI\n\u039d\u03a6\u0392\n=\n   \n2\n=\u00b50n Al\n(6"}, {"Chapter": "1", "sentence_range": "5214-5217", "Text": "The magnetic\nfield due to a current I flowing in the solenoid is B = m0 n I  (neglecting edge\neffects, as before) The total flux linked with the solenoid is\n(\n)(\n)( )\n0\nNB\nnl\nn I\nA\n\u03a6\n\u00b5\n=\n\uf06d0n2AlI\n\uf03d\nwhere nl is the total number of turns Thus, the self-inductance is,\nL\nI\n\u039d\u03a6\u0392\n=\n   \n2\n=\u00b50n Al\n(6 15)\nIf we fill the inside of the solenoid with a material of relative permeability\nmr (for example soft iron, which has a high value of relative permeability),\nthen,\n2\n0\nr\nL\n=\u00b5 \u00b5n Al\n(6"}, {"Chapter": "1", "sentence_range": "5215-5218", "Text": "The total flux linked with the solenoid is\n(\n)(\n)( )\n0\nNB\nnl\nn I\nA\n\u03a6\n\u00b5\n=\n\uf06d0n2AlI\n\uf03d\nwhere nl is the total number of turns Thus, the self-inductance is,\nL\nI\n\u039d\u03a6\u0392\n=\n   \n2\n=\u00b50n Al\n(6 15)\nIf we fill the inside of the solenoid with a material of relative permeability\nmr (for example soft iron, which has a high value of relative permeability),\nthen,\n2\n0\nr\nL\n=\u00b5 \u00b5n Al\n(6 16)\nThe self-inductance of the coil depends on its geometry and on the\npermeability of the medium"}, {"Chapter": "1", "sentence_range": "5216-5219", "Text": "Thus, the self-inductance is,\nL\nI\n\u039d\u03a6\u0392\n=\n   \n2\n=\u00b50n Al\n(6 15)\nIf we fill the inside of the solenoid with a material of relative permeability\nmr (for example soft iron, which has a high value of relative permeability),\nthen,\n2\n0\nr\nL\n=\u00b5 \u00b5n Al\n(6 16)\nThe self-inductance of the coil depends on its geometry and on the\npermeability of the medium The self-induced emf is also called the back emf as it opposes any\nchange in the current in a circuit"}, {"Chapter": "1", "sentence_range": "5217-5220", "Text": "15)\nIf we fill the inside of the solenoid with a material of relative permeability\nmr (for example soft iron, which has a high value of relative permeability),\nthen,\n2\n0\nr\nL\n=\u00b5 \u00b5n Al\n(6 16)\nThe self-inductance of the coil depends on its geometry and on the\npermeability of the medium The self-induced emf is also called the back emf as it opposes any\nchange in the current in a circuit Physically, the self-inductance plays\nRationalised 2023-24\nElectromagnetic\nInduction\n169\nthe role of inertia"}, {"Chapter": "1", "sentence_range": "5218-5221", "Text": "16)\nThe self-inductance of the coil depends on its geometry and on the\npermeability of the medium The self-induced emf is also called the back emf as it opposes any\nchange in the current in a circuit Physically, the self-inductance plays\nRationalised 2023-24\nElectromagnetic\nInduction\n169\nthe role of inertia It is the electromagnetic analogue of mass in mechanics"}, {"Chapter": "1", "sentence_range": "5219-5222", "Text": "The self-induced emf is also called the back emf as it opposes any\nchange in the current in a circuit Physically, the self-inductance plays\nRationalised 2023-24\nElectromagnetic\nInduction\n169\nthe role of inertia It is the electromagnetic analogue of mass in mechanics So, work needs to be done against the back emf (e) in establishing the\ncurrent"}, {"Chapter": "1", "sentence_range": "5220-5223", "Text": "Physically, the self-inductance plays\nRationalised 2023-24\nElectromagnetic\nInduction\n169\nthe role of inertia It is the electromagnetic analogue of mass in mechanics So, work needs to be done against the back emf (e) in establishing the\ncurrent This work done is stored as magnetic potential energy"}, {"Chapter": "1", "sentence_range": "5221-5224", "Text": "It is the electromagnetic analogue of mass in mechanics So, work needs to be done against the back emf (e) in establishing the\ncurrent This work done is stored as magnetic potential energy For the\ncurrent I at an instant in a circuit, the rate of work done is\nd\nd\nW\nI\nt\n\u03b5\n=\nIf we ignore the resistive losses and consider only inductive effect,\nthen using Eq"}, {"Chapter": "1", "sentence_range": "5222-5225", "Text": "So, work needs to be done against the back emf (e) in establishing the\ncurrent This work done is stored as magnetic potential energy For the\ncurrent I at an instant in a circuit, the rate of work done is\nd\nd\nW\nI\nt\n\u03b5\n=\nIf we ignore the resistive losses and consider only inductive effect,\nthen using Eq (6"}, {"Chapter": "1", "sentence_range": "5223-5226", "Text": "This work done is stored as magnetic potential energy For the\ncurrent I at an instant in a circuit, the rate of work done is\nd\nd\nW\nI\nt\n\u03b5\n=\nIf we ignore the resistive losses and consider only inductive effect,\nthen using Eq (6 16),\nd\nd\nd\nd\nW\nI\nL I\nt\nt\n=\nTotal amount of work done in establishing the current I is\nW\nW\nL I\nI\nI\n=\n=\n\u222b\n\u222b\nd\nd\n0\nThus, the energy required to build up the current I is,\n2\n21\nW\nLI\n=\n(6"}, {"Chapter": "1", "sentence_range": "5224-5227", "Text": "For the\ncurrent I at an instant in a circuit, the rate of work done is\nd\nd\nW\nI\nt\n\u03b5\n=\nIf we ignore the resistive losses and consider only inductive effect,\nthen using Eq (6 16),\nd\nd\nd\nd\nW\nI\nL I\nt\nt\n=\nTotal amount of work done in establishing the current I is\nW\nW\nL I\nI\nI\n=\n=\n\u222b\n\u222b\nd\nd\n0\nThus, the energy required to build up the current I is,\n2\n21\nW\nLI\n=\n(6 17)\nThis expression reminds us of mv 2/2 for the (mechanical) kinetic energy\nof a particle of mass m, and shows that L is analogous to m (i"}, {"Chapter": "1", "sentence_range": "5225-5228", "Text": "(6 16),\nd\nd\nd\nd\nW\nI\nL I\nt\nt\n=\nTotal amount of work done in establishing the current I is\nW\nW\nL I\nI\nI\n=\n=\n\u222b\n\u222b\nd\nd\n0\nThus, the energy required to build up the current I is,\n2\n21\nW\nLI\n=\n(6 17)\nThis expression reminds us of mv 2/2 for the (mechanical) kinetic energy\nof a particle of mass m, and shows that L is analogous to m (i e"}, {"Chapter": "1", "sentence_range": "5226-5229", "Text": "16),\nd\nd\nd\nd\nW\nI\nL I\nt\nt\n=\nTotal amount of work done in establishing the current I is\nW\nW\nL I\nI\nI\n=\n=\n\u222b\n\u222b\nd\nd\n0\nThus, the energy required to build up the current I is,\n2\n21\nW\nLI\n=\n(6 17)\nThis expression reminds us of mv 2/2 for the (mechanical) kinetic energy\nof a particle of mass m, and shows that L is analogous to m (i e , L is\nelectrical inertia and opposes growth and decay of current in the circuit)"}, {"Chapter": "1", "sentence_range": "5227-5230", "Text": "17)\nThis expression reminds us of mv 2/2 for the (mechanical) kinetic energy\nof a particle of mass m, and shows that L is analogous to m (i e , L is\nelectrical inertia and opposes growth and decay of current in the circuit) Consider the general case of currents flowing simultaneously in two\nnearby coils"}, {"Chapter": "1", "sentence_range": "5228-5231", "Text": "e , L is\nelectrical inertia and opposes growth and decay of current in the circuit) Consider the general case of currents flowing simultaneously in two\nnearby coils The flux linked with one coil will be the sum of two fluxes\nwhich exist independently"}, {"Chapter": "1", "sentence_range": "5229-5232", "Text": ", L is\nelectrical inertia and opposes growth and decay of current in the circuit) Consider the general case of currents flowing simultaneously in two\nnearby coils The flux linked with one coil will be the sum of two fluxes\nwhich exist independently Equation (6"}, {"Chapter": "1", "sentence_range": "5230-5233", "Text": "Consider the general case of currents flowing simultaneously in two\nnearby coils The flux linked with one coil will be the sum of two fluxes\nwhich exist independently Equation (6 7) would be modified into\nN1\n1\n11\n1\n12\n2\nM\nI\nM\nI\n\u03a6 =\n+\nwhere M11 represents inductance due to the same coil"}, {"Chapter": "1", "sentence_range": "5231-5234", "Text": "The flux linked with one coil will be the sum of two fluxes\nwhich exist independently Equation (6 7) would be modified into\nN1\n1\n11\n1\n12\n2\nM\nI\nM\nI\n\u03a6 =\n+\nwhere M11 represents inductance due to the same coil Therefore, using Faraday\u2019s law,\n1\n2\n1\n11\n12\nd\nd\nd\nd\nI\nI\nM\nM\nt\nt\n\u03b5 = \u2212\n\u2212\nM11 is the self-inductance and is written as L1"}, {"Chapter": "1", "sentence_range": "5232-5235", "Text": "Equation (6 7) would be modified into\nN1\n1\n11\n1\n12\n2\nM\nI\nM\nI\n\u03a6 =\n+\nwhere M11 represents inductance due to the same coil Therefore, using Faraday\u2019s law,\n1\n2\n1\n11\n12\nd\nd\nd\nd\nI\nI\nM\nM\nt\nt\n\u03b5 = \u2212\n\u2212\nM11 is the self-inductance and is written as L1 Therefore,\n1\n2\n1\n1\n12\nd\nd\nd\nd\nI\nI\nL\nM\nt\nt\n\u03b5 = \u2212\n\u2212\nExample 6"}, {"Chapter": "1", "sentence_range": "5233-5236", "Text": "7) would be modified into\nN1\n1\n11\n1\n12\n2\nM\nI\nM\nI\n\u03a6 =\n+\nwhere M11 represents inductance due to the same coil Therefore, using Faraday\u2019s law,\n1\n2\n1\n11\n12\nd\nd\nd\nd\nI\nI\nM\nM\nt\nt\n\u03b5 = \u2212\n\u2212\nM11 is the self-inductance and is written as L1 Therefore,\n1\n2\n1\n1\n12\nd\nd\nd\nd\nI\nI\nL\nM\nt\nt\n\u03b5 = \u2212\n\u2212\nExample 6 9 (a) Obtain the expression for the magnetic energy stored\nin a solenoid in terms of magnetic field B, area A and length l of the\nsolenoid"}, {"Chapter": "1", "sentence_range": "5234-5237", "Text": "Therefore, using Faraday\u2019s law,\n1\n2\n1\n11\n12\nd\nd\nd\nd\nI\nI\nM\nM\nt\nt\n\u03b5 = \u2212\n\u2212\nM11 is the self-inductance and is written as L1 Therefore,\n1\n2\n1\n1\n12\nd\nd\nd\nd\nI\nI\nL\nM\nt\nt\n\u03b5 = \u2212\n\u2212\nExample 6 9 (a) Obtain the expression for the magnetic energy stored\nin a solenoid in terms of magnetic field B, area A and length l of the\nsolenoid (b) How does this magnetic energy compare with the\nelectrostatic energy stored in a capacitor"}, {"Chapter": "1", "sentence_range": "5235-5238", "Text": "Therefore,\n1\n2\n1\n1\n12\nd\nd\nd\nd\nI\nI\nL\nM\nt\nt\n\u03b5 = \u2212\n\u2212\nExample 6 9 (a) Obtain the expression for the magnetic energy stored\nin a solenoid in terms of magnetic field B, area A and length l of the\nsolenoid (b) How does this magnetic energy compare with the\nelectrostatic energy stored in a capacitor Solution\n(a)\nFrom Eq"}, {"Chapter": "1", "sentence_range": "5236-5239", "Text": "9 (a) Obtain the expression for the magnetic energy stored\nin a solenoid in terms of magnetic field B, area A and length l of the\nsolenoid (b) How does this magnetic energy compare with the\nelectrostatic energy stored in a capacitor Solution\n(a)\nFrom Eq (6"}, {"Chapter": "1", "sentence_range": "5237-5240", "Text": "(b) How does this magnetic energy compare with the\nelectrostatic energy stored in a capacitor Solution\n(a)\nFrom Eq (6 17), the magnetic energy is\n2\n21\nUB\nLI\n=\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n=\n(\n)\n1\n2\n2\nL\nB\nn\nnI\n\u00b5\n\u00b5\n0\n0\nsinceB\n \nfor a solenoid\n,\n EXAMPLE 6"}, {"Chapter": "1", "sentence_range": "5238-5241", "Text": "Solution\n(a)\nFrom Eq (6 17), the magnetic energy is\n2\n21\nUB\nLI\n=\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n=\n(\n)\n1\n2\n2\nL\nB\nn\nnI\n\u00b5\n\u00b5\n0\n0\nsinceB\n \nfor a solenoid\n,\n EXAMPLE 6 9\nRationalised 2023-24\nPhysics\n170\n EXAMPLE 6"}, {"Chapter": "1", "sentence_range": "5239-5242", "Text": "(6 17), the magnetic energy is\n2\n21\nUB\nLI\n=\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n=\n(\n)\n1\n2\n2\nL\nB\nn\nnI\n\u00b5\n\u00b5\n0\n0\nsinceB\n \nfor a solenoid\n,\n EXAMPLE 6 9\nRationalised 2023-24\nPhysics\n170\n EXAMPLE 6 9\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n21\n0\n2\n0\n2\n(\n)\n\u00b5\n\u00b5\nn Al\nB\nn\n        [from Eq"}, {"Chapter": "1", "sentence_range": "5240-5243", "Text": "17), the magnetic energy is\n2\n21\nUB\nLI\n=\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n=\n(\n)\n1\n2\n2\nL\nB\nn\nnI\n\u00b5\n\u00b5\n0\n0\nsinceB\n \nfor a solenoid\n,\n EXAMPLE 6 9\nRationalised 2023-24\nPhysics\n170\n EXAMPLE 6 9\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n21\n0\n2\n0\n2\n(\n)\n\u00b5\n\u00b5\nn Al\nB\nn\n        [from Eq (6"}, {"Chapter": "1", "sentence_range": "5241-5244", "Text": "9\nRationalised 2023-24\nPhysics\n170\n EXAMPLE 6 9\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n21\n0\n2\n0\n2\n(\n)\n\u00b5\n\u00b5\nn Al\nB\nn\n        [from Eq (6 15)]\n2\n0\n21\n\u00b5B Al\n=\n(b)\nThe magnetic energy per unit volume is,\nB\nB\nU\nu\n=V\n            (where V is volume that contains flux)\n      \nUB\nAl\n=\n      \n2\n20\nB\n\u00b5\n=\n(6"}, {"Chapter": "1", "sentence_range": "5242-5245", "Text": "9\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n21\n0\n2\n0\n2\n(\n)\n\u00b5\n\u00b5\nn Al\nB\nn\n        [from Eq (6 15)]\n2\n0\n21\n\u00b5B Al\n=\n(b)\nThe magnetic energy per unit volume is,\nB\nB\nU\nu\n=V\n            (where V is volume that contains flux)\n      \nUB\nAl\n=\n      \n2\n20\nB\n\u00b5\n=\n(6 18)\nWe have already obtained the relation for the electrostatic energy\nstored per unit volume in a parallel plate capacitor (refer to Chapter 2,\nEq"}, {"Chapter": "1", "sentence_range": "5243-5246", "Text": "(6 15)]\n2\n0\n21\n\u00b5B Al\n=\n(b)\nThe magnetic energy per unit volume is,\nB\nB\nU\nu\n=V\n            (where V is volume that contains flux)\n      \nUB\nAl\n=\n      \n2\n20\nB\n\u00b5\n=\n(6 18)\nWe have already obtained the relation for the electrostatic energy\nstored per unit volume in a parallel plate capacitor (refer to Chapter 2,\nEq 2"}, {"Chapter": "1", "sentence_range": "5244-5247", "Text": "15)]\n2\n0\n21\n\u00b5B Al\n=\n(b)\nThe magnetic energy per unit volume is,\nB\nB\nU\nu\n=V\n            (where V is volume that contains flux)\n      \nUB\nAl\n=\n      \n2\n20\nB\n\u00b5\n=\n(6 18)\nWe have already obtained the relation for the electrostatic energy\nstored per unit volume in a parallel plate capacitor (refer to Chapter 2,\nEq 2 73),\n2\n210\nu\nE\n\u0395\n\u03b5\n=\n(2"}, {"Chapter": "1", "sentence_range": "5245-5248", "Text": "18)\nWe have already obtained the relation for the electrostatic energy\nstored per unit volume in a parallel plate capacitor (refer to Chapter 2,\nEq 2 73),\n2\n210\nu\nE\n\u0395\n\u03b5\n=\n(2 73)\nIn both the cases energy is proportional to the square of the field\nstrength"}, {"Chapter": "1", "sentence_range": "5246-5249", "Text": "2 73),\n2\n210\nu\nE\n\u0395\n\u03b5\n=\n(2 73)\nIn both the cases energy is proportional to the square of the field\nstrength Equations (6"}, {"Chapter": "1", "sentence_range": "5247-5250", "Text": "73),\n2\n210\nu\nE\n\u0395\n\u03b5\n=\n(2 73)\nIn both the cases energy is proportional to the square of the field\nstrength Equations (6 18) and (2"}, {"Chapter": "1", "sentence_range": "5248-5251", "Text": "73)\nIn both the cases energy is proportional to the square of the field\nstrength Equations (6 18) and (2 73) have been derived for special\ncases: a solenoid and a parallel plate capacitor, respectively"}, {"Chapter": "1", "sentence_range": "5249-5252", "Text": "Equations (6 18) and (2 73) have been derived for special\ncases: a solenoid and a parallel plate capacitor, respectively But they\nare general and valid for any region of space in which a magnetic field\nor/and an electric field exist"}, {"Chapter": "1", "sentence_range": "5250-5253", "Text": "18) and (2 73) have been derived for special\ncases: a solenoid and a parallel plate capacitor, respectively But they\nare general and valid for any region of space in which a magnetic field\nor/and an electric field exist FIGURE 6"}, {"Chapter": "1", "sentence_range": "5251-5254", "Text": "73) have been derived for special\ncases: a solenoid and a parallel plate capacitor, respectively But they\nare general and valid for any region of space in which a magnetic field\nor/and an electric field exist FIGURE 6 13 AC Generator\nInteractive animation on ac generator:\nhttp://micro"}, {"Chapter": "1", "sentence_range": "5252-5255", "Text": "But they\nare general and valid for any region of space in which a magnetic field\nor/and an electric field exist FIGURE 6 13 AC Generator\nInteractive animation on ac generator:\nhttp://micro magnet"}, {"Chapter": "1", "sentence_range": "5253-5256", "Text": "FIGURE 6 13 AC Generator\nInteractive animation on ac generator:\nhttp://micro magnet fsu"}, {"Chapter": "1", "sentence_range": "5254-5257", "Text": "13 AC Generator\nInteractive animation on ac generator:\nhttp://micro magnet fsu edu/electromag/java/generator/ac"}, {"Chapter": "1", "sentence_range": "5255-5258", "Text": "magnet fsu edu/electromag/java/generator/ac html\n6"}, {"Chapter": "1", "sentence_range": "5256-5259", "Text": "fsu edu/electromag/java/generator/ac html\n6 8  AC GENERATOR\nThe phenomenon of electromagnetic induction\nhas been technologically exploited in many ways"}, {"Chapter": "1", "sentence_range": "5257-5260", "Text": "edu/electromag/java/generator/ac html\n6 8  AC GENERATOR\nThe phenomenon of electromagnetic induction\nhas been technologically exploited in many ways An exceptionally important application is the\ngeneration of alternating currents (ac)"}, {"Chapter": "1", "sentence_range": "5258-5261", "Text": "html\n6 8  AC GENERATOR\nThe phenomenon of electromagnetic induction\nhas been technologically exploited in many ways An exceptionally important application is the\ngeneration of alternating currents (ac) The\nmodern ac generator with a typical output\ncapacity of 100 MW is a highly evolved machine"}, {"Chapter": "1", "sentence_range": "5259-5262", "Text": "8  AC GENERATOR\nThe phenomenon of electromagnetic induction\nhas been technologically exploited in many ways An exceptionally important application is the\ngeneration of alternating currents (ac) The\nmodern ac generator with a typical output\ncapacity of 100 MW is a highly evolved machine In this section, we shall describe the basic\nprinciples behind this machine"}, {"Chapter": "1", "sentence_range": "5260-5263", "Text": "An exceptionally important application is the\ngeneration of alternating currents (ac) The\nmodern ac generator with a typical output\ncapacity of 100 MW is a highly evolved machine In this section, we shall describe the basic\nprinciples behind this machine The Yugoslav\ninventor Nicola Tesla is credited with the\ndevelopment of the machine"}, {"Chapter": "1", "sentence_range": "5261-5264", "Text": "The\nmodern ac generator with a typical output\ncapacity of 100 MW is a highly evolved machine In this section, we shall describe the basic\nprinciples behind this machine The Yugoslav\ninventor Nicola Tesla is credited with the\ndevelopment of the machine As was pointed out\nin Section 6"}, {"Chapter": "1", "sentence_range": "5262-5265", "Text": "In this section, we shall describe the basic\nprinciples behind this machine The Yugoslav\ninventor Nicola Tesla is credited with the\ndevelopment of the machine As was pointed out\nin Section 6 3, one method to induce an emf or\ncurrent in a loop is through a change in the\nloop\u2019s orientation or a change in its effective area"}, {"Chapter": "1", "sentence_range": "5263-5266", "Text": "The Yugoslav\ninventor Nicola Tesla is credited with the\ndevelopment of the machine As was pointed out\nin Section 6 3, one method to induce an emf or\ncurrent in a loop is through a change in the\nloop\u2019s orientation or a change in its effective area As the coil rotates in a magnetic field B, the\neffective area of the loop (the face perpendicular\nto the field) is A cos q, where q is the angle\nbetween A and B"}, {"Chapter": "1", "sentence_range": "5264-5267", "Text": "As was pointed out\nin Section 6 3, one method to induce an emf or\ncurrent in a loop is through a change in the\nloop\u2019s orientation or a change in its effective area As the coil rotates in a magnetic field B, the\neffective area of the loop (the face perpendicular\nto the field) is A cos q, where q is the angle\nbetween A and B This method of producing a\nflux change is the principle of operation of a\nRationalised 2023-24\nElectromagnetic\nInduction\n171\nsimple ac generator"}, {"Chapter": "1", "sentence_range": "5265-5268", "Text": "3, one method to induce an emf or\ncurrent in a loop is through a change in the\nloop\u2019s orientation or a change in its effective area As the coil rotates in a magnetic field B, the\neffective area of the loop (the face perpendicular\nto the field) is A cos q, where q is the angle\nbetween A and B This method of producing a\nflux change is the principle of operation of a\nRationalised 2023-24\nElectromagnetic\nInduction\n171\nsimple ac generator An ac generator converts mechanical energy into\nelectrical energy"}, {"Chapter": "1", "sentence_range": "5266-5269", "Text": "As the coil rotates in a magnetic field B, the\neffective area of the loop (the face perpendicular\nto the field) is A cos q, where q is the angle\nbetween A and B This method of producing a\nflux change is the principle of operation of a\nRationalised 2023-24\nElectromagnetic\nInduction\n171\nsimple ac generator An ac generator converts mechanical energy into\nelectrical energy The basic elements of an ac generator are shown in Fig"}, {"Chapter": "1", "sentence_range": "5267-5270", "Text": "This method of producing a\nflux change is the principle of operation of a\nRationalised 2023-24\nElectromagnetic\nInduction\n171\nsimple ac generator An ac generator converts mechanical energy into\nelectrical energy The basic elements of an ac generator are shown in Fig 6"}, {"Chapter": "1", "sentence_range": "5268-5271", "Text": "An ac generator converts mechanical energy into\nelectrical energy The basic elements of an ac generator are shown in Fig 6 13"}, {"Chapter": "1", "sentence_range": "5269-5272", "Text": "The basic elements of an ac generator are shown in Fig 6 13 It consists\nof a coil mounted on a rotor shaft"}, {"Chapter": "1", "sentence_range": "5270-5273", "Text": "6 13 It consists\nof a coil mounted on a rotor shaft The axis of rotation of the coil is\nperpendicular to the direction of the magnetic field"}, {"Chapter": "1", "sentence_range": "5271-5274", "Text": "13 It consists\nof a coil mounted on a rotor shaft The axis of rotation of the coil is\nperpendicular to the direction of the magnetic field The coil (called\narmature) is mechanically rotated in the  uniform magnetic field by some\nexternal means"}, {"Chapter": "1", "sentence_range": "5272-5275", "Text": "It consists\nof a coil mounted on a rotor shaft The axis of rotation of the coil is\nperpendicular to the direction of the magnetic field The coil (called\narmature) is mechanically rotated in the  uniform magnetic field by some\nexternal means The rotation of the coil causes the magnetic flux through\nit to change, so an emf is induced in the coil"}, {"Chapter": "1", "sentence_range": "5273-5276", "Text": "The axis of rotation of the coil is\nperpendicular to the direction of the magnetic field The coil (called\narmature) is mechanically rotated in the  uniform magnetic field by some\nexternal means The rotation of the coil causes the magnetic flux through\nit to change, so an emf is induced in the coil The ends of the\ncoil are connected to an external circuit by means of slip rings\nand brushes"}, {"Chapter": "1", "sentence_range": "5274-5277", "Text": "The coil (called\narmature) is mechanically rotated in the  uniform magnetic field by some\nexternal means The rotation of the coil causes the magnetic flux through\nit to change, so an emf is induced in the coil The ends of the\ncoil are connected to an external circuit by means of slip rings\nand brushes When the coil is rotated with a constant angular speed w, the angle q\nbetween the magnetic field vector B and the area vector A of the coil at any\ninstant t is q  = wt (assuming q = 0\u00b0 at t = 0)"}, {"Chapter": "1", "sentence_range": "5275-5278", "Text": "The rotation of the coil causes the magnetic flux through\nit to change, so an emf is induced in the coil The ends of the\ncoil are connected to an external circuit by means of slip rings\nand brushes When the coil is rotated with a constant angular speed w, the angle q\nbetween the magnetic field vector B and the area vector A of the coil at any\ninstant t is q  = wt (assuming q = 0\u00b0 at t = 0) As a result, the effective area\nof the coil exposed to the magnetic field lines changes with time, and from\nEq"}, {"Chapter": "1", "sentence_range": "5276-5279", "Text": "The ends of the\ncoil are connected to an external circuit by means of slip rings\nand brushes When the coil is rotated with a constant angular speed w, the angle q\nbetween the magnetic field vector B and the area vector A of the coil at any\ninstant t is q  = wt (assuming q = 0\u00b0 at t = 0) As a result, the effective area\nof the coil exposed to the magnetic field lines changes with time, and from\nEq (6"}, {"Chapter": "1", "sentence_range": "5277-5280", "Text": "When the coil is rotated with a constant angular speed w, the angle q\nbetween the magnetic field vector B and the area vector A of the coil at any\ninstant t is q  = wt (assuming q = 0\u00b0 at t = 0) As a result, the effective area\nof the coil exposed to the magnetic field lines changes with time, and from\nEq (6 1), the flux at any time t is\nFB = BA cos q = BA cos wt\nFrom Faraday\u2019s law, the induced emf for the rotating coil of N turns\nis then,\nd\nd\n\u2013\n\u2013\n(cos\n)\ndt\nd\nB\nN\nNBA\nt\nt\n\u03a6\n\u03b5\n\u03c9\n=\n=\n\u03b5Thus, the instantaneous value of the emf is\n\u03c9\n\u03c9\n= NBA\nsin\nt\n(6"}, {"Chapter": "1", "sentence_range": "5278-5281", "Text": "As a result, the effective area\nof the coil exposed to the magnetic field lines changes with time, and from\nEq (6 1), the flux at any time t is\nFB = BA cos q = BA cos wt\nFrom Faraday\u2019s law, the induced emf for the rotating coil of N turns\nis then,\nd\nd\n\u2013\n\u2013\n(cos\n)\ndt\nd\nB\nN\nNBA\nt\nt\n\u03a6\n\u03b5\n\u03c9\n=\n=\n\u03b5Thus, the instantaneous value of the emf is\n\u03c9\n\u03c9\n= NBA\nsin\nt\n(6 19)\nwhere NBAw is the maximum value of the emf, which occurs when\nsin wt = \u00b11"}, {"Chapter": "1", "sentence_range": "5279-5282", "Text": "(6 1), the flux at any time t is\nFB = BA cos q = BA cos wt\nFrom Faraday\u2019s law, the induced emf for the rotating coil of N turns\nis then,\nd\nd\n\u2013\n\u2013\n(cos\n)\ndt\nd\nB\nN\nNBA\nt\nt\n\u03a6\n\u03b5\n\u03c9\n=\n=\n\u03b5Thus, the instantaneous value of the emf is\n\u03c9\n\u03c9\n= NBA\nsin\nt\n(6 19)\nwhere NBAw is the maximum value of the emf, which occurs when\nsin wt = \u00b11 If we denote NBAw as e0, then\ne = e0 sin wt\n(6"}, {"Chapter": "1", "sentence_range": "5280-5283", "Text": "1), the flux at any time t is\nFB = BA cos q = BA cos wt\nFrom Faraday\u2019s law, the induced emf for the rotating coil of N turns\nis then,\nd\nd\n\u2013\n\u2013\n(cos\n)\ndt\nd\nB\nN\nNBA\nt\nt\n\u03a6\n\u03b5\n\u03c9\n=\n=\n\u03b5Thus, the instantaneous value of the emf is\n\u03c9\n\u03c9\n= NBA\nsin\nt\n(6 19)\nwhere NBAw is the maximum value of the emf, which occurs when\nsin wt = \u00b11 If we denote NBAw as e0, then\ne = e0 sin wt\n(6 20)\nSince the value of the sine fuction varies between +1 and \u20131, the sign, or\npolarity of the emf changes with time"}, {"Chapter": "1", "sentence_range": "5281-5284", "Text": "19)\nwhere NBAw is the maximum value of the emf, which occurs when\nsin wt = \u00b11 If we denote NBAw as e0, then\ne = e0 sin wt\n(6 20)\nSince the value of the sine fuction varies between +1 and \u20131, the sign, or\npolarity of the emf changes with time Note from Fig"}, {"Chapter": "1", "sentence_range": "5282-5285", "Text": "If we denote NBAw as e0, then\ne = e0 sin wt\n(6 20)\nSince the value of the sine fuction varies between +1 and \u20131, the sign, or\npolarity of the emf changes with time Note from Fig 6"}, {"Chapter": "1", "sentence_range": "5283-5286", "Text": "20)\nSince the value of the sine fuction varies between +1 and \u20131, the sign, or\npolarity of the emf changes with time Note from Fig 6 14 that the emf\nhas its extremum value when q = 90\u00b0 or q = 270\u00b0, as the change of flux is\ngreatest at these points"}, {"Chapter": "1", "sentence_range": "5284-5287", "Text": "Note from Fig 6 14 that the emf\nhas its extremum value when q = 90\u00b0 or q = 270\u00b0, as the change of flux is\ngreatest at these points The direction of the current changes periodically and therefore the current\nis called alternating current (ac)"}, {"Chapter": "1", "sentence_range": "5285-5288", "Text": "6 14 that the emf\nhas its extremum value when q = 90\u00b0 or q = 270\u00b0, as the change of flux is\ngreatest at these points The direction of the current changes periodically and therefore the current\nis called alternating current (ac) Since w = 2pn, Eq (6"}, {"Chapter": "1", "sentence_range": "5286-5289", "Text": "14 that the emf\nhas its extremum value when q = 90\u00b0 or q = 270\u00b0, as the change of flux is\ngreatest at these points The direction of the current changes periodically and therefore the current\nis called alternating current (ac) Since w = 2pn, Eq (6 20) can be written as\ne = e0sin 2p n t\n(6"}, {"Chapter": "1", "sentence_range": "5287-5290", "Text": "The direction of the current changes periodically and therefore the current\nis called alternating current (ac) Since w = 2pn, Eq (6 20) can be written as\ne = e0sin 2p n t\n(6 21)\nwhere n is the frequency of revolution of the generator\u2019s coil"}, {"Chapter": "1", "sentence_range": "5288-5291", "Text": "Since w = 2pn, Eq (6 20) can be written as\ne = e0sin 2p n t\n(6 21)\nwhere n is the frequency of revolution of the generator\u2019s coil Note that Eq"}, {"Chapter": "1", "sentence_range": "5289-5292", "Text": "20) can be written as\ne = e0sin 2p n t\n(6 21)\nwhere n is the frequency of revolution of the generator\u2019s coil Note that Eq (6"}, {"Chapter": "1", "sentence_range": "5290-5293", "Text": "21)\nwhere n is the frequency of revolution of the generator\u2019s coil Note that Eq (6 20) and (6"}, {"Chapter": "1", "sentence_range": "5291-5294", "Text": "Note that Eq (6 20) and (6 21) give the instantaneous value of the emf\nand e varies between +e0 and \u2013e0 periodically"}, {"Chapter": "1", "sentence_range": "5292-5295", "Text": "(6 20) and (6 21) give the instantaneous value of the emf\nand e varies between +e0 and \u2013e0 periodically We shall learn how to\ndetermine the time-averaged value for the alternating voltage and current\nin the next chapter"}, {"Chapter": "1", "sentence_range": "5293-5296", "Text": "20) and (6 21) give the instantaneous value of the emf\nand e varies between +e0 and \u2013e0 periodically We shall learn how to\ndetermine the time-averaged value for the alternating voltage and current\nin the next chapter In commercial generators, the mechanical energy required for\nrotation of the armature is provided by water falling from a height, for\nexample, from dams"}, {"Chapter": "1", "sentence_range": "5294-5297", "Text": "21) give the instantaneous value of the emf\nand e varies between +e0 and \u2013e0 periodically We shall learn how to\ndetermine the time-averaged value for the alternating voltage and current\nin the next chapter In commercial generators, the mechanical energy required for\nrotation of the armature is provided by water falling from a height, for\nexample, from dams These are called hydro-electric generators"}, {"Chapter": "1", "sentence_range": "5295-5298", "Text": "We shall learn how to\ndetermine the time-averaged value for the alternating voltage and current\nin the next chapter In commercial generators, the mechanical energy required for\nrotation of the armature is provided by water falling from a height, for\nexample, from dams These are called hydro-electric generators Alternatively, water is heated to produce steam using coal or other\nsources"}, {"Chapter": "1", "sentence_range": "5296-5299", "Text": "In commercial generators, the mechanical energy required for\nrotation of the armature is provided by water falling from a height, for\nexample, from dams These are called hydro-electric generators Alternatively, water is heated to produce steam using coal or other\nsources The steam at high pressure produces the rotation of the\narmature"}, {"Chapter": "1", "sentence_range": "5297-5300", "Text": "These are called hydro-electric generators Alternatively, water is heated to produce steam using coal or other\nsources The steam at high pressure produces the rotation of the\narmature These are called thermal generators"}, {"Chapter": "1", "sentence_range": "5298-5301", "Text": "Alternatively, water is heated to produce steam using coal or other\nsources The steam at high pressure produces the rotation of the\narmature These are called thermal generators Instead of coal, if a\nnuclear fuel is used, we get nuclear power generators"}, {"Chapter": "1", "sentence_range": "5299-5302", "Text": "The steam at high pressure produces the rotation of the\narmature These are called thermal generators Instead of coal, if a\nnuclear fuel is used, we get nuclear power generators Modern day\ngenerators produce electric power as high as 500 MW, i"}, {"Chapter": "1", "sentence_range": "5300-5303", "Text": "These are called thermal generators Instead of coal, if a\nnuclear fuel is used, we get nuclear power generators Modern day\ngenerators produce electric power as high as 500 MW, i e"}, {"Chapter": "1", "sentence_range": "5301-5304", "Text": "Instead of coal, if a\nnuclear fuel is used, we get nuclear power generators Modern day\ngenerators produce electric power as high as 500 MW, i e , one can light\nRationalised 2023-24\nPhysics\n172\n EXAMPLE 6"}, {"Chapter": "1", "sentence_range": "5302-5305", "Text": "Modern day\ngenerators produce electric power as high as 500 MW, i e , one can light\nRationalised 2023-24\nPhysics\n172\n EXAMPLE 6 10\nExample 6"}, {"Chapter": "1", "sentence_range": "5303-5306", "Text": "e , one can light\nRationalised 2023-24\nPhysics\n172\n EXAMPLE 6 10\nExample 6 10 Kamla peddles a stationary bicycle"}, {"Chapter": "1", "sentence_range": "5304-5307", "Text": ", one can light\nRationalised 2023-24\nPhysics\n172\n EXAMPLE 6 10\nExample 6 10 Kamla peddles a stationary bicycle The pedals of the\nbicycle are attached to a 100 turn coil of area 0"}, {"Chapter": "1", "sentence_range": "5305-5308", "Text": "10\nExample 6 10 Kamla peddles a stationary bicycle The pedals of the\nbicycle are attached to a 100 turn coil of area 0 10 m2"}, {"Chapter": "1", "sentence_range": "5306-5309", "Text": "10 Kamla peddles a stationary bicycle The pedals of the\nbicycle are attached to a 100 turn coil of area 0 10 m2 The coil rotates\nat half a revolution per second and it is placed in a uniform magnetic\nfield of 0"}, {"Chapter": "1", "sentence_range": "5307-5310", "Text": "The pedals of the\nbicycle are attached to a 100 turn coil of area 0 10 m2 The coil rotates\nat half a revolution per second and it is placed in a uniform magnetic\nfield of 0 01 T perpendicular to the axis of rotation of the coil"}, {"Chapter": "1", "sentence_range": "5308-5311", "Text": "10 m2 The coil rotates\nat half a revolution per second and it is placed in a uniform magnetic\nfield of 0 01 T perpendicular to the axis of rotation of the coil What is\nthe maximum voltage generated in the coil"}, {"Chapter": "1", "sentence_range": "5309-5312", "Text": "The coil rotates\nat half a revolution per second and it is placed in a uniform magnetic\nfield of 0 01 T perpendicular to the axis of rotation of the coil What is\nthe maximum voltage generated in the coil Solution  Here n = 0"}, {"Chapter": "1", "sentence_range": "5310-5313", "Text": "01 T perpendicular to the axis of rotation of the coil What is\nthe maximum voltage generated in the coil Solution  Here n = 0 5 Hz; N =100, A = 0"}, {"Chapter": "1", "sentence_range": "5311-5314", "Text": "What is\nthe maximum voltage generated in the coil Solution  Here n = 0 5 Hz; N =100, A = 0 1 m2 and B = 0"}, {"Chapter": "1", "sentence_range": "5312-5315", "Text": "Solution  Here n = 0 5 Hz; N =100, A = 0 1 m2 and B = 0 01 T"}, {"Chapter": "1", "sentence_range": "5313-5316", "Text": "5 Hz; N =100, A = 0 1 m2 and B = 0 01 T Employing\nEq"}, {"Chapter": "1", "sentence_range": "5314-5317", "Text": "1 m2 and B = 0 01 T Employing\nEq (6"}, {"Chapter": "1", "sentence_range": "5315-5318", "Text": "01 T Employing\nEq (6 19)\ne0 = NBA (2 p n)\n   = 100 \u00d7 0"}, {"Chapter": "1", "sentence_range": "5316-5319", "Text": "Employing\nEq (6 19)\ne0 = NBA (2 p n)\n   = 100 \u00d7 0 01 \u00d7 0"}, {"Chapter": "1", "sentence_range": "5317-5320", "Text": "(6 19)\ne0 = NBA (2 p n)\n   = 100 \u00d7 0 01 \u00d7 0 1 \u00d7 2 \u00d7 3"}, {"Chapter": "1", "sentence_range": "5318-5321", "Text": "19)\ne0 = NBA (2 p n)\n   = 100 \u00d7 0 01 \u00d7 0 1 \u00d7 2 \u00d7 3 14 \u00d7 0"}, {"Chapter": "1", "sentence_range": "5319-5322", "Text": "01 \u00d7 0 1 \u00d7 2 \u00d7 3 14 \u00d7 0 5\n   = 0"}, {"Chapter": "1", "sentence_range": "5320-5323", "Text": "1 \u00d7 2 \u00d7 3 14 \u00d7 0 5\n   = 0 314 V\nThe maximum voltage is 0"}, {"Chapter": "1", "sentence_range": "5321-5324", "Text": "14 \u00d7 0 5\n   = 0 314 V\nThe maximum voltage is 0 314 V"}, {"Chapter": "1", "sentence_range": "5322-5325", "Text": "5\n   = 0 314 V\nThe maximum voltage is 0 314 V We urge you to explore such alternative possibilities for power\ngeneration"}, {"Chapter": "1", "sentence_range": "5323-5326", "Text": "314 V\nThe maximum voltage is 0 314 V We urge you to explore such alternative possibilities for power\ngeneration FIGURE 6"}, {"Chapter": "1", "sentence_range": "5324-5327", "Text": "314 V We urge you to explore such alternative possibilities for power\ngeneration FIGURE 6 14 An alternating emf is generated by a loop of wire rotating in a magnetic field"}, {"Chapter": "1", "sentence_range": "5325-5328", "Text": "We urge you to explore such alternative possibilities for power\ngeneration FIGURE 6 14 An alternating emf is generated by a loop of wire rotating in a magnetic field up 5 million 100 W bulbs"}, {"Chapter": "1", "sentence_range": "5326-5329", "Text": "FIGURE 6 14 An alternating emf is generated by a loop of wire rotating in a magnetic field up 5 million 100 W bulbs In most generators, the coils are held\nstationary and it is the electromagnets which are rotated"}, {"Chapter": "1", "sentence_range": "5327-5330", "Text": "14 An alternating emf is generated by a loop of wire rotating in a magnetic field up 5 million 100 W bulbs In most generators, the coils are held\nstationary and it is the electromagnets which are rotated The frequency\nof rotation is 50 Hz in India"}, {"Chapter": "1", "sentence_range": "5328-5331", "Text": "up 5 million 100 W bulbs In most generators, the coils are held\nstationary and it is the electromagnets which are rotated The frequency\nof rotation is 50 Hz in India In certain countries such as USA, it is\n60 Hz"}, {"Chapter": "1", "sentence_range": "5329-5332", "Text": "In most generators, the coils are held\nstationary and it is the electromagnets which are rotated The frequency\nof rotation is 50 Hz in India In certain countries such as USA, it is\n60 Hz Rationalised 2023-24\nElectromagnetic\nInduction\n173\nSUMMARY\n1"}, {"Chapter": "1", "sentence_range": "5330-5333", "Text": "The frequency\nof rotation is 50 Hz in India In certain countries such as USA, it is\n60 Hz Rationalised 2023-24\nElectromagnetic\nInduction\n173\nSUMMARY\n1 The magnetic flux through a surface of area A placed in a uniform magnetic\nfield B is defined as,\nFB = B"}, {"Chapter": "1", "sentence_range": "5331-5334", "Text": "In certain countries such as USA, it is\n60 Hz Rationalised 2023-24\nElectromagnetic\nInduction\n173\nSUMMARY\n1 The magnetic flux through a surface of area A placed in a uniform magnetic\nfield B is defined as,\nFB = B A = BA cos q\nwhere q is the angle between B and A"}, {"Chapter": "1", "sentence_range": "5332-5335", "Text": "Rationalised 2023-24\nElectromagnetic\nInduction\n173\nSUMMARY\n1 The magnetic flux through a surface of area A placed in a uniform magnetic\nfield B is defined as,\nFB = B A = BA cos q\nwhere q is the angle between B and A 2"}, {"Chapter": "1", "sentence_range": "5333-5336", "Text": "The magnetic flux through a surface of area A placed in a uniform magnetic\nfield B is defined as,\nFB = B A = BA cos q\nwhere q is the angle between B and A 2 Faraday\u2019s laws of induction imply that the emf induced in a coil of N\nturns is directly related to the rate of change of flux through it,\ndB\nNd\n\u03a6t\n\u03b5 = \u2212\nHere FB is the flux linked with one turn of the coil"}, {"Chapter": "1", "sentence_range": "5334-5337", "Text": "A = BA cos q\nwhere q is the angle between B and A 2 Faraday\u2019s laws of induction imply that the emf induced in a coil of N\nturns is directly related to the rate of change of flux through it,\ndB\nNd\n\u03a6t\n\u03b5 = \u2212\nHere FB is the flux linked with one turn of the coil If the circuit is\nclosed, a current I = e/R is set up in it, where R is the resistance of the\ncircuit"}, {"Chapter": "1", "sentence_range": "5335-5338", "Text": "2 Faraday\u2019s laws of induction imply that the emf induced in a coil of N\nturns is directly related to the rate of change of flux through it,\ndB\nNd\n\u03a6t\n\u03b5 = \u2212\nHere FB is the flux linked with one turn of the coil If the circuit is\nclosed, a current I = e/R is set up in it, where R is the resistance of the\ncircuit 3"}, {"Chapter": "1", "sentence_range": "5336-5339", "Text": "Faraday\u2019s laws of induction imply that the emf induced in a coil of N\nturns is directly related to the rate of change of flux through it,\ndB\nNd\n\u03a6t\n\u03b5 = \u2212\nHere FB is the flux linked with one turn of the coil If the circuit is\nclosed, a current I = e/R is set up in it, where R is the resistance of the\ncircuit 3 Lenz\u2019s law states that the polarity of the induced emf is such that it\ntends to produce a current which opposes the change in magnetic flux\nthat produces it"}, {"Chapter": "1", "sentence_range": "5337-5340", "Text": "If the circuit is\nclosed, a current I = e/R is set up in it, where R is the resistance of the\ncircuit 3 Lenz\u2019s law states that the polarity of the induced emf is such that it\ntends to produce a current which opposes the change in magnetic flux\nthat produces it The negative sign in the expression for Faraday\u2019s law\nindicates this fact"}, {"Chapter": "1", "sentence_range": "5338-5341", "Text": "3 Lenz\u2019s law states that the polarity of the induced emf is such that it\ntends to produce a current which opposes the change in magnetic flux\nthat produces it The negative sign in the expression for Faraday\u2019s law\nindicates this fact 4"}, {"Chapter": "1", "sentence_range": "5339-5342", "Text": "Lenz\u2019s law states that the polarity of the induced emf is such that it\ntends to produce a current which opposes the change in magnetic flux\nthat produces it The negative sign in the expression for Faraday\u2019s law\nindicates this fact 4 When a metal rod of length l is placed normal to a uniform magnetic\nfield B and moved with a velocity v perpendicular to the field, the\ninduced emf (called motional emf) across its ends is\ne = Bl v\n5"}, {"Chapter": "1", "sentence_range": "5340-5343", "Text": "The negative sign in the expression for Faraday\u2019s law\nindicates this fact 4 When a metal rod of length l is placed normal to a uniform magnetic\nfield B and moved with a velocity v perpendicular to the field, the\ninduced emf (called motional emf) across its ends is\ne = Bl v\n5 Inductance is the ratio of the flux-linkage to current"}, {"Chapter": "1", "sentence_range": "5341-5344", "Text": "4 When a metal rod of length l is placed normal to a uniform magnetic\nfield B and moved with a velocity v perpendicular to the field, the\ninduced emf (called motional emf) across its ends is\ne = Bl v\n5 Inductance is the ratio of the flux-linkage to current It is equal to NF/I"}, {"Chapter": "1", "sentence_range": "5342-5345", "Text": "When a metal rod of length l is placed normal to a uniform magnetic\nfield B and moved with a velocity v perpendicular to the field, the\ninduced emf (called motional emf) across its ends is\ne = Bl v\n5 Inductance is the ratio of the flux-linkage to current It is equal to NF/I 6"}, {"Chapter": "1", "sentence_range": "5343-5346", "Text": "Inductance is the ratio of the flux-linkage to current It is equal to NF/I 6 A changing current in a coil (coil 2) can induce an emf in a nearby coil\n(coil 1)"}, {"Chapter": "1", "sentence_range": "5344-5347", "Text": "It is equal to NF/I 6 A changing current in a coil (coil 2) can induce an emf in a nearby coil\n(coil 1) This relation is given by,\n2\n1\n12\nd\nId\nM\nt\n\u03b5 = \u2212\nThe quantity M12 is called mutual inductance of coil 1 with respect to\ncoil 2"}, {"Chapter": "1", "sentence_range": "5345-5348", "Text": "6 A changing current in a coil (coil 2) can induce an emf in a nearby coil\n(coil 1) This relation is given by,\n2\n1\n12\nd\nId\nM\nt\n\u03b5 = \u2212\nThe quantity M12 is called mutual inductance of coil 1 with respect to\ncoil 2 One can similarly define M21"}, {"Chapter": "1", "sentence_range": "5346-5349", "Text": "A changing current in a coil (coil 2) can induce an emf in a nearby coil\n(coil 1) This relation is given by,\n2\n1\n12\nd\nId\nM\nt\n\u03b5 = \u2212\nThe quantity M12 is called mutual inductance of coil 1 with respect to\ncoil 2 One can similarly define M21 There exists a general equality,\nM12 = M21\n7"}, {"Chapter": "1", "sentence_range": "5347-5350", "Text": "This relation is given by,\n2\n1\n12\nd\nId\nM\nt\n\u03b5 = \u2212\nThe quantity M12 is called mutual inductance of coil 1 with respect to\ncoil 2 One can similarly define M21 There exists a general equality,\nM12 = M21\n7 When a current in a coil changes, it induces a back emf in the same\ncoil"}, {"Chapter": "1", "sentence_range": "5348-5351", "Text": "One can similarly define M21 There exists a general equality,\nM12 = M21\n7 When a current in a coil changes, it induces a back emf in the same\ncoil The self-induced emf is given by,\nd\nd\nI\nL\nt\n\u03b5 = \u2212\nL is the self-inductance of the coil"}, {"Chapter": "1", "sentence_range": "5349-5352", "Text": "There exists a general equality,\nM12 = M21\n7 When a current in a coil changes, it induces a back emf in the same\ncoil The self-induced emf is given by,\nd\nd\nI\nL\nt\n\u03b5 = \u2212\nL is the self-inductance of the coil It is a measure of the inertia of the\ncoil against the change of current through it"}, {"Chapter": "1", "sentence_range": "5350-5353", "Text": "When a current in a coil changes, it induces a back emf in the same\ncoil The self-induced emf is given by,\nd\nd\nI\nL\nt\n\u03b5 = \u2212\nL is the self-inductance of the coil It is a measure of the inertia of the\ncoil against the change of current through it 8"}, {"Chapter": "1", "sentence_range": "5351-5354", "Text": "The self-induced emf is given by,\nd\nd\nI\nL\nt\n\u03b5 = \u2212\nL is the self-inductance of the coil It is a measure of the inertia of the\ncoil against the change of current through it 8 The self-inductance of a long solenoid, the core of which consists of a\nmagnetic material of relative permeability mr, is given by\nL = mr  m0 n2 Al\nwhere A is the area of cross-section of the solenoid, l its length and n\nthe number of turns per unit length"}, {"Chapter": "1", "sentence_range": "5352-5355", "Text": "It is a measure of the inertia of the\ncoil against the change of current through it 8 The self-inductance of a long solenoid, the core of which consists of a\nmagnetic material of relative permeability mr, is given by\nL = mr  m0 n2 Al\nwhere A is the area of cross-section of the solenoid, l its length and n\nthe number of turns per unit length 9"}, {"Chapter": "1", "sentence_range": "5353-5356", "Text": "8 The self-inductance of a long solenoid, the core of which consists of a\nmagnetic material of relative permeability mr, is given by\nL = mr  m0 n2 Al\nwhere A is the area of cross-section of the solenoid, l its length and n\nthe number of turns per unit length 9 In an ac generator, mechanical energy is converted to electrical energy\nby virtue of electromagnetic induction"}, {"Chapter": "1", "sentence_range": "5354-5357", "Text": "The self-inductance of a long solenoid, the core of which consists of a\nmagnetic material of relative permeability mr, is given by\nL = mr  m0 n2 Al\nwhere A is the area of cross-section of the solenoid, l its length and n\nthe number of turns per unit length 9 In an ac generator, mechanical energy is converted to electrical energy\nby virtue of electromagnetic induction If coil of N turn and area A is\nrotated at n revolutions per second in a uniform magnetic field B, then\nthe motional emf produced is\ne = NBA (2pn) sin (2pnt)\nwhere we have assumed that at time t = 0 s, the coil is perpendicular to\nthe field"}, {"Chapter": "1", "sentence_range": "5355-5358", "Text": "9 In an ac generator, mechanical energy is converted to electrical energy\nby virtue of electromagnetic induction If coil of N turn and area A is\nrotated at n revolutions per second in a uniform magnetic field B, then\nthe motional emf produced is\ne = NBA (2pn) sin (2pnt)\nwhere we have assumed that at time t = 0 s, the coil is perpendicular to\nthe field Rationalised 2023-24\nPhysics\n174\nPOINTS TO PONDER\n1"}, {"Chapter": "1", "sentence_range": "5356-5359", "Text": "In an ac generator, mechanical energy is converted to electrical energy\nby virtue of electromagnetic induction If coil of N turn and area A is\nrotated at n revolutions per second in a uniform magnetic field B, then\nthe motional emf produced is\ne = NBA (2pn) sin (2pnt)\nwhere we have assumed that at time t = 0 s, the coil is perpendicular to\nthe field Rationalised 2023-24\nPhysics\n174\nPOINTS TO PONDER\n1 Electricity and magnetism are intimately related"}, {"Chapter": "1", "sentence_range": "5357-5360", "Text": "If coil of N turn and area A is\nrotated at n revolutions per second in a uniform magnetic field B, then\nthe motional emf produced is\ne = NBA (2pn) sin (2pnt)\nwhere we have assumed that at time t = 0 s, the coil is perpendicular to\nthe field Rationalised 2023-24\nPhysics\n174\nPOINTS TO PONDER\n1 Electricity and magnetism are intimately related In the early part of the\nnineteenth century, the experiments of Oersted, Ampere and others\nestablished that moving charges (currents) produce a magnetic field"}, {"Chapter": "1", "sentence_range": "5358-5361", "Text": "Rationalised 2023-24\nPhysics\n174\nPOINTS TO PONDER\n1 Electricity and magnetism are intimately related In the early part of the\nnineteenth century, the experiments of Oersted, Ampere and others\nestablished that moving charges (currents) produce a magnetic field Somewhat later, around 1830, the experiments of Faraday and Henry\ndemonstrated that a moving magnet can induce electric current"}, {"Chapter": "1", "sentence_range": "5359-5362", "Text": "Electricity and magnetism are intimately related In the early part of the\nnineteenth century, the experiments of Oersted, Ampere and others\nestablished that moving charges (currents) produce a magnetic field Somewhat later, around 1830, the experiments of Faraday and Henry\ndemonstrated that a moving magnet can induce electric current 2"}, {"Chapter": "1", "sentence_range": "5360-5363", "Text": "In the early part of the\nnineteenth century, the experiments of Oersted, Ampere and others\nestablished that moving charges (currents) produce a magnetic field Somewhat later, around 1830, the experiments of Faraday and Henry\ndemonstrated that a moving magnet can induce electric current 2 In a closed circuit, electric currents are induced so as to oppose the\nchanging magnetic flux"}, {"Chapter": "1", "sentence_range": "5361-5364", "Text": "Somewhat later, around 1830, the experiments of Faraday and Henry\ndemonstrated that a moving magnet can induce electric current 2 In a closed circuit, electric currents are induced so as to oppose the\nchanging magnetic flux It is as per the law of conservation of energy"}, {"Chapter": "1", "sentence_range": "5362-5365", "Text": "2 In a closed circuit, electric currents are induced so as to oppose the\nchanging magnetic flux It is as per the law of conservation of energy However, in case of an open circuit, an emf is induced across its ends"}, {"Chapter": "1", "sentence_range": "5363-5366", "Text": "In a closed circuit, electric currents are induced so as to oppose the\nchanging magnetic flux It is as per the law of conservation of energy However, in case of an open circuit, an emf is induced across its ends How is it related to the flux change"}, {"Chapter": "1", "sentence_range": "5364-5367", "Text": "It is as per the law of conservation of energy However, in case of an open circuit, an emf is induced across its ends How is it related to the flux change 3"}, {"Chapter": "1", "sentence_range": "5365-5368", "Text": "However, in case of an open circuit, an emf is induced across its ends How is it related to the flux change 3 The motional emf discussed in Section 6"}, {"Chapter": "1", "sentence_range": "5366-5369", "Text": "How is it related to the flux change 3 The motional emf discussed in Section 6 5 can be argued independently\nfrom Faraday\u2019s law using the Lorentz force on moving charges"}, {"Chapter": "1", "sentence_range": "5367-5370", "Text": "3 The motional emf discussed in Section 6 5 can be argued independently\nfrom Faraday\u2019s law using the Lorentz force on moving charges However,\neven if the charges are stationary [and the q (v \u00d7 B) term of the Lorentz\nforce is not operative], an emf is nevertheless induced in the presence of a\ntime-varying magnetic field"}, {"Chapter": "1", "sentence_range": "5368-5371", "Text": "The motional emf discussed in Section 6 5 can be argued independently\nfrom Faraday\u2019s law using the Lorentz force on moving charges However,\neven if the charges are stationary [and the q (v \u00d7 B) term of the Lorentz\nforce is not operative], an emf is nevertheless induced in the presence of a\ntime-varying magnetic field Thus, moving charges in static field and static\ncharges in a time-varying field seem to be symmetric situation for Faraday\u2019s\nlaw"}, {"Chapter": "1", "sentence_range": "5369-5372", "Text": "5 can be argued independently\nfrom Faraday\u2019s law using the Lorentz force on moving charges However,\neven if the charges are stationary [and the q (v \u00d7 B) term of the Lorentz\nforce is not operative], an emf is nevertheless induced in the presence of a\ntime-varying magnetic field Thus, moving charges in static field and static\ncharges in a time-varying field seem to be symmetric situation for Faraday\u2019s\nlaw This gives a tantalising hint on the relevance of the principle of\nrelativity for Faraday\u2019s law"}, {"Chapter": "1", "sentence_range": "5370-5373", "Text": "However,\neven if the charges are stationary [and the q (v \u00d7 B) term of the Lorentz\nforce is not operative], an emf is nevertheless induced in the presence of a\ntime-varying magnetic field Thus, moving charges in static field and static\ncharges in a time-varying field seem to be symmetric situation for Faraday\u2019s\nlaw This gives a tantalising hint on the relevance of the principle of\nrelativity for Faraday\u2019s law EXERCISES\n6"}, {"Chapter": "1", "sentence_range": "5371-5374", "Text": "Thus, moving charges in static field and static\ncharges in a time-varying field seem to be symmetric situation for Faraday\u2019s\nlaw This gives a tantalising hint on the relevance of the principle of\nrelativity for Faraday\u2019s law EXERCISES\n6 1\nPredict the direction of induced current in the situations described\nby the following Figs"}, {"Chapter": "1", "sentence_range": "5372-5375", "Text": "This gives a tantalising hint on the relevance of the principle of\nrelativity for Faraday\u2019s law EXERCISES\n6 1\nPredict the direction of induced current in the situations described\nby the following Figs 6"}, {"Chapter": "1", "sentence_range": "5373-5376", "Text": "EXERCISES\n6 1\nPredict the direction of induced current in the situations described\nby the following Figs 6 15(a) to (f )"}, {"Chapter": "1", "sentence_range": "5374-5377", "Text": "1\nPredict the direction of induced current in the situations described\nby the following Figs 6 15(a) to (f ) Quantity\nSymbol\nUnits\nDimensions\nEquations\nMagnetic Flux\nFB\nWb (weber)\n[M L2 T \u20132 A\u20131]\nFB = B\nA\ni\nEMF\ne\nV (volt)\n[M L2 T \u20133 A\u20131]\ne = \nB\nd(\n)/d\nN\nt\n\u03a6\n\u2212\nMutual Inductance\nM\nH (henry)\n[M L2 T \u20132 A\u20132]\ne1 \n(\n)\n12\nd2\n/d\nM\nI\nt\n= \u2212\nSelf Inductance\nL\nH (henry)\n[M L2 T \u20132 A\u20132]\n(\nLd /d)\nI\nt\n\u03b5 = \u2212\nRationalised 2023-24\nElectromagnetic\nInduction\n175\nFIGURE 6"}, {"Chapter": "1", "sentence_range": "5375-5378", "Text": "6 15(a) to (f ) Quantity\nSymbol\nUnits\nDimensions\nEquations\nMagnetic Flux\nFB\nWb (weber)\n[M L2 T \u20132 A\u20131]\nFB = B\nA\ni\nEMF\ne\nV (volt)\n[M L2 T \u20133 A\u20131]\ne = \nB\nd(\n)/d\nN\nt\n\u03a6\n\u2212\nMutual Inductance\nM\nH (henry)\n[M L2 T \u20132 A\u20132]\ne1 \n(\n)\n12\nd2\n/d\nM\nI\nt\n= \u2212\nSelf Inductance\nL\nH (henry)\n[M L2 T \u20132 A\u20132]\n(\nLd /d)\nI\nt\n\u03b5 = \u2212\nRationalised 2023-24\nElectromagnetic\nInduction\n175\nFIGURE 6 15\n6"}, {"Chapter": "1", "sentence_range": "5376-5379", "Text": "15(a) to (f ) Quantity\nSymbol\nUnits\nDimensions\nEquations\nMagnetic Flux\nFB\nWb (weber)\n[M L2 T \u20132 A\u20131]\nFB = B\nA\ni\nEMF\ne\nV (volt)\n[M L2 T \u20133 A\u20131]\ne = \nB\nd(\n)/d\nN\nt\n\u03a6\n\u2212\nMutual Inductance\nM\nH (henry)\n[M L2 T \u20132 A\u20132]\ne1 \n(\n)\n12\nd2\n/d\nM\nI\nt\n= \u2212\nSelf Inductance\nL\nH (henry)\n[M L2 T \u20132 A\u20132]\n(\nLd /d)\nI\nt\n\u03b5 = \u2212\nRationalised 2023-24\nElectromagnetic\nInduction\n175\nFIGURE 6 15\n6 2\nUse Lenz\u2019s law to determine the direction of induced current in the\nsituations described by Fig"}, {"Chapter": "1", "sentence_range": "5377-5380", "Text": "Quantity\nSymbol\nUnits\nDimensions\nEquations\nMagnetic Flux\nFB\nWb (weber)\n[M L2 T \u20132 A\u20131]\nFB = B\nA\ni\nEMF\ne\nV (volt)\n[M L2 T \u20133 A\u20131]\ne = \nB\nd(\n)/d\nN\nt\n\u03a6\n\u2212\nMutual Inductance\nM\nH (henry)\n[M L2 T \u20132 A\u20132]\ne1 \n(\n)\n12\nd2\n/d\nM\nI\nt\n= \u2212\nSelf Inductance\nL\nH (henry)\n[M L2 T \u20132 A\u20132]\n(\nLd /d)\nI\nt\n\u03b5 = \u2212\nRationalised 2023-24\nElectromagnetic\nInduction\n175\nFIGURE 6 15\n6 2\nUse Lenz\u2019s law to determine the direction of induced current in the\nsituations described by Fig 6"}, {"Chapter": "1", "sentence_range": "5378-5381", "Text": "15\n6 2\nUse Lenz\u2019s law to determine the direction of induced current in the\nsituations described by Fig 6 16:\n(a) A wire of irregular shape turning into a circular shape;\nRationalised 2023-24\nPhysics\n176\n(b) A circular loop being deformed into a narrow straight wire"}, {"Chapter": "1", "sentence_range": "5379-5382", "Text": "2\nUse Lenz\u2019s law to determine the direction of induced current in the\nsituations described by Fig 6 16:\n(a) A wire of irregular shape turning into a circular shape;\nRationalised 2023-24\nPhysics\n176\n(b) A circular loop being deformed into a narrow straight wire FIGURE 6"}, {"Chapter": "1", "sentence_range": "5380-5383", "Text": "6 16:\n(a) A wire of irregular shape turning into a circular shape;\nRationalised 2023-24\nPhysics\n176\n(b) A circular loop being deformed into a narrow straight wire FIGURE 6 16\n6"}, {"Chapter": "1", "sentence_range": "5381-5384", "Text": "16:\n(a) A wire of irregular shape turning into a circular shape;\nRationalised 2023-24\nPhysics\n176\n(b) A circular loop being deformed into a narrow straight wire FIGURE 6 16\n6 3\nA long solenoid with 15 turns per cm has a small loop of area 2"}, {"Chapter": "1", "sentence_range": "5382-5385", "Text": "FIGURE 6 16\n6 3\nA long solenoid with 15 turns per cm has a small loop of area 2 0 cm2\nplaced inside the solenoid normal to its axis"}, {"Chapter": "1", "sentence_range": "5383-5386", "Text": "16\n6 3\nA long solenoid with 15 turns per cm has a small loop of area 2 0 cm2\nplaced inside the solenoid normal to its axis If the current carried\nby the solenoid changes steadily from 2"}, {"Chapter": "1", "sentence_range": "5384-5387", "Text": "3\nA long solenoid with 15 turns per cm has a small loop of area 2 0 cm2\nplaced inside the solenoid normal to its axis If the current carried\nby the solenoid changes steadily from 2 0 A to 4"}, {"Chapter": "1", "sentence_range": "5385-5388", "Text": "0 cm2\nplaced inside the solenoid normal to its axis If the current carried\nby the solenoid changes steadily from 2 0 A to 4 0 A in 0"}, {"Chapter": "1", "sentence_range": "5386-5389", "Text": "If the current carried\nby the solenoid changes steadily from 2 0 A to 4 0 A in 0 1 s, what is\nthe induced emf in the loop while the current is changing"}, {"Chapter": "1", "sentence_range": "5387-5390", "Text": "0 A to 4 0 A in 0 1 s, what is\nthe induced emf in the loop while the current is changing 6"}, {"Chapter": "1", "sentence_range": "5388-5391", "Text": "0 A in 0 1 s, what is\nthe induced emf in the loop while the current is changing 6 4\nA rectangular wire loop of sides 8 cm and 2 cm with a small cut is\nmoving out of a region of uniform magnetic field of magnitude 0"}, {"Chapter": "1", "sentence_range": "5389-5392", "Text": "1 s, what is\nthe induced emf in the loop while the current is changing 6 4\nA rectangular wire loop of sides 8 cm and 2 cm with a small cut is\nmoving out of a region of uniform magnetic field of magnitude 0 3 T\ndirected normal to the loop"}, {"Chapter": "1", "sentence_range": "5390-5393", "Text": "6 4\nA rectangular wire loop of sides 8 cm and 2 cm with a small cut is\nmoving out of a region of uniform magnetic field of magnitude 0 3 T\ndirected normal to the loop What is the emf developed across the\ncut if the velocity of the loop is 1 cm s\u20131 in a direction normal to the\n(a) longer side, (b) shorter side of the loop"}, {"Chapter": "1", "sentence_range": "5391-5394", "Text": "4\nA rectangular wire loop of sides 8 cm and 2 cm with a small cut is\nmoving out of a region of uniform magnetic field of magnitude 0 3 T\ndirected normal to the loop What is the emf developed across the\ncut if the velocity of the loop is 1 cm s\u20131 in a direction normal to the\n(a) longer side, (b) shorter side of the loop For how long does the\ninduced voltage last in each case"}, {"Chapter": "1", "sentence_range": "5392-5395", "Text": "3 T\ndirected normal to the loop What is the emf developed across the\ncut if the velocity of the loop is 1 cm s\u20131 in a direction normal to the\n(a) longer side, (b) shorter side of the loop For how long does the\ninduced voltage last in each case 6"}, {"Chapter": "1", "sentence_range": "5393-5396", "Text": "What is the emf developed across the\ncut if the velocity of the loop is 1 cm s\u20131 in a direction normal to the\n(a) longer side, (b) shorter side of the loop For how long does the\ninduced voltage last in each case 6 5\nA 1"}, {"Chapter": "1", "sentence_range": "5394-5397", "Text": "For how long does the\ninduced voltage last in each case 6 5\nA 1 0 m long metallic rod is rotated with an angular frequency of\n400 rad s\u20131\n about an axis normal to the rod passing through its one\nend"}, {"Chapter": "1", "sentence_range": "5395-5398", "Text": "6 5\nA 1 0 m long metallic rod is rotated with an angular frequency of\n400 rad s\u20131\n about an axis normal to the rod passing through its one\nend The other end of the rod is in contact with a circular metallic\nring"}, {"Chapter": "1", "sentence_range": "5396-5399", "Text": "5\nA 1 0 m long metallic rod is rotated with an angular frequency of\n400 rad s\u20131\n about an axis normal to the rod passing through its one\nend The other end of the rod is in contact with a circular metallic\nring A constant and uniform magnetic field of 0"}, {"Chapter": "1", "sentence_range": "5397-5400", "Text": "0 m long metallic rod is rotated with an angular frequency of\n400 rad s\u20131\n about an axis normal to the rod passing through its one\nend The other end of the rod is in contact with a circular metallic\nring A constant and uniform magnetic field of 0 5 T parallel to the\naxis exists everywhere"}, {"Chapter": "1", "sentence_range": "5398-5401", "Text": "The other end of the rod is in contact with a circular metallic\nring A constant and uniform magnetic field of 0 5 T parallel to the\naxis exists everywhere Calculate the emf developed between the\ncentre and the ring"}, {"Chapter": "1", "sentence_range": "5399-5402", "Text": "A constant and uniform magnetic field of 0 5 T parallel to the\naxis exists everywhere Calculate the emf developed between the\ncentre and the ring 6"}, {"Chapter": "1", "sentence_range": "5400-5403", "Text": "5 T parallel to the\naxis exists everywhere Calculate the emf developed between the\ncentre and the ring 6 6\nA horizontal straight wire 10 m long extending from east to west is\nfalling with a speed of 5"}, {"Chapter": "1", "sentence_range": "5401-5404", "Text": "Calculate the emf developed between the\ncentre and the ring 6 6\nA horizontal straight wire 10 m long extending from east to west is\nfalling with a speed of 5 0 m s\u20131, at right angles to the horizontal\ncomponent of the earth\u2019s magnetic field, 0"}, {"Chapter": "1", "sentence_range": "5402-5405", "Text": "6 6\nA horizontal straight wire 10 m long extending from east to west is\nfalling with a speed of 5 0 m s\u20131, at right angles to the horizontal\ncomponent of the earth\u2019s magnetic field, 0 30 \u00b4 10\u20134 Wb m\u20132"}, {"Chapter": "1", "sentence_range": "5403-5406", "Text": "6\nA horizontal straight wire 10 m long extending from east to west is\nfalling with a speed of 5 0 m s\u20131, at right angles to the horizontal\ncomponent of the earth\u2019s magnetic field, 0 30 \u00b4 10\u20134 Wb m\u20132 (a) What is the instantaneous value of the emf induced in the wire"}, {"Chapter": "1", "sentence_range": "5404-5407", "Text": "0 m s\u20131, at right angles to the horizontal\ncomponent of the earth\u2019s magnetic field, 0 30 \u00b4 10\u20134 Wb m\u20132 (a) What is the instantaneous value of the emf induced in the wire (b) What is the direction of the emf"}, {"Chapter": "1", "sentence_range": "5405-5408", "Text": "30 \u00b4 10\u20134 Wb m\u20132 (a) What is the instantaneous value of the emf induced in the wire (b) What is the direction of the emf (c) Which end of the wire is at the higher electrical potential"}, {"Chapter": "1", "sentence_range": "5406-5409", "Text": "(a) What is the instantaneous value of the emf induced in the wire (b) What is the direction of the emf (c) Which end of the wire is at the higher electrical potential 6"}, {"Chapter": "1", "sentence_range": "5407-5410", "Text": "(b) What is the direction of the emf (c) Which end of the wire is at the higher electrical potential 6 7\nCurrent in a circuit falls from 5"}, {"Chapter": "1", "sentence_range": "5408-5411", "Text": "(c) Which end of the wire is at the higher electrical potential 6 7\nCurrent in a circuit falls from 5 0 A to 0"}, {"Chapter": "1", "sentence_range": "5409-5412", "Text": "6 7\nCurrent in a circuit falls from 5 0 A to 0 0 A in 0"}, {"Chapter": "1", "sentence_range": "5410-5413", "Text": "7\nCurrent in a circuit falls from 5 0 A to 0 0 A in 0 1 s"}, {"Chapter": "1", "sentence_range": "5411-5414", "Text": "0 A to 0 0 A in 0 1 s If an average emf\nof 200 V induced, give an estimate of  the self-inductance of the circuit"}, {"Chapter": "1", "sentence_range": "5412-5415", "Text": "0 A in 0 1 s If an average emf\nof 200 V induced, give an estimate of  the self-inductance of the circuit 6"}, {"Chapter": "1", "sentence_range": "5413-5416", "Text": "1 s If an average emf\nof 200 V induced, give an estimate of  the self-inductance of the circuit 6 8\nA pair of adjacent coils has a mutual inductance of 1"}, {"Chapter": "1", "sentence_range": "5414-5417", "Text": "If an average emf\nof 200 V induced, give an estimate of  the self-inductance of the circuit 6 8\nA pair of adjacent coils has a mutual inductance of 1 5 H"}, {"Chapter": "1", "sentence_range": "5415-5418", "Text": "6 8\nA pair of adjacent coils has a mutual inductance of 1 5 H If the\ncurrent in one coil changes from 0 to 20 A in 0"}, {"Chapter": "1", "sentence_range": "5416-5419", "Text": "8\nA pair of adjacent coils has a mutual inductance of 1 5 H If the\ncurrent in one coil changes from 0 to 20 A in 0 5 s, what is the\nchange of flux linkage with the other coil"}, {"Chapter": "1", "sentence_range": "5417-5420", "Text": "5 H If the\ncurrent in one coil changes from 0 to 20 A in 0 5 s, what is the\nchange of flux linkage with the other coil Rationalised 2023-24\n7"}, {"Chapter": "1", "sentence_range": "5418-5421", "Text": "If the\ncurrent in one coil changes from 0 to 20 A in 0 5 s, what is the\nchange of flux linkage with the other coil Rationalised 2023-24\n7 1  INTRODUCTION\nWe have so far considered direct current (dc) sources and circuits with dc\nsources"}, {"Chapter": "1", "sentence_range": "5419-5422", "Text": "5 s, what is the\nchange of flux linkage with the other coil Rationalised 2023-24\n7 1  INTRODUCTION\nWe have so far considered direct current (dc) sources and circuits with dc\nsources These currents do not change direction with time"}, {"Chapter": "1", "sentence_range": "5420-5423", "Text": "Rationalised 2023-24\n7 1  INTRODUCTION\nWe have so far considered direct current (dc) sources and circuits with dc\nsources These currents do not change direction with time But voltages\nand currents that vary with time are very common"}, {"Chapter": "1", "sentence_range": "5421-5424", "Text": "1  INTRODUCTION\nWe have so far considered direct current (dc) sources and circuits with dc\nsources These currents do not change direction with time But voltages\nand currents that vary with time are very common The electric mains\nsupply in our homes and offices is a voltage that varies like a sine function\nwith time"}, {"Chapter": "1", "sentence_range": "5422-5425", "Text": "These currents do not change direction with time But voltages\nand currents that vary with time are very common The electric mains\nsupply in our homes and offices is a voltage that varies like a sine function\nwith time Such a voltage is called alternating voltage (ac voltage) and\nthe current driven by it in a circuit is called the alternating current (ac\ncurrent)*"}, {"Chapter": "1", "sentence_range": "5423-5426", "Text": "But voltages\nand currents that vary with time are very common The electric mains\nsupply in our homes and offices is a voltage that varies like a sine function\nwith time Such a voltage is called alternating voltage (ac voltage) and\nthe current driven by it in a circuit is called the alternating current (ac\ncurrent)* Today, most of the electrical devices we use require ac voltage"}, {"Chapter": "1", "sentence_range": "5424-5427", "Text": "The electric mains\nsupply in our homes and offices is a voltage that varies like a sine function\nwith time Such a voltage is called alternating voltage (ac voltage) and\nthe current driven by it in a circuit is called the alternating current (ac\ncurrent)* Today, most of the electrical devices we use require ac voltage This is mainly because most of the electrical energy sold by power\ncompanies is transmitted and distributed as alternating current"}, {"Chapter": "1", "sentence_range": "5425-5428", "Text": "Such a voltage is called alternating voltage (ac voltage) and\nthe current driven by it in a circuit is called the alternating current (ac\ncurrent)* Today, most of the electrical devices we use require ac voltage This is mainly because most of the electrical energy sold by power\ncompanies is transmitted and distributed as alternating current The main\nreason for preferring use of ac voltage over dc voltage is that ac voltages\ncan be easily and efficiently converted from one voltage to the other by\nmeans of transformers"}, {"Chapter": "1", "sentence_range": "5426-5429", "Text": "Today, most of the electrical devices we use require ac voltage This is mainly because most of the electrical energy sold by power\ncompanies is transmitted and distributed as alternating current The main\nreason for preferring use of ac voltage over dc voltage is that ac voltages\ncan be easily and efficiently converted from one voltage to the other by\nmeans of transformers Further, electrical energy can also be transmitted\neconomically over long distances"}, {"Chapter": "1", "sentence_range": "5427-5430", "Text": "This is mainly because most of the electrical energy sold by power\ncompanies is transmitted and distributed as alternating current The main\nreason for preferring use of ac voltage over dc voltage is that ac voltages\ncan be easily and efficiently converted from one voltage to the other by\nmeans of transformers Further, electrical energy can also be transmitted\neconomically over long distances AC circuits exhibit characteristics which\nare exploited in many devices of daily use"}, {"Chapter": "1", "sentence_range": "5428-5431", "Text": "The main\nreason for preferring use of ac voltage over dc voltage is that ac voltages\ncan be easily and efficiently converted from one voltage to the other by\nmeans of transformers Further, electrical energy can also be transmitted\neconomically over long distances AC circuits exhibit characteristics which\nare exploited in many devices of daily use For example, whenever we\ntune our radio to a favourite station, we are taking advantage of a special\nproperty of ac circuits \u2013 one of many that you will study in this chapter"}, {"Chapter": "1", "sentence_range": "5429-5432", "Text": "Further, electrical energy can also be transmitted\neconomically over long distances AC circuits exhibit characteristics which\nare exploited in many devices of daily use For example, whenever we\ntune our radio to a favourite station, we are taking advantage of a special\nproperty of ac circuits \u2013 one of many that you will study in this chapter Chapter Seven\nALTERNATING\nCURRENT\n*\nThe phrases ac voltage and ac current are contradictory and redundant,\nrespectively, since they mean, literally, alternating current voltage and alternating\ncurrent current"}, {"Chapter": "1", "sentence_range": "5430-5433", "Text": "AC circuits exhibit characteristics which\nare exploited in many devices of daily use For example, whenever we\ntune our radio to a favourite station, we are taking advantage of a special\nproperty of ac circuits \u2013 one of many that you will study in this chapter Chapter Seven\nALTERNATING\nCURRENT\n*\nThe phrases ac voltage and ac current are contradictory and redundant,\nrespectively, since they mean, literally, alternating current voltage and alternating\ncurrent current Still, the abbreviation ac to designate an electrical quantity\ndisplaying simple harmonic time dependance has become so universally accepted\nthat we follow others in its use"}, {"Chapter": "1", "sentence_range": "5431-5434", "Text": "For example, whenever we\ntune our radio to a favourite station, we are taking advantage of a special\nproperty of ac circuits \u2013 one of many that you will study in this chapter Chapter Seven\nALTERNATING\nCURRENT\n*\nThe phrases ac voltage and ac current are contradictory and redundant,\nrespectively, since they mean, literally, alternating current voltage and alternating\ncurrent current Still, the abbreviation ac to designate an electrical quantity\ndisplaying simple harmonic time dependance has become so universally accepted\nthat we follow others in its use Further, voltage \u2013 another phrase commonly\nused means potential difference between two points"}, {"Chapter": "1", "sentence_range": "5432-5435", "Text": "Chapter Seven\nALTERNATING\nCURRENT\n*\nThe phrases ac voltage and ac current are contradictory and redundant,\nrespectively, since they mean, literally, alternating current voltage and alternating\ncurrent current Still, the abbreviation ac to designate an electrical quantity\ndisplaying simple harmonic time dependance has become so universally accepted\nthat we follow others in its use Further, voltage \u2013 another phrase commonly\nused means potential difference between two points Rationalised 2023-24\nPhysics\n178\nNICOLA TESLA (1856 \u2013 1943)\nNicola Tesla  (1856 \u2013\n1943) Serbian-American\nscientist, inventor and\ngenius"}, {"Chapter": "1", "sentence_range": "5433-5436", "Text": "Still, the abbreviation ac to designate an electrical quantity\ndisplaying simple harmonic time dependance has become so universally accepted\nthat we follow others in its use Further, voltage \u2013 another phrase commonly\nused means potential difference between two points Rationalised 2023-24\nPhysics\n178\nNICOLA TESLA (1856 \u2013 1943)\nNicola Tesla  (1856 \u2013\n1943) Serbian-American\nscientist, inventor and\ngenius He conceived the\nidea \nof \nthe \nrotating\nmagnetic field, which is the\nbasis of practically all\nalternating \ncurrent\nmachinery, and which\nhelped usher in the age of\nelectric power"}, {"Chapter": "1", "sentence_range": "5434-5437", "Text": "Further, voltage \u2013 another phrase commonly\nused means potential difference between two points Rationalised 2023-24\nPhysics\n178\nNICOLA TESLA (1856 \u2013 1943)\nNicola Tesla  (1856 \u2013\n1943) Serbian-American\nscientist, inventor and\ngenius He conceived the\nidea \nof \nthe \nrotating\nmagnetic field, which is the\nbasis of practically all\nalternating \ncurrent\nmachinery, and which\nhelped usher in the age of\nelectric power He also\ninvented among other\nthings the induction motor,\nthe polyphase system of ac\npower, \nand \nthe \nhigh\nfrequency induction coil\n(the Tesla coil) used in radio\nand television sets and\nother electronic equipment"}, {"Chapter": "1", "sentence_range": "5435-5438", "Text": "Rationalised 2023-24\nPhysics\n178\nNICOLA TESLA (1856 \u2013 1943)\nNicola Tesla  (1856 \u2013\n1943) Serbian-American\nscientist, inventor and\ngenius He conceived the\nidea \nof \nthe \nrotating\nmagnetic field, which is the\nbasis of practically all\nalternating \ncurrent\nmachinery, and which\nhelped usher in the age of\nelectric power He also\ninvented among other\nthings the induction motor,\nthe polyphase system of ac\npower, \nand \nthe \nhigh\nfrequency induction coil\n(the Tesla coil) used in radio\nand television sets and\nother electronic equipment The SI unit of magnetic field\nis named in his honour"}, {"Chapter": "1", "sentence_range": "5436-5439", "Text": "He conceived the\nidea \nof \nthe \nrotating\nmagnetic field, which is the\nbasis of practically all\nalternating \ncurrent\nmachinery, and which\nhelped usher in the age of\nelectric power He also\ninvented among other\nthings the induction motor,\nthe polyphase system of ac\npower, \nand \nthe \nhigh\nfrequency induction coil\n(the Tesla coil) used in radio\nand television sets and\nother electronic equipment The SI unit of magnetic field\nis named in his honour 7"}, {"Chapter": "1", "sentence_range": "5437-5440", "Text": "He also\ninvented among other\nthings the induction motor,\nthe polyphase system of ac\npower, \nand \nthe \nhigh\nfrequency induction coil\n(the Tesla coil) used in radio\nand television sets and\nother electronic equipment The SI unit of magnetic field\nis named in his honour 7 2  AC VOLTAGE APPLIED TO A RESISTOR\nFigure 7"}, {"Chapter": "1", "sentence_range": "5438-5441", "Text": "The SI unit of magnetic field\nis named in his honour 7 2  AC VOLTAGE APPLIED TO A RESISTOR\nFigure 7 1 shows a resistor connected to a source e of\nac voltage"}, {"Chapter": "1", "sentence_range": "5439-5442", "Text": "7 2  AC VOLTAGE APPLIED TO A RESISTOR\nFigure 7 1 shows a resistor connected to a source e of\nac voltage The symbol for an ac source in a circuit\ndiagram is"}, {"Chapter": "1", "sentence_range": "5440-5443", "Text": "2  AC VOLTAGE APPLIED TO A RESISTOR\nFigure 7 1 shows a resistor connected to a source e of\nac voltage The symbol for an ac source in a circuit\ndiagram is We consider a source which produces\nsinusoidally varying potential difference across its\nterminals"}, {"Chapter": "1", "sentence_range": "5441-5444", "Text": "1 shows a resistor connected to a source e of\nac voltage The symbol for an ac source in a circuit\ndiagram is We consider a source which produces\nsinusoidally varying potential difference across its\nterminals Let this potential difference, also called ac\nvoltage, be given by\nmsin\nv\nv\n\u03c9t\n=\n(7"}, {"Chapter": "1", "sentence_range": "5442-5445", "Text": "The symbol for an ac source in a circuit\ndiagram is We consider a source which produces\nsinusoidally varying potential difference across its\nterminals Let this potential difference, also called ac\nvoltage, be given by\nmsin\nv\nv\n\u03c9t\n=\n(7 1)\nwhere vm is the amplitude of the oscillating potential\ndifference and w  is its angular frequency"}, {"Chapter": "1", "sentence_range": "5443-5446", "Text": "We consider a source which produces\nsinusoidally varying potential difference across its\nterminals Let this potential difference, also called ac\nvoltage, be given by\nmsin\nv\nv\n\u03c9t\n=\n(7 1)\nwhere vm is the amplitude of the oscillating potential\ndifference and w  is its angular frequency To find the value of current through the resistor, we\napply Kirchhoff\u2019s loop rule \n\u2211\u03b5( )t =\n0 (refer to Section\n3"}, {"Chapter": "1", "sentence_range": "5444-5447", "Text": "Let this potential difference, also called ac\nvoltage, be given by\nmsin\nv\nv\n\u03c9t\n=\n(7 1)\nwhere vm is the amplitude of the oscillating potential\ndifference and w  is its angular frequency To find the value of current through the resistor, we\napply Kirchhoff\u2019s loop rule \n\u2211\u03b5( )t =\n0 (refer to Section\n3 13), to the circuit shown in Fig"}, {"Chapter": "1", "sentence_range": "5445-5448", "Text": "1)\nwhere vm is the amplitude of the oscillating potential\ndifference and w  is its angular frequency To find the value of current through the resistor, we\napply Kirchhoff\u2019s loop rule \n\u2211\u03b5( )t =\n0 (refer to Section\n3 13), to the circuit shown in Fig 7"}, {"Chapter": "1", "sentence_range": "5446-5449", "Text": "To find the value of current through the resistor, we\napply Kirchhoff\u2019s loop rule \n\u2211\u03b5( )t =\n0 (refer to Section\n3 13), to the circuit shown in Fig 7 1 to get\n=\nvmsin\nt\ni R\n\u03c9\nor   \nvmsin\ni\nt\nR\n\u03c9\n=\nSince R is a constant, we can write this equation as\nmsin\ni\ni\n\u03c9t\n=\n(7"}, {"Chapter": "1", "sentence_range": "5447-5450", "Text": "13), to the circuit shown in Fig 7 1 to get\n=\nvmsin\nt\ni R\n\u03c9\nor   \nvmsin\ni\nt\nR\n\u03c9\n=\nSince R is a constant, we can write this equation as\nmsin\ni\ni\n\u03c9t\n=\n(7 2)\nwhere the current amplitude im is given by\nm\nm\nv\ni\n=R\n(7"}, {"Chapter": "1", "sentence_range": "5448-5451", "Text": "7 1 to get\n=\nvmsin\nt\ni R\n\u03c9\nor   \nvmsin\ni\nt\nR\n\u03c9\n=\nSince R is a constant, we can write this equation as\nmsin\ni\ni\n\u03c9t\n=\n(7 2)\nwhere the current amplitude im is given by\nm\nm\nv\ni\n=R\n(7 3)\nEquation (7"}, {"Chapter": "1", "sentence_range": "5449-5452", "Text": "1 to get\n=\nvmsin\nt\ni R\n\u03c9\nor   \nvmsin\ni\nt\nR\n\u03c9\n=\nSince R is a constant, we can write this equation as\nmsin\ni\ni\n\u03c9t\n=\n(7 2)\nwhere the current amplitude im is given by\nm\nm\nv\ni\n=R\n(7 3)\nEquation (7 3) is Ohm\u2019s law, which for resistors, works equally\nwell for both ac and dc voltages"}, {"Chapter": "1", "sentence_range": "5450-5453", "Text": "2)\nwhere the current amplitude im is given by\nm\nm\nv\ni\n=R\n(7 3)\nEquation (7 3) is Ohm\u2019s law, which for resistors, works equally\nwell for both ac and dc voltages The voltage across a pure resistor\nand the current through it, given by Eqs"}, {"Chapter": "1", "sentence_range": "5451-5454", "Text": "3)\nEquation (7 3) is Ohm\u2019s law, which for resistors, works equally\nwell for both ac and dc voltages The voltage across a pure resistor\nand the current through it, given by Eqs (7"}, {"Chapter": "1", "sentence_range": "5452-5455", "Text": "3) is Ohm\u2019s law, which for resistors, works equally\nwell for both ac and dc voltages The voltage across a pure resistor\nand the current through it, given by Eqs (7 1) and (7"}, {"Chapter": "1", "sentence_range": "5453-5456", "Text": "The voltage across a pure resistor\nand the current through it, given by Eqs (7 1) and (7 2) are\nplotted as a function of time in Fig"}, {"Chapter": "1", "sentence_range": "5454-5457", "Text": "(7 1) and (7 2) are\nplotted as a function of time in Fig 7"}, {"Chapter": "1", "sentence_range": "5455-5458", "Text": "1) and (7 2) are\nplotted as a function of time in Fig 7 2"}, {"Chapter": "1", "sentence_range": "5456-5459", "Text": "2) are\nplotted as a function of time in Fig 7 2 Note, in particular that\nboth v and i reach zero, minimum and maximum values at the\nsame time"}, {"Chapter": "1", "sentence_range": "5457-5460", "Text": "7 2 Note, in particular that\nboth v and i reach zero, minimum and maximum values at the\nsame time Clearly, the voltage and current are in phase with\neach other"}, {"Chapter": "1", "sentence_range": "5458-5461", "Text": "2 Note, in particular that\nboth v and i reach zero, minimum and maximum values at the\nsame time Clearly, the voltage and current are in phase with\neach other We see that, like the applied voltage, the current varies\nsinusoidally and has corresponding positive and negative values\nduring each cycle"}, {"Chapter": "1", "sentence_range": "5459-5462", "Text": "Note, in particular that\nboth v and i reach zero, minimum and maximum values at the\nsame time Clearly, the voltage and current are in phase with\neach other We see that, like the applied voltage, the current varies\nsinusoidally and has corresponding positive and negative values\nduring each cycle Thus, the sum of the instantaneous current\nvalues over one complete cycle is zero, and the average current\nis zero"}, {"Chapter": "1", "sentence_range": "5460-5463", "Text": "Clearly, the voltage and current are in phase with\neach other We see that, like the applied voltage, the current varies\nsinusoidally and has corresponding positive and negative values\nduring each cycle Thus, the sum of the instantaneous current\nvalues over one complete cycle is zero, and the average current\nis zero The fact that the average current is zero, however, does\nFIGURE 7"}, {"Chapter": "1", "sentence_range": "5461-5464", "Text": "We see that, like the applied voltage, the current varies\nsinusoidally and has corresponding positive and negative values\nduring each cycle Thus, the sum of the instantaneous current\nvalues over one complete cycle is zero, and the average current\nis zero The fact that the average current is zero, however, does\nFIGURE 7 1  AC voltage applied to a resistor"}, {"Chapter": "1", "sentence_range": "5462-5465", "Text": "Thus, the sum of the instantaneous current\nvalues over one complete cycle is zero, and the average current\nis zero The fact that the average current is zero, however, does\nFIGURE 7 1  AC voltage applied to a resistor FIGURE 7"}, {"Chapter": "1", "sentence_range": "5463-5466", "Text": "The fact that the average current is zero, however, does\nFIGURE 7 1  AC voltage applied to a resistor FIGURE 7 2 In a pure\nresistor, the voltage and\ncurrent are in phase"}, {"Chapter": "1", "sentence_range": "5464-5467", "Text": "1  AC voltage applied to a resistor FIGURE 7 2 In a pure\nresistor, the voltage and\ncurrent are in phase The\nminima, zero and maxima\noccur at the same\nrespective times"}, {"Chapter": "1", "sentence_range": "5465-5468", "Text": "FIGURE 7 2 In a pure\nresistor, the voltage and\ncurrent are in phase The\nminima, zero and maxima\noccur at the same\nrespective times Rationalised 2023-24\n179\nAlternating Current\nGEORGE WESTINGHOUSE (1846 \u2013 1914)\nGeorge \nWestinghouse\n(1846 \u2013 1914) A leading\nproponent of the use of\nalternating current over\ndirect \ncurrent"}, {"Chapter": "1", "sentence_range": "5466-5469", "Text": "2 In a pure\nresistor, the voltage and\ncurrent are in phase The\nminima, zero and maxima\noccur at the same\nrespective times Rationalised 2023-24\n179\nAlternating Current\nGEORGE WESTINGHOUSE (1846 \u2013 1914)\nGeorge \nWestinghouse\n(1846 \u2013 1914) A leading\nproponent of the use of\nalternating current over\ndirect \ncurrent Thus,\nhe came into conflict\nwith Thomas Alva Edison,\nan advocate of direct\ncurrent"}, {"Chapter": "1", "sentence_range": "5467-5470", "Text": "The\nminima, zero and maxima\noccur at the same\nrespective times Rationalised 2023-24\n179\nAlternating Current\nGEORGE WESTINGHOUSE (1846 \u2013 1914)\nGeorge \nWestinghouse\n(1846 \u2013 1914) A leading\nproponent of the use of\nalternating current over\ndirect \ncurrent Thus,\nhe came into conflict\nwith Thomas Alva Edison,\nan advocate of direct\ncurrent Westinghouse\nwas convinced that the\ntechnology of alternating\ncurrent was the key to\nthe \nelectrical \nfuture"}, {"Chapter": "1", "sentence_range": "5468-5471", "Text": "Rationalised 2023-24\n179\nAlternating Current\nGEORGE WESTINGHOUSE (1846 \u2013 1914)\nGeorge \nWestinghouse\n(1846 \u2013 1914) A leading\nproponent of the use of\nalternating current over\ndirect \ncurrent Thus,\nhe came into conflict\nwith Thomas Alva Edison,\nan advocate of direct\ncurrent Westinghouse\nwas convinced that the\ntechnology of alternating\ncurrent was the key to\nthe \nelectrical \nfuture He founded the famous\nCompany named after him\nand enlisted the services\nof \nNicola \nTesla \nand\nother inventors in the\ndevelopment of alternating\ncurrent \nmotors \nand\napparatus \nfor \nthe\ntransmission \nof \nhigh\ntension current, pioneering\nin large scale lighting"}, {"Chapter": "1", "sentence_range": "5469-5472", "Text": "Thus,\nhe came into conflict\nwith Thomas Alva Edison,\nan advocate of direct\ncurrent Westinghouse\nwas convinced that the\ntechnology of alternating\ncurrent was the key to\nthe \nelectrical \nfuture He founded the famous\nCompany named after him\nand enlisted the services\nof \nNicola \nTesla \nand\nother inventors in the\ndevelopment of alternating\ncurrent \nmotors \nand\napparatus \nfor \nthe\ntransmission \nof \nhigh\ntension current, pioneering\nin large scale lighting not mean that the average power consumed is zero and\nthat there is no dissipation of electrical energy"}, {"Chapter": "1", "sentence_range": "5470-5473", "Text": "Westinghouse\nwas convinced that the\ntechnology of alternating\ncurrent was the key to\nthe \nelectrical \nfuture He founded the famous\nCompany named after him\nand enlisted the services\nof \nNicola \nTesla \nand\nother inventors in the\ndevelopment of alternating\ncurrent \nmotors \nand\napparatus \nfor \nthe\ntransmission \nof \nhigh\ntension current, pioneering\nin large scale lighting not mean that the average power consumed is zero and\nthat there is no dissipation of electrical energy As you\nknow, Joule heating is given by i2R and depends on i2\n(which is always positive whether i is positive or negative)\nand not on i"}, {"Chapter": "1", "sentence_range": "5471-5474", "Text": "He founded the famous\nCompany named after him\nand enlisted the services\nof \nNicola \nTesla \nand\nother inventors in the\ndevelopment of alternating\ncurrent \nmotors \nand\napparatus \nfor \nthe\ntransmission \nof \nhigh\ntension current, pioneering\nin large scale lighting not mean that the average power consumed is zero and\nthat there is no dissipation of electrical energy As you\nknow, Joule heating is given by i2R and depends on i2\n(which is always positive whether i is positive or negative)\nand not on i Thus, there is Joule heating and\ndissipation \nof \nelectrical \nenergy \nwhen \nan\nac current passes through a resistor"}, {"Chapter": "1", "sentence_range": "5472-5475", "Text": "not mean that the average power consumed is zero and\nthat there is no dissipation of electrical energy As you\nknow, Joule heating is given by i2R and depends on i2\n(which is always positive whether i is positive or negative)\nand not on i Thus, there is Joule heating and\ndissipation \nof \nelectrical \nenergy \nwhen \nan\nac current passes through a resistor The instantaneous power dissipated in the resistor is\n2\n2\nsin2\nm\np\ni R\ni R\n\u03c9t\n=\n=\n(7"}, {"Chapter": "1", "sentence_range": "5473-5476", "Text": "As you\nknow, Joule heating is given by i2R and depends on i2\n(which is always positive whether i is positive or negative)\nand not on i Thus, there is Joule heating and\ndissipation \nof \nelectrical \nenergy \nwhen \nan\nac current passes through a resistor The instantaneous power dissipated in the resistor is\n2\n2\nsin2\nm\np\ni R\ni R\n\u03c9t\n=\n=\n(7 4)\nThe average value of p over a cycle is*\n2\n2\nsin2\nm\np\ni R\ni R\n\u03c9t\n= <\n> = <\n>\n  [7"}, {"Chapter": "1", "sentence_range": "5474-5477", "Text": "Thus, there is Joule heating and\ndissipation \nof \nelectrical \nenergy \nwhen \nan\nac current passes through a resistor The instantaneous power dissipated in the resistor is\n2\n2\nsin2\nm\np\ni R\ni R\n\u03c9t\n=\n=\n(7 4)\nThe average value of p over a cycle is*\n2\n2\nsin2\nm\np\ni R\ni R\n\u03c9t\n= <\n> = <\n>\n  [7 5(a)]\nwhere the bar over a letter (here, p) denotes its average\nvalue and <"}, {"Chapter": "1", "sentence_range": "5475-5478", "Text": "The instantaneous power dissipated in the resistor is\n2\n2\nsin2\nm\np\ni R\ni R\n\u03c9t\n=\n=\n(7 4)\nThe average value of p over a cycle is*\n2\n2\nsin2\nm\np\ni R\ni R\n\u03c9t\n= <\n> = <\n>\n  [7 5(a)]\nwhere the bar over a letter (here, p) denotes its average\nvalue and < > denotes taking average of the quantity\ninside the bracket"}, {"Chapter": "1", "sentence_range": "5476-5479", "Text": "4)\nThe average value of p over a cycle is*\n2\n2\nsin2\nm\np\ni R\ni R\n\u03c9t\n= <\n> = <\n>\n  [7 5(a)]\nwhere the bar over a letter (here, p) denotes its average\nvalue and < > denotes taking average of the quantity\ninside the bracket Since,  i2\nm and R are constants,\n2\nsin2\nm\np\ni R\n\u03c9t\n=\n<\n>\n[7"}, {"Chapter": "1", "sentence_range": "5477-5480", "Text": "5(a)]\nwhere the bar over a letter (here, p) denotes its average\nvalue and < > denotes taking average of the quantity\ninside the bracket Since,  i2\nm and R are constants,\n2\nsin2\nm\np\ni R\n\u03c9t\n=\n<\n>\n[7 5(b)]\nUsing the trigonometric identity, sin2 wt =\n1/2 (1\u2013 cos 2wt), we have < sin2 wt > = (1/2) (1\u2013 < cos 2wt >)\nand since < cos2wt > = 0**, we have,\n2\n1\nsin\n2\n\u03c9t\n<\n> =\nThus,\n212\nm\np\ni R\n=\n[7"}, {"Chapter": "1", "sentence_range": "5478-5481", "Text": "> denotes taking average of the quantity\ninside the bracket Since,  i2\nm and R are constants,\n2\nsin2\nm\np\ni R\n\u03c9t\n=\n<\n>\n[7 5(b)]\nUsing the trigonometric identity, sin2 wt =\n1/2 (1\u2013 cos 2wt), we have < sin2 wt > = (1/2) (1\u2013 < cos 2wt >)\nand since < cos2wt > = 0**, we have,\n2\n1\nsin\n2\n\u03c9t\n<\n> =\nThus,\n212\nm\np\ni R\n=\n[7 5(c)]\nTo express ac power in the same form as dc power\n(P = I2R), a special value of current is defined and used"}, {"Chapter": "1", "sentence_range": "5479-5482", "Text": "Since,  i2\nm and R are constants,\n2\nsin2\nm\np\ni R\n\u03c9t\n=\n<\n>\n[7 5(b)]\nUsing the trigonometric identity, sin2 wt =\n1/2 (1\u2013 cos 2wt), we have < sin2 wt > = (1/2) (1\u2013 < cos 2wt >)\nand since < cos2wt > = 0**, we have,\n2\n1\nsin\n2\n\u03c9t\n<\n> =\nThus,\n212\nm\np\ni R\n=\n[7 5(c)]\nTo express ac power in the same form as dc power\n(P = I2R), a special value of current is defined and used It is called, root mean square (rms) or effective current\n(Fig"}, {"Chapter": "1", "sentence_range": "5480-5483", "Text": "5(b)]\nUsing the trigonometric identity, sin2 wt =\n1/2 (1\u2013 cos 2wt), we have < sin2 wt > = (1/2) (1\u2013 < cos 2wt >)\nand since < cos2wt > = 0**, we have,\n2\n1\nsin\n2\n\u03c9t\n<\n> =\nThus,\n212\nm\np\ni R\n=\n[7 5(c)]\nTo express ac power in the same form as dc power\n(P = I2R), a special value of current is defined and used It is called, root mean square (rms) or effective current\n(Fig 7"}, {"Chapter": "1", "sentence_range": "5481-5484", "Text": "5(c)]\nTo express ac power in the same form as dc power\n(P = I2R), a special value of current is defined and used It is called, root mean square (rms) or effective current\n(Fig 7 3) and is denoted by Irms or I"}, {"Chapter": "1", "sentence_range": "5482-5485", "Text": "It is called, root mean square (rms) or effective current\n(Fig 7 3) and is denoted by Irms or I *\nThe average value of a function F (t) over a period T is given by F t\nT\nF t\nt\nT\n( )\n( )\n=\n\u222b\n1\n0\nd\n** <\n> =\n\u222b\n=\n\uf8f0\uf8ef\uf8ee\n\uf8fb\uf8fa\uf8f9\n=\n\u2212\n[\n] =\ncos\ncos\nsin\nsin\n2\n1\n2\n1\n22\n21\n2\n0\n0\n0\n0\n\u03c9\n\u03c9\n\u03c9\u03c9\n\u03c9\n\u03c9\nt\nT\nt dt\nT\nt\nT\nT\nT\nT\nFIGURE 7"}, {"Chapter": "1", "sentence_range": "5483-5486", "Text": "7 3) and is denoted by Irms or I *\nThe average value of a function F (t) over a period T is given by F t\nT\nF t\nt\nT\n( )\n( )\n=\n\u222b\n1\n0\nd\n** <\n> =\n\u222b\n=\n\uf8f0\uf8ef\uf8ee\n\uf8fb\uf8fa\uf8f9\n=\n\u2212\n[\n] =\ncos\ncos\nsin\nsin\n2\n1\n2\n1\n22\n21\n2\n0\n0\n0\n0\n\u03c9\n\u03c9\n\u03c9\u03c9\n\u03c9\n\u03c9\nt\nT\nt dt\nT\nt\nT\nT\nT\nT\nFIGURE 7 3 The rms current I is related to the\npeak  current im by I = \nmi/ 2\n = 0"}, {"Chapter": "1", "sentence_range": "5484-5487", "Text": "3) and is denoted by Irms or I *\nThe average value of a function F (t) over a period T is given by F t\nT\nF t\nt\nT\n( )\n( )\n=\n\u222b\n1\n0\nd\n** <\n> =\n\u222b\n=\n\uf8f0\uf8ef\uf8ee\n\uf8fb\uf8fa\uf8f9\n=\n\u2212\n[\n] =\ncos\ncos\nsin\nsin\n2\n1\n2\n1\n22\n21\n2\n0\n0\n0\n0\n\u03c9\n\u03c9\n\u03c9\u03c9\n\u03c9\n\u03c9\nt\nT\nt dt\nT\nt\nT\nT\nT\nT\nFIGURE 7 3 The rms current I is related to the\npeak  current im by I = \nmi/ 2\n = 0 707 im"}, {"Chapter": "1", "sentence_range": "5485-5488", "Text": "*\nThe average value of a function F (t) over a period T is given by F t\nT\nF t\nt\nT\n( )\n( )\n=\n\u222b\n1\n0\nd\n** <\n> =\n\u222b\n=\n\uf8f0\uf8ef\uf8ee\n\uf8fb\uf8fa\uf8f9\n=\n\u2212\n[\n] =\ncos\ncos\nsin\nsin\n2\n1\n2\n1\n22\n21\n2\n0\n0\n0\n0\n\u03c9\n\u03c9\n\u03c9\u03c9\n\u03c9\n\u03c9\nt\nT\nt dt\nT\nt\nT\nT\nT\nT\nFIGURE 7 3 The rms current I is related to the\npeak  current im by I = \nmi/ 2\n = 0 707 im Rationalised 2023-24\nPhysics\n180\nIt is defined by\n2\n212\n2\nm\nm\ni\nI\ni\ni\n=\n=\n=\n= 0"}, {"Chapter": "1", "sentence_range": "5486-5489", "Text": "3 The rms current I is related to the\npeak  current im by I = \nmi/ 2\n = 0 707 im Rationalised 2023-24\nPhysics\n180\nIt is defined by\n2\n212\n2\nm\nm\ni\nI\ni\ni\n=\n=\n=\n= 0 707 im\n(7"}, {"Chapter": "1", "sentence_range": "5487-5490", "Text": "707 im Rationalised 2023-24\nPhysics\n180\nIt is defined by\n2\n212\n2\nm\nm\ni\nI\ni\ni\n=\n=\n=\n= 0 707 im\n(7 6)\nIn terms of I, the average power, denoted by P is\n2\n2\n21\nm\np\nP\ni R\nI R\n=\n=\n=\n(7"}, {"Chapter": "1", "sentence_range": "5488-5491", "Text": "Rationalised 2023-24\nPhysics\n180\nIt is defined by\n2\n212\n2\nm\nm\ni\nI\ni\ni\n=\n=\n=\n= 0 707 im\n(7 6)\nIn terms of I, the average power, denoted by P is\n2\n2\n21\nm\np\nP\ni R\nI R\n=\n=\n=\n(7 7)\nSimilarly, we define the rms voltage or effective voltage by\nV = \n2\nm\nv\n = 0"}, {"Chapter": "1", "sentence_range": "5489-5492", "Text": "707 im\n(7 6)\nIn terms of I, the average power, denoted by P is\n2\n2\n21\nm\np\nP\ni R\nI R\n=\n=\n=\n(7 7)\nSimilarly, we define the rms voltage or effective voltage by\nV = \n2\nm\nv\n = 0 707 vm\n(7"}, {"Chapter": "1", "sentence_range": "5490-5493", "Text": "6)\nIn terms of I, the average power, denoted by P is\n2\n2\n21\nm\np\nP\ni R\nI R\n=\n=\n=\n(7 7)\nSimilarly, we define the rms voltage or effective voltage by\nV = \n2\nm\nv\n = 0 707 vm\n(7 8)\nFrom Eq"}, {"Chapter": "1", "sentence_range": "5491-5494", "Text": "7)\nSimilarly, we define the rms voltage or effective voltage by\nV = \n2\nm\nv\n = 0 707 vm\n(7 8)\nFrom Eq (7"}, {"Chapter": "1", "sentence_range": "5492-5495", "Text": "707 vm\n(7 8)\nFrom Eq (7 3), we have\nvm = imR\nor,  \n2\n2\nm\nm\nv\ni\nR\n=\nor,  V = IR\n(7"}, {"Chapter": "1", "sentence_range": "5493-5496", "Text": "8)\nFrom Eq (7 3), we have\nvm = imR\nor,  \n2\n2\nm\nm\nv\ni\nR\n=\nor,  V = IR\n(7 9)\nEquation (7"}, {"Chapter": "1", "sentence_range": "5494-5497", "Text": "(7 3), we have\nvm = imR\nor,  \n2\n2\nm\nm\nv\ni\nR\n=\nor,  V = IR\n(7 9)\nEquation (7 9) gives the relation between ac current and ac voltage\nand is similar to that in the dc case"}, {"Chapter": "1", "sentence_range": "5495-5498", "Text": "3), we have\nvm = imR\nor,  \n2\n2\nm\nm\nv\ni\nR\n=\nor,  V = IR\n(7 9)\nEquation (7 9) gives the relation between ac current and ac voltage\nand is similar to that in the dc case This shows the advantage of\nintroducing the concept of rms values"}, {"Chapter": "1", "sentence_range": "5496-5499", "Text": "9)\nEquation (7 9) gives the relation between ac current and ac voltage\nand is similar to that in the dc case This shows the advantage of\nintroducing the concept of rms values In terms of rms values, the equation\nfor power [Eq"}, {"Chapter": "1", "sentence_range": "5497-5500", "Text": "9) gives the relation between ac current and ac voltage\nand is similar to that in the dc case This shows the advantage of\nintroducing the concept of rms values In terms of rms values, the equation\nfor power [Eq (7"}, {"Chapter": "1", "sentence_range": "5498-5501", "Text": "This shows the advantage of\nintroducing the concept of rms values In terms of rms values, the equation\nfor power [Eq (7 7)] and relation between current and voltage in ac circuits\nare essentially the same as those for the dc case"}, {"Chapter": "1", "sentence_range": "5499-5502", "Text": "In terms of rms values, the equation\nfor power [Eq (7 7)] and relation between current and voltage in ac circuits\nare essentially the same as those for the dc case It is customary to measure and specify rms values for ac quantities"}, {"Chapter": "1", "sentence_range": "5500-5503", "Text": "(7 7)] and relation between current and voltage in ac circuits\nare essentially the same as those for the dc case It is customary to measure and specify rms values for ac quantities For\nexample, the household line voltage of 220 V is an rms value with a peak\nvoltage of\nvm = 2  V =  (1"}, {"Chapter": "1", "sentence_range": "5501-5504", "Text": "7)] and relation between current and voltage in ac circuits\nare essentially the same as those for the dc case It is customary to measure and specify rms values for ac quantities For\nexample, the household line voltage of 220 V is an rms value with a peak\nvoltage of\nvm = 2  V =  (1 414)(220 V) = 311 V\nIn fact, the I or rms current is the equivalent dc current that would\nproduce the same average power loss as the alternating current"}, {"Chapter": "1", "sentence_range": "5502-5505", "Text": "It is customary to measure and specify rms values for ac quantities For\nexample, the household line voltage of 220 V is an rms value with a peak\nvoltage of\nvm = 2  V =  (1 414)(220 V) = 311 V\nIn fact, the I or rms current is the equivalent dc current that would\nproduce the same average power loss as the alternating current Equation\n(7"}, {"Chapter": "1", "sentence_range": "5503-5506", "Text": "For\nexample, the household line voltage of 220 V is an rms value with a peak\nvoltage of\nvm = 2  V =  (1 414)(220 V) = 311 V\nIn fact, the I or rms current is the equivalent dc current that would\nproduce the same average power loss as the alternating current Equation\n(7 7) can also be written as\nP = V2 / R = I V    (since V = I R)\nExample 7"}, {"Chapter": "1", "sentence_range": "5504-5507", "Text": "414)(220 V) = 311 V\nIn fact, the I or rms current is the equivalent dc current that would\nproduce the same average power loss as the alternating current Equation\n(7 7) can also be written as\nP = V2 / R = I V    (since V = I R)\nExample 7 1 A light  bulb is rated at 100W for a 220 V supply"}, {"Chapter": "1", "sentence_range": "5505-5508", "Text": "Equation\n(7 7) can also be written as\nP = V2 / R = I V    (since V = I R)\nExample 7 1 A light  bulb is rated at 100W for a 220 V supply Find\n(a) the resistance of the bulb; (b) the peak voltage of the source; and\n(c) the rms current through the bulb"}, {"Chapter": "1", "sentence_range": "5506-5509", "Text": "7) can also be written as\nP = V2 / R = I V    (since V = I R)\nExample 7 1 A light  bulb is rated at 100W for a 220 V supply Find\n(a) the resistance of the bulb; (b) the peak voltage of the source; and\n(c) the rms current through the bulb Solution\n(a) We are given P = 100 W and V = 220 V"}, {"Chapter": "1", "sentence_range": "5507-5510", "Text": "1 A light  bulb is rated at 100W for a 220 V supply Find\n(a) the resistance of the bulb; (b) the peak voltage of the source; and\n(c) the rms current through the bulb Solution\n(a) We are given P = 100 W and V = 220 V The resistance of the\nbulb is\n(\n)\n2\n2\n220 V\n484\n100 W\nV\nR\n=P\n=\n=\n\u2126\n(b) The peak voltage of the source is\nV\n2\n311\nvm\nV\n=\n=\n(c) Since, P = I V\n100 W\n0"}, {"Chapter": "1", "sentence_range": "5508-5511", "Text": "Find\n(a) the resistance of the bulb; (b) the peak voltage of the source; and\n(c) the rms current through the bulb Solution\n(a) We are given P = 100 W and V = 220 V The resistance of the\nbulb is\n(\n)\n2\n2\n220 V\n484\n100 W\nV\nR\n=P\n=\n=\n\u2126\n(b) The peak voltage of the source is\nV\n2\n311\nvm\nV\n=\n=\n(c) Since, P = I V\n100 W\n0 454A\n220 V\n\ufffd\n\ufffd\n\ufffd\nP\nI\nV\n EXAMPLE 7"}, {"Chapter": "1", "sentence_range": "5509-5512", "Text": "Solution\n(a) We are given P = 100 W and V = 220 V The resistance of the\nbulb is\n(\n)\n2\n2\n220 V\n484\n100 W\nV\nR\n=P\n=\n=\n\u2126\n(b) The peak voltage of the source is\nV\n2\n311\nvm\nV\n=\n=\n(c) Since, P = I V\n100 W\n0 454A\n220 V\n\ufffd\n\ufffd\n\ufffd\nP\nI\nV\n EXAMPLE 7 1\nRationalised 2023-24\n181\nAlternating Current\n7"}, {"Chapter": "1", "sentence_range": "5510-5513", "Text": "The resistance of the\nbulb is\n(\n)\n2\n2\n220 V\n484\n100 W\nV\nR\n=P\n=\n=\n\u2126\n(b) The peak voltage of the source is\nV\n2\n311\nvm\nV\n=\n=\n(c) Since, P = I V\n100 W\n0 454A\n220 V\n\ufffd\n\ufffd\n\ufffd\nP\nI\nV\n EXAMPLE 7 1\nRationalised 2023-24\n181\nAlternating Current\n7 3 REPRESENTATION OF AC CURRENT AND VOLTAGE\nBY ROTATING VECTORS \u2014 PHASORS\nIn the previous section, we learnt that the current through  a resistor is\nin phase with the ac voltage"}, {"Chapter": "1", "sentence_range": "5511-5514", "Text": "454A\n220 V\n\ufffd\n\ufffd\n\ufffd\nP\nI\nV\n EXAMPLE 7 1\nRationalised 2023-24\n181\nAlternating Current\n7 3 REPRESENTATION OF AC CURRENT AND VOLTAGE\nBY ROTATING VECTORS \u2014 PHASORS\nIn the previous section, we learnt that the current through  a resistor is\nin phase with the ac voltage But this is not so in the case  of an inductor,\na capacitor or a combination  of these circuit elements"}, {"Chapter": "1", "sentence_range": "5512-5515", "Text": "1\nRationalised 2023-24\n181\nAlternating Current\n7 3 REPRESENTATION OF AC CURRENT AND VOLTAGE\nBY ROTATING VECTORS \u2014 PHASORS\nIn the previous section, we learnt that the current through  a resistor is\nin phase with the ac voltage But this is not so in the case  of an inductor,\na capacitor or a combination  of these circuit elements In order to show\nphase relationship between voltage and current\nin an ac circuit, we use the notion of phasors"}, {"Chapter": "1", "sentence_range": "5513-5516", "Text": "3 REPRESENTATION OF AC CURRENT AND VOLTAGE\nBY ROTATING VECTORS \u2014 PHASORS\nIn the previous section, we learnt that the current through  a resistor is\nin phase with the ac voltage But this is not so in the case  of an inductor,\na capacitor or a combination  of these circuit elements In order to show\nphase relationship between voltage and current\nin an ac circuit, we use the notion of phasors The analysis of an ac circuit is facilitated by the\nuse of a phasor diagram"}, {"Chapter": "1", "sentence_range": "5514-5517", "Text": "But this is not so in the case  of an inductor,\na capacitor or a combination  of these circuit elements In order to show\nphase relationship between voltage and current\nin an ac circuit, we use the notion of phasors The analysis of an ac circuit is facilitated by the\nuse of a phasor diagram A phasor* is a vector\nwhich rotates about the origin with angular\nspeed w, as shown in Fig"}, {"Chapter": "1", "sentence_range": "5515-5518", "Text": "In order to show\nphase relationship between voltage and current\nin an ac circuit, we use the notion of phasors The analysis of an ac circuit is facilitated by the\nuse of a phasor diagram A phasor* is a vector\nwhich rotates about the origin with angular\nspeed w, as shown in Fig 7"}, {"Chapter": "1", "sentence_range": "5516-5519", "Text": "The analysis of an ac circuit is facilitated by the\nuse of a phasor diagram A phasor* is a vector\nwhich rotates about the origin with angular\nspeed w, as shown in Fig 7 4"}, {"Chapter": "1", "sentence_range": "5517-5520", "Text": "A phasor* is a vector\nwhich rotates about the origin with angular\nspeed w, as shown in Fig 7 4 The vertical\ncomponents of phasors V and I represent the\nsinusoidally varying quantities v and i"}, {"Chapter": "1", "sentence_range": "5518-5521", "Text": "7 4 The vertical\ncomponents of phasors V and I represent the\nsinusoidally varying quantities v and i The\nmagnitudes of phasors V and I represent  the\namplitudes or the peak values vm and im of these\noscillating quantities"}, {"Chapter": "1", "sentence_range": "5519-5522", "Text": "4 The vertical\ncomponents of phasors V and I represent the\nsinusoidally varying quantities v and i The\nmagnitudes of phasors V and I represent  the\namplitudes or the peak values vm and im of these\noscillating quantities Figure 7"}, {"Chapter": "1", "sentence_range": "5520-5523", "Text": "The vertical\ncomponents of phasors V and I represent the\nsinusoidally varying quantities v and i The\nmagnitudes of phasors V and I represent  the\namplitudes or the peak values vm and im of these\noscillating quantities Figure 7 4(a) shows the\nvoltage and current phasors and their\nrelationship at time t1 for the case of an ac source\nconnected to a resistor i"}, {"Chapter": "1", "sentence_range": "5521-5524", "Text": "The\nmagnitudes of phasors V and I represent  the\namplitudes or the peak values vm and im of these\noscillating quantities Figure 7 4(a) shows the\nvoltage and current phasors and their\nrelationship at time t1 for the case of an ac source\nconnected to a resistor i e"}, {"Chapter": "1", "sentence_range": "5522-5525", "Text": "Figure 7 4(a) shows the\nvoltage and current phasors and their\nrelationship at time t1 for the case of an ac source\nconnected to a resistor i e , corresponding to the\ncircuit shown in Fig"}, {"Chapter": "1", "sentence_range": "5523-5526", "Text": "4(a) shows the\nvoltage and current phasors and their\nrelationship at time t1 for the case of an ac source\nconnected to a resistor i e , corresponding to the\ncircuit shown in Fig 7"}, {"Chapter": "1", "sentence_range": "5524-5527", "Text": "e , corresponding to the\ncircuit shown in Fig 7 1"}, {"Chapter": "1", "sentence_range": "5525-5528", "Text": ", corresponding to the\ncircuit shown in Fig 7 1 The projection of\nvoltage and current phasors on vertical axis, i"}, {"Chapter": "1", "sentence_range": "5526-5529", "Text": "7 1 The projection of\nvoltage and current phasors on vertical axis, i e"}, {"Chapter": "1", "sentence_range": "5527-5530", "Text": "1 The projection of\nvoltage and current phasors on vertical axis, i e , vm sinw t and im sinw t,\nrespectively represent the value of voltage and current at that instant"}, {"Chapter": "1", "sentence_range": "5528-5531", "Text": "The projection of\nvoltage and current phasors on vertical axis, i e , vm sinw t and im sinw t,\nrespectively represent the value of voltage and current at that instant As\nthey rotate with frequency w, curves in Fig"}, {"Chapter": "1", "sentence_range": "5529-5532", "Text": "e , vm sinw t and im sinw t,\nrespectively represent the value of voltage and current at that instant As\nthey rotate with frequency w, curves in Fig 7"}, {"Chapter": "1", "sentence_range": "5530-5533", "Text": ", vm sinw t and im sinw t,\nrespectively represent the value of voltage and current at that instant As\nthey rotate with frequency w, curves in Fig 7 4(b) are generated"}, {"Chapter": "1", "sentence_range": "5531-5534", "Text": "As\nthey rotate with frequency w, curves in Fig 7 4(b) are generated From Fig"}, {"Chapter": "1", "sentence_range": "5532-5535", "Text": "7 4(b) are generated From Fig 7"}, {"Chapter": "1", "sentence_range": "5533-5536", "Text": "4(b) are generated From Fig 7 4(a) we see that phasors V and I for the case of a resistor are\nin the same direction"}, {"Chapter": "1", "sentence_range": "5534-5537", "Text": "From Fig 7 4(a) we see that phasors V and I for the case of a resistor are\nin the same direction This is so for all times"}, {"Chapter": "1", "sentence_range": "5535-5538", "Text": "7 4(a) we see that phasors V and I for the case of a resistor are\nin the same direction This is so for all times This means that the phase\nangle between the voltage and the current is zero"}, {"Chapter": "1", "sentence_range": "5536-5539", "Text": "4(a) we see that phasors V and I for the case of a resistor are\nin the same direction This is so for all times This means that the phase\nangle between the voltage and the current is zero 7"}, {"Chapter": "1", "sentence_range": "5537-5540", "Text": "This is so for all times This means that the phase\nangle between the voltage and the current is zero 7 4  AC VOLTAGE APPLIED TO AN INDUCTOR\nFigure 7"}, {"Chapter": "1", "sentence_range": "5538-5541", "Text": "This means that the phase\nangle between the voltage and the current is zero 7 4  AC VOLTAGE APPLIED TO AN INDUCTOR\nFigure 7 5 shows an ac source connected to an inductor"}, {"Chapter": "1", "sentence_range": "5539-5542", "Text": "7 4  AC VOLTAGE APPLIED TO AN INDUCTOR\nFigure 7 5 shows an ac source connected to an inductor Usually,\ninductors have appreciable resistance in their windings, but we shall\nassume that this inductor has negligible resistance"}, {"Chapter": "1", "sentence_range": "5540-5543", "Text": "4  AC VOLTAGE APPLIED TO AN INDUCTOR\nFigure 7 5 shows an ac source connected to an inductor Usually,\ninductors have appreciable resistance in their windings, but we shall\nassume that this inductor has negligible resistance Thus, the circuit is a purely inductive ac circuit"}, {"Chapter": "1", "sentence_range": "5541-5544", "Text": "5 shows an ac source connected to an inductor Usually,\ninductors have appreciable resistance in their windings, but we shall\nassume that this inductor has negligible resistance Thus, the circuit is a purely inductive ac circuit Let\nthe voltage across the source be v = vm sinw t"}, {"Chapter": "1", "sentence_range": "5542-5545", "Text": "Usually,\ninductors have appreciable resistance in their windings, but we shall\nassume that this inductor has negligible resistance Thus, the circuit is a purely inductive ac circuit Let\nthe voltage across the source be v = vm sinw t Using\nthe Kirchhoff\u2019s loop rule, \n\u2211\u03b5 ( )t =\n0 , and  since there\nis no resistor in the circuit,\nd\n0\nd\ni\nv\nL\nt\n\u2212\n=\n(7"}, {"Chapter": "1", "sentence_range": "5543-5546", "Text": "Thus, the circuit is a purely inductive ac circuit Let\nthe voltage across the source be v = vm sinw t Using\nthe Kirchhoff\u2019s loop rule, \n\u2211\u03b5 ( )t =\n0 , and  since there\nis no resistor in the circuit,\nd\n0\nd\ni\nv\nL\nt\n\u2212\n=\n(7 10)\nwhere the second term is the self-induced Faraday\nemf in the inductor; and L is the self-inductance of\nFIGURE 7"}, {"Chapter": "1", "sentence_range": "5544-5547", "Text": "Let\nthe voltage across the source be v = vm sinw t Using\nthe Kirchhoff\u2019s loop rule, \n\u2211\u03b5 ( )t =\n0 , and  since there\nis no resistor in the circuit,\nd\n0\nd\ni\nv\nL\nt\n\u2212\n=\n(7 10)\nwhere the second term is the self-induced Faraday\nemf in the inductor; and L is the self-inductance of\nFIGURE 7 4 (a) A phasor diagram for the\ncircuit in Fig 7"}, {"Chapter": "1", "sentence_range": "5545-5548", "Text": "Using\nthe Kirchhoff\u2019s loop rule, \n\u2211\u03b5 ( )t =\n0 , and  since there\nis no resistor in the circuit,\nd\n0\nd\ni\nv\nL\nt\n\u2212\n=\n(7 10)\nwhere the second term is the self-induced Faraday\nemf in the inductor; and L is the self-inductance of\nFIGURE 7 4 (a) A phasor diagram for the\ncircuit in Fig 7 1"}, {"Chapter": "1", "sentence_range": "5546-5549", "Text": "10)\nwhere the second term is the self-induced Faraday\nemf in the inductor; and L is the self-inductance of\nFIGURE 7 4 (a) A phasor diagram for the\ncircuit in Fig 7 1 (b) Graph of v and\ni versus wt"}, {"Chapter": "1", "sentence_range": "5547-5550", "Text": "4 (a) A phasor diagram for the\ncircuit in Fig 7 1 (b) Graph of v and\ni versus wt FIGURE 7"}, {"Chapter": "1", "sentence_range": "5548-5551", "Text": "1 (b) Graph of v and\ni versus wt FIGURE 7 5  An ac source\nconnected to an inductor"}, {"Chapter": "1", "sentence_range": "5549-5552", "Text": "(b) Graph of v and\ni versus wt FIGURE 7 5  An ac source\nconnected to an inductor *\nThough voltage and current in ac circuit are represented by phasors \u2013 rotating\nvectors, they are not vectors themselves"}, {"Chapter": "1", "sentence_range": "5550-5553", "Text": "FIGURE 7 5  An ac source\nconnected to an inductor *\nThough voltage and current in ac circuit are represented by phasors \u2013 rotating\nvectors, they are not vectors themselves They are scalar quantities"}, {"Chapter": "1", "sentence_range": "5551-5554", "Text": "5  An ac source\nconnected to an inductor *\nThough voltage and current in ac circuit are represented by phasors \u2013 rotating\nvectors, they are not vectors themselves They are scalar quantities It so happens\nthat the amplitudes and phases of harmonically varying scalars combine\nmathematically in the same way as do the projections of rotating vectors of\ncorresponding magnitudes and directions"}, {"Chapter": "1", "sentence_range": "5552-5555", "Text": "*\nThough voltage and current in ac circuit are represented by phasors \u2013 rotating\nvectors, they are not vectors themselves They are scalar quantities It so happens\nthat the amplitudes and phases of harmonically varying scalars combine\nmathematically in the same way as do the projections of rotating vectors of\ncorresponding magnitudes and directions The rotating vectors that represent\nharmonically varying scalar quantities are introduced only to provide us with a\nsimple way of adding these quantities using a rule that we already know"}, {"Chapter": "1", "sentence_range": "5553-5556", "Text": "They are scalar quantities It so happens\nthat the amplitudes and phases of harmonically varying scalars combine\nmathematically in the same way as do the projections of rotating vectors of\ncorresponding magnitudes and directions The rotating vectors that represent\nharmonically varying scalar quantities are introduced only to provide us with a\nsimple way of adding these quantities using a rule that we already know Rationalised 2023-24\nPhysics\n182\nthe inductor"}, {"Chapter": "1", "sentence_range": "5554-5557", "Text": "It so happens\nthat the amplitudes and phases of harmonically varying scalars combine\nmathematically in the same way as do the projections of rotating vectors of\ncorresponding magnitudes and directions The rotating vectors that represent\nharmonically varying scalar quantities are introduced only to provide us with a\nsimple way of adding these quantities using a rule that we already know Rationalised 2023-24\nPhysics\n182\nthe inductor The negative sign follows from Lenz\u2019s law (Chapter 6)"}, {"Chapter": "1", "sentence_range": "5555-5558", "Text": "The rotating vectors that represent\nharmonically varying scalar quantities are introduced only to provide us with a\nsimple way of adding these quantities using a rule that we already know Rationalised 2023-24\nPhysics\n182\nthe inductor The negative sign follows from Lenz\u2019s law (Chapter 6) Combining Eqs"}, {"Chapter": "1", "sentence_range": "5556-5559", "Text": "Rationalised 2023-24\nPhysics\n182\nthe inductor The negative sign follows from Lenz\u2019s law (Chapter 6) Combining Eqs (7"}, {"Chapter": "1", "sentence_range": "5557-5560", "Text": "The negative sign follows from Lenz\u2019s law (Chapter 6) Combining Eqs (7 1) and (7"}, {"Chapter": "1", "sentence_range": "5558-5561", "Text": "Combining Eqs (7 1) and (7 10), we have\nd\nsin\nd\nvm\ni\nv\nt\nt\nL\nL\n\u03c9\n=\n=\n(7"}, {"Chapter": "1", "sentence_range": "5559-5562", "Text": "(7 1) and (7 10), we have\nd\nsin\nd\nvm\ni\nv\nt\nt\nL\nL\n\u03c9\n=\n=\n(7 11)\nEquation (7"}, {"Chapter": "1", "sentence_range": "5560-5563", "Text": "1) and (7 10), we have\nd\nsin\nd\nvm\ni\nv\nt\nt\nL\nL\n\u03c9\n=\n=\n(7 11)\nEquation (7 11) implies that the equation for i(t), the current as a\nfunction of time, must be such that its slope di/dt is a sinusoidally varying\nquantity, with the same phase as the source voltage and an amplitude\ngiven by vm/L"}, {"Chapter": "1", "sentence_range": "5561-5564", "Text": "10), we have\nd\nsin\nd\nvm\ni\nv\nt\nt\nL\nL\n\u03c9\n=\n=\n(7 11)\nEquation (7 11) implies that the equation for i(t), the current as a\nfunction of time, must be such that its slope di/dt is a sinusoidally varying\nquantity, with the same phase as the source voltage and an amplitude\ngiven by vm/L To obtain the current, we integrate di/dt with respect to\ntime:\nd\nd d\nd\nti\nt\nv\nL\nt\nt\nm\n\u222b\n\u222b\n=\nsin(\n)\n\u03c9\nand get,\ncos(\n)\nconstant\nvm\ni\nt\nL\n= \u2212\n\u03c9\n+\n\u03c9\nThe integration constant has the dimension of current and is time-\nindependent"}, {"Chapter": "1", "sentence_range": "5562-5565", "Text": "11)\nEquation (7 11) implies that the equation for i(t), the current as a\nfunction of time, must be such that its slope di/dt is a sinusoidally varying\nquantity, with the same phase as the source voltage and an amplitude\ngiven by vm/L To obtain the current, we integrate di/dt with respect to\ntime:\nd\nd d\nd\nti\nt\nv\nL\nt\nt\nm\n\u222b\n\u222b\n=\nsin(\n)\n\u03c9\nand get,\ncos(\n)\nconstant\nvm\ni\nt\nL\n= \u2212\n\u03c9\n+\n\u03c9\nThe integration constant has the dimension of current and is time-\nindependent Since the source has an emf which oscillates symmetrically\nabout zero, the current it sustains also oscillates symmetrically about\nzero, so that no constant or time-independent component of the current\nexists"}, {"Chapter": "1", "sentence_range": "5563-5566", "Text": "11) implies that the equation for i(t), the current as a\nfunction of time, must be such that its slope di/dt is a sinusoidally varying\nquantity, with the same phase as the source voltage and an amplitude\ngiven by vm/L To obtain the current, we integrate di/dt with respect to\ntime:\nd\nd d\nd\nti\nt\nv\nL\nt\nt\nm\n\u222b\n\u222b\n=\nsin(\n)\n\u03c9\nand get,\ncos(\n)\nconstant\nvm\ni\nt\nL\n= \u2212\n\u03c9\n+\n\u03c9\nThe integration constant has the dimension of current and is time-\nindependent Since the source has an emf which oscillates symmetrically\nabout zero, the current it sustains also oscillates symmetrically about\nzero, so that no constant or time-independent component of the current\nexists Therefore, the integration constant is zero"}, {"Chapter": "1", "sentence_range": "5564-5567", "Text": "To obtain the current, we integrate di/dt with respect to\ntime:\nd\nd d\nd\nti\nt\nv\nL\nt\nt\nm\n\u222b\n\u222b\n=\nsin(\n)\n\u03c9\nand get,\ncos(\n)\nconstant\nvm\ni\nt\nL\n= \u2212\n\u03c9\n+\n\u03c9\nThe integration constant has the dimension of current and is time-\nindependent Since the source has an emf which oscillates symmetrically\nabout zero, the current it sustains also oscillates symmetrically about\nzero, so that no constant or time-independent component of the current\nexists Therefore, the integration constant is zero Using\n\u2212\n=\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\ncos(\n)\nsin\n\u03c9\n\u03c9\nt\nt\n2\u03c0\n, we have\ni\ni\nt\n=m\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nsin \u03c9\n2\u03c0\n(7"}, {"Chapter": "1", "sentence_range": "5565-5568", "Text": "Since the source has an emf which oscillates symmetrically\nabout zero, the current it sustains also oscillates symmetrically about\nzero, so that no constant or time-independent component of the current\nexists Therefore, the integration constant is zero Using\n\u2212\n=\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\ncos(\n)\nsin\n\u03c9\n\u03c9\nt\nt\n2\u03c0\n, we have\ni\ni\nt\n=m\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nsin \u03c9\n2\u03c0\n(7 12)\nwhere  \nm\nm\nv\ni\n= \u03c9L\nis the amplitude of the current"}, {"Chapter": "1", "sentence_range": "5566-5569", "Text": "Therefore, the integration constant is zero Using\n\u2212\n=\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\ncos(\n)\nsin\n\u03c9\n\u03c9\nt\nt\n2\u03c0\n, we have\ni\ni\nt\n=m\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nsin \u03c9\n2\u03c0\n(7 12)\nwhere  \nm\nm\nv\ni\n= \u03c9L\nis the amplitude of the current The quantity w L is\nanalogous to the resistance and is called inductive reactance, denoted\nby XL:\nXL = w L\n(7"}, {"Chapter": "1", "sentence_range": "5567-5570", "Text": "Using\n\u2212\n=\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\ncos(\n)\nsin\n\u03c9\n\u03c9\nt\nt\n2\u03c0\n, we have\ni\ni\nt\n=m\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nsin \u03c9\n2\u03c0\n(7 12)\nwhere  \nm\nm\nv\ni\n= \u03c9L\nis the amplitude of the current The quantity w L is\nanalogous to the resistance and is called inductive reactance, denoted\nby XL:\nXL = w L\n(7 13)\nThe amplitude of the current is, then\nm\nm\nL\nv\ni\n=X\n(7"}, {"Chapter": "1", "sentence_range": "5568-5571", "Text": "12)\nwhere  \nm\nm\nv\ni\n= \u03c9L\nis the amplitude of the current The quantity w L is\nanalogous to the resistance and is called inductive reactance, denoted\nby XL:\nXL = w L\n(7 13)\nThe amplitude of the current is, then\nm\nm\nL\nv\ni\n=X\n(7 14)\nThe dimension of inductive reactance is the same as that of resistance\nand its SI unit is ohm (W)"}, {"Chapter": "1", "sentence_range": "5569-5572", "Text": "The quantity w L is\nanalogous to the resistance and is called inductive reactance, denoted\nby XL:\nXL = w L\n(7 13)\nThe amplitude of the current is, then\nm\nm\nL\nv\ni\n=X\n(7 14)\nThe dimension of inductive reactance is the same as that of resistance\nand its SI unit is ohm (W) The inductive reactance limits the current in a\npurely inductive circuit in the same way as the resistance limits the\ncurrent in a purely resistive circuit"}, {"Chapter": "1", "sentence_range": "5570-5573", "Text": "13)\nThe amplitude of the current is, then\nm\nm\nL\nv\ni\n=X\n(7 14)\nThe dimension of inductive reactance is the same as that of resistance\nand its SI unit is ohm (W) The inductive reactance limits the current in a\npurely inductive circuit in the same way as the resistance limits the\ncurrent in a purely resistive circuit The inductive reactance is directly\nproportional to the inductance and to the frequency of the current"}, {"Chapter": "1", "sentence_range": "5571-5574", "Text": "14)\nThe dimension of inductive reactance is the same as that of resistance\nand its SI unit is ohm (W) The inductive reactance limits the current in a\npurely inductive circuit in the same way as the resistance limits the\ncurrent in a purely resistive circuit The inductive reactance is directly\nproportional to the inductance and to the frequency of the current A comparison of Eqs"}, {"Chapter": "1", "sentence_range": "5572-5575", "Text": "The inductive reactance limits the current in a\npurely inductive circuit in the same way as the resistance limits the\ncurrent in a purely resistive circuit The inductive reactance is directly\nproportional to the inductance and to the frequency of the current A comparison of Eqs (7"}, {"Chapter": "1", "sentence_range": "5573-5576", "Text": "The inductive reactance is directly\nproportional to the inductance and to the frequency of the current A comparison of Eqs (7 1) and (7"}, {"Chapter": "1", "sentence_range": "5574-5577", "Text": "A comparison of Eqs (7 1) and (7 12) for the source voltage and the\ncurrent in an inductor shows that the current lags the voltage by p/2 or\none-quarter (1/4) cycle"}, {"Chapter": "1", "sentence_range": "5575-5578", "Text": "(7 1) and (7 12) for the source voltage and the\ncurrent in an inductor shows that the current lags the voltage by p/2 or\none-quarter (1/4) cycle Figure 7"}, {"Chapter": "1", "sentence_range": "5576-5579", "Text": "1) and (7 12) for the source voltage and the\ncurrent in an inductor shows that the current lags the voltage by p/2 or\none-quarter (1/4) cycle Figure 7 6 (a) shows the voltage and the current\nphasors in the present case at instant t1"}, {"Chapter": "1", "sentence_range": "5577-5580", "Text": "12) for the source voltage and the\ncurrent in an inductor shows that the current lags the voltage by p/2 or\none-quarter (1/4) cycle Figure 7 6 (a) shows the voltage and the current\nphasors in the present case at instant t1 The current phasor I is p/2\nbehind the voltage phasor V"}, {"Chapter": "1", "sentence_range": "5578-5581", "Text": "Figure 7 6 (a) shows the voltage and the current\nphasors in the present case at instant t1 The current phasor I is p/2\nbehind the voltage phasor V When rotated with frequency w counter-\nclockwise, they generate the voltage and current given by Eqs"}, {"Chapter": "1", "sentence_range": "5579-5582", "Text": "6 (a) shows the voltage and the current\nphasors in the present case at instant t1 The current phasor I is p/2\nbehind the voltage phasor V When rotated with frequency w counter-\nclockwise, they generate the voltage and current given by Eqs (7"}, {"Chapter": "1", "sentence_range": "5580-5583", "Text": "The current phasor I is p/2\nbehind the voltage phasor V When rotated with frequency w counter-\nclockwise, they generate the voltage and current given by Eqs (7 1) and\n(7"}, {"Chapter": "1", "sentence_range": "5581-5584", "Text": "When rotated with frequency w counter-\nclockwise, they generate the voltage and current given by Eqs (7 1) and\n(7 12), respectively and as shown in Fig"}, {"Chapter": "1", "sentence_range": "5582-5585", "Text": "(7 1) and\n(7 12), respectively and as shown in Fig 7"}, {"Chapter": "1", "sentence_range": "5583-5586", "Text": "1) and\n(7 12), respectively and as shown in Fig 7 6(b)"}, {"Chapter": "1", "sentence_range": "5584-5587", "Text": "12), respectively and as shown in Fig 7 6(b) Interactive animation on Phasor diagrams of ac circuits containing, R, L, C and RLC series circuits:\nhttp://www"}, {"Chapter": "1", "sentence_range": "5585-5588", "Text": "7 6(b) Interactive animation on Phasor diagrams of ac circuits containing, R, L, C and RLC series circuits:\nhttp://www animations"}, {"Chapter": "1", "sentence_range": "5586-5589", "Text": "6(b) Interactive animation on Phasor diagrams of ac circuits containing, R, L, C and RLC series circuits:\nhttp://www animations physics"}, {"Chapter": "1", "sentence_range": "5587-5590", "Text": "Interactive animation on Phasor diagrams of ac circuits containing, R, L, C and RLC series circuits:\nhttp://www animations physics unsw"}, {"Chapter": "1", "sentence_range": "5588-5591", "Text": "animations physics unsw edu"}, {"Chapter": "1", "sentence_range": "5589-5592", "Text": "physics unsw edu au//jw/AC"}, {"Chapter": "1", "sentence_range": "5590-5593", "Text": "unsw edu au//jw/AC html\nRationalised 2023-24\n183\nAlternating Current\n EXAMPLE 7"}, {"Chapter": "1", "sentence_range": "5591-5594", "Text": "edu au//jw/AC html\nRationalised 2023-24\n183\nAlternating Current\n EXAMPLE 7 2\nWe see that the current reaches its maximum value later than the\nvoltage by one-fourth of a period \n4T\n2\n=\n\uf8f0\uf8ef\uf8ee\n\uf8fb\uf8fa\uf8f9\n\u03c0/\n\u03c9"}, {"Chapter": "1", "sentence_range": "5592-5595", "Text": "au//jw/AC html\nRationalised 2023-24\n183\nAlternating Current\n EXAMPLE 7 2\nWe see that the current reaches its maximum value later than the\nvoltage by one-fourth of a period \n4T\n2\n=\n\uf8f0\uf8ef\uf8ee\n\uf8fb\uf8fa\uf8f9\n\u03c0/\n\u03c9 You have seen that an\ninductor has reactance that limits current similar to resistance in a\ndc circuit"}, {"Chapter": "1", "sentence_range": "5593-5596", "Text": "html\nRationalised 2023-24\n183\nAlternating Current\n EXAMPLE 7 2\nWe see that the current reaches its maximum value later than the\nvoltage by one-fourth of a period \n4T\n2\n=\n\uf8f0\uf8ef\uf8ee\n\uf8fb\uf8fa\uf8f9\n\u03c0/\n\u03c9 You have seen that an\ninductor has reactance that limits current similar to resistance in a\ndc circuit Does it also consume power like a resistance"}, {"Chapter": "1", "sentence_range": "5594-5597", "Text": "2\nWe see that the current reaches its maximum value later than the\nvoltage by one-fourth of a period \n4T\n2\n=\n\uf8f0\uf8ef\uf8ee\n\uf8fb\uf8fa\uf8f9\n\u03c0/\n\u03c9 You have seen that an\ninductor has reactance that limits current similar to resistance in a\ndc circuit Does it also consume power like a resistance Let us try to\nfind out"}, {"Chapter": "1", "sentence_range": "5595-5598", "Text": "You have seen that an\ninductor has reactance that limits current similar to resistance in a\ndc circuit Does it also consume power like a resistance Let us try to\nfind out The instantaneous power supplied to the inductor is\np\ni v\ni\nt\nv\nt\nL\nm\nm\n=\n=\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n(\n)\nsin\nsin\n\u03c9\n\u03c9\n\u03c0\n2 \u00d7\n(\n)\n(\n)\ncos\nsin\nm\ni vm\nt\nt\n\u03c9\n\u03c9\n= \u2212\n(\n)\nsin 2\n2\nm\ni vm\n\u03c9t\n= \u2212\nSo, the average power over a complete cycle is\n(\n)\nL\nsin 2\n2\nm\ni vm\nP\n\u03c9t\n=\n\u2212\n(\n)\nsin 2\n2\nm\ni vm\n\u03c9t\n= \u2212\n= 0,\nsince the average of sin (2wt) over a complete cycle is  zero"}, {"Chapter": "1", "sentence_range": "5596-5599", "Text": "Does it also consume power like a resistance Let us try to\nfind out The instantaneous power supplied to the inductor is\np\ni v\ni\nt\nv\nt\nL\nm\nm\n=\n=\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n(\n)\nsin\nsin\n\u03c9\n\u03c9\n\u03c0\n2 \u00d7\n(\n)\n(\n)\ncos\nsin\nm\ni vm\nt\nt\n\u03c9\n\u03c9\n= \u2212\n(\n)\nsin 2\n2\nm\ni vm\n\u03c9t\n= \u2212\nSo, the average power over a complete cycle is\n(\n)\nL\nsin 2\n2\nm\ni vm\nP\n\u03c9t\n=\n\u2212\n(\n)\nsin 2\n2\nm\ni vm\n\u03c9t\n= \u2212\n= 0,\nsince the average of sin (2wt) over a complete cycle is  zero Thus, the average power supplied to an inductor over one complete\ncycle is zero"}, {"Chapter": "1", "sentence_range": "5597-5600", "Text": "Let us try to\nfind out The instantaneous power supplied to the inductor is\np\ni v\ni\nt\nv\nt\nL\nm\nm\n=\n=\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n(\n)\nsin\nsin\n\u03c9\n\u03c9\n\u03c0\n2 \u00d7\n(\n)\n(\n)\ncos\nsin\nm\ni vm\nt\nt\n\u03c9\n\u03c9\n= \u2212\n(\n)\nsin 2\n2\nm\ni vm\n\u03c9t\n= \u2212\nSo, the average power over a complete cycle is\n(\n)\nL\nsin 2\n2\nm\ni vm\nP\n\u03c9t\n=\n\u2212\n(\n)\nsin 2\n2\nm\ni vm\n\u03c9t\n= \u2212\n= 0,\nsince the average of sin (2wt) over a complete cycle is  zero Thus, the average power supplied to an inductor over one complete\ncycle is zero Example 7"}, {"Chapter": "1", "sentence_range": "5598-5601", "Text": "The instantaneous power supplied to the inductor is\np\ni v\ni\nt\nv\nt\nL\nm\nm\n=\n=\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n(\n)\nsin\nsin\n\u03c9\n\u03c9\n\u03c0\n2 \u00d7\n(\n)\n(\n)\ncos\nsin\nm\ni vm\nt\nt\n\u03c9\n\u03c9\n= \u2212\n(\n)\nsin 2\n2\nm\ni vm\n\u03c9t\n= \u2212\nSo, the average power over a complete cycle is\n(\n)\nL\nsin 2\n2\nm\ni vm\nP\n\u03c9t\n=\n\u2212\n(\n)\nsin 2\n2\nm\ni vm\n\u03c9t\n= \u2212\n= 0,\nsince the average of sin (2wt) over a complete cycle is  zero Thus, the average power supplied to an inductor over one complete\ncycle is zero Example 7 2 A pure inductor of 25"}, {"Chapter": "1", "sentence_range": "5599-5602", "Text": "Thus, the average power supplied to an inductor over one complete\ncycle is zero Example 7 2 A pure inductor of 25 0 mH is connected to a source of\n220 V"}, {"Chapter": "1", "sentence_range": "5600-5603", "Text": "Example 7 2 A pure inductor of 25 0 mH is connected to a source of\n220 V Find the inductive reactance and rms current in the circuit if\nthe frequency of the source is 50 Hz"}, {"Chapter": "1", "sentence_range": "5601-5604", "Text": "2 A pure inductor of 25 0 mH is connected to a source of\n220 V Find the inductive reactance and rms current in the circuit if\nthe frequency of the source is 50 Hz Solution The inductive reactance,\n\u2013\n="}, {"Chapter": "1", "sentence_range": "5602-5605", "Text": "0 mH is connected to a source of\n220 V Find the inductive reactance and rms current in the circuit if\nthe frequency of the source is 50 Hz Solution The inductive reactance,\n\u2013\n= 3\n2\n2\n3 14\n50\n25\n10\n\u03c0\u03bd\n\u00d7\n\u00d7\n\u00d7\n\u00d7\n\u2126\nXL\nL =\n      = 7"}, {"Chapter": "1", "sentence_range": "5603-5606", "Text": "Find the inductive reactance and rms current in the circuit if\nthe frequency of the source is 50 Hz Solution The inductive reactance,\n\u2013\n= 3\n2\n2\n3 14\n50\n25\n10\n\u03c0\u03bd\n\u00d7\n\u00d7\n\u00d7\n\u00d7\n\u2126\nXL\nL =\n      = 7 85W\nThe rms current in the circuit is\nV\nA\n220\n28\n7"}, {"Chapter": "1", "sentence_range": "5604-5607", "Text": "Solution The inductive reactance,\n\u2013\n= 3\n2\n2\n3 14\n50\n25\n10\n\u03c0\u03bd\n\u00d7\n\u00d7\n\u00d7\n\u00d7\n\u2126\nXL\nL =\n      = 7 85W\nThe rms current in the circuit is\nV\nA\n220\n28\n7 85\nL\nV\nI\n=X\n=\n=\n\u2126\nFIGURE 7"}, {"Chapter": "1", "sentence_range": "5605-5608", "Text": "3\n2\n2\n3 14\n50\n25\n10\n\u03c0\u03bd\n\u00d7\n\u00d7\n\u00d7\n\u00d7\n\u2126\nXL\nL =\n      = 7 85W\nThe rms current in the circuit is\nV\nA\n220\n28\n7 85\nL\nV\nI\n=X\n=\n=\n\u2126\nFIGURE 7 6 (a) A Phasor diagram for the circuit in Fig"}, {"Chapter": "1", "sentence_range": "5606-5609", "Text": "85W\nThe rms current in the circuit is\nV\nA\n220\n28\n7 85\nL\nV\nI\n=X\n=\n=\n\u2126\nFIGURE 7 6 (a) A Phasor diagram for the circuit in Fig 7"}, {"Chapter": "1", "sentence_range": "5607-5610", "Text": "85\nL\nV\nI\n=X\n=\n=\n\u2126\nFIGURE 7 6 (a) A Phasor diagram for the circuit in Fig 7 5"}, {"Chapter": "1", "sentence_range": "5608-5611", "Text": "6 (a) A Phasor diagram for the circuit in Fig 7 5 (b) Graph of v and i versus wt"}, {"Chapter": "1", "sentence_range": "5609-5612", "Text": "7 5 (b) Graph of v and i versus wt Rationalised 2023-24\nPhysics\n184\n7"}, {"Chapter": "1", "sentence_range": "5610-5613", "Text": "5 (b) Graph of v and i versus wt Rationalised 2023-24\nPhysics\n184\n7 5   AC VOLTAGE APPLIED TO A CAPACITOR\nFigure 7"}, {"Chapter": "1", "sentence_range": "5611-5614", "Text": "(b) Graph of v and i versus wt Rationalised 2023-24\nPhysics\n184\n7 5   AC VOLTAGE APPLIED TO A CAPACITOR\nFigure 7 7 shows an ac source e generating ac voltage v = vm sin wt\nconnected to a capacitor only, a purely capacitive ac circuit"}, {"Chapter": "1", "sentence_range": "5612-5615", "Text": "Rationalised 2023-24\nPhysics\n184\n7 5   AC VOLTAGE APPLIED TO A CAPACITOR\nFigure 7 7 shows an ac source e generating ac voltage v = vm sin wt\nconnected to a capacitor only, a purely capacitive ac circuit When a capacitor is connected to a voltage source\nin a dc circuit, current will flow for the short time\nrequired to charge the capacitor"}, {"Chapter": "1", "sentence_range": "5613-5616", "Text": "5   AC VOLTAGE APPLIED TO A CAPACITOR\nFigure 7 7 shows an ac source e generating ac voltage v = vm sin wt\nconnected to a capacitor only, a purely capacitive ac circuit When a capacitor is connected to a voltage source\nin a dc circuit, current will flow for the short time\nrequired to charge the capacitor As charge\naccumulates on the capacitor plates, the voltage\nacross them increases, opposing the current"}, {"Chapter": "1", "sentence_range": "5614-5617", "Text": "7 shows an ac source e generating ac voltage v = vm sin wt\nconnected to a capacitor only, a purely capacitive ac circuit When a capacitor is connected to a voltage source\nin a dc circuit, current will flow for the short time\nrequired to charge the capacitor As charge\naccumulates on the capacitor plates, the voltage\nacross them increases, opposing the current That is,\na capacitor in a dc circuit will limit or oppose the\ncurrent as it charges"}, {"Chapter": "1", "sentence_range": "5615-5618", "Text": "When a capacitor is connected to a voltage source\nin a dc circuit, current will flow for the short time\nrequired to charge the capacitor As charge\naccumulates on the capacitor plates, the voltage\nacross them increases, opposing the current That is,\na capacitor in a dc circuit will limit or oppose the\ncurrent as it charges When the capacitor is fully\ncharged, the current in the circuit falls to zero"}, {"Chapter": "1", "sentence_range": "5616-5619", "Text": "As charge\naccumulates on the capacitor plates, the voltage\nacross them increases, opposing the current That is,\na capacitor in a dc circuit will limit or oppose the\ncurrent as it charges When the capacitor is fully\ncharged, the current in the circuit falls to zero When the capacitor is connected to an ac source,\nas in Fig"}, {"Chapter": "1", "sentence_range": "5617-5620", "Text": "That is,\na capacitor in a dc circuit will limit or oppose the\ncurrent as it charges When the capacitor is fully\ncharged, the current in the circuit falls to zero When the capacitor is connected to an ac source,\nas in Fig 7"}, {"Chapter": "1", "sentence_range": "5618-5621", "Text": "When the capacitor is fully\ncharged, the current in the circuit falls to zero When the capacitor is connected to an ac source,\nas in Fig 7 7, it limits or regulates the current, but\ndoes not completely prevent the flow of charge"}, {"Chapter": "1", "sentence_range": "5619-5622", "Text": "When the capacitor is connected to an ac source,\nas in Fig 7 7, it limits or regulates the current, but\ndoes not completely prevent the flow of charge The\ncapacitor is alternately charged and discharged as\nthe current reverses each half cycle"}, {"Chapter": "1", "sentence_range": "5620-5623", "Text": "7 7, it limits or regulates the current, but\ndoes not completely prevent the flow of charge The\ncapacitor is alternately charged and discharged as\nthe current reverses each half cycle Let q  be the\ncharge on the capacitor at any time t"}, {"Chapter": "1", "sentence_range": "5621-5624", "Text": "7, it limits or regulates the current, but\ndoes not completely prevent the flow of charge The\ncapacitor is alternately charged and discharged as\nthe current reverses each half cycle Let q  be the\ncharge on the capacitor at any time t The instantaneous voltage v across\nthe capacitor is\nq\nv\n=C\n(7"}, {"Chapter": "1", "sentence_range": "5622-5625", "Text": "The\ncapacitor is alternately charged and discharged as\nthe current reverses each half cycle Let q  be the\ncharge on the capacitor at any time t The instantaneous voltage v across\nthe capacitor is\nq\nv\n=C\n(7 15)\nFrom the Kirchhoff\u2019s loop rule, the voltage across the source and the\ncapacitor are equal,\nsin\nm\nq\nv\nt\nC\n\u03c9\n=\nTo find the current, we use the relation \ndd\nq\ni\nt\n=\n(\n)\ndd\nsin\ncos(\n)\nm\nm\ni\nv C\nt\nC v\nt\nt\n\u03c9\n\u03c9\n\u03c9\n=\n=\nUsing the relation, cos(\n)\nsin\n\u03c9\n\u03c9\nt\nt\n=\n+\n\uf8ed\uf8ec\uf8eb\n\u03c0\uf8f8\uf8f7\uf8f6\n2\n, we have\ni\ni\nt\n=m\n+\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nsin \u03c9\n2\u03c0\n(7"}, {"Chapter": "1", "sentence_range": "5623-5626", "Text": "Let q  be the\ncharge on the capacitor at any time t The instantaneous voltage v across\nthe capacitor is\nq\nv\n=C\n(7 15)\nFrom the Kirchhoff\u2019s loop rule, the voltage across the source and the\ncapacitor are equal,\nsin\nm\nq\nv\nt\nC\n\u03c9\n=\nTo find the current, we use the relation \ndd\nq\ni\nt\n=\n(\n)\ndd\nsin\ncos(\n)\nm\nm\ni\nv C\nt\nC v\nt\nt\n\u03c9\n\u03c9\n\u03c9\n=\n=\nUsing the relation, cos(\n)\nsin\n\u03c9\n\u03c9\nt\nt\n=\n+\n\uf8ed\uf8ec\uf8eb\n\u03c0\uf8f8\uf8f7\uf8f6\n2\n, we have\ni\ni\nt\n=m\n+\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nsin \u03c9\n2\u03c0\n(7 16)\nwhere the amplitude of the oscillating current is im = w Cvm"}, {"Chapter": "1", "sentence_range": "5624-5627", "Text": "The instantaneous voltage v across\nthe capacitor is\nq\nv\n=C\n(7 15)\nFrom the Kirchhoff\u2019s loop rule, the voltage across the source and the\ncapacitor are equal,\nsin\nm\nq\nv\nt\nC\n\u03c9\n=\nTo find the current, we use the relation \ndd\nq\ni\nt\n=\n(\n)\ndd\nsin\ncos(\n)\nm\nm\ni\nv C\nt\nC v\nt\nt\n\u03c9\n\u03c9\n\u03c9\n=\n=\nUsing the relation, cos(\n)\nsin\n\u03c9\n\u03c9\nt\nt\n=\n+\n\uf8ed\uf8ec\uf8eb\n\u03c0\uf8f8\uf8f7\uf8f6\n2\n, we have\ni\ni\nt\n=m\n+\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nsin \u03c9\n2\u03c0\n(7 16)\nwhere the amplitude of the oscillating current is im = w Cvm We can rewrite\nit as\n(1/\n)\nm\nm\nv\ni\n\u03c9C\n=\nComparing it to  im= vm/R for a purely resistive circuit, we find that\n(1/wC)  plays the role of resistance"}, {"Chapter": "1", "sentence_range": "5625-5628", "Text": "15)\nFrom the Kirchhoff\u2019s loop rule, the voltage across the source and the\ncapacitor are equal,\nsin\nm\nq\nv\nt\nC\n\u03c9\n=\nTo find the current, we use the relation \ndd\nq\ni\nt\n=\n(\n)\ndd\nsin\ncos(\n)\nm\nm\ni\nv C\nt\nC v\nt\nt\n\u03c9\n\u03c9\n\u03c9\n=\n=\nUsing the relation, cos(\n)\nsin\n\u03c9\n\u03c9\nt\nt\n=\n+\n\uf8ed\uf8ec\uf8eb\n\u03c0\uf8f8\uf8f7\uf8f6\n2\n, we have\ni\ni\nt\n=m\n+\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nsin \u03c9\n2\u03c0\n(7 16)\nwhere the amplitude of the oscillating current is im = w Cvm We can rewrite\nit as\n(1/\n)\nm\nm\nv\ni\n\u03c9C\n=\nComparing it to  im= vm/R for a purely resistive circuit, we find that\n(1/wC)  plays the role of resistance It is called capacitive reactance and\nis denoted by Xc,\nXc= 1/wC\n(7"}, {"Chapter": "1", "sentence_range": "5626-5629", "Text": "16)\nwhere the amplitude of the oscillating current is im = w Cvm We can rewrite\nit as\n(1/\n)\nm\nm\nv\ni\n\u03c9C\n=\nComparing it to  im= vm/R for a purely resistive circuit, we find that\n(1/wC)  plays the role of resistance It is called capacitive reactance and\nis denoted by Xc,\nXc= 1/wC\n(7 17)\nso that the amplitude of the current is\nm\nm\nC\nv\ni\n=X\n(7"}, {"Chapter": "1", "sentence_range": "5627-5630", "Text": "We can rewrite\nit as\n(1/\n)\nm\nm\nv\ni\n\u03c9C\n=\nComparing it to  im= vm/R for a purely resistive circuit, we find that\n(1/wC)  plays the role of resistance It is called capacitive reactance and\nis denoted by Xc,\nXc= 1/wC\n(7 17)\nso that the amplitude of the current is\nm\nm\nC\nv\ni\n=X\n(7 18)\nFIGURE 7"}, {"Chapter": "1", "sentence_range": "5628-5631", "Text": "It is called capacitive reactance and\nis denoted by Xc,\nXc= 1/wC\n(7 17)\nso that the amplitude of the current is\nm\nm\nC\nv\ni\n=X\n(7 18)\nFIGURE 7 7  An ac source\nconnected to a capacitor"}, {"Chapter": "1", "sentence_range": "5629-5632", "Text": "17)\nso that the amplitude of the current is\nm\nm\nC\nv\ni\n=X\n(7 18)\nFIGURE 7 7  An ac source\nconnected to a capacitor Rationalised 2023-24\n185\nAlternating Current\nFIGURE  7"}, {"Chapter": "1", "sentence_range": "5630-5633", "Text": "18)\nFIGURE 7 7  An ac source\nconnected to a capacitor Rationalised 2023-24\n185\nAlternating Current\nFIGURE  7 8 (a) A Phasor diagram for the circuit\nin Fig"}, {"Chapter": "1", "sentence_range": "5631-5634", "Text": "7  An ac source\nconnected to a capacitor Rationalised 2023-24\n185\nAlternating Current\nFIGURE  7 8 (a) A Phasor diagram for the circuit\nin Fig 7"}, {"Chapter": "1", "sentence_range": "5632-5635", "Text": "Rationalised 2023-24\n185\nAlternating Current\nFIGURE  7 8 (a) A Phasor diagram for the circuit\nin Fig 7 8"}, {"Chapter": "1", "sentence_range": "5633-5636", "Text": "8 (a) A Phasor diagram for the circuit\nin Fig 7 8 (b) Graph of v and i versus wt"}, {"Chapter": "1", "sentence_range": "5634-5637", "Text": "7 8 (b) Graph of v and i versus wt The dimension of capacitive reactance is the\nsame as that of resistance and its SI unit is\nohm (W)"}, {"Chapter": "1", "sentence_range": "5635-5638", "Text": "8 (b) Graph of v and i versus wt The dimension of capacitive reactance is the\nsame as that of resistance and its SI unit is\nohm (W) The capacitive reactance limits the\namplitude of the current in a purely capacitive\ncircuit in the same way as the resistance limits\nthe current in a purely resistive circuit"}, {"Chapter": "1", "sentence_range": "5636-5639", "Text": "(b) Graph of v and i versus wt The dimension of capacitive reactance is the\nsame as that of resistance and its SI unit is\nohm (W) The capacitive reactance limits the\namplitude of the current in a purely capacitive\ncircuit in the same way as the resistance limits\nthe current in a purely resistive circuit But it\nis inversely proportional to the frequency and\nthe capacitance"}, {"Chapter": "1", "sentence_range": "5637-5640", "Text": "The dimension of capacitive reactance is the\nsame as that of resistance and its SI unit is\nohm (W) The capacitive reactance limits the\namplitude of the current in a purely capacitive\ncircuit in the same way as the resistance limits\nthe current in a purely resistive circuit But it\nis inversely proportional to the frequency and\nthe capacitance A comparison of Eq"}, {"Chapter": "1", "sentence_range": "5638-5641", "Text": "The capacitive reactance limits the\namplitude of the current in a purely capacitive\ncircuit in the same way as the resistance limits\nthe current in a purely resistive circuit But it\nis inversely proportional to the frequency and\nthe capacitance A comparison of Eq (7"}, {"Chapter": "1", "sentence_range": "5639-5642", "Text": "But it\nis inversely proportional to the frequency and\nthe capacitance A comparison of Eq (7 16) with the\nequation of source voltage, Eq"}, {"Chapter": "1", "sentence_range": "5640-5643", "Text": "A comparison of Eq (7 16) with the\nequation of source voltage, Eq (7"}, {"Chapter": "1", "sentence_range": "5641-5644", "Text": "(7 16) with the\nequation of source voltage, Eq (7 1) shows that\nthe current is p/2 ahead of voltage"}, {"Chapter": "1", "sentence_range": "5642-5645", "Text": "16) with the\nequation of source voltage, Eq (7 1) shows that\nthe current is p/2 ahead of voltage Figure 7"}, {"Chapter": "1", "sentence_range": "5643-5646", "Text": "(7 1) shows that\nthe current is p/2 ahead of voltage Figure 7 8(a) shows the phasor diagram at an instant t1"}, {"Chapter": "1", "sentence_range": "5644-5647", "Text": "1) shows that\nthe current is p/2 ahead of voltage Figure 7 8(a) shows the phasor diagram at an instant t1 Here the current\nphasor I is p/2 ahead of the voltage phasor V as they rotate\ncounterclockwise"}, {"Chapter": "1", "sentence_range": "5645-5648", "Text": "Figure 7 8(a) shows the phasor diagram at an instant t1 Here the current\nphasor I is p/2 ahead of the voltage phasor V as they rotate\ncounterclockwise Figure 7"}, {"Chapter": "1", "sentence_range": "5646-5649", "Text": "8(a) shows the phasor diagram at an instant t1 Here the current\nphasor I is p/2 ahead of the voltage phasor V as they rotate\ncounterclockwise Figure 7 8(b) shows the variation of voltage and current\nwith time"}, {"Chapter": "1", "sentence_range": "5647-5650", "Text": "Here the current\nphasor I is p/2 ahead of the voltage phasor V as they rotate\ncounterclockwise Figure 7 8(b) shows the variation of voltage and current\nwith time We see that the current reaches its maximum value earlier than\nthe voltage by one-fourth of a period"}, {"Chapter": "1", "sentence_range": "5648-5651", "Text": "Figure 7 8(b) shows the variation of voltage and current\nwith time We see that the current reaches its maximum value earlier than\nthe voltage by one-fourth of a period The instantaneous power supplied to the capacitor is\npc = i v = im cos(wt)vm sin(wt)\n    = imvm cos(wt) sin(wt)\n    \nsin(2\n)\n2\nm\ni vm\n\u03c9t\n=\n(7"}, {"Chapter": "1", "sentence_range": "5649-5652", "Text": "8(b) shows the variation of voltage and current\nwith time We see that the current reaches its maximum value earlier than\nthe voltage by one-fourth of a period The instantaneous power supplied to the capacitor is\npc = i v = im cos(wt)vm sin(wt)\n    = imvm cos(wt) sin(wt)\n    \nsin(2\n)\n2\nm\ni vm\n\u03c9t\n=\n(7 19)\nSo, as in the case of an inductor, the average power\nsin(2\n)\nsin(2\n)\n0\n2\n2\nm\nm\nm\nm\nC\ni v\ni v\nP\nt\nt\n\u03c9\n\u03c9\n=\n=\n=\nsince <sin (2wt)> = 0 over a complete cycle"}, {"Chapter": "1", "sentence_range": "5650-5653", "Text": "We see that the current reaches its maximum value earlier than\nthe voltage by one-fourth of a period The instantaneous power supplied to the capacitor is\npc = i v = im cos(wt)vm sin(wt)\n    = imvm cos(wt) sin(wt)\n    \nsin(2\n)\n2\nm\ni vm\n\u03c9t\n=\n(7 19)\nSo, as in the case of an inductor, the average power\nsin(2\n)\nsin(2\n)\n0\n2\n2\nm\nm\nm\nm\nC\ni v\ni v\nP\nt\nt\n\u03c9\n\u03c9\n=\n=\n=\nsince <sin (2wt)> = 0 over a complete cycle Thus, we see that in the case of an inductor, the current lags the voltage\nby p/2 and in the case of a capacitor, the current leads the voltage by p/2"}, {"Chapter": "1", "sentence_range": "5651-5654", "Text": "The instantaneous power supplied to the capacitor is\npc = i v = im cos(wt)vm sin(wt)\n    = imvm cos(wt) sin(wt)\n    \nsin(2\n)\n2\nm\ni vm\n\u03c9t\n=\n(7 19)\nSo, as in the case of an inductor, the average power\nsin(2\n)\nsin(2\n)\n0\n2\n2\nm\nm\nm\nm\nC\ni v\ni v\nP\nt\nt\n\u03c9\n\u03c9\n=\n=\n=\nsince <sin (2wt)> = 0 over a complete cycle Thus, we see that in the case of an inductor, the current lags the voltage\nby p/2 and in the case of a capacitor, the current leads the voltage by p/2 Example 7"}, {"Chapter": "1", "sentence_range": "5652-5655", "Text": "19)\nSo, as in the case of an inductor, the average power\nsin(2\n)\nsin(2\n)\n0\n2\n2\nm\nm\nm\nm\nC\ni v\ni v\nP\nt\nt\n\u03c9\n\u03c9\n=\n=\n=\nsince <sin (2wt)> = 0 over a complete cycle Thus, we see that in the case of an inductor, the current lags the voltage\nby p/2 and in the case of a capacitor, the current leads the voltage by p/2 Example 7 3 A lamp is connected in series with a capacitor"}, {"Chapter": "1", "sentence_range": "5653-5656", "Text": "Thus, we see that in the case of an inductor, the current lags the voltage\nby p/2 and in the case of a capacitor, the current leads the voltage by p/2 Example 7 3 A lamp is connected in series with a capacitor Predict\nyour observations for dc and ac connections"}, {"Chapter": "1", "sentence_range": "5654-5657", "Text": "Example 7 3 A lamp is connected in series with a capacitor Predict\nyour observations for dc and ac connections What happens in each\ncase if the capacitance of the capacitor is reduced"}, {"Chapter": "1", "sentence_range": "5655-5658", "Text": "3 A lamp is connected in series with a capacitor Predict\nyour observations for dc and ac connections What happens in each\ncase if the capacitance of the capacitor is reduced Solution When a dc source is connected to a capacitor, the capacitor\ngets charged and after charging no current flows in the circuit and\nthe lamp will not glow"}, {"Chapter": "1", "sentence_range": "5656-5659", "Text": "Predict\nyour observations for dc and ac connections What happens in each\ncase if the capacitance of the capacitor is reduced Solution When a dc source is connected to a capacitor, the capacitor\ngets charged and after charging no current flows in the circuit and\nthe lamp will not glow There will be no change even if C is reduced"}, {"Chapter": "1", "sentence_range": "5657-5660", "Text": "What happens in each\ncase if the capacitance of the capacitor is reduced Solution When a dc source is connected to a capacitor, the capacitor\ngets charged and after charging no current flows in the circuit and\nthe lamp will not glow There will be no change even if C is reduced With ac source, the capacitor offers capacitative reactance (1/wC )\nand the current flows in the circuit"}, {"Chapter": "1", "sentence_range": "5658-5661", "Text": "Solution When a dc source is connected to a capacitor, the capacitor\ngets charged and after charging no current flows in the circuit and\nthe lamp will not glow There will be no change even if C is reduced With ac source, the capacitor offers capacitative reactance (1/wC )\nand the current flows in the circuit Consequently, the lamp will shine"}, {"Chapter": "1", "sentence_range": "5659-5662", "Text": "There will be no change even if C is reduced With ac source, the capacitor offers capacitative reactance (1/wC )\nand the current flows in the circuit Consequently, the lamp will shine Reducing C will increase reactance and the lamp will shine less brightly\nthan before"}, {"Chapter": "1", "sentence_range": "5660-5663", "Text": "With ac source, the capacitor offers capacitative reactance (1/wC )\nand the current flows in the circuit Consequently, the lamp will shine Reducing C will increase reactance and the lamp will shine less brightly\nthan before Example 7"}, {"Chapter": "1", "sentence_range": "5661-5664", "Text": "Consequently, the lamp will shine Reducing C will increase reactance and the lamp will shine less brightly\nthan before Example 7 4 A 15"}, {"Chapter": "1", "sentence_range": "5662-5665", "Text": "Reducing C will increase reactance and the lamp will shine less brightly\nthan before Example 7 4 A 15 0 mF capacitor is connected to a 220 V, 50 Hz source"}, {"Chapter": "1", "sentence_range": "5663-5666", "Text": "Example 7 4 A 15 0 mF capacitor is connected to a 220 V, 50 Hz source Find the capacitive reactance and the current (rms and peak) in the\ncircuit"}, {"Chapter": "1", "sentence_range": "5664-5667", "Text": "4 A 15 0 mF capacitor is connected to a 220 V, 50 Hz source Find the capacitive reactance and the current (rms and peak) in the\ncircuit If the frequency is doubled, what happens to the capacitive\nreactance and the current"}, {"Chapter": "1", "sentence_range": "5665-5668", "Text": "0 mF capacitor is connected to a 220 V, 50 Hz source Find the capacitive reactance and the current (rms and peak) in the\ncircuit If the frequency is doubled, what happens to the capacitive\nreactance and the current Solution  The capacitive reactance is\n6F\n1\n1\n212\n2\n2 (50Hz)(15"}, {"Chapter": "1", "sentence_range": "5666-5669", "Text": "Find the capacitive reactance and the current (rms and peak) in the\ncircuit If the frequency is doubled, what happens to the capacitive\nreactance and the current Solution  The capacitive reactance is\n6F\n1\n1\n212\n2\n2 (50Hz)(15 0\n10\n)\nXC\n\u03bdC\n\u2212\n=\n=\n=\n\u2126\n\u03c0\n\u03c0\n\u00d7\nThe rms current is\n EXAMPLE 7"}, {"Chapter": "1", "sentence_range": "5667-5670", "Text": "If the frequency is doubled, what happens to the capacitive\nreactance and the current Solution  The capacitive reactance is\n6F\n1\n1\n212\n2\n2 (50Hz)(15 0\n10\n)\nXC\n\u03bdC\n\u2212\n=\n=\n=\n\u2126\n\u03c0\n\u03c0\n\u00d7\nThe rms current is\n EXAMPLE 7 3\n EXAMPLE 7"}, {"Chapter": "1", "sentence_range": "5668-5671", "Text": "Solution  The capacitive reactance is\n6F\n1\n1\n212\n2\n2 (50Hz)(15 0\n10\n)\nXC\n\u03bdC\n\u2212\n=\n=\n=\n\u2126\n\u03c0\n\u03c0\n\u00d7\nThe rms current is\n EXAMPLE 7 3\n EXAMPLE 7 4\nRationalised 2023-24\nPhysics\n186\n EXAMPLE 7"}, {"Chapter": "1", "sentence_range": "5669-5672", "Text": "0\n10\n)\nXC\n\u03bdC\n\u2212\n=\n=\n=\n\u2126\n\u03c0\n\u03c0\n\u00d7\nThe rms current is\n EXAMPLE 7 3\n EXAMPLE 7 4\nRationalised 2023-24\nPhysics\n186\n EXAMPLE 7 5\n EXAMPLE 7"}, {"Chapter": "1", "sentence_range": "5670-5673", "Text": "3\n EXAMPLE 7 4\nRationalised 2023-24\nPhysics\n186\n EXAMPLE 7 5\n EXAMPLE 7 4\nV\nA\n220\n1"}, {"Chapter": "1", "sentence_range": "5671-5674", "Text": "4\nRationalised 2023-24\nPhysics\n186\n EXAMPLE 7 5\n EXAMPLE 7 4\nV\nA\n220\n1 04\n212\nC\nV\nI\n=X\n=\n=\n\u2126\nThe peak current is\n2\n(1"}, {"Chapter": "1", "sentence_range": "5672-5675", "Text": "5\n EXAMPLE 7 4\nV\nA\n220\n1 04\n212\nC\nV\nI\n=X\n=\n=\n\u2126\nThe peak current is\n2\n(1 41)(1"}, {"Chapter": "1", "sentence_range": "5673-5676", "Text": "4\nV\nA\n220\n1 04\n212\nC\nV\nI\n=X\n=\n=\n\u2126\nThe peak current is\n2\n(1 41)(1 04\n)\n1"}, {"Chapter": "1", "sentence_range": "5674-5677", "Text": "04\n212\nC\nV\nI\n=X\n=\n=\n\u2126\nThe peak current is\n2\n(1 41)(1 04\n)\n1 47\nim\nI\nA\nA\n=\n=\n=\nThis current oscillates between +1"}, {"Chapter": "1", "sentence_range": "5675-5678", "Text": "41)(1 04\n)\n1 47\nim\nI\nA\nA\n=\n=\n=\nThis current oscillates between +1 47A and \u20131"}, {"Chapter": "1", "sentence_range": "5676-5679", "Text": "04\n)\n1 47\nim\nI\nA\nA\n=\n=\n=\nThis current oscillates between +1 47A and \u20131 47 A, and is ahead of\nthe voltage by p/2"}, {"Chapter": "1", "sentence_range": "5677-5680", "Text": "47\nim\nI\nA\nA\n=\n=\n=\nThis current oscillates between +1 47A and \u20131 47 A, and is ahead of\nthe voltage by p/2 If the frequency is doubled, the capacitive reactance is halved and\nconsequently, the current is doubled"}, {"Chapter": "1", "sentence_range": "5678-5681", "Text": "47A and \u20131 47 A, and is ahead of\nthe voltage by p/2 If the frequency is doubled, the capacitive reactance is halved and\nconsequently, the current is doubled Example 7"}, {"Chapter": "1", "sentence_range": "5679-5682", "Text": "47 A, and is ahead of\nthe voltage by p/2 If the frequency is doubled, the capacitive reactance is halved and\nconsequently, the current is doubled Example 7 5 A light bulb and an open coil inductor are connected to\nan ac source through a key as shown in Fig"}, {"Chapter": "1", "sentence_range": "5680-5683", "Text": "If the frequency is doubled, the capacitive reactance is halved and\nconsequently, the current is doubled Example 7 5 A light bulb and an open coil inductor are connected to\nan ac source through a key as shown in Fig 7"}, {"Chapter": "1", "sentence_range": "5681-5684", "Text": "Example 7 5 A light bulb and an open coil inductor are connected to\nan ac source through a key as shown in Fig 7 9"}, {"Chapter": "1", "sentence_range": "5682-5685", "Text": "5 A light bulb and an open coil inductor are connected to\nan ac source through a key as shown in Fig 7 9 FIGURE 7"}, {"Chapter": "1", "sentence_range": "5683-5686", "Text": "7 9 FIGURE 7 9\nThe switch is closed and after sometime, an iron rod is inserted into\nthe interior of the inductor"}, {"Chapter": "1", "sentence_range": "5684-5687", "Text": "9 FIGURE 7 9\nThe switch is closed and after sometime, an iron rod is inserted into\nthe interior of the inductor The glow of the light bulb (a) increases; (b)\ndecreases; (c) is unchanged, as the iron rod is inserted"}, {"Chapter": "1", "sentence_range": "5685-5688", "Text": "FIGURE 7 9\nThe switch is closed and after sometime, an iron rod is inserted into\nthe interior of the inductor The glow of the light bulb (a) increases; (b)\ndecreases; (c) is unchanged, as the iron rod is inserted Give your\nanswer with reasons"}, {"Chapter": "1", "sentence_range": "5686-5689", "Text": "9\nThe switch is closed and after sometime, an iron rod is inserted into\nthe interior of the inductor The glow of the light bulb (a) increases; (b)\ndecreases; (c) is unchanged, as the iron rod is inserted Give your\nanswer with reasons Solution  As the iron rod is inserted, the magnetic field inside the coil\nmagnetizes the iron increasing the magnetic field inside it"}, {"Chapter": "1", "sentence_range": "5687-5690", "Text": "The glow of the light bulb (a) increases; (b)\ndecreases; (c) is unchanged, as the iron rod is inserted Give your\nanswer with reasons Solution  As the iron rod is inserted, the magnetic field inside the coil\nmagnetizes the iron increasing the magnetic field inside it Hence,\nthe inductance of the coil increases"}, {"Chapter": "1", "sentence_range": "5688-5691", "Text": "Give your\nanswer with reasons Solution  As the iron rod is inserted, the magnetic field inside the coil\nmagnetizes the iron increasing the magnetic field inside it Hence,\nthe inductance of the coil increases Consequently, the inductive\nreactance of the coil increases"}, {"Chapter": "1", "sentence_range": "5689-5692", "Text": "Solution  As the iron rod is inserted, the magnetic field inside the coil\nmagnetizes the iron increasing the magnetic field inside it Hence,\nthe inductance of the coil increases Consequently, the inductive\nreactance of the coil increases As a result, a larger fraction of the\napplied ac voltage appears across the inductor, leaving less voltage\nacross the bulb"}, {"Chapter": "1", "sentence_range": "5690-5693", "Text": "Hence,\nthe inductance of the coil increases Consequently, the inductive\nreactance of the coil increases As a result, a larger fraction of the\napplied ac voltage appears across the inductor, leaving less voltage\nacross the bulb Therefore, the glow of the light bulb decreases"}, {"Chapter": "1", "sentence_range": "5691-5694", "Text": "Consequently, the inductive\nreactance of the coil increases As a result, a larger fraction of the\napplied ac voltage appears across the inductor, leaving less voltage\nacross the bulb Therefore, the glow of the light bulb decreases 7"}, {"Chapter": "1", "sentence_range": "5692-5695", "Text": "As a result, a larger fraction of the\napplied ac voltage appears across the inductor, leaving less voltage\nacross the bulb Therefore, the glow of the light bulb decreases 7 6  AC VOLTAGE APPLIED TO A SERIES  LCR CIRCUIT\nFigure 7"}, {"Chapter": "1", "sentence_range": "5693-5696", "Text": "Therefore, the glow of the light bulb decreases 7 6  AC VOLTAGE APPLIED TO A SERIES  LCR CIRCUIT\nFigure 7 10 shows a series LCR circuit connected to an ac source e"}, {"Chapter": "1", "sentence_range": "5694-5697", "Text": "7 6  AC VOLTAGE APPLIED TO A SERIES  LCR CIRCUIT\nFigure 7 10 shows a series LCR circuit connected to an ac source e As\nusual, we take the voltage of the source to be v = vm sin wt"}, {"Chapter": "1", "sentence_range": "5695-5698", "Text": "6  AC VOLTAGE APPLIED TO A SERIES  LCR CIRCUIT\nFigure 7 10 shows a series LCR circuit connected to an ac source e As\nusual, we take the voltage of the source to be v = vm sin wt If q is the charge on the capacitor and i the\ncurrent, at time t, we have, from Kirchhoff\u2019s loop\nrule:\ndd\ni\nq\nL\ni R\nv\nt\nC\n+\n+\n=\n(7"}, {"Chapter": "1", "sentence_range": "5696-5699", "Text": "10 shows a series LCR circuit connected to an ac source e As\nusual, we take the voltage of the source to be v = vm sin wt If q is the charge on the capacitor and i the\ncurrent, at time t, we have, from Kirchhoff\u2019s loop\nrule:\ndd\ni\nq\nL\ni R\nv\nt\nC\n+\n+\n=\n(7 20)\nWe want to determine the instantaneous\ncurrent i and its phase relationship to the applied\nalternating voltage v"}, {"Chapter": "1", "sentence_range": "5697-5700", "Text": "As\nusual, we take the voltage of the source to be v = vm sin wt If q is the charge on the capacitor and i the\ncurrent, at time t, we have, from Kirchhoff\u2019s loop\nrule:\ndd\ni\nq\nL\ni R\nv\nt\nC\n+\n+\n=\n(7 20)\nWe want to determine the instantaneous\ncurrent i and its phase relationship to the applied\nalternating voltage v We shall solve this problem\nby two methods"}, {"Chapter": "1", "sentence_range": "5698-5701", "Text": "If q is the charge on the capacitor and i the\ncurrent, at time t, we have, from Kirchhoff\u2019s loop\nrule:\ndd\ni\nq\nL\ni R\nv\nt\nC\n+\n+\n=\n(7 20)\nWe want to determine the instantaneous\ncurrent i and its phase relationship to the applied\nalternating voltage v We shall solve this problem\nby two methods First, we use the technique of\nphasors and in the second method, we solve\nEq"}, {"Chapter": "1", "sentence_range": "5699-5702", "Text": "20)\nWe want to determine the instantaneous\ncurrent i and its phase relationship to the applied\nalternating voltage v We shall solve this problem\nby two methods First, we use the technique of\nphasors and in the second method, we solve\nEq (7"}, {"Chapter": "1", "sentence_range": "5700-5703", "Text": "We shall solve this problem\nby two methods First, we use the technique of\nphasors and in the second method, we solve\nEq (7 20) analytically to obtain the time\u2013\ndependence of i"}, {"Chapter": "1", "sentence_range": "5701-5704", "Text": "First, we use the technique of\nphasors and in the second method, we solve\nEq (7 20) analytically to obtain the time\u2013\ndependence of i FIGURE 7"}, {"Chapter": "1", "sentence_range": "5702-5705", "Text": "(7 20) analytically to obtain the time\u2013\ndependence of i FIGURE 7 10 A series LCR circuit\nconnected to an ac source"}, {"Chapter": "1", "sentence_range": "5703-5706", "Text": "20) analytically to obtain the time\u2013\ndependence of i FIGURE 7 10 A series LCR circuit\nconnected to an ac source Rationalised 2023-24\n187\nAlternating Current\n7"}, {"Chapter": "1", "sentence_range": "5704-5707", "Text": "FIGURE 7 10 A series LCR circuit\nconnected to an ac source Rationalised 2023-24\n187\nAlternating Current\n7 6"}, {"Chapter": "1", "sentence_range": "5705-5708", "Text": "10 A series LCR circuit\nconnected to an ac source Rationalised 2023-24\n187\nAlternating Current\n7 6 1  Phasor-diagram solution\nFrom the circuit shown in Fig"}, {"Chapter": "1", "sentence_range": "5706-5709", "Text": "Rationalised 2023-24\n187\nAlternating Current\n7 6 1  Phasor-diagram solution\nFrom the circuit shown in Fig 7"}, {"Chapter": "1", "sentence_range": "5707-5710", "Text": "6 1  Phasor-diagram solution\nFrom the circuit shown in Fig 7 10, we see that the resistor, inductor\nand capacitor are in series"}, {"Chapter": "1", "sentence_range": "5708-5711", "Text": "1  Phasor-diagram solution\nFrom the circuit shown in Fig 7 10, we see that the resistor, inductor\nand capacitor are in series Therefore, the ac current in each element is\nthe same at any time, having the same amplitude and phase"}, {"Chapter": "1", "sentence_range": "5709-5712", "Text": "7 10, we see that the resistor, inductor\nand capacitor are in series Therefore, the ac current in each element is\nthe same at any time, having the same amplitude and phase Let it be\ni = im sin(wt+f)\n(7"}, {"Chapter": "1", "sentence_range": "5710-5713", "Text": "10, we see that the resistor, inductor\nand capacitor are in series Therefore, the ac current in each element is\nthe same at any time, having the same amplitude and phase Let it be\ni = im sin(wt+f)\n(7 21)\nwhere f is the phase difference between the voltage across the source and\nthe current in the circuit"}, {"Chapter": "1", "sentence_range": "5711-5714", "Text": "Therefore, the ac current in each element is\nthe same at any time, having the same amplitude and phase Let it be\ni = im sin(wt+f)\n(7 21)\nwhere f is the phase difference between the voltage across the source and\nthe current in the circuit On the basis of what we have learnt in the previous\nsections, we shall construct a phasor diagram for the present case"}, {"Chapter": "1", "sentence_range": "5712-5715", "Text": "Let it be\ni = im sin(wt+f)\n(7 21)\nwhere f is the phase difference between the voltage across the source and\nthe current in the circuit On the basis of what we have learnt in the previous\nsections, we shall construct a phasor diagram for the present case Let I be the phasor representing the current in the circuit as given by\nEq"}, {"Chapter": "1", "sentence_range": "5713-5716", "Text": "21)\nwhere f is the phase difference between the voltage across the source and\nthe current in the circuit On the basis of what we have learnt in the previous\nsections, we shall construct a phasor diagram for the present case Let I be the phasor representing the current in the circuit as given by\nEq (7"}, {"Chapter": "1", "sentence_range": "5714-5717", "Text": "On the basis of what we have learnt in the previous\nsections, we shall construct a phasor diagram for the present case Let I be the phasor representing the current in the circuit as given by\nEq (7 21)"}, {"Chapter": "1", "sentence_range": "5715-5718", "Text": "Let I be the phasor representing the current in the circuit as given by\nEq (7 21) Further, let VL, VR, VC, and V represent the voltage across the\ninductor, resistor, capacitor and the source, respectively"}, {"Chapter": "1", "sentence_range": "5716-5719", "Text": "(7 21) Further, let VL, VR, VC, and V represent the voltage across the\ninductor, resistor, capacitor and the source, respectively From previous\nsection, we know that VR is parallel to I, VC is p/2\nbehind I and VL is  p/2 ahead of I"}, {"Chapter": "1", "sentence_range": "5717-5720", "Text": "21) Further, let VL, VR, VC, and V represent the voltage across the\ninductor, resistor, capacitor and the source, respectively From previous\nsection, we know that VR is parallel to I, VC is p/2\nbehind I and VL is  p/2 ahead of I VL, VR, VC and I\nare shown in Fig"}, {"Chapter": "1", "sentence_range": "5718-5721", "Text": "Further, let VL, VR, VC, and V represent the voltage across the\ninductor, resistor, capacitor and the source, respectively From previous\nsection, we know that VR is parallel to I, VC is p/2\nbehind I and VL is  p/2 ahead of I VL, VR, VC and I\nare shown in Fig 7"}, {"Chapter": "1", "sentence_range": "5719-5722", "Text": "From previous\nsection, we know that VR is parallel to I, VC is p/2\nbehind I and VL is  p/2 ahead of I VL, VR, VC and I\nare shown in Fig 7 11(a) with apppropriate phase-\nrelations"}, {"Chapter": "1", "sentence_range": "5720-5723", "Text": "VL, VR, VC and I\nare shown in Fig 7 11(a) with apppropriate phase-\nrelations The length of these phasors or the amplitude\nof VR, VC and VL are:\nvRm = im R, vCm = im XC, vLm = im XL\n(7"}, {"Chapter": "1", "sentence_range": "5721-5724", "Text": "7 11(a) with apppropriate phase-\nrelations The length of these phasors or the amplitude\nof VR, VC and VL are:\nvRm = im R, vCm = im XC, vLm = im XL\n(7 22)\nThe voltage Equation (7"}, {"Chapter": "1", "sentence_range": "5722-5725", "Text": "11(a) with apppropriate phase-\nrelations The length of these phasors or the amplitude\nof VR, VC and VL are:\nvRm = im R, vCm = im XC, vLm = im XL\n(7 22)\nThe voltage Equation (7 20) for the circuit can\nbe written as\nvL + vR + vC = v\n(7"}, {"Chapter": "1", "sentence_range": "5723-5726", "Text": "The length of these phasors or the amplitude\nof VR, VC and VL are:\nvRm = im R, vCm = im XC, vLm = im XL\n(7 22)\nThe voltage Equation (7 20) for the circuit can\nbe written as\nvL + vR + vC = v\n(7 23)\nThe phasor relation whose vertical component\ngives the above equation is\nVL + VR + VC = V\n(7"}, {"Chapter": "1", "sentence_range": "5724-5727", "Text": "22)\nThe voltage Equation (7 20) for the circuit can\nbe written as\nvL + vR + vC = v\n(7 23)\nThe phasor relation whose vertical component\ngives the above equation is\nVL + VR + VC = V\n(7 24)\nThis relation is represented in Fig"}, {"Chapter": "1", "sentence_range": "5725-5728", "Text": "20) for the circuit can\nbe written as\nvL + vR + vC = v\n(7 23)\nThe phasor relation whose vertical component\ngives the above equation is\nVL + VR + VC = V\n(7 24)\nThis relation is represented in Fig 7"}, {"Chapter": "1", "sentence_range": "5726-5729", "Text": "23)\nThe phasor relation whose vertical component\ngives the above equation is\nVL + VR + VC = V\n(7 24)\nThis relation is represented in Fig 7 11(b)"}, {"Chapter": "1", "sentence_range": "5727-5730", "Text": "24)\nThis relation is represented in Fig 7 11(b) Since\nVC  and VL are always along the same line and in\nopposite directions, they can be combined into a single phasor (VC + VL)\nwhich has a magnitude \u00bdvCm \u2013 vLm\u00bd"}, {"Chapter": "1", "sentence_range": "5728-5731", "Text": "7 11(b) Since\nVC  and VL are always along the same line and in\nopposite directions, they can be combined into a single phasor (VC + VL)\nwhich has a magnitude \u00bdvCm \u2013 vLm\u00bd Since V is represented as the\nhypotenuse of a right-triangle whose sides are VR and (VC + VL), the\npythagorean theorem gives:\n(\n)\n2\n2\n2\nm\nRm\nCm\nLm\nv\nv\nv\nv\n=\n+\n\u2212\nSubstituting the values of vRm, vCm, and vLm from Eq"}, {"Chapter": "1", "sentence_range": "5729-5732", "Text": "11(b) Since\nVC  and VL are always along the same line and in\nopposite directions, they can be combined into a single phasor (VC + VL)\nwhich has a magnitude \u00bdvCm \u2013 vLm\u00bd Since V is represented as the\nhypotenuse of a right-triangle whose sides are VR and (VC + VL), the\npythagorean theorem gives:\n(\n)\n2\n2\n2\nm\nRm\nCm\nLm\nv\nv\nv\nv\n=\n+\n\u2212\nSubstituting the values of vRm, vCm, and vLm from Eq (7"}, {"Chapter": "1", "sentence_range": "5730-5733", "Text": "Since\nVC  and VL are always along the same line and in\nopposite directions, they can be combined into a single phasor (VC + VL)\nwhich has a magnitude \u00bdvCm \u2013 vLm\u00bd Since V is represented as the\nhypotenuse of a right-triangle whose sides are VR and (VC + VL), the\npythagorean theorem gives:\n(\n)\n2\n2\n2\nm\nRm\nCm\nLm\nv\nv\nv\nv\n=\n+\n\u2212\nSubstituting the values of vRm, vCm, and vLm from Eq (7 22) into the above\nequation, we have\n2\n2\n2\n(\n)\n(\n)\nm\nm\nm\nC\nm\nL\nv\ni R\ni X\ni X\n=\n+\n\u2212\n     =\n+\n\u2212\n\uf8ee\uf8f0\n\uf8f9\uf8fb\ni\nR\nX\nX\nm\nC\nL\n2\n2\n2\n(\n)\nor,  \n2\n2\n(\n)\nm\nm\nC\nL\nv\ni\nR\nX\nX\n=\n+\n\u2212\n[7"}, {"Chapter": "1", "sentence_range": "5731-5734", "Text": "Since V is represented as the\nhypotenuse of a right-triangle whose sides are VR and (VC + VL), the\npythagorean theorem gives:\n(\n)\n2\n2\n2\nm\nRm\nCm\nLm\nv\nv\nv\nv\n=\n+\n\u2212\nSubstituting the values of vRm, vCm, and vLm from Eq (7 22) into the above\nequation, we have\n2\n2\n2\n(\n)\n(\n)\nm\nm\nm\nC\nm\nL\nv\ni R\ni X\ni X\n=\n+\n\u2212\n     =\n+\n\u2212\n\uf8ee\uf8f0\n\uf8f9\uf8fb\ni\nR\nX\nX\nm\nC\nL\n2\n2\n2\n(\n)\nor,  \n2\n2\n(\n)\nm\nm\nC\nL\nv\ni\nR\nX\nX\n=\n+\n\u2212\n[7 25(a)]\nBy analogy to the resistance in a circuit, we introduce the impedance Z\nin an ac circuit:\nm\nm\nv\ni\n=Z\n[7"}, {"Chapter": "1", "sentence_range": "5732-5735", "Text": "(7 22) into the above\nequation, we have\n2\n2\n2\n(\n)\n(\n)\nm\nm\nm\nC\nm\nL\nv\ni R\ni X\ni X\n=\n+\n\u2212\n     =\n+\n\u2212\n\uf8ee\uf8f0\n\uf8f9\uf8fb\ni\nR\nX\nX\nm\nC\nL\n2\n2\n2\n(\n)\nor,  \n2\n2\n(\n)\nm\nm\nC\nL\nv\ni\nR\nX\nX\n=\n+\n\u2212\n[7 25(a)]\nBy analogy to the resistance in a circuit, we introduce the impedance Z\nin an ac circuit:\nm\nm\nv\ni\n=Z\n[7 25(b)]\nwhere \n2\n2\n(\n)\nC\nL\nZ\nR\nX\nX\n=\n+\n\u2212\n(7"}, {"Chapter": "1", "sentence_range": "5733-5736", "Text": "22) into the above\nequation, we have\n2\n2\n2\n(\n)\n(\n)\nm\nm\nm\nC\nm\nL\nv\ni R\ni X\ni X\n=\n+\n\u2212\n     =\n+\n\u2212\n\uf8ee\uf8f0\n\uf8f9\uf8fb\ni\nR\nX\nX\nm\nC\nL\n2\n2\n2\n(\n)\nor,  \n2\n2\n(\n)\nm\nm\nC\nL\nv\ni\nR\nX\nX\n=\n+\n\u2212\n[7 25(a)]\nBy analogy to the resistance in a circuit, we introduce the impedance Z\nin an ac circuit:\nm\nm\nv\ni\n=Z\n[7 25(b)]\nwhere \n2\n2\n(\n)\nC\nL\nZ\nR\nX\nX\n=\n+\n\u2212\n(7 26)\nFIGURE 7"}, {"Chapter": "1", "sentence_range": "5734-5737", "Text": "25(a)]\nBy analogy to the resistance in a circuit, we introduce the impedance Z\nin an ac circuit:\nm\nm\nv\ni\n=Z\n[7 25(b)]\nwhere \n2\n2\n(\n)\nC\nL\nZ\nR\nX\nX\n=\n+\n\u2212\n(7 26)\nFIGURE 7 11 (a) Relation between the\nphasors VL, VR, VC, and I, (b) Relation\nbetween the  phasors VL, VR, and (VL + VC)\nfor the circuit in Fig"}, {"Chapter": "1", "sentence_range": "5735-5738", "Text": "25(b)]\nwhere \n2\n2\n(\n)\nC\nL\nZ\nR\nX\nX\n=\n+\n\u2212\n(7 26)\nFIGURE 7 11 (a) Relation between the\nphasors VL, VR, VC, and I, (b) Relation\nbetween the  phasors VL, VR, and (VL + VC)\nfor the circuit in Fig 7"}, {"Chapter": "1", "sentence_range": "5736-5739", "Text": "26)\nFIGURE 7 11 (a) Relation between the\nphasors VL, VR, VC, and I, (b) Relation\nbetween the  phasors VL, VR, and (VL + VC)\nfor the circuit in Fig 7 10"}, {"Chapter": "1", "sentence_range": "5737-5740", "Text": "11 (a) Relation between the\nphasors VL, VR, VC, and I, (b) Relation\nbetween the  phasors VL, VR, and (VL + VC)\nfor the circuit in Fig 7 10 Rationalised 2023-24\nPhysics\n188\nSince phasor I is always parallel to phasor VR, the phase angle f\nis the angle between VR and V and can be determined from\nFig"}, {"Chapter": "1", "sentence_range": "5738-5741", "Text": "7 10 Rationalised 2023-24\nPhysics\n188\nSince phasor I is always parallel to phasor VR, the phase angle f\nis the angle between VR and V and can be determined from\nFig 7"}, {"Chapter": "1", "sentence_range": "5739-5742", "Text": "10 Rationalised 2023-24\nPhysics\n188\nSince phasor I is always parallel to phasor VR, the phase angle f\nis the angle between VR and V and can be determined from\nFig 7 12:\ntan\nCm\nLm\nRm\nv\nv\nv\n\u03c6\n\u2212\n=\nUsing Eq"}, {"Chapter": "1", "sentence_range": "5740-5743", "Text": "Rationalised 2023-24\nPhysics\n188\nSince phasor I is always parallel to phasor VR, the phase angle f\nis the angle between VR and V and can be determined from\nFig 7 12:\ntan\nCm\nLm\nRm\nv\nv\nv\n\u03c6\n\u2212\n=\nUsing Eq (7"}, {"Chapter": "1", "sentence_range": "5741-5744", "Text": "7 12:\ntan\nCm\nLm\nRm\nv\nv\nv\n\u03c6\n\u2212\n=\nUsing Eq (7 22), we have\ntan\nC\nL\nX\nX\nR\n\u03c6\n\u2212\n=\n(7"}, {"Chapter": "1", "sentence_range": "5742-5745", "Text": "12:\ntan\nCm\nLm\nRm\nv\nv\nv\n\u03c6\n\u2212\n=\nUsing Eq (7 22), we have\ntan\nC\nL\nX\nX\nR\n\u03c6\n\u2212\n=\n(7 27)\nEquations (7"}, {"Chapter": "1", "sentence_range": "5743-5746", "Text": "(7 22), we have\ntan\nC\nL\nX\nX\nR\n\u03c6\n\u2212\n=\n(7 27)\nEquations (7 26) and (7"}, {"Chapter": "1", "sentence_range": "5744-5747", "Text": "22), we have\ntan\nC\nL\nX\nX\nR\n\u03c6\n\u2212\n=\n(7 27)\nEquations (7 26) and (7 27) are graphically shown in Fig"}, {"Chapter": "1", "sentence_range": "5745-5748", "Text": "27)\nEquations (7 26) and (7 27) are graphically shown in Fig (7"}, {"Chapter": "1", "sentence_range": "5746-5749", "Text": "26) and (7 27) are graphically shown in Fig (7 12)"}, {"Chapter": "1", "sentence_range": "5747-5750", "Text": "27) are graphically shown in Fig (7 12) This is called Impedance diagram which is a right-triangle with\nZ as its hypotenuse"}, {"Chapter": "1", "sentence_range": "5748-5751", "Text": "(7 12) This is called Impedance diagram which is a right-triangle with\nZ as its hypotenuse Equation 7"}, {"Chapter": "1", "sentence_range": "5749-5752", "Text": "12) This is called Impedance diagram which is a right-triangle with\nZ as its hypotenuse Equation 7 25(a) gives the amplitude of the current and Eq"}, {"Chapter": "1", "sentence_range": "5750-5753", "Text": "This is called Impedance diagram which is a right-triangle with\nZ as its hypotenuse Equation 7 25(a) gives the amplitude of the current and Eq (7"}, {"Chapter": "1", "sentence_range": "5751-5754", "Text": "Equation 7 25(a) gives the amplitude of the current and Eq (7 27)\ngives the phase angle"}, {"Chapter": "1", "sentence_range": "5752-5755", "Text": "25(a) gives the amplitude of the current and Eq (7 27)\ngives the phase angle With these, Eq"}, {"Chapter": "1", "sentence_range": "5753-5756", "Text": "(7 27)\ngives the phase angle With these, Eq (7"}, {"Chapter": "1", "sentence_range": "5754-5757", "Text": "27)\ngives the phase angle With these, Eq (7 21) is completely specified"}, {"Chapter": "1", "sentence_range": "5755-5758", "Text": "With these, Eq (7 21) is completely specified If XC > XL, f is positive and the circuit is predominantly capacitive"}, {"Chapter": "1", "sentence_range": "5756-5759", "Text": "(7 21) is completely specified If XC > XL, f is positive and the circuit is predominantly capacitive Consequently, the current in the circuit leads the source voltage"}, {"Chapter": "1", "sentence_range": "5757-5760", "Text": "21) is completely specified If XC > XL, f is positive and the circuit is predominantly capacitive Consequently, the current in the circuit leads the source voltage If\nXC < XL, f is negative and the circuit is predominantly inductive"}, {"Chapter": "1", "sentence_range": "5758-5761", "Text": "If XC > XL, f is positive and the circuit is predominantly capacitive Consequently, the current in the circuit leads the source voltage If\nXC < XL, f is negative and the circuit is predominantly inductive Consequently, the current in the circuit lags  the source voltage"}, {"Chapter": "1", "sentence_range": "5759-5762", "Text": "Consequently, the current in the circuit leads the source voltage If\nXC < XL, f is negative and the circuit is predominantly inductive Consequently, the current in the circuit lags  the source voltage Figure 7"}, {"Chapter": "1", "sentence_range": "5760-5763", "Text": "If\nXC < XL, f is negative and the circuit is predominantly inductive Consequently, the current in the circuit lags  the source voltage Figure 7 13 shows the phasor diagram and variation of v and i with w t\nfor the case XC > XL"}, {"Chapter": "1", "sentence_range": "5761-5764", "Text": "Consequently, the current in the circuit lags  the source voltage Figure 7 13 shows the phasor diagram and variation of v and i with w t\nfor the case XC > XL Thus, we have obtained the amplitude\nand phase of current for an LCR series circuit\nusing the technique of phasors"}, {"Chapter": "1", "sentence_range": "5762-5765", "Text": "Figure 7 13 shows the phasor diagram and variation of v and i with w t\nfor the case XC > XL Thus, we have obtained the amplitude\nand phase of current for an LCR series circuit\nusing the technique of phasors But this\nmethod of analysing ac circuits suffers from\ncertain disadvantages"}, {"Chapter": "1", "sentence_range": "5763-5766", "Text": "13 shows the phasor diagram and variation of v and i with w t\nfor the case XC > XL Thus, we have obtained the amplitude\nand phase of current for an LCR series circuit\nusing the technique of phasors But this\nmethod of analysing ac circuits suffers from\ncertain disadvantages First, the phasor\ndiagram say nothing about the initial\ncondition"}, {"Chapter": "1", "sentence_range": "5764-5767", "Text": "Thus, we have obtained the amplitude\nand phase of current for an LCR series circuit\nusing the technique of phasors But this\nmethod of analysing ac circuits suffers from\ncertain disadvantages First, the phasor\ndiagram say nothing about the initial\ncondition One can take any arbitrary value\nof t (say, t1, as done throughout this chapter)\nand draw different phasors which show the\nrelative angle between different phasors"}, {"Chapter": "1", "sentence_range": "5765-5768", "Text": "But this\nmethod of analysing ac circuits suffers from\ncertain disadvantages First, the phasor\ndiagram say nothing about the initial\ncondition One can take any arbitrary value\nof t (say, t1, as done throughout this chapter)\nand draw different phasors which show the\nrelative angle between different phasors The solution so obtained is called the\nsteady-state solution"}, {"Chapter": "1", "sentence_range": "5766-5769", "Text": "First, the phasor\ndiagram say nothing about the initial\ncondition One can take any arbitrary value\nof t (say, t1, as done throughout this chapter)\nand draw different phasors which show the\nrelative angle between different phasors The solution so obtained is called the\nsteady-state solution This is not a general\nsolution"}, {"Chapter": "1", "sentence_range": "5767-5770", "Text": "One can take any arbitrary value\nof t (say, t1, as done throughout this chapter)\nand draw different phasors which show the\nrelative angle between different phasors The solution so obtained is called the\nsteady-state solution This is not a general\nsolution Additionally, we do have a\ntransient solution which exists even for\nv = 0"}, {"Chapter": "1", "sentence_range": "5768-5771", "Text": "The solution so obtained is called the\nsteady-state solution This is not a general\nsolution Additionally, we do have a\ntransient solution which exists even for\nv = 0 The general solution is the sum of the\ntransient solution and the steady-state\nsolution"}, {"Chapter": "1", "sentence_range": "5769-5772", "Text": "This is not a general\nsolution Additionally, we do have a\ntransient solution which exists even for\nv = 0 The general solution is the sum of the\ntransient solution and the steady-state\nsolution After a sufficiently long time, the effects of the transient solution\ndie out and the behaviour of the circuit is described by the steady-state\nsolution"}, {"Chapter": "1", "sentence_range": "5770-5773", "Text": "Additionally, we do have a\ntransient solution which exists even for\nv = 0 The general solution is the sum of the\ntransient solution and the steady-state\nsolution After a sufficiently long time, the effects of the transient solution\ndie out and the behaviour of the circuit is described by the steady-state\nsolution 7"}, {"Chapter": "1", "sentence_range": "5771-5774", "Text": "The general solution is the sum of the\ntransient solution and the steady-state\nsolution After a sufficiently long time, the effects of the transient solution\ndie out and the behaviour of the circuit is described by the steady-state\nsolution 7 6"}, {"Chapter": "1", "sentence_range": "5772-5775", "Text": "After a sufficiently long time, the effects of the transient solution\ndie out and the behaviour of the circuit is described by the steady-state\nsolution 7 6 2  Resonance\nAn interesting characteristic of the series RLC circuit is the phenomenon\nof resonance"}, {"Chapter": "1", "sentence_range": "5773-5776", "Text": "7 6 2  Resonance\nAn interesting characteristic of the series RLC circuit is the phenomenon\nof resonance The phenomenon of resonance is common among systems\nthat have a tendency to oscillate at a particular frequency"}, {"Chapter": "1", "sentence_range": "5774-5777", "Text": "6 2  Resonance\nAn interesting characteristic of the series RLC circuit is the phenomenon\nof resonance The phenomenon of resonance is common among systems\nthat have a tendency to oscillate at a particular frequency This frequency\nis called the system\u2019s natural frequency"}, {"Chapter": "1", "sentence_range": "5775-5778", "Text": "2  Resonance\nAn interesting characteristic of the series RLC circuit is the phenomenon\nof resonance The phenomenon of resonance is common among systems\nthat have a tendency to oscillate at a particular frequency This frequency\nis called the system\u2019s natural frequency If such a system is driven by an\nenergy source at a frequency that is near the natural frequency, the\namplitude of oscillation is found to be large"}, {"Chapter": "1", "sentence_range": "5776-5779", "Text": "The phenomenon of resonance is common among systems\nthat have a tendency to oscillate at a particular frequency This frequency\nis called the system\u2019s natural frequency If such a system is driven by an\nenergy source at a frequency that is near the natural frequency, the\namplitude of oscillation is found to be large A familiar example of this\nphenomenon is a child on a swing"}, {"Chapter": "1", "sentence_range": "5777-5780", "Text": "This frequency\nis called the system\u2019s natural frequency If such a system is driven by an\nenergy source at a frequency that is near the natural frequency, the\namplitude of oscillation is found to be large A familiar example of this\nphenomenon is a child on a swing The swing has a natural frequency\nfor swinging back and forth like a pendulum"}, {"Chapter": "1", "sentence_range": "5778-5781", "Text": "If such a system is driven by an\nenergy source at a frequency that is near the natural frequency, the\namplitude of oscillation is found to be large A familiar example of this\nphenomenon is a child on a swing The swing has a natural frequency\nfor swinging back and forth like a pendulum If the child pulls on the\nFIGURE 7"}, {"Chapter": "1", "sentence_range": "5779-5782", "Text": "A familiar example of this\nphenomenon is a child on a swing The swing has a natural frequency\nfor swinging back and forth like a pendulum If the child pulls on the\nFIGURE 7 12  Impedance\ndiagram"}, {"Chapter": "1", "sentence_range": "5780-5783", "Text": "The swing has a natural frequency\nfor swinging back and forth like a pendulum If the child pulls on the\nFIGURE 7 12  Impedance\ndiagram FIGURE 7"}, {"Chapter": "1", "sentence_range": "5781-5784", "Text": "If the child pulls on the\nFIGURE 7 12  Impedance\ndiagram FIGURE 7 13 (a) Phasor diagram of V and I"}, {"Chapter": "1", "sentence_range": "5782-5785", "Text": "12  Impedance\ndiagram FIGURE 7 13 (a) Phasor diagram of V and I (b) Graphs of v and i versus w t for a series LCR\ncircuit where XC > XL"}, {"Chapter": "1", "sentence_range": "5783-5786", "Text": "FIGURE 7 13 (a) Phasor diagram of V and I (b) Graphs of v and i versus w t for a series LCR\ncircuit where XC > XL Rationalised 2023-24\n189\nAlternating Current\nrope at regular intervals and the frequency of the pulls is almost the\nsame as the frequency of swinging, the amplitude of the swinging will be\nlarge (Chapter 13, Class XI)"}, {"Chapter": "1", "sentence_range": "5784-5787", "Text": "13 (a) Phasor diagram of V and I (b) Graphs of v and i versus w t for a series LCR\ncircuit where XC > XL Rationalised 2023-24\n189\nAlternating Current\nrope at regular intervals and the frequency of the pulls is almost the\nsame as the frequency of swinging, the amplitude of the swinging will be\nlarge (Chapter 13, Class XI) For an RLC circuit driven with voltage of amplitude vm and frequency\nw, we found that the current amplitude is given by\n2\n2\n(\n)\nm\nm\nm\nC\nL\nv\nv\ni\nZ\nR\nX\nX\n=\n=\n+\n\u2212\nwith Xc = 1/wC and XL = w L"}, {"Chapter": "1", "sentence_range": "5785-5788", "Text": "(b) Graphs of v and i versus w t for a series LCR\ncircuit where XC > XL Rationalised 2023-24\n189\nAlternating Current\nrope at regular intervals and the frequency of the pulls is almost the\nsame as the frequency of swinging, the amplitude of the swinging will be\nlarge (Chapter 13, Class XI) For an RLC circuit driven with voltage of amplitude vm and frequency\nw, we found that the current amplitude is given by\n2\n2\n(\n)\nm\nm\nm\nC\nL\nv\nv\ni\nZ\nR\nX\nX\n=\n=\n+\n\u2212\nwith Xc = 1/wC and XL = w L So if w is varied, then at a particular frequency\nw0, Xc = XL, and the impedance is minimum (\n)\n2\n02\nZ\nR\nR\n=\n+\n="}, {"Chapter": "1", "sentence_range": "5786-5789", "Text": "Rationalised 2023-24\n189\nAlternating Current\nrope at regular intervals and the frequency of the pulls is almost the\nsame as the frequency of swinging, the amplitude of the swinging will be\nlarge (Chapter 13, Class XI) For an RLC circuit driven with voltage of amplitude vm and frequency\nw, we found that the current amplitude is given by\n2\n2\n(\n)\nm\nm\nm\nC\nL\nv\nv\ni\nZ\nR\nX\nX\n=\n=\n+\n\u2212\nwith Xc = 1/wC and XL = w L So if w is varied, then at a particular frequency\nw0, Xc = XL, and the impedance is minimum (\n)\n2\n02\nZ\nR\nR\n=\n+\n= This\nfrequency is called the resonant frequency:\n0\n0\n1\nor\nc\nL\nX\nX\nL\nC\n\u03c9\n\u03c9\n=\n=\nor   \n0\n1\nLC\n\u03c9\n=\n(7"}, {"Chapter": "1", "sentence_range": "5787-5790", "Text": "For an RLC circuit driven with voltage of amplitude vm and frequency\nw, we found that the current amplitude is given by\n2\n2\n(\n)\nm\nm\nm\nC\nL\nv\nv\ni\nZ\nR\nX\nX\n=\n=\n+\n\u2212\nwith Xc = 1/wC and XL = w L So if w is varied, then at a particular frequency\nw0, Xc = XL, and the impedance is minimum (\n)\n2\n02\nZ\nR\nR\n=\n+\n= This\nfrequency is called the resonant frequency:\n0\n0\n1\nor\nc\nL\nX\nX\nL\nC\n\u03c9\n\u03c9\n=\n=\nor   \n0\n1\nLC\n\u03c9\n=\n(7 28)\nAt resonant frequency, the current amplitude\nis maximum; im = vm/R"}, {"Chapter": "1", "sentence_range": "5788-5791", "Text": "So if w is varied, then at a particular frequency\nw0, Xc = XL, and the impedance is minimum (\n)\n2\n02\nZ\nR\nR\n=\n+\n= This\nfrequency is called the resonant frequency:\n0\n0\n1\nor\nc\nL\nX\nX\nL\nC\n\u03c9\n\u03c9\n=\n=\nor   \n0\n1\nLC\n\u03c9\n=\n(7 28)\nAt resonant frequency, the current amplitude\nis maximum; im = vm/R Figure 7"}, {"Chapter": "1", "sentence_range": "5789-5792", "Text": "This\nfrequency is called the resonant frequency:\n0\n0\n1\nor\nc\nL\nX\nX\nL\nC\n\u03c9\n\u03c9\n=\n=\nor   \n0\n1\nLC\n\u03c9\n=\n(7 28)\nAt resonant frequency, the current amplitude\nis maximum; im = vm/R Figure 7 16 shows the variation of im with w\nin a RLC series circuit with L = 1"}, {"Chapter": "1", "sentence_range": "5790-5793", "Text": "28)\nAt resonant frequency, the current amplitude\nis maximum; im = vm/R Figure 7 16 shows the variation of im with w\nin a RLC series circuit with L = 1 00 mH, C =\n1"}, {"Chapter": "1", "sentence_range": "5791-5794", "Text": "Figure 7 16 shows the variation of im with w\nin a RLC series circuit with L = 1 00 mH, C =\n1 00 nF for two values of R: (i) R = 100 W\nand (ii) R = 200 W"}, {"Chapter": "1", "sentence_range": "5792-5795", "Text": "16 shows the variation of im with w\nin a RLC series circuit with L = 1 00 mH, C =\n1 00 nF for two values of R: (i) R = 100 W\nand (ii) R = 200 W For the source applied vm =\n100 V"}, {"Chapter": "1", "sentence_range": "5793-5796", "Text": "00 mH, C =\n1 00 nF for two values of R: (i) R = 100 W\nand (ii) R = 200 W For the source applied vm =\n100 V w0 for this case is \n1\nLC\n \n \n \n \n \n  = 1"}, {"Chapter": "1", "sentence_range": "5794-5797", "Text": "00 nF for two values of R: (i) R = 100 W\nand (ii) R = 200 W For the source applied vm =\n100 V w0 for this case is \n1\nLC\n \n \n \n \n \n  = 1 00\u00d7106\nrad/s"}, {"Chapter": "1", "sentence_range": "5795-5798", "Text": "For the source applied vm =\n100 V w0 for this case is \n1\nLC\n \n \n \n \n \n  = 1 00\u00d7106\nrad/s We see that the current amplitude is\nmaximum at the resonant frequency"}, {"Chapter": "1", "sentence_range": "5796-5799", "Text": "w0 for this case is \n1\nLC\n \n \n \n \n \n  = 1 00\u00d7106\nrad/s We see that the current amplitude is\nmaximum at the resonant frequency Since im =\nvm / R at resonance, the current amplitude for\ncase (i) is twice to that for case (ii)"}, {"Chapter": "1", "sentence_range": "5797-5800", "Text": "00\u00d7106\nrad/s We see that the current amplitude is\nmaximum at the resonant frequency Since im =\nvm / R at resonance, the current amplitude for\ncase (i) is twice to that for case (ii) Resonant circuits have a variety of applications, for example, in the\ntuning  mechanism of a radio or a TV set"}, {"Chapter": "1", "sentence_range": "5798-5801", "Text": "We see that the current amplitude is\nmaximum at the resonant frequency Since im =\nvm / R at resonance, the current amplitude for\ncase (i) is twice to that for case (ii) Resonant circuits have a variety of applications, for example, in the\ntuning  mechanism of a radio or a TV set The antenna of a radio accepts\nsignals from many broadcasting stations"}, {"Chapter": "1", "sentence_range": "5799-5802", "Text": "Since im =\nvm / R at resonance, the current amplitude for\ncase (i) is twice to that for case (ii) Resonant circuits have a variety of applications, for example, in the\ntuning  mechanism of a radio or a TV set The antenna of a radio accepts\nsignals from many broadcasting stations The signals picked up in the\nantenna acts as a source in the tuning circuit of the radio, so the circuit\ncan be driven at many frequencies"}, {"Chapter": "1", "sentence_range": "5800-5803", "Text": "Resonant circuits have a variety of applications, for example, in the\ntuning  mechanism of a radio or a TV set The antenna of a radio accepts\nsignals from many broadcasting stations The signals picked up in the\nantenna acts as a source in the tuning circuit of the radio, so the circuit\ncan be driven at many frequencies But to hear one particular radio\nstation, we tune the radio"}, {"Chapter": "1", "sentence_range": "5801-5804", "Text": "The antenna of a radio accepts\nsignals from many broadcasting stations The signals picked up in the\nantenna acts as a source in the tuning circuit of the radio, so the circuit\ncan be driven at many frequencies But to hear one particular radio\nstation, we tune the radio In tuning, we vary the capacitance of a\ncapacitor in the tuning circuit such that the resonant frequency of the\ncircuit becomes nearly equal to the frequency of the radio signal received"}, {"Chapter": "1", "sentence_range": "5802-5805", "Text": "The signals picked up in the\nantenna acts as a source in the tuning circuit of the radio, so the circuit\ncan be driven at many frequencies But to hear one particular radio\nstation, we tune the radio In tuning, we vary the capacitance of a\ncapacitor in the tuning circuit such that the resonant frequency of the\ncircuit becomes nearly equal to the frequency of the radio signal received When this happens, the amplitude of the current with the frequency of\nthe signal of the particular radio station in the circuit is maximum"}, {"Chapter": "1", "sentence_range": "5803-5806", "Text": "But to hear one particular radio\nstation, we tune the radio In tuning, we vary the capacitance of a\ncapacitor in the tuning circuit such that the resonant frequency of the\ncircuit becomes nearly equal to the frequency of the radio signal received When this happens, the amplitude of the current with the frequency of\nthe signal of the particular radio station in the circuit is maximum It is important to note that resonance phenomenon is exhibited by a\ncircuit only if both L and C are present in the circuit"}, {"Chapter": "1", "sentence_range": "5804-5807", "Text": "In tuning, we vary the capacitance of a\ncapacitor in the tuning circuit such that the resonant frequency of the\ncircuit becomes nearly equal to the frequency of the radio signal received When this happens, the amplitude of the current with the frequency of\nthe signal of the particular radio station in the circuit is maximum It is important to note that resonance phenomenon is exhibited by a\ncircuit only if both L and C are present in the circuit Only then do the\nvoltages across L and C cancel each other (both being out of phase)\nand the current amplitude is vm/R, the total source voltage appearing\nacross R"}, {"Chapter": "1", "sentence_range": "5805-5808", "Text": "When this happens, the amplitude of the current with the frequency of\nthe signal of the particular radio station in the circuit is maximum It is important to note that resonance phenomenon is exhibited by a\ncircuit only if both L and C are present in the circuit Only then do the\nvoltages across L and C cancel each other (both being out of phase)\nand the current amplitude is vm/R, the total source voltage appearing\nacross R This means that we cannot have resonance in a RL or\nRC circuit"}, {"Chapter": "1", "sentence_range": "5806-5809", "Text": "It is important to note that resonance phenomenon is exhibited by a\ncircuit only if both L and C are present in the circuit Only then do the\nvoltages across L and C cancel each other (both being out of phase)\nand the current amplitude is vm/R, the total source voltage appearing\nacross R This means that we cannot have resonance in a RL or\nRC circuit FIGURE 7"}, {"Chapter": "1", "sentence_range": "5807-5810", "Text": "Only then do the\nvoltages across L and C cancel each other (both being out of phase)\nand the current amplitude is vm/R, the total source voltage appearing\nacross R This means that we cannot have resonance in a RL or\nRC circuit FIGURE 7 14 Variation of im with w for two\ncases: (i) R = 100 W, (ii) R = 200 W,\nL = 1"}, {"Chapter": "1", "sentence_range": "5808-5811", "Text": "This means that we cannot have resonance in a RL or\nRC circuit FIGURE 7 14 Variation of im with w for two\ncases: (i) R = 100 W, (ii) R = 200 W,\nL = 1 00 mH"}, {"Chapter": "1", "sentence_range": "5809-5812", "Text": "FIGURE 7 14 Variation of im with w for two\ncases: (i) R = 100 W, (ii) R = 200 W,\nL = 1 00 mH Rationalised 2023-24\nPhysics\n190\nExample 7"}, {"Chapter": "1", "sentence_range": "5810-5813", "Text": "14 Variation of im with w for two\ncases: (i) R = 100 W, (ii) R = 200 W,\nL = 1 00 mH Rationalised 2023-24\nPhysics\n190\nExample 7 6 A resistor of 200 W and a capacitor of 15"}, {"Chapter": "1", "sentence_range": "5811-5814", "Text": "00 mH Rationalised 2023-24\nPhysics\n190\nExample 7 6 A resistor of 200 W and a capacitor of 15 0 mF are\nconnected in series to a 220 V, 50 Hz ac source"}, {"Chapter": "1", "sentence_range": "5812-5815", "Text": "Rationalised 2023-24\nPhysics\n190\nExample 7 6 A resistor of 200 W and a capacitor of 15 0 mF are\nconnected in series to a 220 V, 50 Hz ac source (a) Calculate the\ncurrent in the circuit; (b) Calculate the voltage (rms) across the\nresistor and the capacitor"}, {"Chapter": "1", "sentence_range": "5813-5816", "Text": "6 A resistor of 200 W and a capacitor of 15 0 mF are\nconnected in series to a 220 V, 50 Hz ac source (a) Calculate the\ncurrent in the circuit; (b) Calculate the voltage (rms) across the\nresistor and the capacitor Is the algebraic sum of these voltages\nmore than the source voltage"}, {"Chapter": "1", "sentence_range": "5814-5817", "Text": "0 mF are\nconnected in series to a 220 V, 50 Hz ac source (a) Calculate the\ncurrent in the circuit; (b) Calculate the voltage (rms) across the\nresistor and the capacitor Is the algebraic sum of these voltages\nmore than the source voltage If yes, resolve the paradox"}, {"Chapter": "1", "sentence_range": "5815-5818", "Text": "(a) Calculate the\ncurrent in the circuit; (b) Calculate the voltage (rms) across the\nresistor and the capacitor Is the algebraic sum of these voltages\nmore than the source voltage If yes, resolve the paradox Solution\nGiven\nF\n6\n200\n,\n15"}, {"Chapter": "1", "sentence_range": "5816-5819", "Text": "Is the algebraic sum of these voltages\nmore than the source voltage If yes, resolve the paradox Solution\nGiven\nF\n6\n200\n,\n15 0\n15"}, {"Chapter": "1", "sentence_range": "5817-5820", "Text": "If yes, resolve the paradox Solution\nGiven\nF\n6\n200\n,\n15 0\n15 0\n10\nF\nR\nC\n\u2212\n=\n\u2126\n=\n\u00b5\n=\n\u00d7\n220 V,\n50Hz\nV\n\u03bd\n=\n=\n(a)\nIn order to calculate the current, we need the impedance of\nthe circuit"}, {"Chapter": "1", "sentence_range": "5818-5821", "Text": "Solution\nGiven\nF\n6\n200\n,\n15 0\n15 0\n10\nF\nR\nC\n\u2212\n=\n\u2126\n=\n\u00b5\n=\n\u00d7\n220 V,\n50Hz\nV\n\u03bd\n=\n=\n(a)\nIn order to calculate the current, we need the impedance of\nthe circuit It is\n2\n2\n2\n2\n(2\n)\nC\nZ\nR\nX\nR\n\u03c0 \u03bdC\n\u2212\n=\n+\n=\n+\n   \nF\n2\n6\n2\n(200\n)\n(2\n3"}, {"Chapter": "1", "sentence_range": "5819-5822", "Text": "0\n15 0\n10\nF\nR\nC\n\u2212\n=\n\u2126\n=\n\u00b5\n=\n\u00d7\n220 V,\n50Hz\nV\n\u03bd\n=\n=\n(a)\nIn order to calculate the current, we need the impedance of\nthe circuit It is\n2\n2\n2\n2\n(2\n)\nC\nZ\nR\nX\nR\n\u03c0 \u03bdC\n\u2212\n=\n+\n=\n+\n   \nF\n2\n6\n2\n(200\n)\n(2\n3 14\n50\n15"}, {"Chapter": "1", "sentence_range": "5820-5823", "Text": "0\n10\nF\nR\nC\n\u2212\n=\n\u2126\n=\n\u00b5\n=\n\u00d7\n220 V,\n50Hz\nV\n\u03bd\n=\n=\n(a)\nIn order to calculate the current, we need the impedance of\nthe circuit It is\n2\n2\n2\n2\n(2\n)\nC\nZ\nR\nX\nR\n\u03c0 \u03bdC\n\u2212\n=\n+\n=\n+\n   \nF\n2\n6\n2\n(200\n)\n(2\n3 14\n50\n15 0\n10\n)\n\u2212\n\u2212\n=\n\u2126\n+\n\u00d7\n\u00d7\n\u00d7\n\u00d7\n   \n2\n2\n(200\n)\n(212"}, {"Chapter": "1", "sentence_range": "5821-5824", "Text": "It is\n2\n2\n2\n2\n(2\n)\nC\nZ\nR\nX\nR\n\u03c0 \u03bdC\n\u2212\n=\n+\n=\n+\n   \nF\n2\n6\n2\n(200\n)\n(2\n3 14\n50\n15 0\n10\n)\n\u2212\n\u2212\n=\n\u2126\n+\n\u00d7\n\u00d7\n\u00d7\n\u00d7\n   \n2\n2\n(200\n)\n(212 3\n)\n=\n\u2126\n+\n\u2126\n   \n=291"}, {"Chapter": "1", "sentence_range": "5822-5825", "Text": "14\n50\n15 0\n10\n)\n\u2212\n\u2212\n=\n\u2126\n+\n\u00d7\n\u00d7\n\u00d7\n\u00d7\n   \n2\n2\n(200\n)\n(212 3\n)\n=\n\u2126\n+\n\u2126\n   \n=291 67\n\u2126\nTherefore, the current in the circuit is\n220V\n0"}, {"Chapter": "1", "sentence_range": "5823-5826", "Text": "0\n10\n)\n\u2212\n\u2212\n=\n\u2126\n+\n\u00d7\n\u00d7\n\u00d7\n\u00d7\n   \n2\n2\n(200\n)\n(212 3\n)\n=\n\u2126\n+\n\u2126\n   \n=291 67\n\u2126\nTherefore, the current in the circuit is\n220V\n0 755 A\n291"}, {"Chapter": "1", "sentence_range": "5824-5827", "Text": "3\n)\n=\n\u2126\n+\n\u2126\n   \n=291 67\n\u2126\nTherefore, the current in the circuit is\n220V\n0 755 A\n291 5\nV\nI\n=Z\n=\n=\n\u2126\n(b)\nSince the current is the same throughout the circuit, we have\n(0"}, {"Chapter": "1", "sentence_range": "5825-5828", "Text": "67\n\u2126\nTherefore, the current in the circuit is\n220V\n0 755 A\n291 5\nV\nI\n=Z\n=\n=\n\u2126\n(b)\nSince the current is the same throughout the circuit, we have\n(0 755 A)(200\n)\n151V\nVR\n=I R\n=\n\u2126 =\n(0"}, {"Chapter": "1", "sentence_range": "5826-5829", "Text": "755 A\n291 5\nV\nI\n=Z\n=\n=\n\u2126\n(b)\nSince the current is the same throughout the circuit, we have\n(0 755 A)(200\n)\n151V\nVR\n=I R\n=\n\u2126 =\n(0 755 A)(212"}, {"Chapter": "1", "sentence_range": "5827-5830", "Text": "5\nV\nI\n=Z\n=\n=\n\u2126\n(b)\nSince the current is the same throughout the circuit, we have\n(0 755 A)(200\n)\n151V\nVR\n=I R\n=\n\u2126 =\n(0 755 A)(212 3\n)\n160"}, {"Chapter": "1", "sentence_range": "5828-5831", "Text": "755 A)(200\n)\n151V\nVR\n=I R\n=\n\u2126 =\n(0 755 A)(212 3\n)\n160 3 V\nC\nC\nV\n=I X\n=\n\u2126 =\nThe algebraic sum of the two voltages, VR and VC is 311"}, {"Chapter": "1", "sentence_range": "5829-5832", "Text": "755 A)(212 3\n)\n160 3 V\nC\nC\nV\n=I X\n=\n\u2126 =\nThe algebraic sum of the two voltages, VR and VC is 311 3 V which\nis more than the source voltage of 220 V"}, {"Chapter": "1", "sentence_range": "5830-5833", "Text": "3\n)\n160 3 V\nC\nC\nV\n=I X\n=\n\u2126 =\nThe algebraic sum of the two voltages, VR and VC is 311 3 V which\nis more than the source voltage of 220 V How to resolve this\nparadox"}, {"Chapter": "1", "sentence_range": "5831-5834", "Text": "3 V\nC\nC\nV\n=I X\n=\n\u2126 =\nThe algebraic sum of the two voltages, VR and VC is 311 3 V which\nis more than the source voltage of 220 V How to resolve this\nparadox As you have learnt in the text, the two voltages are not\nin the same phase"}, {"Chapter": "1", "sentence_range": "5832-5835", "Text": "3 V which\nis more than the source voltage of 220 V How to resolve this\nparadox As you have learnt in the text, the two voltages are not\nin the same phase Therefore, they cannot be added like ordinary\nnumbers"}, {"Chapter": "1", "sentence_range": "5833-5836", "Text": "How to resolve this\nparadox As you have learnt in the text, the two voltages are not\nin the same phase Therefore, they cannot be added like ordinary\nnumbers The two voltages are out of phase by ninety degrees"}, {"Chapter": "1", "sentence_range": "5834-5837", "Text": "As you have learnt in the text, the two voltages are not\nin the same phase Therefore, they cannot be added like ordinary\nnumbers The two voltages are out of phase by ninety degrees Therefore, the total of these voltages must be obtained using the\nPythagorean theorem:\n2\n2\nR C\nR\nC\nV\nV\nV\n+\n=\n+\n= 220 V\nThus, if the phase difference between two voltages is properly taken\ninto account, the total voltage across the resistor and the capacitor\nis equal to the voltage of the source"}, {"Chapter": "1", "sentence_range": "5835-5838", "Text": "Therefore, they cannot be added like ordinary\nnumbers The two voltages are out of phase by ninety degrees Therefore, the total of these voltages must be obtained using the\nPythagorean theorem:\n2\n2\nR C\nR\nC\nV\nV\nV\n+\n=\n+\n= 220 V\nThus, if the phase difference between two voltages is properly taken\ninto account, the total voltage across the resistor and the capacitor\nis equal to the voltage of the source EXAMPLE 7"}, {"Chapter": "1", "sentence_range": "5836-5839", "Text": "The two voltages are out of phase by ninety degrees Therefore, the total of these voltages must be obtained using the\nPythagorean theorem:\n2\n2\nR C\nR\nC\nV\nV\nV\n+\n=\n+\n= 220 V\nThus, if the phase difference between two voltages is properly taken\ninto account, the total voltage across the resistor and the capacitor\nis equal to the voltage of the source EXAMPLE 7 6\n7"}, {"Chapter": "1", "sentence_range": "5837-5840", "Text": "Therefore, the total of these voltages must be obtained using the\nPythagorean theorem:\n2\n2\nR C\nR\nC\nV\nV\nV\n+\n=\n+\n= 220 V\nThus, if the phase difference between two voltages is properly taken\ninto account, the total voltage across the resistor and the capacitor\nis equal to the voltage of the source EXAMPLE 7 6\n7 7  POWER IN AC CIRCUIT: THE POWER FACTOR\nWe have seen that a voltage v = vm sinwt applied to a series RLC circuit\ndrives a current in the circuit given by i = im sin(wt + f) where\nm\nm\nv\ni\n=Z\n  and \u03c6 =\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\u2212\ntan 1\nX\nX\nR\nC\nL\nTherefore, the instantaneous power p supplied by the source is\nRationalised 2023-24\n191\nAlternating Current\n(\n) [\n]\nsin\nsin(\n)\nm\nm\np\nv i\nv\nt\ni\nt\n\u03c9\n\u03c9\n\u03c6\n=\n=\n\u00d7\n+\n[\n]\ncos\ncos(2\n)\n2\nm m\nv i\nt\n\u03c6\n\u03c9\n\u03c6\n=\n\u2212\n+\n(7"}, {"Chapter": "1", "sentence_range": "5838-5841", "Text": "EXAMPLE 7 6\n7 7  POWER IN AC CIRCUIT: THE POWER FACTOR\nWe have seen that a voltage v = vm sinwt applied to a series RLC circuit\ndrives a current in the circuit given by i = im sin(wt + f) where\nm\nm\nv\ni\n=Z\n  and \u03c6 =\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\u2212\ntan 1\nX\nX\nR\nC\nL\nTherefore, the instantaneous power p supplied by the source is\nRationalised 2023-24\n191\nAlternating Current\n(\n) [\n]\nsin\nsin(\n)\nm\nm\np\nv i\nv\nt\ni\nt\n\u03c9\n\u03c9\n\u03c6\n=\n=\n\u00d7\n+\n[\n]\ncos\ncos(2\n)\n2\nm m\nv i\nt\n\u03c6\n\u03c9\n\u03c6\n=\n\u2212\n+\n(7 29)\nThe average power over a cycle is given by the average of the two terms in\nR"}, {"Chapter": "1", "sentence_range": "5839-5842", "Text": "6\n7 7  POWER IN AC CIRCUIT: THE POWER FACTOR\nWe have seen that a voltage v = vm sinwt applied to a series RLC circuit\ndrives a current in the circuit given by i = im sin(wt + f) where\nm\nm\nv\ni\n=Z\n  and \u03c6 =\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\u2212\ntan 1\nX\nX\nR\nC\nL\nTherefore, the instantaneous power p supplied by the source is\nRationalised 2023-24\n191\nAlternating Current\n(\n) [\n]\nsin\nsin(\n)\nm\nm\np\nv i\nv\nt\ni\nt\n\u03c9\n\u03c9\n\u03c6\n=\n=\n\u00d7\n+\n[\n]\ncos\ncos(2\n)\n2\nm m\nv i\nt\n\u03c6\n\u03c9\n\u03c6\n=\n\u2212\n+\n(7 29)\nThe average power over a cycle is given by the average of the two terms in\nR H"}, {"Chapter": "1", "sentence_range": "5840-5843", "Text": "7  POWER IN AC CIRCUIT: THE POWER FACTOR\nWe have seen that a voltage v = vm sinwt applied to a series RLC circuit\ndrives a current in the circuit given by i = im sin(wt + f) where\nm\nm\nv\ni\n=Z\n  and \u03c6 =\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\u2212\ntan 1\nX\nX\nR\nC\nL\nTherefore, the instantaneous power p supplied by the source is\nRationalised 2023-24\n191\nAlternating Current\n(\n) [\n]\nsin\nsin(\n)\nm\nm\np\nv i\nv\nt\ni\nt\n\u03c9\n\u03c9\n\u03c6\n=\n=\n\u00d7\n+\n[\n]\ncos\ncos(2\n)\n2\nm m\nv i\nt\n\u03c6\n\u03c9\n\u03c6\n=\n\u2212\n+\n(7 29)\nThe average power over a cycle is given by the average of the two terms in\nR H S"}, {"Chapter": "1", "sentence_range": "5841-5844", "Text": "29)\nThe average power over a cycle is given by the average of the two terms in\nR H S of Eq"}, {"Chapter": "1", "sentence_range": "5842-5845", "Text": "H S of Eq (7"}, {"Chapter": "1", "sentence_range": "5843-5846", "Text": "S of Eq (7 37)"}, {"Chapter": "1", "sentence_range": "5844-5847", "Text": "of Eq (7 37) It is only the second term which is time-dependent"}, {"Chapter": "1", "sentence_range": "5845-5848", "Text": "(7 37) It is only the second term which is time-dependent Its average is zero (the positive half of the cosine cancels the negative\nhalf)"}, {"Chapter": "1", "sentence_range": "5846-5849", "Text": "37) It is only the second term which is time-dependent Its average is zero (the positive half of the cosine cancels the negative\nhalf) Therefore,\ncos\n2\nm m\nv i\nP\n\u03c6\n=\ncos\n2\n2\nm\nm\nv\ni\n\u03c6\n=\nV Icos\n\u03c6\n=\n[7"}, {"Chapter": "1", "sentence_range": "5847-5850", "Text": "It is only the second term which is time-dependent Its average is zero (the positive half of the cosine cancels the negative\nhalf) Therefore,\ncos\n2\nm m\nv i\nP\n\u03c6\n=\ncos\n2\n2\nm\nm\nv\ni\n\u03c6\n=\nV Icos\n\u03c6\n=\n[7 30(a)]\nThis can also be written as,\n2\ncos\nP\nI Z\n\u03c6\n=\n[7"}, {"Chapter": "1", "sentence_range": "5848-5851", "Text": "Its average is zero (the positive half of the cosine cancels the negative\nhalf) Therefore,\ncos\n2\nm m\nv i\nP\n\u03c6\n=\ncos\n2\n2\nm\nm\nv\ni\n\u03c6\n=\nV Icos\n\u03c6\n=\n[7 30(a)]\nThis can also be written as,\n2\ncos\nP\nI Z\n\u03c6\n=\n[7 30(b)]\nSo, the average power dissipated depends not only on the voltage and\ncurrent but also on the cosine of the phase angle f between them"}, {"Chapter": "1", "sentence_range": "5849-5852", "Text": "Therefore,\ncos\n2\nm m\nv i\nP\n\u03c6\n=\ncos\n2\n2\nm\nm\nv\ni\n\u03c6\n=\nV Icos\n\u03c6\n=\n[7 30(a)]\nThis can also be written as,\n2\ncos\nP\nI Z\n\u03c6\n=\n[7 30(b)]\nSo, the average power dissipated depends not only on the voltage and\ncurrent but also on the cosine of the phase angle f between them The\nquantity cosf is called the power factor"}, {"Chapter": "1", "sentence_range": "5850-5853", "Text": "30(a)]\nThis can also be written as,\n2\ncos\nP\nI Z\n\u03c6\n=\n[7 30(b)]\nSo, the average power dissipated depends not only on the voltage and\ncurrent but also on the cosine of the phase angle f between them The\nquantity cosf is called the power factor Let us discuss the following\ncases:\nCase (i) Resistive circuit: If the circuit contains only pure R, it is called\nresistive"}, {"Chapter": "1", "sentence_range": "5851-5854", "Text": "30(b)]\nSo, the average power dissipated depends not only on the voltage and\ncurrent but also on the cosine of the phase angle f between them The\nquantity cosf is called the power factor Let us discuss the following\ncases:\nCase (i) Resistive circuit: If the circuit contains only pure R, it is called\nresistive In that case f = 0, cos f = 1"}, {"Chapter": "1", "sentence_range": "5852-5855", "Text": "The\nquantity cosf is called the power factor Let us discuss the following\ncases:\nCase (i) Resistive circuit: If the circuit contains only pure R, it is called\nresistive In that case f = 0, cos f = 1 There is maximum power dissipation"}, {"Chapter": "1", "sentence_range": "5853-5856", "Text": "Let us discuss the following\ncases:\nCase (i) Resistive circuit: If the circuit contains only pure R, it is called\nresistive In that case f = 0, cos f = 1 There is maximum power dissipation Case (ii) Purely inductive or capacitive circuit: If the circuit contains\nonly an inductor or capacitor, we know that the phase difference between\nvoltage and current is p/2"}, {"Chapter": "1", "sentence_range": "5854-5857", "Text": "In that case f = 0, cos f = 1 There is maximum power dissipation Case (ii) Purely inductive or capacitive circuit: If the circuit contains\nonly an inductor or capacitor, we know that the phase difference between\nvoltage and current is p/2 Therefore, cos f = 0, and no power is dissipated\neven though a current is flowing in the circuit"}, {"Chapter": "1", "sentence_range": "5855-5858", "Text": "There is maximum power dissipation Case (ii) Purely inductive or capacitive circuit: If the circuit contains\nonly an inductor or capacitor, we know that the phase difference between\nvoltage and current is p/2 Therefore, cos f = 0, and no power is dissipated\neven though a current is flowing in the circuit This current is sometimes\nreferred to as wattless current"}, {"Chapter": "1", "sentence_range": "5856-5859", "Text": "Case (ii) Purely inductive or capacitive circuit: If the circuit contains\nonly an inductor or capacitor, we know that the phase difference between\nvoltage and current is p/2 Therefore, cos f = 0, and no power is dissipated\neven though a current is flowing in the circuit This current is sometimes\nreferred to as wattless current Case (iii) LCR series circuit: In an LCR series circuit, power dissipated is\ngiven by Eq"}, {"Chapter": "1", "sentence_range": "5857-5860", "Text": "Therefore, cos f = 0, and no power is dissipated\neven though a current is flowing in the circuit This current is sometimes\nreferred to as wattless current Case (iii) LCR series circuit: In an LCR series circuit, power dissipated is\ngiven by Eq (7"}, {"Chapter": "1", "sentence_range": "5858-5861", "Text": "This current is sometimes\nreferred to as wattless current Case (iii) LCR series circuit: In an LCR series circuit, power dissipated is\ngiven by Eq (7 30) where f = tan\u20131 (Xc \u2013 XL )/ R"}, {"Chapter": "1", "sentence_range": "5859-5862", "Text": "Case (iii) LCR series circuit: In an LCR series circuit, power dissipated is\ngiven by Eq (7 30) where f = tan\u20131 (Xc \u2013 XL )/ R So,  f may be non-zero  in\na RL or RC or RCL circuit"}, {"Chapter": "1", "sentence_range": "5860-5863", "Text": "(7 30) where f = tan\u20131 (Xc \u2013 XL )/ R So,  f may be non-zero  in\na RL or RC or RCL circuit Even in such cases, power is dissipated only in\nthe resistor"}, {"Chapter": "1", "sentence_range": "5861-5864", "Text": "30) where f = tan\u20131 (Xc \u2013 XL )/ R So,  f may be non-zero  in\na RL or RC or RCL circuit Even in such cases, power is dissipated only in\nthe resistor Case (iv) Power dissipated at resonance in LCR circuit: At resonance\nXc \u2013 XL= 0, and f = 0"}, {"Chapter": "1", "sentence_range": "5862-5865", "Text": "So,  f may be non-zero  in\na RL or RC or RCL circuit Even in such cases, power is dissipated only in\nthe resistor Case (iv) Power dissipated at resonance in LCR circuit: At resonance\nXc \u2013 XL= 0, and f = 0 Therefore, cosf = 1 and P = I 2Z = I 2 R"}, {"Chapter": "1", "sentence_range": "5863-5866", "Text": "Even in such cases, power is dissipated only in\nthe resistor Case (iv) Power dissipated at resonance in LCR circuit: At resonance\nXc \u2013 XL= 0, and f = 0 Therefore, cosf = 1 and P = I 2Z = I 2 R That is,\nmaximum power is dissipated in a circuit (through R) at resonance"}, {"Chapter": "1", "sentence_range": "5864-5867", "Text": "Case (iv) Power dissipated at resonance in LCR circuit: At resonance\nXc \u2013 XL= 0, and f = 0 Therefore, cosf = 1 and P = I 2Z = I 2 R That is,\nmaximum power is dissipated in a circuit (through R) at resonance Example 7"}, {"Chapter": "1", "sentence_range": "5865-5868", "Text": "Therefore, cosf = 1 and P = I 2Z = I 2 R That is,\nmaximum power is dissipated in a circuit (through R) at resonance Example 7 7 (a) For circuits used for transporting electric power, a\nlow power factor implies large power loss in transmission"}, {"Chapter": "1", "sentence_range": "5866-5869", "Text": "That is,\nmaximum power is dissipated in a circuit (through R) at resonance Example 7 7 (a) For circuits used for transporting electric power, a\nlow power factor implies large power loss in transmission Explain"}, {"Chapter": "1", "sentence_range": "5867-5870", "Text": "Example 7 7 (a) For circuits used for transporting electric power, a\nlow power factor implies large power loss in transmission Explain (b) Power factor can often be improved by the use of a capacitor of\nappropriate capacitance in the circuit"}, {"Chapter": "1", "sentence_range": "5868-5871", "Text": "7 (a) For circuits used for transporting electric power, a\nlow power factor implies large power loss in transmission Explain (b) Power factor can often be improved by the use of a capacitor of\nappropriate capacitance in the circuit Explain"}, {"Chapter": "1", "sentence_range": "5869-5872", "Text": "Explain (b) Power factor can often be improved by the use of a capacitor of\nappropriate capacitance in the circuit Explain Solution (a) We know that P = I V cosf where cosf is the power factor"}, {"Chapter": "1", "sentence_range": "5870-5873", "Text": "(b) Power factor can often be improved by the use of a capacitor of\nappropriate capacitance in the circuit Explain Solution (a) We know that P = I V cosf where cosf is the power factor To supply a given power at a given voltage, if cosf is small, we have to\nincrease current accordingly"}, {"Chapter": "1", "sentence_range": "5871-5874", "Text": "Explain Solution (a) We know that P = I V cosf where cosf is the power factor To supply a given power at a given voltage, if cosf is small, we have to\nincrease current accordingly But this will lead to large power loss\n(I2R) in transmission"}, {"Chapter": "1", "sentence_range": "5872-5875", "Text": "Solution (a) We know that P = I V cosf where cosf is the power factor To supply a given power at a given voltage, if cosf is small, we have to\nincrease current accordingly But this will lead to large power loss\n(I2R) in transmission (b)Suppose in a circuit, current I lags the voltage by an angle f"}, {"Chapter": "1", "sentence_range": "5873-5876", "Text": "To supply a given power at a given voltage, if cosf is small, we have to\nincrease current accordingly But this will lead to large power loss\n(I2R) in transmission (b)Suppose in a circuit, current I lags the voltage by an angle f Then\npower factor cosf =R/Z"}, {"Chapter": "1", "sentence_range": "5874-5877", "Text": "But this will lead to large power loss\n(I2R) in transmission (b)Suppose in a circuit, current I lags the voltage by an angle f Then\npower factor cosf =R/Z We can improve the power factor (tending to 1) by making Z tend to\nR"}, {"Chapter": "1", "sentence_range": "5875-5878", "Text": "(b)Suppose in a circuit, current I lags the voltage by an angle f Then\npower factor cosf =R/Z We can improve the power factor (tending to 1) by making Z tend to\nR Let us understand, with the help of a phasor diagram (Fig"}, {"Chapter": "1", "sentence_range": "5876-5879", "Text": "Then\npower factor cosf =R/Z We can improve the power factor (tending to 1) by making Z tend to\nR Let us understand, with the help of a phasor diagram (Fig 7"}, {"Chapter": "1", "sentence_range": "5877-5880", "Text": "We can improve the power factor (tending to 1) by making Z tend to\nR Let us understand, with the help of a phasor diagram (Fig 7 15)\n EXAMPLE 7"}, {"Chapter": "1", "sentence_range": "5878-5881", "Text": "Let us understand, with the help of a phasor diagram (Fig 7 15)\n EXAMPLE 7 7\nRationalised 2023-24\nPhysics\n192\nhow this can be achieved"}, {"Chapter": "1", "sentence_range": "5879-5882", "Text": "7 15)\n EXAMPLE 7 7\nRationalised 2023-24\nPhysics\n192\nhow this can be achieved Let us resolve I into two components"}, {"Chapter": "1", "sentence_range": "5880-5883", "Text": "15)\n EXAMPLE 7 7\nRationalised 2023-24\nPhysics\n192\nhow this can be achieved Let us resolve I into two components Ip\nalong the applied voltage V and Iq perpendicular to the applied\nvoltage"}, {"Chapter": "1", "sentence_range": "5881-5884", "Text": "7\nRationalised 2023-24\nPhysics\n192\nhow this can be achieved Let us resolve I into two components Ip\nalong the applied voltage V and Iq perpendicular to the applied\nvoltage Iq as you have learnt in Section 7"}, {"Chapter": "1", "sentence_range": "5882-5885", "Text": "Let us resolve I into two components Ip\nalong the applied voltage V and Iq perpendicular to the applied\nvoltage Iq as you have learnt in Section 7 7, is called the wattless\ncomponent since corresponding to this component of current, there\nis no power loss"}, {"Chapter": "1", "sentence_range": "5883-5886", "Text": "Ip\nalong the applied voltage V and Iq perpendicular to the applied\nvoltage Iq as you have learnt in Section 7 7, is called the wattless\ncomponent since corresponding to this component of current, there\nis no power loss IP  is known as the power component because it is\nin phase with the voltage and corresponds to power loss in the circuit"}, {"Chapter": "1", "sentence_range": "5884-5887", "Text": "Iq as you have learnt in Section 7 7, is called the wattless\ncomponent since corresponding to this component of current, there\nis no power loss IP  is known as the power component because it is\nin phase with the voltage and corresponds to power loss in the circuit It\u2019s clear from this analysis that if we want to improve power factor,\nwe must completely neutralize the lagging wattless current Iq by an\nequal leading wattless current I\u00a2q"}, {"Chapter": "1", "sentence_range": "5885-5888", "Text": "7, is called the wattless\ncomponent since corresponding to this component of current, there\nis no power loss IP  is known as the power component because it is\nin phase with the voltage and corresponds to power loss in the circuit It\u2019s clear from this analysis that if we want to improve power factor,\nwe must completely neutralize the lagging wattless current Iq by an\nequal leading wattless current I\u00a2q This can be done by connecting\na capacitor of appropriate value in parallel so that Iq and I\u00a2q cancel\neach other and P is effectively Ip V"}, {"Chapter": "1", "sentence_range": "5886-5889", "Text": "IP  is known as the power component because it is\nin phase with the voltage and corresponds to power loss in the circuit It\u2019s clear from this analysis that if we want to improve power factor,\nwe must completely neutralize the lagging wattless current Iq by an\nequal leading wattless current I\u00a2q This can be done by connecting\na capacitor of appropriate value in parallel so that Iq and I\u00a2q cancel\neach other and P is effectively Ip V Example 7"}, {"Chapter": "1", "sentence_range": "5887-5890", "Text": "It\u2019s clear from this analysis that if we want to improve power factor,\nwe must completely neutralize the lagging wattless current Iq by an\nequal leading wattless current I\u00a2q This can be done by connecting\na capacitor of appropriate value in parallel so that Iq and I\u00a2q cancel\neach other and P is effectively Ip V Example 7 8 A sinusoidal voltage of peak value 283 V and frequency\n50 Hz is applied to a series LCR circuit in which\nR = 3 W, L = 25"}, {"Chapter": "1", "sentence_range": "5888-5891", "Text": "This can be done by connecting\na capacitor of appropriate value in parallel so that Iq and I\u00a2q cancel\neach other and P is effectively Ip V Example 7 8 A sinusoidal voltage of peak value 283 V and frequency\n50 Hz is applied to a series LCR circuit in which\nR = 3 W, L = 25 48 mH, and C = 796 mF"}, {"Chapter": "1", "sentence_range": "5889-5892", "Text": "Example 7 8 A sinusoidal voltage of peak value 283 V and frequency\n50 Hz is applied to a series LCR circuit in which\nR = 3 W, L = 25 48 mH, and C = 796 mF Find (a) the impedance of the\ncircuit; (b) the phase difference between the voltage across the source\nand the current; (c) the power dissipated in the circuit; and (d) the\npower factor"}, {"Chapter": "1", "sentence_range": "5890-5893", "Text": "8 A sinusoidal voltage of peak value 283 V and frequency\n50 Hz is applied to a series LCR circuit in which\nR = 3 W, L = 25 48 mH, and C = 796 mF Find (a) the impedance of the\ncircuit; (b) the phase difference between the voltage across the source\nand the current; (c) the power dissipated in the circuit; and (d) the\npower factor Solution\n(a) To find the impedance of the circuit, we first calculate XL and XC"}, {"Chapter": "1", "sentence_range": "5891-5894", "Text": "48 mH, and C = 796 mF Find (a) the impedance of the\ncircuit; (b) the phase difference between the voltage across the source\nand the current; (c) the power dissipated in the circuit; and (d) the\npower factor Solution\n(a) To find the impedance of the circuit, we first calculate XL and XC XL = 2 pnL\n    = 2 \u00d7 3"}, {"Chapter": "1", "sentence_range": "5892-5895", "Text": "Find (a) the impedance of the\ncircuit; (b) the phase difference between the voltage across the source\nand the current; (c) the power dissipated in the circuit; and (d) the\npower factor Solution\n(a) To find the impedance of the circuit, we first calculate XL and XC XL = 2 pnL\n    = 2 \u00d7 3 14 \u00d7 50 \u00d7 25"}, {"Chapter": "1", "sentence_range": "5893-5896", "Text": "Solution\n(a) To find the impedance of the circuit, we first calculate XL and XC XL = 2 pnL\n    = 2 \u00d7 3 14 \u00d7 50 \u00d7 25 48 \u00d7 10\u20133 W = 8 W\n21\nXC\n\u03bdC\n=\n\u03c0\n6\n1\n4\n2\n3"}, {"Chapter": "1", "sentence_range": "5894-5897", "Text": "XL = 2 pnL\n    = 2 \u00d7 3 14 \u00d7 50 \u00d7 25 48 \u00d7 10\u20133 W = 8 W\n21\nXC\n\u03bdC\n=\n\u03c0\n6\n1\n4\n2\n3 14\n50\n796\n10\u2212\n=\n=\n\u2126\n\u00d7\n\u00d7\n\u00d7\n\u00d7\nTherefore,\n2\n2\n2\n2\n(\n)\n3\n(8\n4)\nL\nC\nZ\nR\nX\nX\n=\n+\n\u2212\n=\n+\n\u2212\n    = 5 W\n(b) Phase difference, f = tan\u20131\nC\nL\nX\nX\nR\n\u2212\n=\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 = \u2212\n\u00b0\ntan\u2212"}, {"Chapter": "1", "sentence_range": "5895-5898", "Text": "14 \u00d7 50 \u00d7 25 48 \u00d7 10\u20133 W = 8 W\n21\nXC\n\u03bdC\n=\n\u03c0\n6\n1\n4\n2\n3 14\n50\n796\n10\u2212\n=\n=\n\u2126\n\u00d7\n\u00d7\n\u00d7\n\u00d7\nTherefore,\n2\n2\n2\n2\n(\n)\n3\n(8\n4)\nL\nC\nZ\nR\nX\nX\n=\n+\n\u2212\n=\n+\n\u2212\n    = 5 W\n(b) Phase difference, f = tan\u20131\nC\nL\nX\nX\nR\n\u2212\n=\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 = \u2212\n\u00b0\ntan\u2212 1\n4\n38\n53 1\n EXAMPLE 7"}, {"Chapter": "1", "sentence_range": "5896-5899", "Text": "48 \u00d7 10\u20133 W = 8 W\n21\nXC\n\u03bdC\n=\n\u03c0\n6\n1\n4\n2\n3 14\n50\n796\n10\u2212\n=\n=\n\u2126\n\u00d7\n\u00d7\n\u00d7\n\u00d7\nTherefore,\n2\n2\n2\n2\n(\n)\n3\n(8\n4)\nL\nC\nZ\nR\nX\nX\n=\n+\n\u2212\n=\n+\n\u2212\n    = 5 W\n(b) Phase difference, f = tan\u20131\nC\nL\nX\nX\nR\n\u2212\n=\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 = \u2212\n\u00b0\ntan\u2212 1\n4\n38\n53 1\n EXAMPLE 7 8\nFIGURE 7"}, {"Chapter": "1", "sentence_range": "5897-5900", "Text": "14\n50\n796\n10\u2212\n=\n=\n\u2126\n\u00d7\n\u00d7\n\u00d7\n\u00d7\nTherefore,\n2\n2\n2\n2\n(\n)\n3\n(8\n4)\nL\nC\nZ\nR\nX\nX\n=\n+\n\u2212\n=\n+\n\u2212\n    = 5 W\n(b) Phase difference, f = tan\u20131\nC\nL\nX\nX\nR\n\u2212\n=\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 = \u2212\n\u00b0\ntan\u2212 1\n4\n38\n53 1\n EXAMPLE 7 8\nFIGURE 7 15\n EXAMPLE 7"}, {"Chapter": "1", "sentence_range": "5898-5901", "Text": "1\n4\n38\n53 1\n EXAMPLE 7 8\nFIGURE 7 15\n EXAMPLE 7 7\nRationalised 2023-24\n193\nAlternating Current\n EXAMPLE 7"}, {"Chapter": "1", "sentence_range": "5899-5902", "Text": "8\nFIGURE 7 15\n EXAMPLE 7 7\nRationalised 2023-24\n193\nAlternating Current\n EXAMPLE 7 9\nSince f is negative, the current in the circuit lags the voltage\nacross the source"}, {"Chapter": "1", "sentence_range": "5900-5903", "Text": "15\n EXAMPLE 7 7\nRationalised 2023-24\n193\nAlternating Current\n EXAMPLE 7 9\nSince f is negative, the current in the circuit lags the voltage\nacross the source (c) The power dissipated in the circuit is\n2\nP\nI R\n=\nNow, I\n=im\n=\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 =\n2\n1\n2\n283\n5\n40A\nTherefore, \nA\nW\n(40 )2\n3\n4800\nP =\n\u00d7\n\u2126 =\n(d) Power factor =\n\ufffd\n\ufffd\ncos\ncos \u201353"}, {"Chapter": "1", "sentence_range": "5901-5904", "Text": "7\nRationalised 2023-24\n193\nAlternating Current\n EXAMPLE 7 9\nSince f is negative, the current in the circuit lags the voltage\nacross the source (c) The power dissipated in the circuit is\n2\nP\nI R\n=\nNow, I\n=im\n=\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 =\n2\n1\n2\n283\n5\n40A\nTherefore, \nA\nW\n(40 )2\n3\n4800\nP =\n\u00d7\n\u2126 =\n(d) Power factor =\n\ufffd\n\ufffd\ncos\ncos \u201353 1\n0"}, {"Chapter": "1", "sentence_range": "5902-5905", "Text": "9\nSince f is negative, the current in the circuit lags the voltage\nacross the source (c) The power dissipated in the circuit is\n2\nP\nI R\n=\nNow, I\n=im\n=\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 =\n2\n1\n2\n283\n5\n40A\nTherefore, \nA\nW\n(40 )2\n3\n4800\nP =\n\u00d7\n\u2126 =\n(d) Power factor =\n\ufffd\n\ufffd\ncos\ncos \u201353 1\n0 6\n\ufffd \ufffd\n\ufffd \ufffd\nExample 7"}, {"Chapter": "1", "sentence_range": "5903-5906", "Text": "(c) The power dissipated in the circuit is\n2\nP\nI R\n=\nNow, I\n=im\n=\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 =\n2\n1\n2\n283\n5\n40A\nTherefore, \nA\nW\n(40 )2\n3\n4800\nP =\n\u00d7\n\u2126 =\n(d) Power factor =\n\ufffd\n\ufffd\ncos\ncos \u201353 1\n0 6\n\ufffd \ufffd\n\ufffd \ufffd\nExample 7 9 Suppose the frequency of the source in the previous\nexample can be varied"}, {"Chapter": "1", "sentence_range": "5904-5907", "Text": "1\n0 6\n\ufffd \ufffd\n\ufffd \ufffd\nExample 7 9 Suppose the frequency of the source in the previous\nexample can be varied (a) What is the frequency of the source at\nwhich resonance occurs"}, {"Chapter": "1", "sentence_range": "5905-5908", "Text": "6\n\ufffd \ufffd\n\ufffd \ufffd\nExample 7 9 Suppose the frequency of the source in the previous\nexample can be varied (a) What is the frequency of the source at\nwhich resonance occurs (b) Calculate the impedance, the current,\nand the power dissipated at the resonant condition"}, {"Chapter": "1", "sentence_range": "5906-5909", "Text": "9 Suppose the frequency of the source in the previous\nexample can be varied (a) What is the frequency of the source at\nwhich resonance occurs (b) Calculate the impedance, the current,\nand the power dissipated at the resonant condition Solution\n(a) The frequency at which the resonance occurs is\n0\n3\n6\n1\n1\n25"}, {"Chapter": "1", "sentence_range": "5907-5910", "Text": "(a) What is the frequency of the source at\nwhich resonance occurs (b) Calculate the impedance, the current,\nand the power dissipated at the resonant condition Solution\n(a) The frequency at which the resonance occurs is\n0\n3\n6\n1\n1\n25 48\n10\n796\n10\nLC\n\u03c9\n\u2212\n\u2212\n=\n=\n\u00d7\n\u00d7\n\u00d7\n     \n222"}, {"Chapter": "1", "sentence_range": "5908-5911", "Text": "(b) Calculate the impedance, the current,\nand the power dissipated at the resonant condition Solution\n(a) The frequency at which the resonance occurs is\n0\n3\n6\n1\n1\n25 48\n10\n796\n10\nLC\n\u03c9\n\u2212\n\u2212\n=\n=\n\u00d7\n\u00d7\n\u00d7\n     \n222 1rad/s\n=\n0\n221"}, {"Chapter": "1", "sentence_range": "5909-5912", "Text": "Solution\n(a) The frequency at which the resonance occurs is\n0\n3\n6\n1\n1\n25 48\n10\n796\n10\nLC\n\u03c9\n\u2212\n\u2212\n=\n=\n\u00d7\n\u00d7\n\u00d7\n     \n222 1rad/s\n=\n0\n221 1 Hz\n35"}, {"Chapter": "1", "sentence_range": "5910-5913", "Text": "48\n10\n796\n10\nLC\n\u03c9\n\u2212\n\u2212\n=\n=\n\u00d7\n\u00d7\n\u00d7\n     \n222 1rad/s\n=\n0\n221 1 Hz\n35 4Hz\n2\n2\n3"}, {"Chapter": "1", "sentence_range": "5911-5914", "Text": "1rad/s\n=\n0\n221 1 Hz\n35 4Hz\n2\n2\n3 14\nr\n\u03bd =\u03c9\n=\n=\n\u03c0\n\u00d7\n(b) The impedance Z at resonant condition is equal to the resistance:\n3\nZ\n=R\n=\n\u2126\nThe rms current at resonance is\n=\n=\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\n=\nZV\nRV\n283\n2\n31\n66 7"}, {"Chapter": "1", "sentence_range": "5912-5915", "Text": "1 Hz\n35 4Hz\n2\n2\n3 14\nr\n\u03bd =\u03c9\n=\n=\n\u03c0\n\u00d7\n(b) The impedance Z at resonant condition is equal to the resistance:\n3\nZ\n=R\n=\n\u2126\nThe rms current at resonance is\n=\n=\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\n=\nZV\nRV\n283\n2\n31\n66 7 A\nThe power dissipated at resonance is\n2\n(66"}, {"Chapter": "1", "sentence_range": "5913-5916", "Text": "4Hz\n2\n2\n3 14\nr\n\u03bd =\u03c9\n=\n=\n\u03c0\n\u00d7\n(b) The impedance Z at resonant condition is equal to the resistance:\n3\nZ\n=R\n=\n\u2126\nThe rms current at resonance is\n=\n=\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\n=\nZV\nRV\n283\n2\n31\n66 7 A\nThe power dissipated at resonance is\n2\n(66 7)2\n3\n13"}, {"Chapter": "1", "sentence_range": "5914-5917", "Text": "14\nr\n\u03bd =\u03c9\n=\n=\n\u03c0\n\u00d7\n(b) The impedance Z at resonant condition is equal to the resistance:\n3\nZ\n=R\n=\n\u2126\nThe rms current at resonance is\n=\n=\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\n=\nZV\nRV\n283\n2\n31\n66 7 A\nThe power dissipated at resonance is\n2\n(66 7)2\n3\n13 35 kW\nP\nI\nR\n=\n\u00d7\n=\n\u00d7\n=\nYou can see that in the present case, power dissipated\nat resonance is more than the power dissipated in Example 7"}, {"Chapter": "1", "sentence_range": "5915-5918", "Text": "A\nThe power dissipated at resonance is\n2\n(66 7)2\n3\n13 35 kW\nP\nI\nR\n=\n\u00d7\n=\n\u00d7\n=\nYou can see that in the present case, power dissipated\nat resonance is more than the power dissipated in Example 7 8"}, {"Chapter": "1", "sentence_range": "5916-5919", "Text": "7)2\n3\n13 35 kW\nP\nI\nR\n=\n\u00d7\n=\n\u00d7\n=\nYou can see that in the present case, power dissipated\nat resonance is more than the power dissipated in Example 7 8 EXAMPLE 7"}, {"Chapter": "1", "sentence_range": "5917-5920", "Text": "35 kW\nP\nI\nR\n=\n\u00d7\n=\n\u00d7\n=\nYou can see that in the present case, power dissipated\nat resonance is more than the power dissipated in Example 7 8 EXAMPLE 7 8\n EXAMPLE 7"}, {"Chapter": "1", "sentence_range": "5918-5921", "Text": "8 EXAMPLE 7 8\n EXAMPLE 7 10\nExample 7"}, {"Chapter": "1", "sentence_range": "5919-5922", "Text": "EXAMPLE 7 8\n EXAMPLE 7 10\nExample 7 10 At an airport, a person is made to walk through the\ndoorway of a metal detector, for security reasons"}, {"Chapter": "1", "sentence_range": "5920-5923", "Text": "8\n EXAMPLE 7 10\nExample 7 10 At an airport, a person is made to walk through the\ndoorway of a metal detector, for security reasons If she/he is carrying\nanything made of metal, the metal detector emits a sound"}, {"Chapter": "1", "sentence_range": "5921-5924", "Text": "10\nExample 7 10 At an airport, a person is made to walk through the\ndoorway of a metal detector, for security reasons If she/he is carrying\nanything made of metal, the metal detector emits a sound On what\nprinciple does this detector work"}, {"Chapter": "1", "sentence_range": "5922-5925", "Text": "10 At an airport, a person is made to walk through the\ndoorway of a metal detector, for security reasons If she/he is carrying\nanything made of metal, the metal detector emits a sound On what\nprinciple does this detector work Solution  The metal detector works on the principle of resonance in\nac circuits"}, {"Chapter": "1", "sentence_range": "5923-5926", "Text": "If she/he is carrying\nanything made of metal, the metal detector emits a sound On what\nprinciple does this detector work Solution  The metal detector works on the principle of resonance in\nac circuits When you walk through a metal detector, you are,\nin fact, walking through a coil of many turns"}, {"Chapter": "1", "sentence_range": "5924-5927", "Text": "On what\nprinciple does this detector work Solution  The metal detector works on the principle of resonance in\nac circuits When you walk through a metal detector, you are,\nin fact, walking through a coil of many turns The coil is connected to\na capacitor tuned so that the circuit is in resonance"}, {"Chapter": "1", "sentence_range": "5925-5928", "Text": "Solution  The metal detector works on the principle of resonance in\nac circuits When you walk through a metal detector, you are,\nin fact, walking through a coil of many turns The coil is connected to\na capacitor tuned so that the circuit is in resonance When\nyou walk through with metal in your pocket, the impedance of the\ncircuit changes \u2013 resulting in significant change in current in the\ncircuit"}, {"Chapter": "1", "sentence_range": "5926-5929", "Text": "When you walk through a metal detector, you are,\nin fact, walking through a coil of many turns The coil is connected to\na capacitor tuned so that the circuit is in resonance When\nyou walk through with metal in your pocket, the impedance of the\ncircuit changes \u2013 resulting in significant change in current in the\ncircuit This change in current is detected and the electronic circuitry\ncauses a sound to be emitted as an alarm"}, {"Chapter": "1", "sentence_range": "5927-5930", "Text": "The coil is connected to\na capacitor tuned so that the circuit is in resonance When\nyou walk through with metal in your pocket, the impedance of the\ncircuit changes \u2013 resulting in significant change in current in the\ncircuit This change in current is detected and the electronic circuitry\ncauses a sound to be emitted as an alarm Rationalised 2023-24\nPhysics\n194\n7"}, {"Chapter": "1", "sentence_range": "5928-5931", "Text": "When\nyou walk through with metal in your pocket, the impedance of the\ncircuit changes \u2013 resulting in significant change in current in the\ncircuit This change in current is detected and the electronic circuitry\ncauses a sound to be emitted as an alarm Rationalised 2023-24\nPhysics\n194\n7 8  TRANSFORMERS\nFor many purposes, it is necessary to change (or transform) an alternating\nvoltage from one to another of greater or smaller value"}, {"Chapter": "1", "sentence_range": "5929-5932", "Text": "This change in current is detected and the electronic circuitry\ncauses a sound to be emitted as an alarm Rationalised 2023-24\nPhysics\n194\n7 8  TRANSFORMERS\nFor many purposes, it is necessary to change (or transform) an alternating\nvoltage from one to another of greater or smaller value This is done with\na device called transformer using the principle of mutual induction"}, {"Chapter": "1", "sentence_range": "5930-5933", "Text": "Rationalised 2023-24\nPhysics\n194\n7 8  TRANSFORMERS\nFor many purposes, it is necessary to change (or transform) an alternating\nvoltage from one to another of greater or smaller value This is done with\na device called transformer using the principle of mutual induction A transformer consists of two sets of coils, insulated from each other"}, {"Chapter": "1", "sentence_range": "5931-5934", "Text": "8  TRANSFORMERS\nFor many purposes, it is necessary to change (or transform) an alternating\nvoltage from one to another of greater or smaller value This is done with\na device called transformer using the principle of mutual induction A transformer consists of two sets of coils, insulated from each other They are wound on a soft-iron core, either one on top of the other as in\nFig"}, {"Chapter": "1", "sentence_range": "5932-5935", "Text": "This is done with\na device called transformer using the principle of mutual induction A transformer consists of two sets of coils, insulated from each other They are wound on a soft-iron core, either one on top of the other as in\nFig 7"}, {"Chapter": "1", "sentence_range": "5933-5936", "Text": "A transformer consists of two sets of coils, insulated from each other They are wound on a soft-iron core, either one on top of the other as in\nFig 7 16(a) or on separate limbs of the core as in Fig"}, {"Chapter": "1", "sentence_range": "5934-5937", "Text": "They are wound on a soft-iron core, either one on top of the other as in\nFig 7 16(a) or on separate limbs of the core as in Fig 7"}, {"Chapter": "1", "sentence_range": "5935-5938", "Text": "7 16(a) or on separate limbs of the core as in Fig 7 16(b)"}, {"Chapter": "1", "sentence_range": "5936-5939", "Text": "16(a) or on separate limbs of the core as in Fig 7 16(b) One of the\ncoils called the primary coil has Np turns"}, {"Chapter": "1", "sentence_range": "5937-5940", "Text": "7 16(b) One of the\ncoils called the primary coil has Np turns The other coil is called the\nsecondary coil; it has Ns turns"}, {"Chapter": "1", "sentence_range": "5938-5941", "Text": "16(b) One of the\ncoils called the primary coil has Np turns The other coil is called the\nsecondary coil; it has Ns turns Often the primary coil is the input coil\nand the secondary coil is the output coil of the transformer"}, {"Chapter": "1", "sentence_range": "5939-5942", "Text": "One of the\ncoils called the primary coil has Np turns The other coil is called the\nsecondary coil; it has Ns turns Often the primary coil is the input coil\nand the secondary coil is the output coil of the transformer FIGURE 7"}, {"Chapter": "1", "sentence_range": "5940-5943", "Text": "The other coil is called the\nsecondary coil; it has Ns turns Often the primary coil is the input coil\nand the secondary coil is the output coil of the transformer FIGURE 7 16 Two arrangements for winding of primary and secondary coil in a transformer:\n(a) two coils on top of each other, (b) two coils on separate limbs of the core"}, {"Chapter": "1", "sentence_range": "5941-5944", "Text": "Often the primary coil is the input coil\nand the secondary coil is the output coil of the transformer FIGURE 7 16 Two arrangements for winding of primary and secondary coil in a transformer:\n(a) two coils on top of each other, (b) two coils on separate limbs of the core When an alternating voltage is applied to the  primary, the resulting\ncurrent produces an alternating magnetic flux which links the secondary\nand induces an emf in it"}, {"Chapter": "1", "sentence_range": "5942-5945", "Text": "FIGURE 7 16 Two arrangements for winding of primary and secondary coil in a transformer:\n(a) two coils on top of each other, (b) two coils on separate limbs of the core When an alternating voltage is applied to the  primary, the resulting\ncurrent produces an alternating magnetic flux which links the secondary\nand induces an emf in it The value of this emf depends on the number of\nturns in the secondary"}, {"Chapter": "1", "sentence_range": "5943-5946", "Text": "16 Two arrangements for winding of primary and secondary coil in a transformer:\n(a) two coils on top of each other, (b) two coils on separate limbs of the core When an alternating voltage is applied to the  primary, the resulting\ncurrent produces an alternating magnetic flux which links the secondary\nand induces an emf in it The value of this emf depends on the number of\nturns in the secondary We consider an ideal transformer in which the\nprimary has negligible resistance and all the flux in the core links both\nprimary and secondary windings"}, {"Chapter": "1", "sentence_range": "5944-5947", "Text": "When an alternating voltage is applied to the  primary, the resulting\ncurrent produces an alternating magnetic flux which links the secondary\nand induces an emf in it The value of this emf depends on the number of\nturns in the secondary We consider an ideal transformer in which the\nprimary has negligible resistance and all the flux in the core links both\nprimary and secondary windings Let f be the flux in each turn in the core\nat time t due to current in the primary when a voltage vp is applied to it"}, {"Chapter": "1", "sentence_range": "5945-5948", "Text": "The value of this emf depends on the number of\nturns in the secondary We consider an ideal transformer in which the\nprimary has negligible resistance and all the flux in the core links both\nprimary and secondary windings Let f be the flux in each turn in the core\nat time t due to current in the primary when a voltage vp is applied to it Then the induced emf or voltage es, in the secondary with Ns turns is\nd\nd\ns\nNs\n\u03c6t\n\u03b5\n= \u2212\n(7"}, {"Chapter": "1", "sentence_range": "5946-5949", "Text": "We consider an ideal transformer in which the\nprimary has negligible resistance and all the flux in the core links both\nprimary and secondary windings Let f be the flux in each turn in the core\nat time t due to current in the primary when a voltage vp is applied to it Then the induced emf or voltage es, in the secondary with Ns turns is\nd\nd\ns\nNs\n\u03c6t\n\u03b5\n= \u2212\n(7 31)\nThe alternating flux f also induces an emf, called back emf in the\nprimary"}, {"Chapter": "1", "sentence_range": "5947-5950", "Text": "Let f be the flux in each turn in the core\nat time t due to current in the primary when a voltage vp is applied to it Then the induced emf or voltage es, in the secondary with Ns turns is\nd\nd\ns\nNs\n\u03c6t\n\u03b5\n= \u2212\n(7 31)\nThe alternating flux f also induces an emf, called back emf in the\nprimary This is\nd\nd\np\nNp\n\u03c6t\n\u03b5\n= \u2212\n(7"}, {"Chapter": "1", "sentence_range": "5948-5951", "Text": "Then the induced emf or voltage es, in the secondary with Ns turns is\nd\nd\ns\nNs\n\u03c6t\n\u03b5\n= \u2212\n(7 31)\nThe alternating flux f also induces an emf, called back emf in the\nprimary This is\nd\nd\np\nNp\n\u03c6t\n\u03b5\n= \u2212\n(7 32)\nBut ep = vp"}, {"Chapter": "1", "sentence_range": "5949-5952", "Text": "31)\nThe alternating flux f also induces an emf, called back emf in the\nprimary This is\nd\nd\np\nNp\n\u03c6t\n\u03b5\n= \u2212\n(7 32)\nBut ep = vp If this were not so, the primary current would be infinite\nsince the  primary has zero resistance (as assumed)"}, {"Chapter": "1", "sentence_range": "5950-5953", "Text": "This is\nd\nd\np\nNp\n\u03c6t\n\u03b5\n= \u2212\n(7 32)\nBut ep = vp If this were not so, the primary current would be infinite\nsince the  primary has zero resistance (as assumed) If the secondary is\nan open circuit or the current taken from it is small, then to a good\napproximation\nes = vs\nRationalised 2023-24\n195\nAlternating Current\nwhere  vs is the voltage across the secondary"}, {"Chapter": "1", "sentence_range": "5951-5954", "Text": "32)\nBut ep = vp If this were not so, the primary current would be infinite\nsince the  primary has zero resistance (as assumed) If the secondary is\nan open circuit or the current taken from it is small, then to a good\napproximation\nes = vs\nRationalised 2023-24\n195\nAlternating Current\nwhere  vs is the voltage across the secondary Therefore, Eqs"}, {"Chapter": "1", "sentence_range": "5952-5955", "Text": "If this were not so, the primary current would be infinite\nsince the  primary has zero resistance (as assumed) If the secondary is\nan open circuit or the current taken from it is small, then to a good\napproximation\nes = vs\nRationalised 2023-24\n195\nAlternating Current\nwhere  vs is the voltage across the secondary Therefore, Eqs (7"}, {"Chapter": "1", "sentence_range": "5953-5956", "Text": "If the secondary is\nan open circuit or the current taken from it is small, then to a good\napproximation\nes = vs\nRationalised 2023-24\n195\nAlternating Current\nwhere  vs is the voltage across the secondary Therefore, Eqs (7 31) and\n(7"}, {"Chapter": "1", "sentence_range": "5954-5957", "Text": "Therefore, Eqs (7 31) and\n(7 32) can be written as\ns\ns\nd\nv\nN\nd t\n\u03c6\n= \u2212\n[7"}, {"Chapter": "1", "sentence_range": "5955-5958", "Text": "(7 31) and\n(7 32) can be written as\ns\ns\nd\nv\nN\nd t\n\u03c6\n= \u2212\n[7 31(a)]\np\np\nd\nv\nN\nd t\n\u03c6\n= \u2212\n[7"}, {"Chapter": "1", "sentence_range": "5956-5959", "Text": "31) and\n(7 32) can be written as\ns\ns\nd\nv\nN\nd t\n\u03c6\n= \u2212\n[7 31(a)]\np\np\nd\nv\nN\nd t\n\u03c6\n= \u2212\n[7 32(a)]\nFrom Eqs"}, {"Chapter": "1", "sentence_range": "5957-5960", "Text": "32) can be written as\ns\ns\nd\nv\nN\nd t\n\u03c6\n= \u2212\n[7 31(a)]\np\np\nd\nv\nN\nd t\n\u03c6\n= \u2212\n[7 32(a)]\nFrom Eqs [7"}, {"Chapter": "1", "sentence_range": "5958-5961", "Text": "31(a)]\np\np\nd\nv\nN\nd t\n\u03c6\n= \u2212\n[7 32(a)]\nFrom Eqs [7 31 (a)] and [7"}, {"Chapter": "1", "sentence_range": "5959-5962", "Text": "32(a)]\nFrom Eqs [7 31 (a)] and [7 32 (a)], we have\ns\ns\np\np\nv\nN\nv\n=N\n(7"}, {"Chapter": "1", "sentence_range": "5960-5963", "Text": "[7 31 (a)] and [7 32 (a)], we have\ns\ns\np\np\nv\nN\nv\n=N\n(7 33)\nNote that the above relation has been obtained using three\nassumptions: (i) the primary resistance and current are small; (ii) the\nsame flux links both the primary and the secondary as very little flux\nescapes from the core, and (iii) the secondary current is small"}, {"Chapter": "1", "sentence_range": "5961-5964", "Text": "31 (a)] and [7 32 (a)], we have\ns\ns\np\np\nv\nN\nv\n=N\n(7 33)\nNote that the above relation has been obtained using three\nassumptions: (i) the primary resistance and current are small; (ii) the\nsame flux links both the primary and the secondary as very little flux\nescapes from the core, and (iii) the secondary current is small If the transformer is assumed to be 100% efficient (no energy losses),\nthe power input is equal to the power output, and since p = i v,\nipvp = isvs\n(7"}, {"Chapter": "1", "sentence_range": "5962-5965", "Text": "32 (a)], we have\ns\ns\np\np\nv\nN\nv\n=N\n(7 33)\nNote that the above relation has been obtained using three\nassumptions: (i) the primary resistance and current are small; (ii) the\nsame flux links both the primary and the secondary as very little flux\nescapes from the core, and (iii) the secondary current is small If the transformer is assumed to be 100% efficient (no energy losses),\nthe power input is equal to the power output, and since p = i v,\nipvp = isvs\n(7 34)\nAlthough some energy is always lost, this is a good approximation,\nsince a well designed transformer may have an efficiency of more than\n95%"}, {"Chapter": "1", "sentence_range": "5963-5966", "Text": "33)\nNote that the above relation has been obtained using three\nassumptions: (i) the primary resistance and current are small; (ii) the\nsame flux links both the primary and the secondary as very little flux\nescapes from the core, and (iii) the secondary current is small If the transformer is assumed to be 100% efficient (no energy losses),\nthe power input is equal to the power output, and since p = i v,\nipvp = isvs\n(7 34)\nAlthough some energy is always lost, this is a good approximation,\nsince a well designed transformer may have an efficiency of more than\n95% Combining Eqs"}, {"Chapter": "1", "sentence_range": "5964-5967", "Text": "If the transformer is assumed to be 100% efficient (no energy losses),\nthe power input is equal to the power output, and since p = i v,\nipvp = isvs\n(7 34)\nAlthough some energy is always lost, this is a good approximation,\nsince a well designed transformer may have an efficiency of more than\n95% Combining Eqs (7"}, {"Chapter": "1", "sentence_range": "5965-5968", "Text": "34)\nAlthough some energy is always lost, this is a good approximation,\nsince a well designed transformer may have an efficiency of more than\n95% Combining Eqs (7 33) and (7"}, {"Chapter": "1", "sentence_range": "5966-5969", "Text": "Combining Eqs (7 33) and (7 34), we have\np\ns\ns\ns\np\np\ni\nv\nN\ni\nv\nN\n=\n=\n(7"}, {"Chapter": "1", "sentence_range": "5967-5970", "Text": "(7 33) and (7 34), we have\np\ns\ns\ns\np\np\ni\nv\nN\ni\nv\nN\n=\n=\n(7 35)\nSince i and v both oscillate with the same frequency as the ac source,\nEq"}, {"Chapter": "1", "sentence_range": "5968-5971", "Text": "33) and (7 34), we have\np\ns\ns\ns\np\np\ni\nv\nN\ni\nv\nN\n=\n=\n(7 35)\nSince i and v both oscillate with the same frequency as the ac source,\nEq (7"}, {"Chapter": "1", "sentence_range": "5969-5972", "Text": "34), we have\np\ns\ns\ns\np\np\ni\nv\nN\ni\nv\nN\n=\n=\n(7 35)\nSince i and v both oscillate with the same frequency as the ac source,\nEq (7 35) also gives the ratio of the amplitudes or rms values of\ncorresponding quantities"}, {"Chapter": "1", "sentence_range": "5970-5973", "Text": "35)\nSince i and v both oscillate with the same frequency as the ac source,\nEq (7 35) also gives the ratio of the amplitudes or rms values of\ncorresponding quantities Now, we can see how a transformer affects the voltage and current"}, {"Chapter": "1", "sentence_range": "5971-5974", "Text": "(7 35) also gives the ratio of the amplitudes or rms values of\ncorresponding quantities Now, we can see how a transformer affects the voltage and current We have:\nV\nNN\nV\ns\ns\np\np\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n   and  I\nN\nN\nI\ns\np\ns\np\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n(7"}, {"Chapter": "1", "sentence_range": "5972-5975", "Text": "35) also gives the ratio of the amplitudes or rms values of\ncorresponding quantities Now, we can see how a transformer affects the voltage and current We have:\nV\nNN\nV\ns\ns\np\np\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n   and  I\nN\nN\nI\ns\np\ns\np\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n(7 36)\nThat is, if the secondary coil has a greater number of turns than the\nprimary (Ns > Np), the voltage is stepped up (Vs > Vp)"}, {"Chapter": "1", "sentence_range": "5973-5976", "Text": "Now, we can see how a transformer affects the voltage and current We have:\nV\nNN\nV\ns\ns\np\np\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n   and  I\nN\nN\nI\ns\np\ns\np\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n(7 36)\nThat is, if the secondary coil has a greater number of turns than the\nprimary (Ns > Np), the voltage is stepped up (Vs > Vp) This type of\narrangement is called a step-up transformer"}, {"Chapter": "1", "sentence_range": "5974-5977", "Text": "We have:\nV\nNN\nV\ns\ns\np\np\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n   and  I\nN\nN\nI\ns\np\ns\np\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\n(7 36)\nThat is, if the secondary coil has a greater number of turns than the\nprimary (Ns > Np), the voltage is stepped up (Vs > Vp) This type of\narrangement is called a step-up transformer However, in this arrangement,\nthere is less current in the secondary than in the primary (Np/Ns < 1 and Is\n< Ip)"}, {"Chapter": "1", "sentence_range": "5975-5978", "Text": "36)\nThat is, if the secondary coil has a greater number of turns than the\nprimary (Ns > Np), the voltage is stepped up (Vs > Vp) This type of\narrangement is called a step-up transformer However, in this arrangement,\nthere is less current in the secondary than in the primary (Np/Ns < 1 and Is\n< Ip) For example, if the primary coil of a transformer has 100 turns and\nthe secondary has 200 turns,  Ns/Np = 2 and Np/Ns=1/2"}, {"Chapter": "1", "sentence_range": "5976-5979", "Text": "This type of\narrangement is called a step-up transformer However, in this arrangement,\nthere is less current in the secondary than in the primary (Np/Ns < 1 and Is\n< Ip) For example, if the primary coil of a transformer has 100 turns and\nthe secondary has 200 turns,  Ns/Np = 2 and Np/Ns=1/2 Thus, a 220V\ninput at 10A will step-up to 440 V output at 5"}, {"Chapter": "1", "sentence_range": "5977-5980", "Text": "However, in this arrangement,\nthere is less current in the secondary than in the primary (Np/Ns < 1 and Is\n< Ip) For example, if the primary coil of a transformer has 100 turns and\nthe secondary has 200 turns,  Ns/Np = 2 and Np/Ns=1/2 Thus, a 220V\ninput at 10A will step-up to 440 V output at 5 0 A"}, {"Chapter": "1", "sentence_range": "5978-5981", "Text": "For example, if the primary coil of a transformer has 100 turns and\nthe secondary has 200 turns,  Ns/Np = 2 and Np/Ns=1/2 Thus, a 220V\ninput at 10A will step-up to 440 V output at 5 0 A If the secondary coil has less turns than the primary (Ns < Np),\nwe have a step-down transformer"}, {"Chapter": "1", "sentence_range": "5979-5982", "Text": "Thus, a 220V\ninput at 10A will step-up to 440 V output at 5 0 A If the secondary coil has less turns than the primary (Ns < Np),\nwe have a step-down transformer In this case, Vs < Vp and Is > Ip"}, {"Chapter": "1", "sentence_range": "5980-5983", "Text": "0 A If the secondary coil has less turns than the primary (Ns < Np),\nwe have a step-down transformer In this case, Vs < Vp and Is > Ip That\nis, the voltage is stepped down, or reduced, and the current\nis increased"}, {"Chapter": "1", "sentence_range": "5981-5984", "Text": "If the secondary coil has less turns than the primary (Ns < Np),\nwe have a step-down transformer In this case, Vs < Vp and Is > Ip That\nis, the voltage is stepped down, or reduced, and the current\nis increased The equations obtained above apply to ideal transformers (without\nany energy losses)"}, {"Chapter": "1", "sentence_range": "5982-5985", "Text": "In this case, Vs < Vp and Is > Ip That\nis, the voltage is stepped down, or reduced, and the current\nis increased The equations obtained above apply to ideal transformers (without\nany energy losses) But in actual transformers, small energy losses do\noccur due to the following reasons:\n(i) Flux Leakage: There is always some flux leakage; that is, not all of\nthe flux due to primary passes through the secondary due to poor\nRationalised 2023-24\nPhysics\n196\nSUMMARY\n1"}, {"Chapter": "1", "sentence_range": "5983-5986", "Text": "That\nis, the voltage is stepped down, or reduced, and the current\nis increased The equations obtained above apply to ideal transformers (without\nany energy losses) But in actual transformers, small energy losses do\noccur due to the following reasons:\n(i) Flux Leakage: There is always some flux leakage; that is, not all of\nthe flux due to primary passes through the secondary due to poor\nRationalised 2023-24\nPhysics\n196\nSUMMARY\n1 An alternating voltage \nsin\n=\n\u03c9\nm\nv\nv\nt  applied to a resistor R drives a\ncurrent i = im sinwt in the resistor, \nm\nm\nv\ni\n=R"}, {"Chapter": "1", "sentence_range": "5984-5987", "Text": "The equations obtained above apply to ideal transformers (without\nany energy losses) But in actual transformers, small energy losses do\noccur due to the following reasons:\n(i) Flux Leakage: There is always some flux leakage; that is, not all of\nthe flux due to primary passes through the secondary due to poor\nRationalised 2023-24\nPhysics\n196\nSUMMARY\n1 An alternating voltage \nsin\n=\n\u03c9\nm\nv\nv\nt  applied to a resistor R drives a\ncurrent i = im sinwt in the resistor, \nm\nm\nv\ni\n=R The current is in phase with\nthe applied voltage"}, {"Chapter": "1", "sentence_range": "5985-5988", "Text": "But in actual transformers, small energy losses do\noccur due to the following reasons:\n(i) Flux Leakage: There is always some flux leakage; that is, not all of\nthe flux due to primary passes through the secondary due to poor\nRationalised 2023-24\nPhysics\n196\nSUMMARY\n1 An alternating voltage \nsin\n=\n\u03c9\nm\nv\nv\nt  applied to a resistor R drives a\ncurrent i = im sinwt in the resistor, \nm\nm\nv\ni\n=R The current is in phase with\nthe applied voltage 2"}, {"Chapter": "1", "sentence_range": "5986-5989", "Text": "An alternating voltage \nsin\n=\n\u03c9\nm\nv\nv\nt  applied to a resistor R drives a\ncurrent i = im sinwt in the resistor, \nm\nm\nv\ni\n=R The current is in phase with\nthe applied voltage 2 For an alternating current i = im sin wt passing through a resistor R, the\naverage power loss P (averaged over a cycle) due to joule heating is\n( 1/2 )i 2\nmR"}, {"Chapter": "1", "sentence_range": "5987-5990", "Text": "The current is in phase with\nthe applied voltage 2 For an alternating current i = im sin wt passing through a resistor R, the\naverage power loss P (averaged over a cycle) due to joule heating is\n( 1/2 )i 2\nmR To express it in the same form as the dc power (P = I 2R), a\nspecial value of current is used"}, {"Chapter": "1", "sentence_range": "5988-5991", "Text": "2 For an alternating current i = im sin wt passing through a resistor R, the\naverage power loss P (averaged over a cycle) due to joule heating is\n( 1/2 )i 2\nmR To express it in the same form as the dc power (P = I 2R), a\nspecial value of current is used It is called root mean square (rms)\ncurrent and is donoted by I:\n0"}, {"Chapter": "1", "sentence_range": "5989-5992", "Text": "For an alternating current i = im sin wt passing through a resistor R, the\naverage power loss P (averaged over a cycle) due to joule heating is\n( 1/2 )i 2\nmR To express it in the same form as the dc power (P = I 2R), a\nspecial value of current is used It is called root mean square (rms)\ncurrent and is donoted by I:\n0 707\n2\nm\nm\ni\nI\ni\n=\n=\nSimilarly, the rms voltage is defined by\n0"}, {"Chapter": "1", "sentence_range": "5990-5993", "Text": "To express it in the same form as the dc power (P = I 2R), a\nspecial value of current is used It is called root mean square (rms)\ncurrent and is donoted by I:\n0 707\n2\nm\nm\ni\nI\ni\n=\n=\nSimilarly, the rms voltage is defined by\n0 707\n2\nm\nm\nv\nV\nv\n=\n=\nWe have P = IV = I 2R\n3"}, {"Chapter": "1", "sentence_range": "5991-5994", "Text": "It is called root mean square (rms)\ncurrent and is donoted by I:\n0 707\n2\nm\nm\ni\nI\ni\n=\n=\nSimilarly, the rms voltage is defined by\n0 707\n2\nm\nm\nv\nV\nv\n=\n=\nWe have P = IV = I 2R\n3 An ac voltage v = vm sin wt applied to a pure inductor L, drives a current\nin the inductor i = im sin (wt \u2013 p/2), where im = vm/XL"}, {"Chapter": "1", "sentence_range": "5992-5995", "Text": "707\n2\nm\nm\ni\nI\ni\n=\n=\nSimilarly, the rms voltage is defined by\n0 707\n2\nm\nm\nv\nV\nv\n=\n=\nWe have P = IV = I 2R\n3 An ac voltage v = vm sin wt applied to a pure inductor L, drives a current\nin the inductor i = im sin (wt \u2013 p/2), where im = vm/XL XL = wL is called\ninductive reactance"}, {"Chapter": "1", "sentence_range": "5993-5996", "Text": "707\n2\nm\nm\nv\nV\nv\n=\n=\nWe have P = IV = I 2R\n3 An ac voltage v = vm sin wt applied to a pure inductor L, drives a current\nin the inductor i = im sin (wt \u2013 p/2), where im = vm/XL XL = wL is called\ninductive reactance The current in the inductor lags the voltage by\np/2"}, {"Chapter": "1", "sentence_range": "5994-5997", "Text": "An ac voltage v = vm sin wt applied to a pure inductor L, drives a current\nin the inductor i = im sin (wt \u2013 p/2), where im = vm/XL XL = wL is called\ninductive reactance The current in the inductor lags the voltage by\np/2 The average power supplied to an inductor over one complete cycle\nis zero"}, {"Chapter": "1", "sentence_range": "5995-5998", "Text": "XL = wL is called\ninductive reactance The current in the inductor lags the voltage by\np/2 The average power supplied to an inductor over one complete cycle\nis zero design of the core or the air gaps in the core"}, {"Chapter": "1", "sentence_range": "5996-5999", "Text": "The current in the inductor lags the voltage by\np/2 The average power supplied to an inductor over one complete cycle\nis zero design of the core or the air gaps in the core It can be reduced by\nwinding the primary and secondary coils one over the other"}, {"Chapter": "1", "sentence_range": "5997-6000", "Text": "The average power supplied to an inductor over one complete cycle\nis zero design of the core or the air gaps in the core It can be reduced by\nwinding the primary and secondary coils one over the other (ii) Resistance of the windings: The wire used for the windings has some\nresistance and so, energy is lost due to heat produced in the wire\n(I 2R)"}, {"Chapter": "1", "sentence_range": "5998-6001", "Text": "design of the core or the air gaps in the core It can be reduced by\nwinding the primary and secondary coils one over the other (ii) Resistance of the windings: The wire used for the windings has some\nresistance and so, energy is lost due to heat produced in the wire\n(I 2R) In high current, low voltage windings, these are minimised by\nusing thick wire"}, {"Chapter": "1", "sentence_range": "5999-6002", "Text": "It can be reduced by\nwinding the primary and secondary coils one over the other (ii) Resistance of the windings: The wire used for the windings has some\nresistance and so, energy is lost due to heat produced in the wire\n(I 2R) In high current, low voltage windings, these are minimised by\nusing thick wire (iii) Eddy currents: The alternating magnetic flux induces eddy currents\nin the iron core and causes heating"}, {"Chapter": "1", "sentence_range": "6000-6003", "Text": "(ii) Resistance of the windings: The wire used for the windings has some\nresistance and so, energy is lost due to heat produced in the wire\n(I 2R) In high current, low voltage windings, these are minimised by\nusing thick wire (iii) Eddy currents: The alternating magnetic flux induces eddy currents\nin the iron core and causes heating The effect is reduced by using a\nlaminated core"}, {"Chapter": "1", "sentence_range": "6001-6004", "Text": "In high current, low voltage windings, these are minimised by\nusing thick wire (iii) Eddy currents: The alternating magnetic flux induces eddy currents\nin the iron core and causes heating The effect is reduced by using a\nlaminated core (iv) Hysteresis: The magnetisation of the core is repeatedly reversed by\nthe alternating magnetic field"}, {"Chapter": "1", "sentence_range": "6002-6005", "Text": "(iii) Eddy currents: The alternating magnetic flux induces eddy currents\nin the iron core and causes heating The effect is reduced by using a\nlaminated core (iv) Hysteresis: The magnetisation of the core is repeatedly reversed by\nthe alternating magnetic field The resulting expenditure of energy in\nthe core appears as heat and is kept to a minimum by using a magnetic\nmaterial which has a low hysteresis loss"}, {"Chapter": "1", "sentence_range": "6003-6006", "Text": "The effect is reduced by using a\nlaminated core (iv) Hysteresis: The magnetisation of the core is repeatedly reversed by\nthe alternating magnetic field The resulting expenditure of energy in\nthe core appears as heat and is kept to a minimum by using a magnetic\nmaterial which has a low hysteresis loss The large scale transmission and distribution of electrical energy over\nlong distances is done with the use of transformers"}, {"Chapter": "1", "sentence_range": "6004-6007", "Text": "(iv) Hysteresis: The magnetisation of the core is repeatedly reversed by\nthe alternating magnetic field The resulting expenditure of energy in\nthe core appears as heat and is kept to a minimum by using a magnetic\nmaterial which has a low hysteresis loss The large scale transmission and distribution of electrical energy over\nlong distances is done with the use of transformers The voltage output\nof the generator is stepped-up (so that current is reduced and\nconsequently, the I 2R loss is cut down)"}, {"Chapter": "1", "sentence_range": "6005-6008", "Text": "The resulting expenditure of energy in\nthe core appears as heat and is kept to a minimum by using a magnetic\nmaterial which has a low hysteresis loss The large scale transmission and distribution of electrical energy over\nlong distances is done with the use of transformers The voltage output\nof the generator is stepped-up (so that current is reduced and\nconsequently, the I 2R loss is cut down) It is then transmitted over long\ndistances to an area sub-station near the consumers"}, {"Chapter": "1", "sentence_range": "6006-6009", "Text": "The large scale transmission and distribution of electrical energy over\nlong distances is done with the use of transformers The voltage output\nof the generator is stepped-up (so that current is reduced and\nconsequently, the I 2R loss is cut down) It is then transmitted over long\ndistances to an area sub-station near the consumers There the voltage\nis stepped down"}, {"Chapter": "1", "sentence_range": "6007-6010", "Text": "The voltage output\nof the generator is stepped-up (so that current is reduced and\nconsequently, the I 2R loss is cut down) It is then transmitted over long\ndistances to an area sub-station near the consumers There the voltage\nis stepped down It is further stepped down at distributing sub-stations\nand utility poles before a power supply of 240 V reaches our homes"}, {"Chapter": "1", "sentence_range": "6008-6011", "Text": "It is then transmitted over long\ndistances to an area sub-station near the consumers There the voltage\nis stepped down It is further stepped down at distributing sub-stations\nand utility poles before a power supply of 240 V reaches our homes Rationalised 2023-24\n197\nAlternating Current\n4"}, {"Chapter": "1", "sentence_range": "6009-6012", "Text": "There the voltage\nis stepped down It is further stepped down at distributing sub-stations\nand utility poles before a power supply of 240 V reaches our homes Rationalised 2023-24\n197\nAlternating Current\n4 An ac voltage v = vm sinwt applied to a capacitor drives a current in the\ncapacitor: i = im sin (wt + p/2)"}, {"Chapter": "1", "sentence_range": "6010-6013", "Text": "It is further stepped down at distributing sub-stations\nand utility poles before a power supply of 240 V reaches our homes Rationalised 2023-24\n197\nAlternating Current\n4 An ac voltage v = vm sinwt applied to a capacitor drives a current in the\ncapacitor: i = im sin (wt + p/2) Here,\n1\nm,\nm\nC\nC\nv\ni\nX\nX\n\u03c9C\n=\n=\n is called capacitive reactance"}, {"Chapter": "1", "sentence_range": "6011-6014", "Text": "Rationalised 2023-24\n197\nAlternating Current\n4 An ac voltage v = vm sinwt applied to a capacitor drives a current in the\ncapacitor: i = im sin (wt + p/2) Here,\n1\nm,\nm\nC\nC\nv\ni\nX\nX\n\u03c9C\n=\n=\n is called capacitive reactance The current through the capacitor is p/2 ahead of the applied voltage"}, {"Chapter": "1", "sentence_range": "6012-6015", "Text": "An ac voltage v = vm sinwt applied to a capacitor drives a current in the\ncapacitor: i = im sin (wt + p/2) Here,\n1\nm,\nm\nC\nC\nv\ni\nX\nX\n\u03c9C\n=\n=\n is called capacitive reactance The current through the capacitor is p/2 ahead of the applied voltage As in the case of inductor, the average power supplied to a capacitor\nover one complete cycle is zero"}, {"Chapter": "1", "sentence_range": "6013-6016", "Text": "Here,\n1\nm,\nm\nC\nC\nv\ni\nX\nX\n\u03c9C\n=\n=\n is called capacitive reactance The current through the capacitor is p/2 ahead of the applied voltage As in the case of inductor, the average power supplied to a capacitor\nover one complete cycle is zero 5"}, {"Chapter": "1", "sentence_range": "6014-6017", "Text": "The current through the capacitor is p/2 ahead of the applied voltage As in the case of inductor, the average power supplied to a capacitor\nover one complete cycle is zero 5 For a series RLC circuit driven by voltage v = vm sin wt, the current is\ngiven by i = im sin (wt + f)\nwhere\n(\n)\n2\n2\nm\nm\nC\nL\nv\ni\nR\nX\nX\n=\n+\n\u2212\nand\ntan1\nC\nL\nX\nX\nR\n\u03c6\n\u2212\n\u2212\n=\n(\n)\n2\n2\nC\nL\nZ\nR\nX\nX\n=\n+\n\u2212\n is called the impedance of the circuit"}, {"Chapter": "1", "sentence_range": "6015-6018", "Text": "As in the case of inductor, the average power supplied to a capacitor\nover one complete cycle is zero 5 For a series RLC circuit driven by voltage v = vm sin wt, the current is\ngiven by i = im sin (wt + f)\nwhere\n(\n)\n2\n2\nm\nm\nC\nL\nv\ni\nR\nX\nX\n=\n+\n\u2212\nand\ntan1\nC\nL\nX\nX\nR\n\u03c6\n\u2212\n\u2212\n=\n(\n)\n2\n2\nC\nL\nZ\nR\nX\nX\n=\n+\n\u2212\n is called the impedance of the circuit The average power loss over a complete cycle is given by\nP = V I cosf\nThe term cosf is called the power factor"}, {"Chapter": "1", "sentence_range": "6016-6019", "Text": "5 For a series RLC circuit driven by voltage v = vm sin wt, the current is\ngiven by i = im sin (wt + f)\nwhere\n(\n)\n2\n2\nm\nm\nC\nL\nv\ni\nR\nX\nX\n=\n+\n\u2212\nand\ntan1\nC\nL\nX\nX\nR\n\u03c6\n\u2212\n\u2212\n=\n(\n)\n2\n2\nC\nL\nZ\nR\nX\nX\n=\n+\n\u2212\n is called the impedance of the circuit The average power loss over a complete cycle is given by\nP = V I cosf\nThe term cosf is called the power factor 6"}, {"Chapter": "1", "sentence_range": "6017-6020", "Text": "For a series RLC circuit driven by voltage v = vm sin wt, the current is\ngiven by i = im sin (wt + f)\nwhere\n(\n)\n2\n2\nm\nm\nC\nL\nv\ni\nR\nX\nX\n=\n+\n\u2212\nand\ntan1\nC\nL\nX\nX\nR\n\u03c6\n\u2212\n\u2212\n=\n(\n)\n2\n2\nC\nL\nZ\nR\nX\nX\n=\n+\n\u2212\n is called the impedance of the circuit The average power loss over a complete cycle is given by\nP = V I cosf\nThe term cosf is called the power factor 6 In a purely inductive or capacitive circuit, cosf = 0 and no power is\ndissipated even though a current is flowing in the circuit"}, {"Chapter": "1", "sentence_range": "6018-6021", "Text": "The average power loss over a complete cycle is given by\nP = V I cosf\nThe term cosf is called the power factor 6 In a purely inductive or capacitive circuit, cosf = 0 and no power is\ndissipated even though a current is flowing in the circuit In such cases,\ncurrent is referred to as a wattless current"}, {"Chapter": "1", "sentence_range": "6019-6022", "Text": "6 In a purely inductive or capacitive circuit, cosf = 0 and no power is\ndissipated even though a current is flowing in the circuit In such cases,\ncurrent is referred to as a wattless current 7"}, {"Chapter": "1", "sentence_range": "6020-6023", "Text": "In a purely inductive or capacitive circuit, cosf = 0 and no power is\ndissipated even though a current is flowing in the circuit In such cases,\ncurrent is referred to as a wattless current 7 The phase relationship between current and voltage in an ac circuit\ncan be shown conveniently by representing voltage and current by\nrotating vectors called phasors"}, {"Chapter": "1", "sentence_range": "6021-6024", "Text": "In such cases,\ncurrent is referred to as a wattless current 7 The phase relationship between current and voltage in an ac circuit\ncan be shown conveniently by representing voltage and current by\nrotating vectors called phasors A phasor is a vector which rotates\nabout the origin with angular speed w"}, {"Chapter": "1", "sentence_range": "6022-6025", "Text": "7 The phase relationship between current and voltage in an ac circuit\ncan be shown conveniently by representing voltage and current by\nrotating vectors called phasors A phasor is a vector which rotates\nabout the origin with angular speed w The magnitude of a phasor\nrepresents the amplitude or peak value of the quantity (voltage or\ncurrent) represented by the phasor"}, {"Chapter": "1", "sentence_range": "6023-6026", "Text": "The phase relationship between current and voltage in an ac circuit\ncan be shown conveniently by representing voltage and current by\nrotating vectors called phasors A phasor is a vector which rotates\nabout the origin with angular speed w The magnitude of a phasor\nrepresents the amplitude or peak value of the quantity (voltage or\ncurrent) represented by the phasor The analysis of an ac circuit is facilitated by the use of a phasor\ndiagram"}, {"Chapter": "1", "sentence_range": "6024-6027", "Text": "A phasor is a vector which rotates\nabout the origin with angular speed w The magnitude of a phasor\nrepresents the amplitude or peak value of the quantity (voltage or\ncurrent) represented by the phasor The analysis of an ac circuit is facilitated by the use of a phasor\ndiagram 8"}, {"Chapter": "1", "sentence_range": "6025-6028", "Text": "The magnitude of a phasor\nrepresents the amplitude or peak value of the quantity (voltage or\ncurrent) represented by the phasor The analysis of an ac circuit is facilitated by the use of a phasor\ndiagram 8 A transformer consists of an iron core on which are bound a primary\ncoil of Np turns and a secondary coil of Ns turns"}, {"Chapter": "1", "sentence_range": "6026-6029", "Text": "The analysis of an ac circuit is facilitated by the use of a phasor\ndiagram 8 A transformer consists of an iron core on which are bound a primary\ncoil of Np turns and a secondary coil of Ns turns If the primary coil is\nconnected to an ac source, the primary and secondary voltages are\nrelated by\nV\nNN\nV\ns\ns\np\np\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\nand the currents are related by\nI\nN\nN\nI\ns\np\ns\np\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\nIf the secondary coil has a greater number of turns than the primary, the\nvoltage is stepped-up (Vs > Vp)"}, {"Chapter": "1", "sentence_range": "6027-6030", "Text": "8 A transformer consists of an iron core on which are bound a primary\ncoil of Np turns and a secondary coil of Ns turns If the primary coil is\nconnected to an ac source, the primary and secondary voltages are\nrelated by\nV\nNN\nV\ns\ns\np\np\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\nand the currents are related by\nI\nN\nN\nI\ns\np\ns\np\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\nIf the secondary coil has a greater number of turns than the primary, the\nvoltage is stepped-up (Vs > Vp) This type of arrangement is called a step-\nup transformer"}, {"Chapter": "1", "sentence_range": "6028-6031", "Text": "A transformer consists of an iron core on which are bound a primary\ncoil of Np turns and a secondary coil of Ns turns If the primary coil is\nconnected to an ac source, the primary and secondary voltages are\nrelated by\nV\nNN\nV\ns\ns\np\np\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\nand the currents are related by\nI\nN\nN\nI\ns\np\ns\np\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\nIf the secondary coil has a greater number of turns than the primary, the\nvoltage is stepped-up (Vs > Vp) This type of arrangement is called a step-\nup transformer If the secondary coil has turns less than the primary, we\nhave a step-down transformer"}, {"Chapter": "1", "sentence_range": "6029-6032", "Text": "If the primary coil is\nconnected to an ac source, the primary and secondary voltages are\nrelated by\nV\nNN\nV\ns\ns\np\np\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\nand the currents are related by\nI\nN\nN\nI\ns\np\ns\np\n= \uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\uf8f8\uf8f7\nIf the secondary coil has a greater number of turns than the primary, the\nvoltage is stepped-up (Vs > Vp) This type of arrangement is called a step-\nup transformer If the secondary coil has turns less than the primary, we\nhave a step-down transformer Rationalised 2023-24\nPhysics\n198\n Physical quantity\nSymbol\nDimensions\nUnit\nRemarks\nrms voltage\nV\n[M L\n2 T\n\u20133 A\n\u20131]\nV\nV \n= \n2\nvm\n, \nv m \nis \nthe\namplitude of the ac voltage"}, {"Chapter": "1", "sentence_range": "6030-6033", "Text": "This type of arrangement is called a step-\nup transformer If the secondary coil has turns less than the primary, we\nhave a step-down transformer Rationalised 2023-24\nPhysics\n198\n Physical quantity\nSymbol\nDimensions\nUnit\nRemarks\nrms voltage\nV\n[M L\n2 T\n\u20133 A\n\u20131]\nV\nV \n= \n2\nvm\n, \nv m \nis \nthe\namplitude of the ac voltage rms current\nI\n[\n A]\nA\nI = \n2\nim\n, im is the amplitude of\nthe ac current"}, {"Chapter": "1", "sentence_range": "6031-6034", "Text": "If the secondary coil has turns less than the primary, we\nhave a step-down transformer Rationalised 2023-24\nPhysics\n198\n Physical quantity\nSymbol\nDimensions\nUnit\nRemarks\nrms voltage\nV\n[M L\n2 T\n\u20133 A\n\u20131]\nV\nV \n= \n2\nvm\n, \nv m \nis \nthe\namplitude of the ac voltage rms current\nI\n[\n A]\nA\nI = \n2\nim\n, im is the amplitude of\nthe ac current Reactance:\n     Inductive\nXL\n[M\n L\n2 T\n\u20133 A\n\u20132]\nXL = \uf077 L\n     Capacitive\nXC\n[M\n L\n2 T\n\u20133 A\n\u20132]\nXC = 1/\uf077 C\nImpedance\nZ\n[M\n L\n2 T\n\u20133 A\n\u20132]\nDepends \non \nelements\npresent in the circuit"}, {"Chapter": "1", "sentence_range": "6032-6035", "Text": "Rationalised 2023-24\nPhysics\n198\n Physical quantity\nSymbol\nDimensions\nUnit\nRemarks\nrms voltage\nV\n[M L\n2 T\n\u20133 A\n\u20131]\nV\nV \n= \n2\nvm\n, \nv m \nis \nthe\namplitude of the ac voltage rms current\nI\n[\n A]\nA\nI = \n2\nim\n, im is the amplitude of\nthe ac current Reactance:\n     Inductive\nXL\n[M\n L\n2 T\n\u20133 A\n\u20132]\nXL = \uf077 L\n     Capacitive\nXC\n[M\n L\n2 T\n\u20133 A\n\u20132]\nXC = 1/\uf077 C\nImpedance\nZ\n[M\n L\n2 T\n\u20133 A\n\u20132]\nDepends \non \nelements\npresent in the circuit Resonant\nwr or w0\n[T\n\u20131]\nHz\nw0 \nLC\n1\n\uf03d\n for a\nfrequency\nseries RLC circuit\nQuality factor\nQ\nDimensionless\n0\n0\n1\nL\nQ\nR\nC R\n\u03c9\n\u03c9\n=\n=\n for a series\nRLC circuit"}, {"Chapter": "1", "sentence_range": "6033-6036", "Text": "rms current\nI\n[\n A]\nA\nI = \n2\nim\n, im is the amplitude of\nthe ac current Reactance:\n     Inductive\nXL\n[M\n L\n2 T\n\u20133 A\n\u20132]\nXL = \uf077 L\n     Capacitive\nXC\n[M\n L\n2 T\n\u20133 A\n\u20132]\nXC = 1/\uf077 C\nImpedance\nZ\n[M\n L\n2 T\n\u20133 A\n\u20132]\nDepends \non \nelements\npresent in the circuit Resonant\nwr or w0\n[T\n\u20131]\nHz\nw0 \nLC\n1\n\uf03d\n for a\nfrequency\nseries RLC circuit\nQuality factor\nQ\nDimensionless\n0\n0\n1\nL\nQ\nR\nC R\n\u03c9\n\u03c9\n=\n=\n for a series\nRLC circuit Power factor\nDimensionless\n= \ncosf, \nf \nis \nthe \nphase\ndifference \nbetween \nvoltage\napplied \nand \ncurrent \nin\nthe circuit"}, {"Chapter": "1", "sentence_range": "6034-6037", "Text": "Reactance:\n     Inductive\nXL\n[M\n L\n2 T\n\u20133 A\n\u20132]\nXL = \uf077 L\n     Capacitive\nXC\n[M\n L\n2 T\n\u20133 A\n\u20132]\nXC = 1/\uf077 C\nImpedance\nZ\n[M\n L\n2 T\n\u20133 A\n\u20132]\nDepends \non \nelements\npresent in the circuit Resonant\nwr or w0\n[T\n\u20131]\nHz\nw0 \nLC\n1\n\uf03d\n for a\nfrequency\nseries RLC circuit\nQuality factor\nQ\nDimensionless\n0\n0\n1\nL\nQ\nR\nC R\n\u03c9\n\u03c9\n=\n=\n for a series\nRLC circuit Power factor\nDimensionless\n= \ncosf, \nf \nis \nthe \nphase\ndifference \nbetween \nvoltage\napplied \nand \ncurrent \nin\nthe circuit POINTS TO PONDER\n1"}, {"Chapter": "1", "sentence_range": "6035-6038", "Text": "Resonant\nwr or w0\n[T\n\u20131]\nHz\nw0 \nLC\n1\n\uf03d\n for a\nfrequency\nseries RLC circuit\nQuality factor\nQ\nDimensionless\n0\n0\n1\nL\nQ\nR\nC R\n\u03c9\n\u03c9\n=\n=\n for a series\nRLC circuit Power factor\nDimensionless\n= \ncosf, \nf \nis \nthe \nphase\ndifference \nbetween \nvoltage\napplied \nand \ncurrent \nin\nthe circuit POINTS TO PONDER\n1 When a value is given for ac voltage or current, it is ordinarily the rms\nvalue"}, {"Chapter": "1", "sentence_range": "6036-6039", "Text": "Power factor\nDimensionless\n= \ncosf, \nf \nis \nthe \nphase\ndifference \nbetween \nvoltage\napplied \nand \ncurrent \nin\nthe circuit POINTS TO PONDER\n1 When a value is given for ac voltage or current, it is ordinarily the rms\nvalue The voltage across the terminals of an outlet in your room is\nnormally 240 V"}, {"Chapter": "1", "sentence_range": "6037-6040", "Text": "POINTS TO PONDER\n1 When a value is given for ac voltage or current, it is ordinarily the rms\nvalue The voltage across the terminals of an outlet in your room is\nnormally 240 V This refers to the rms value of the voltage"}, {"Chapter": "1", "sentence_range": "6038-6041", "Text": "When a value is given for ac voltage or current, it is ordinarily the rms\nvalue The voltage across the terminals of an outlet in your room is\nnormally 240 V This refers to the rms value of the voltage The amplitude\nof this voltage is\nV\n2\n2(240)\n340\nvm\nV\n=\n=\n=\n2"}, {"Chapter": "1", "sentence_range": "6039-6042", "Text": "The voltage across the terminals of an outlet in your room is\nnormally 240 V This refers to the rms value of the voltage The amplitude\nof this voltage is\nV\n2\n2(240)\n340\nvm\nV\n=\n=\n=\n2 The power rating of an element used in ac circuits refers to its average\npower rating"}, {"Chapter": "1", "sentence_range": "6040-6043", "Text": "This refers to the rms value of the voltage The amplitude\nof this voltage is\nV\n2\n2(240)\n340\nvm\nV\n=\n=\n=\n2 The power rating of an element used in ac circuits refers to its average\npower rating 3"}, {"Chapter": "1", "sentence_range": "6041-6044", "Text": "The amplitude\nof this voltage is\nV\n2\n2(240)\n340\nvm\nV\n=\n=\n=\n2 The power rating of an element used in ac circuits refers to its average\npower rating 3 The power consumed in an ac circuit is never negative"}, {"Chapter": "1", "sentence_range": "6042-6045", "Text": "The power rating of an element used in ac circuits refers to its average\npower rating 3 The power consumed in an ac circuit is never negative 4"}, {"Chapter": "1", "sentence_range": "6043-6046", "Text": "3 The power consumed in an ac circuit is never negative 4 Both alternating current and direct current are measured in amperes"}, {"Chapter": "1", "sentence_range": "6044-6047", "Text": "The power consumed in an ac circuit is never negative 4 Both alternating current and direct current are measured in amperes But how is the ampere defined for an alternating current"}, {"Chapter": "1", "sentence_range": "6045-6048", "Text": "4 Both alternating current and direct current are measured in amperes But how is the ampere defined for an alternating current It cannot be\nderived from the mutual attraction of two parallel wires carrying ac\ncurrents, as the dc ampere is derived"}, {"Chapter": "1", "sentence_range": "6046-6049", "Text": "Both alternating current and direct current are measured in amperes But how is the ampere defined for an alternating current It cannot be\nderived from the mutual attraction of two parallel wires carrying ac\ncurrents, as the dc ampere is derived An ac current changes direction\nRationalised 2023-24\n199\nAlternating Current\nwith the source frequency and the attractive force would average to\nzero"}, {"Chapter": "1", "sentence_range": "6047-6050", "Text": "But how is the ampere defined for an alternating current It cannot be\nderived from the mutual attraction of two parallel wires carrying ac\ncurrents, as the dc ampere is derived An ac current changes direction\nRationalised 2023-24\n199\nAlternating Current\nwith the source frequency and the attractive force would average to\nzero Thus, the ac ampere must be defined in terms of some property\nthat is independent of the direction of the current"}, {"Chapter": "1", "sentence_range": "6048-6051", "Text": "It cannot be\nderived from the mutual attraction of two parallel wires carrying ac\ncurrents, as the dc ampere is derived An ac current changes direction\nRationalised 2023-24\n199\nAlternating Current\nwith the source frequency and the attractive force would average to\nzero Thus, the ac ampere must be defined in terms of some property\nthat is independent of the direction of the current Joule heating\nis such a property, and there is one ampere of rms value of\nalternating current in a circuit if the current produces the same\naverage heating effect as one ampere of dc current would produce\nunder the same conditions"}, {"Chapter": "1", "sentence_range": "6049-6052", "Text": "An ac current changes direction\nRationalised 2023-24\n199\nAlternating Current\nwith the source frequency and the attractive force would average to\nzero Thus, the ac ampere must be defined in terms of some property\nthat is independent of the direction of the current Joule heating\nis such a property, and there is one ampere of rms value of\nalternating current in a circuit if the current produces the same\naverage heating effect as one ampere of dc current would produce\nunder the same conditions 5"}, {"Chapter": "1", "sentence_range": "6050-6053", "Text": "Thus, the ac ampere must be defined in terms of some property\nthat is independent of the direction of the current Joule heating\nis such a property, and there is one ampere of rms value of\nalternating current in a circuit if the current produces the same\naverage heating effect as one ampere of dc current would produce\nunder the same conditions 5 In an ac circuit, while adding voltages across different elements, one\nshould take care of their phases properly"}, {"Chapter": "1", "sentence_range": "6051-6054", "Text": "Joule heating\nis such a property, and there is one ampere of rms value of\nalternating current in a circuit if the current produces the same\naverage heating effect as one ampere of dc current would produce\nunder the same conditions 5 In an ac circuit, while adding voltages across different elements, one\nshould take care of their phases properly For example, if VR and VC\nare voltages across R and C, respectively in an RC circuit, then the\ntotal voltage across RC combination is \n2\n2\nRC\nR\nC\nV\nV\nV\n=\n+\n and not\nVR + VC since VC is p/2 out of phase of VR"}, {"Chapter": "1", "sentence_range": "6052-6055", "Text": "5 In an ac circuit, while adding voltages across different elements, one\nshould take care of their phases properly For example, if VR and VC\nare voltages across R and C, respectively in an RC circuit, then the\ntotal voltage across RC combination is \n2\n2\nRC\nR\nC\nV\nV\nV\n=\n+\n and not\nVR + VC since VC is p/2 out of phase of VR 6"}, {"Chapter": "1", "sentence_range": "6053-6056", "Text": "In an ac circuit, while adding voltages across different elements, one\nshould take care of their phases properly For example, if VR and VC\nare voltages across R and C, respectively in an RC circuit, then the\ntotal voltage across RC combination is \n2\n2\nRC\nR\nC\nV\nV\nV\n=\n+\n and not\nVR + VC since VC is p/2 out of phase of VR 6 Though in a phasor diagram, voltage and current are represented by\nvectors, these quantities are not really vectors themselves"}, {"Chapter": "1", "sentence_range": "6054-6057", "Text": "For example, if VR and VC\nare voltages across R and C, respectively in an RC circuit, then the\ntotal voltage across RC combination is \n2\n2\nRC\nR\nC\nV\nV\nV\n=\n+\n and not\nVR + VC since VC is p/2 out of phase of VR 6 Though in a phasor diagram, voltage and current are represented by\nvectors, these quantities are not really vectors themselves They are\nscalar quantities"}, {"Chapter": "1", "sentence_range": "6055-6058", "Text": "6 Though in a phasor diagram, voltage and current are represented by\nvectors, these quantities are not really vectors themselves They are\nscalar quantities It so happens that the amplitudes and phases of\nharmonically varying scalars combine mathematically in the same\nway as do the projections of rotating vectors of corresponding\nmagnitudes and directions"}, {"Chapter": "1", "sentence_range": "6056-6059", "Text": "Though in a phasor diagram, voltage and current are represented by\nvectors, these quantities are not really vectors themselves They are\nscalar quantities It so happens that the amplitudes and phases of\nharmonically varying scalars combine mathematically in the same\nway as do the projections of rotating vectors of corresponding\nmagnitudes and directions The \u2018rotating vectors\u2019 that represent\nharmonically varying scalar quantities are introduced only to provide\nus with a simple way of adding these quantities using a rule that\nwe already know as the law of vector addition"}, {"Chapter": "1", "sentence_range": "6057-6060", "Text": "They are\nscalar quantities It so happens that the amplitudes and phases of\nharmonically varying scalars combine mathematically in the same\nway as do the projections of rotating vectors of corresponding\nmagnitudes and directions The \u2018rotating vectors\u2019 that represent\nharmonically varying scalar quantities are introduced only to provide\nus with a simple way of adding these quantities using a rule that\nwe already know as the law of vector addition 7"}, {"Chapter": "1", "sentence_range": "6058-6061", "Text": "It so happens that the amplitudes and phases of\nharmonically varying scalars combine mathematically in the same\nway as do the projections of rotating vectors of corresponding\nmagnitudes and directions The \u2018rotating vectors\u2019 that represent\nharmonically varying scalar quantities are introduced only to provide\nus with a simple way of adding these quantities using a rule that\nwe already know as the law of vector addition 7 There are no power losses associated with pure capacitances and pure\ninductances in an ac circuit"}, {"Chapter": "1", "sentence_range": "6059-6062", "Text": "The \u2018rotating vectors\u2019 that represent\nharmonically varying scalar quantities are introduced only to provide\nus with a simple way of adding these quantities using a rule that\nwe already know as the law of vector addition 7 There are no power losses associated with pure capacitances and pure\ninductances in an ac circuit The only element that dissipates energy\nin an ac circuit is the resistive element"}, {"Chapter": "1", "sentence_range": "6060-6063", "Text": "7 There are no power losses associated with pure capacitances and pure\ninductances in an ac circuit The only element that dissipates energy\nin an ac circuit is the resistive element 8"}, {"Chapter": "1", "sentence_range": "6061-6064", "Text": "There are no power losses associated with pure capacitances and pure\ninductances in an ac circuit The only element that dissipates energy\nin an ac circuit is the resistive element 8 In a RLC circuit, resonance phenomenon occur when XL = XC or\n0\n1\nLC\n\u03c9\n="}, {"Chapter": "1", "sentence_range": "6062-6065", "Text": "The only element that dissipates energy\nin an ac circuit is the resistive element 8 In a RLC circuit, resonance phenomenon occur when XL = XC or\n0\n1\nLC\n\u03c9\n= For resonance to occur, the presence of both L and C\nelements in the circuit is a must"}, {"Chapter": "1", "sentence_range": "6063-6066", "Text": "8 In a RLC circuit, resonance phenomenon occur when XL = XC or\n0\n1\nLC\n\u03c9\n= For resonance to occur, the presence of both L and C\nelements in the circuit is a must With only one of these (L or C )\nelements, there is no possibility of voltage cancellation and hence,\nno resonance is possible"}, {"Chapter": "1", "sentence_range": "6064-6067", "Text": "In a RLC circuit, resonance phenomenon occur when XL = XC or\n0\n1\nLC\n\u03c9\n= For resonance to occur, the presence of both L and C\nelements in the circuit is a must With only one of these (L or C )\nelements, there is no possibility of voltage cancellation and hence,\nno resonance is possible 9"}, {"Chapter": "1", "sentence_range": "6065-6068", "Text": "For resonance to occur, the presence of both L and C\nelements in the circuit is a must With only one of these (L or C )\nelements, there is no possibility of voltage cancellation and hence,\nno resonance is possible 9 The power factor in a RLC circuit is a measure of how close the\ncircuit is to expending the maximum power"}, {"Chapter": "1", "sentence_range": "6066-6069", "Text": "With only one of these (L or C )\nelements, there is no possibility of voltage cancellation and hence,\nno resonance is possible 9 The power factor in a RLC circuit is a measure of how close the\ncircuit is to expending the maximum power 10"}, {"Chapter": "1", "sentence_range": "6067-6070", "Text": "9 The power factor in a RLC circuit is a measure of how close the\ncircuit is to expending the maximum power 10 In generators and motors, the roles of input and output are\nreversed"}, {"Chapter": "1", "sentence_range": "6068-6071", "Text": "The power factor in a RLC circuit is a measure of how close the\ncircuit is to expending the maximum power 10 In generators and motors, the roles of input and output are\nreversed In a motor, electric energy is the input and mechanical\nenergy is the output"}, {"Chapter": "1", "sentence_range": "6069-6072", "Text": "10 In generators and motors, the roles of input and output are\nreversed In a motor, electric energy is the input and mechanical\nenergy is the output In a generator, mechanical energy is the\ninput and electric energy is the output"}, {"Chapter": "1", "sentence_range": "6070-6073", "Text": "In generators and motors, the roles of input and output are\nreversed In a motor, electric energy is the input and mechanical\nenergy is the output In a generator, mechanical energy is the\ninput and electric energy is the output Both devices simply\ntransform energy from one form to another"}, {"Chapter": "1", "sentence_range": "6071-6074", "Text": "In a motor, electric energy is the input and mechanical\nenergy is the output In a generator, mechanical energy is the\ninput and electric energy is the output Both devices simply\ntransform energy from one form to another 11"}, {"Chapter": "1", "sentence_range": "6072-6075", "Text": "In a generator, mechanical energy is the\ninput and electric energy is the output Both devices simply\ntransform energy from one form to another 11 A transformer (step-up) changes a low-voltage into a high-voltage"}, {"Chapter": "1", "sentence_range": "6073-6076", "Text": "Both devices simply\ntransform energy from one form to another 11 A transformer (step-up) changes a low-voltage into a high-voltage This does not violate the law of conservation of energy"}, {"Chapter": "1", "sentence_range": "6074-6077", "Text": "11 A transformer (step-up) changes a low-voltage into a high-voltage This does not violate the law of conservation of energy The\ncurrent is reduced by the same proportion"}, {"Chapter": "1", "sentence_range": "6075-6078", "Text": "A transformer (step-up) changes a low-voltage into a high-voltage This does not violate the law of conservation of energy The\ncurrent is reduced by the same proportion Rationalised 2023-24\nPhysics\n200\nEXERCISES\n7"}, {"Chapter": "1", "sentence_range": "6076-6079", "Text": "This does not violate the law of conservation of energy The\ncurrent is reduced by the same proportion Rationalised 2023-24\nPhysics\n200\nEXERCISES\n7 1\nA 100 W resistor is connected to a 220 V, 50 Hz ac supply"}, {"Chapter": "1", "sentence_range": "6077-6080", "Text": "The\ncurrent is reduced by the same proportion Rationalised 2023-24\nPhysics\n200\nEXERCISES\n7 1\nA 100 W resistor is connected to a 220 V, 50 Hz ac supply (a) What is the rms value of current in the circuit"}, {"Chapter": "1", "sentence_range": "6078-6081", "Text": "Rationalised 2023-24\nPhysics\n200\nEXERCISES\n7 1\nA 100 W resistor is connected to a 220 V, 50 Hz ac supply (a) What is the rms value of current in the circuit (b) What is the net power consumed over a full cycle"}, {"Chapter": "1", "sentence_range": "6079-6082", "Text": "1\nA 100 W resistor is connected to a 220 V, 50 Hz ac supply (a) What is the rms value of current in the circuit (b) What is the net power consumed over a full cycle 7"}, {"Chapter": "1", "sentence_range": "6080-6083", "Text": "(a) What is the rms value of current in the circuit (b) What is the net power consumed over a full cycle 7 2\n(a) The peak voltage of an ac supply is 300 V"}, {"Chapter": "1", "sentence_range": "6081-6084", "Text": "(b) What is the net power consumed over a full cycle 7 2\n(a) The peak voltage of an ac supply is 300 V What is the rms voltage"}, {"Chapter": "1", "sentence_range": "6082-6085", "Text": "7 2\n(a) The peak voltage of an ac supply is 300 V What is the rms voltage (b) The rms value of current in an ac circuit is 10 A"}, {"Chapter": "1", "sentence_range": "6083-6086", "Text": "2\n(a) The peak voltage of an ac supply is 300 V What is the rms voltage (b) The rms value of current in an ac circuit is 10 A What is the\npeak current"}, {"Chapter": "1", "sentence_range": "6084-6087", "Text": "What is the rms voltage (b) The rms value of current in an ac circuit is 10 A What is the\npeak current 7"}, {"Chapter": "1", "sentence_range": "6085-6088", "Text": "(b) The rms value of current in an ac circuit is 10 A What is the\npeak current 7 3\nA 44 mH inductor is connected to 220 V, 50 Hz ac supply"}, {"Chapter": "1", "sentence_range": "6086-6089", "Text": "What is the\npeak current 7 3\nA 44 mH inductor is connected to 220 V, 50 Hz ac supply Determine\nthe rms value of the current in the circuit"}, {"Chapter": "1", "sentence_range": "6087-6090", "Text": "7 3\nA 44 mH inductor is connected to 220 V, 50 Hz ac supply Determine\nthe rms value of the current in the circuit 7"}, {"Chapter": "1", "sentence_range": "6088-6091", "Text": "3\nA 44 mH inductor is connected to 220 V, 50 Hz ac supply Determine\nthe rms value of the current in the circuit 7 4\n A 60 mF capacitor is connected to a 110 V, 60 Hz ac supply"}, {"Chapter": "1", "sentence_range": "6089-6092", "Text": "Determine\nthe rms value of the current in the circuit 7 4\n A 60 mF capacitor is connected to a 110 V, 60 Hz ac supply Determine\nthe rms value of the current in the circuit"}, {"Chapter": "1", "sentence_range": "6090-6093", "Text": "7 4\n A 60 mF capacitor is connected to a 110 V, 60 Hz ac supply Determine\nthe rms value of the current in the circuit 7"}, {"Chapter": "1", "sentence_range": "6091-6094", "Text": "4\n A 60 mF capacitor is connected to a 110 V, 60 Hz ac supply Determine\nthe rms value of the current in the circuit 7 5\nIn Exercises 7"}, {"Chapter": "1", "sentence_range": "6092-6095", "Text": "Determine\nthe rms value of the current in the circuit 7 5\nIn Exercises 7 3 and 7"}, {"Chapter": "1", "sentence_range": "6093-6096", "Text": "7 5\nIn Exercises 7 3 and 7 4, what is the net power absorbed by each\ncircuit over a complete cycle"}, {"Chapter": "1", "sentence_range": "6094-6097", "Text": "5\nIn Exercises 7 3 and 7 4, what is the net power absorbed by each\ncircuit over a complete cycle Explain your answer"}, {"Chapter": "1", "sentence_range": "6095-6098", "Text": "3 and 7 4, what is the net power absorbed by each\ncircuit over a complete cycle Explain your answer 7"}, {"Chapter": "1", "sentence_range": "6096-6099", "Text": "4, what is the net power absorbed by each\ncircuit over a complete cycle Explain your answer 7 6\nA charged 30 mF capacitor is connected to a 27 mH inductor"}, {"Chapter": "1", "sentence_range": "6097-6100", "Text": "Explain your answer 7 6\nA charged 30 mF capacitor is connected to a 27 mH inductor What is\nthe angular frequency of free oscillations of the circuit"}, {"Chapter": "1", "sentence_range": "6098-6101", "Text": "7 6\nA charged 30 mF capacitor is connected to a 27 mH inductor What is\nthe angular frequency of free oscillations of the circuit 7"}, {"Chapter": "1", "sentence_range": "6099-6102", "Text": "6\nA charged 30 mF capacitor is connected to a 27 mH inductor What is\nthe angular frequency of free oscillations of the circuit 7 7\nA series LCR circuit with R = 20 W, L = 1"}, {"Chapter": "1", "sentence_range": "6100-6103", "Text": "What is\nthe angular frequency of free oscillations of the circuit 7 7\nA series LCR circuit with R = 20 W, L = 1 5 H and C = 35 mF is connected\nto a variable-frequency 200 V ac supply"}, {"Chapter": "1", "sentence_range": "6101-6104", "Text": "7 7\nA series LCR circuit with R = 20 W, L = 1 5 H and C = 35 mF is connected\nto a variable-frequency 200 V ac supply When the frequency of the\nsupply equals the natural frequency of the circuit, what is the average\npower transferred to the circuit in one complete cycle"}, {"Chapter": "1", "sentence_range": "6102-6105", "Text": "7\nA series LCR circuit with R = 20 W, L = 1 5 H and C = 35 mF is connected\nto a variable-frequency 200 V ac supply When the frequency of the\nsupply equals the natural frequency of the circuit, what is the average\npower transferred to the circuit in one complete cycle 7"}, {"Chapter": "1", "sentence_range": "6103-6106", "Text": "5 H and C = 35 mF is connected\nto a variable-frequency 200 V ac supply When the frequency of the\nsupply equals the natural frequency of the circuit, what is the average\npower transferred to the circuit in one complete cycle 7 8\nFigure 7"}, {"Chapter": "1", "sentence_range": "6104-6107", "Text": "When the frequency of the\nsupply equals the natural frequency of the circuit, what is the average\npower transferred to the circuit in one complete cycle 7 8\nFigure 7 17 shows a series LCR circuit connected to a variable\nfrequency 230 V source"}, {"Chapter": "1", "sentence_range": "6105-6108", "Text": "7 8\nFigure 7 17 shows a series LCR circuit connected to a variable\nfrequency 230 V source L = 5"}, {"Chapter": "1", "sentence_range": "6106-6109", "Text": "8\nFigure 7 17 shows a series LCR circuit connected to a variable\nfrequency 230 V source L = 5 0 H, C = 80mF, R = 40 W"}, {"Chapter": "1", "sentence_range": "6107-6110", "Text": "17 shows a series LCR circuit connected to a variable\nfrequency 230 V source L = 5 0 H, C = 80mF, R = 40 W (a) Determine the source frequency which drives the circuit in\nresonance"}, {"Chapter": "1", "sentence_range": "6108-6111", "Text": "L = 5 0 H, C = 80mF, R = 40 W (a) Determine the source frequency which drives the circuit in\nresonance (b) Obtain the impedance of the circuit and the amplitude of current\nat the resonating frequency"}, {"Chapter": "1", "sentence_range": "6109-6112", "Text": "0 H, C = 80mF, R = 40 W (a) Determine the source frequency which drives the circuit in\nresonance (b) Obtain the impedance of the circuit and the amplitude of current\nat the resonating frequency (c) Determine the rms potential drops across the three elements of\nthe circuit"}, {"Chapter": "1", "sentence_range": "6110-6113", "Text": "(a) Determine the source frequency which drives the circuit in\nresonance (b) Obtain the impedance of the circuit and the amplitude of current\nat the resonating frequency (c) Determine the rms potential drops across the three elements of\nthe circuit Show that the potential drop across the LC\ncombination is zero at the resonating frequency"}, {"Chapter": "1", "sentence_range": "6111-6114", "Text": "(b) Obtain the impedance of the circuit and the amplitude of current\nat the resonating frequency (c) Determine the rms potential drops across the three elements of\nthe circuit Show that the potential drop across the LC\ncombination is zero at the resonating frequency FIGURE 7"}, {"Chapter": "1", "sentence_range": "6112-6115", "Text": "(c) Determine the rms potential drops across the three elements of\nthe circuit Show that the potential drop across the LC\ncombination is zero at the resonating frequency FIGURE 7 17\nRationalised 2023-24\nChapter Eight\nELECTROMAGNETIC\nWAVES\n8"}, {"Chapter": "1", "sentence_range": "6113-6116", "Text": "Show that the potential drop across the LC\ncombination is zero at the resonating frequency FIGURE 7 17\nRationalised 2023-24\nChapter Eight\nELECTROMAGNETIC\nWAVES\n8 1  INTRODUCTION\nIn Chapter 4, we learnt that an electric current produces magnetic field\nand that two current-carrying wires exert a magnetic force on each other"}, {"Chapter": "1", "sentence_range": "6114-6117", "Text": "FIGURE 7 17\nRationalised 2023-24\nChapter Eight\nELECTROMAGNETIC\nWAVES\n8 1  INTRODUCTION\nIn Chapter 4, we learnt that an electric current produces magnetic field\nand that two current-carrying wires exert a magnetic force on each other Further, in Chapter 6, we have seen that a magnetic field changing with\ntime gives rise to an electric field"}, {"Chapter": "1", "sentence_range": "6115-6118", "Text": "17\nRationalised 2023-24\nChapter Eight\nELECTROMAGNETIC\nWAVES\n8 1  INTRODUCTION\nIn Chapter 4, we learnt that an electric current produces magnetic field\nand that two current-carrying wires exert a magnetic force on each other Further, in Chapter 6, we have seen that a magnetic field changing with\ntime gives rise to an electric field Is the converse also true"}, {"Chapter": "1", "sentence_range": "6116-6119", "Text": "1  INTRODUCTION\nIn Chapter 4, we learnt that an electric current produces magnetic field\nand that two current-carrying wires exert a magnetic force on each other Further, in Chapter 6, we have seen that a magnetic field changing with\ntime gives rise to an electric field Is the converse also true Does an\nelectric field changing with time give rise to a magnetic field"}, {"Chapter": "1", "sentence_range": "6117-6120", "Text": "Further, in Chapter 6, we have seen that a magnetic field changing with\ntime gives rise to an electric field Is the converse also true Does an\nelectric field changing with time give rise to a magnetic field James Clerk\nMaxwell (1831-1879), argued that this was indeed the case \u2013 not only\nan electric current but also a time-varying electric field generates magnetic\nfield"}, {"Chapter": "1", "sentence_range": "6118-6121", "Text": "Is the converse also true Does an\nelectric field changing with time give rise to a magnetic field James Clerk\nMaxwell (1831-1879), argued that this was indeed the case \u2013 not only\nan electric current but also a time-varying electric field generates magnetic\nfield While applying the Ampere\u2019s circuital law to find magnetic field at a\npoint outside a capacitor connected to a time-varying current, Maxwell\nnoticed an inconsistency in the Ampere\u2019s circuital law"}, {"Chapter": "1", "sentence_range": "6119-6122", "Text": "Does an\nelectric field changing with time give rise to a magnetic field James Clerk\nMaxwell (1831-1879), argued that this was indeed the case \u2013 not only\nan electric current but also a time-varying electric field generates magnetic\nfield While applying the Ampere\u2019s circuital law to find magnetic field at a\npoint outside a capacitor connected to a time-varying current, Maxwell\nnoticed an inconsistency in the Ampere\u2019s circuital law He suggested the\nexistence of an additional current, called by him, the displacement\ncurrent to remove this inconsistency"}, {"Chapter": "1", "sentence_range": "6120-6123", "Text": "James Clerk\nMaxwell (1831-1879), argued that this was indeed the case \u2013 not only\nan electric current but also a time-varying electric field generates magnetic\nfield While applying the Ampere\u2019s circuital law to find magnetic field at a\npoint outside a capacitor connected to a time-varying current, Maxwell\nnoticed an inconsistency in the Ampere\u2019s circuital law He suggested the\nexistence of an additional current, called by him, the displacement\ncurrent to remove this inconsistency Maxwell formulated a set of equations involving electric and magnetic\nfields, and their sources, the charge and current densities"}, {"Chapter": "1", "sentence_range": "6121-6124", "Text": "While applying the Ampere\u2019s circuital law to find magnetic field at a\npoint outside a capacitor connected to a time-varying current, Maxwell\nnoticed an inconsistency in the Ampere\u2019s circuital law He suggested the\nexistence of an additional current, called by him, the displacement\ncurrent to remove this inconsistency Maxwell formulated a set of equations involving electric and magnetic\nfields, and their sources, the charge and current densities These\nequations are known as Maxwell\u2019s equations"}, {"Chapter": "1", "sentence_range": "6122-6125", "Text": "He suggested the\nexistence of an additional current, called by him, the displacement\ncurrent to remove this inconsistency Maxwell formulated a set of equations involving electric and magnetic\nfields, and their sources, the charge and current densities These\nequations are known as Maxwell\u2019s equations Together with the Lorentz\nforce formula (Chapter 4), they mathematically express all the basic laws\nof electromagnetism"}, {"Chapter": "1", "sentence_range": "6123-6126", "Text": "Maxwell formulated a set of equations involving electric and magnetic\nfields, and their sources, the charge and current densities These\nequations are known as Maxwell\u2019s equations Together with the Lorentz\nforce formula (Chapter 4), they mathematically express all the basic laws\nof electromagnetism The most important prediction to emerge from Maxwell\u2019s equations\nis the existence of electromagnetic waves, which are (coupled) time-\nvarying electric and magnetic fields that propagate in space"}, {"Chapter": "1", "sentence_range": "6124-6127", "Text": "These\nequations are known as Maxwell\u2019s equations Together with the Lorentz\nforce formula (Chapter 4), they mathematically express all the basic laws\nof electromagnetism The most important prediction to emerge from Maxwell\u2019s equations\nis the existence of electromagnetic waves, which are (coupled) time-\nvarying electric and magnetic fields that propagate in space The speed\nof the waves,  according to these equations, turned out to be very close to\nRationalised 2023-24\nPhysics\n202\nthe speed of light( 3 \u00d7108 m/s), obtained from optical\nmeasurements"}, {"Chapter": "1", "sentence_range": "6125-6128", "Text": "Together with the Lorentz\nforce formula (Chapter 4), they mathematically express all the basic laws\nof electromagnetism The most important prediction to emerge from Maxwell\u2019s equations\nis the existence of electromagnetic waves, which are (coupled) time-\nvarying electric and magnetic fields that propagate in space The speed\nof the waves,  according to these equations, turned out to be very close to\nRationalised 2023-24\nPhysics\n202\nthe speed of light( 3 \u00d7108 m/s), obtained from optical\nmeasurements This led to the remarkable conclusion\nthat light is an electromagnetic wave"}, {"Chapter": "1", "sentence_range": "6126-6129", "Text": "The most important prediction to emerge from Maxwell\u2019s equations\nis the existence of electromagnetic waves, which are (coupled) time-\nvarying electric and magnetic fields that propagate in space The speed\nof the waves,  according to these equations, turned out to be very close to\nRationalised 2023-24\nPhysics\n202\nthe speed of light( 3 \u00d7108 m/s), obtained from optical\nmeasurements This led to the remarkable conclusion\nthat light is an electromagnetic wave Maxwell\u2019s work\nthus unified the domain of electricity, magnetism and\nlight"}, {"Chapter": "1", "sentence_range": "6127-6130", "Text": "The speed\nof the waves,  according to these equations, turned out to be very close to\nRationalised 2023-24\nPhysics\n202\nthe speed of light( 3 \u00d7108 m/s), obtained from optical\nmeasurements This led to the remarkable conclusion\nthat light is an electromagnetic wave Maxwell\u2019s work\nthus unified the domain of electricity, magnetism and\nlight Hertz, in 1885, experimentally demonstrated the\nexistence of electromagnetic waves"}, {"Chapter": "1", "sentence_range": "6128-6131", "Text": "This led to the remarkable conclusion\nthat light is an electromagnetic wave Maxwell\u2019s work\nthus unified the domain of electricity, magnetism and\nlight Hertz, in 1885, experimentally demonstrated the\nexistence of electromagnetic waves Its technological use\nby Marconi and others led in due course to the\nrevolution in communication that we are witnessing\ntoday"}, {"Chapter": "1", "sentence_range": "6129-6132", "Text": "Maxwell\u2019s work\nthus unified the domain of electricity, magnetism and\nlight Hertz, in 1885, experimentally demonstrated the\nexistence of electromagnetic waves Its technological use\nby Marconi and others led in due course to the\nrevolution in communication that we are witnessing\ntoday In this chapter, we first discuss the need for\ndisplacement current and its consequences"}, {"Chapter": "1", "sentence_range": "6130-6133", "Text": "Hertz, in 1885, experimentally demonstrated the\nexistence of electromagnetic waves Its technological use\nby Marconi and others led in due course to the\nrevolution in communication that we are witnessing\ntoday In this chapter, we first discuss the need for\ndisplacement current and its consequences Then we\npresent a descriptive account of electromagnetic waves"}, {"Chapter": "1", "sentence_range": "6131-6134", "Text": "Its technological use\nby Marconi and others led in due course to the\nrevolution in communication that we are witnessing\ntoday In this chapter, we first discuss the need for\ndisplacement current and its consequences Then we\npresent a descriptive account of electromagnetic waves The broad spectrum of electromagnetic waves,\nstretching from g rays (wavelength ~10\u201312 m) to long\nradio waves (wavelength ~106 m) is described"}, {"Chapter": "1", "sentence_range": "6132-6135", "Text": "In this chapter, we first discuss the need for\ndisplacement current and its consequences Then we\npresent a descriptive account of electromagnetic waves The broad spectrum of electromagnetic waves,\nstretching from g rays (wavelength ~10\u201312 m) to long\nradio waves (wavelength ~106 m) is described 8"}, {"Chapter": "1", "sentence_range": "6133-6136", "Text": "Then we\npresent a descriptive account of electromagnetic waves The broad spectrum of electromagnetic waves,\nstretching from g rays (wavelength ~10\u201312 m) to long\nradio waves (wavelength ~106 m) is described 8 2  DISPLACEMENT CURRENT\nWe have seen in Chapter 4 that an electrical current\nproduces a magnetic field around it"}, {"Chapter": "1", "sentence_range": "6134-6137", "Text": "The broad spectrum of electromagnetic waves,\nstretching from g rays (wavelength ~10\u201312 m) to long\nradio waves (wavelength ~106 m) is described 8 2  DISPLACEMENT CURRENT\nWe have seen in Chapter 4 that an electrical current\nproduces a magnetic field around it Maxwell showed\nthat for logical consistency, a changing electric field must\nalso produce a magnetic field"}, {"Chapter": "1", "sentence_range": "6135-6138", "Text": "8 2  DISPLACEMENT CURRENT\nWe have seen in Chapter 4 that an electrical current\nproduces a magnetic field around it Maxwell showed\nthat for logical consistency, a changing electric field must\nalso produce a magnetic field This effect is of great\nimportance because it explains the existence of radio\nwaves, gamma rays and visible light, as well as all other\nforms of electromagnetic waves"}, {"Chapter": "1", "sentence_range": "6136-6139", "Text": "2  DISPLACEMENT CURRENT\nWe have seen in Chapter 4 that an electrical current\nproduces a magnetic field around it Maxwell showed\nthat for logical consistency, a changing electric field must\nalso produce a magnetic field This effect is of great\nimportance because it explains the existence of radio\nwaves, gamma rays and visible light, as well as all other\nforms of electromagnetic waves To see how a changing electric field gives rise to\na magnetic field, let us consider the process of\ncharging of a capacitor and apply Ampere\u2019s circuital\nlaw given by (Chapter 4)\n\u201cB"}, {"Chapter": "1", "sentence_range": "6137-6140", "Text": "Maxwell showed\nthat for logical consistency, a changing electric field must\nalso produce a magnetic field This effect is of great\nimportance because it explains the existence of radio\nwaves, gamma rays and visible light, as well as all other\nforms of electromagnetic waves To see how a changing electric field gives rise to\na magnetic field, let us consider the process of\ncharging of a capacitor and apply Ampere\u2019s circuital\nlaw given by (Chapter 4)\n\u201cB dl = m0 i (t)\n          (8"}, {"Chapter": "1", "sentence_range": "6138-6141", "Text": "This effect is of great\nimportance because it explains the existence of radio\nwaves, gamma rays and visible light, as well as all other\nforms of electromagnetic waves To see how a changing electric field gives rise to\na magnetic field, let us consider the process of\ncharging of a capacitor and apply Ampere\u2019s circuital\nlaw given by (Chapter 4)\n\u201cB dl = m0 i (t)\n          (8 1)\nto find magnetic  field at a point outside the capacitor"}, {"Chapter": "1", "sentence_range": "6139-6142", "Text": "To see how a changing electric field gives rise to\na magnetic field, let us consider the process of\ncharging of a capacitor and apply Ampere\u2019s circuital\nlaw given by (Chapter 4)\n\u201cB dl = m0 i (t)\n          (8 1)\nto find magnetic  field at a point outside the capacitor Figure 8"}, {"Chapter": "1", "sentence_range": "6140-6143", "Text": "dl = m0 i (t)\n          (8 1)\nto find magnetic  field at a point outside the capacitor Figure 8 1(a)  shows a parallel plate capacitor C which\nis a part of circuit through which a time-dependent\ncurrent i (t) flows"}, {"Chapter": "1", "sentence_range": "6141-6144", "Text": "1)\nto find magnetic  field at a point outside the capacitor Figure 8 1(a)  shows a parallel plate capacitor C which\nis a part of circuit through which a time-dependent\ncurrent i (t) flows Let us find the magnetic field at a\npoint such as P, in a region outside the parallel plate\ncapacitor"}, {"Chapter": "1", "sentence_range": "6142-6145", "Text": "Figure 8 1(a)  shows a parallel plate capacitor C which\nis a part of circuit through which a time-dependent\ncurrent i (t) flows Let us find the magnetic field at a\npoint such as P, in a region outside the parallel plate\ncapacitor For this, we consider a plane circular loop of\nradius r whose plane is perpendicular to the direction\nof the current-carrying wire, and which is centred\nsymmetrically with respect to the wire [Fig"}, {"Chapter": "1", "sentence_range": "6143-6146", "Text": "1(a)  shows a parallel plate capacitor C which\nis a part of circuit through which a time-dependent\ncurrent i (t) flows Let us find the magnetic field at a\npoint such as P, in a region outside the parallel plate\ncapacitor For this, we consider a plane circular loop of\nradius r whose plane is perpendicular to the direction\nof the current-carrying wire, and which is centred\nsymmetrically with respect to the wire [Fig 8"}, {"Chapter": "1", "sentence_range": "6144-6147", "Text": "Let us find the magnetic field at a\npoint such as P, in a region outside the parallel plate\ncapacitor For this, we consider a plane circular loop of\nradius r whose plane is perpendicular to the direction\nof the current-carrying wire, and which is centred\nsymmetrically with respect to the wire [Fig 8 1(a)]"}, {"Chapter": "1", "sentence_range": "6145-6148", "Text": "For this, we consider a plane circular loop of\nradius r whose plane is perpendicular to the direction\nof the current-carrying wire, and which is centred\nsymmetrically with respect to the wire [Fig 8 1(a)] From\nsymmetry, the magnetic field is directed along the\ncircumference of the circular loop and is the same in\nmagnitude at all points on the loop so that if B is the\nmagnitude of the field, the left side of Eq"}, {"Chapter": "1", "sentence_range": "6146-6149", "Text": "8 1(a)] From\nsymmetry, the magnetic field is directed along the\ncircumference of the circular loop and is the same in\nmagnitude at all points on the loop so that if B is the\nmagnitude of the field, the left side of Eq (8"}, {"Chapter": "1", "sentence_range": "6147-6150", "Text": "1(a)] From\nsymmetry, the magnetic field is directed along the\ncircumference of the circular loop and is the same in\nmagnitude at all points on the loop so that if B is the\nmagnitude of the field, the left side of Eq (8 1) is B (2p r)"}, {"Chapter": "1", "sentence_range": "6148-6151", "Text": "From\nsymmetry, the magnetic field is directed along the\ncircumference of the circular loop and is the same in\nmagnitude at all points on the loop so that if B is the\nmagnitude of the field, the left side of Eq (8 1) is B (2p r) So we have\nB (2pr) = m0i (t)\n   (8"}, {"Chapter": "1", "sentence_range": "6149-6152", "Text": "(8 1) is B (2p r) So we have\nB (2pr) = m0i (t)\n   (8 2)\nJAMES CLERK MAXWELL (1831\u20131879)\nJames Clerk Maxwell\n(1831 \u2013 1879) Born in\nEdinburgh, Scotland,\nwas among the greatest\nphysicists \nof \nthe\nnineteenth century"}, {"Chapter": "1", "sentence_range": "6150-6153", "Text": "1) is B (2p r) So we have\nB (2pr) = m0i (t)\n   (8 2)\nJAMES CLERK MAXWELL (1831\u20131879)\nJames Clerk Maxwell\n(1831 \u2013 1879) Born in\nEdinburgh, Scotland,\nwas among the greatest\nphysicists \nof \nthe\nnineteenth century He\nderived the thermal\nvelocity distribution of\nmolecules in a gas and\nwas among the first to\nobtain \nreliable\nestimates of molecular\nparameters \nfrom\nmeasurable quantities\nlike \nviscosity, \netc"}, {"Chapter": "1", "sentence_range": "6151-6154", "Text": "So we have\nB (2pr) = m0i (t)\n   (8 2)\nJAMES CLERK MAXWELL (1831\u20131879)\nJames Clerk Maxwell\n(1831 \u2013 1879) Born in\nEdinburgh, Scotland,\nwas among the greatest\nphysicists \nof \nthe\nnineteenth century He\nderived the thermal\nvelocity distribution of\nmolecules in a gas and\nwas among the first to\nobtain \nreliable\nestimates of molecular\nparameters \nfrom\nmeasurable quantities\nlike \nviscosity, \netc Maxwell\u2019s \ngreatest\nacheivement was the\nunification of the laws of\nelectricity \nand\nmagnetism (discovered\nby Coulomb, Oersted,\nAmpere and Faraday)\ninto a consistent set of\nequations now called\nMaxwell\u2019s equations"}, {"Chapter": "1", "sentence_range": "6152-6155", "Text": "2)\nJAMES CLERK MAXWELL (1831\u20131879)\nJames Clerk Maxwell\n(1831 \u2013 1879) Born in\nEdinburgh, Scotland,\nwas among the greatest\nphysicists \nof \nthe\nnineteenth century He\nderived the thermal\nvelocity distribution of\nmolecules in a gas and\nwas among the first to\nobtain \nreliable\nestimates of molecular\nparameters \nfrom\nmeasurable quantities\nlike \nviscosity, \netc Maxwell\u2019s \ngreatest\nacheivement was the\nunification of the laws of\nelectricity \nand\nmagnetism (discovered\nby Coulomb, Oersted,\nAmpere and Faraday)\ninto a consistent set of\nequations now called\nMaxwell\u2019s equations From these he arrived at\nthe most important\nconclusion that light is\nan \nwave"}, {"Chapter": "1", "sentence_range": "6153-6156", "Text": "He\nderived the thermal\nvelocity distribution of\nmolecules in a gas and\nwas among the first to\nobtain \nreliable\nestimates of molecular\nparameters \nfrom\nmeasurable quantities\nlike \nviscosity, \netc Maxwell\u2019s \ngreatest\nacheivement was the\nunification of the laws of\nelectricity \nand\nmagnetism (discovered\nby Coulomb, Oersted,\nAmpere and Faraday)\ninto a consistent set of\nequations now called\nMaxwell\u2019s equations From these he arrived at\nthe most important\nconclusion that light is\nan \nwave electromagnetic\nInterestingly,\nMaxwell did not agree\nwith the idea (strongly\nsuggested \nby \nthe\nFaraday\u2019s \nlaws \nof\nelectrolysis) \nthat\nelectricity \nwas\nparticulate in nature"}, {"Chapter": "1", "sentence_range": "6154-6157", "Text": "Maxwell\u2019s \ngreatest\nacheivement was the\nunification of the laws of\nelectricity \nand\nmagnetism (discovered\nby Coulomb, Oersted,\nAmpere and Faraday)\ninto a consistent set of\nequations now called\nMaxwell\u2019s equations From these he arrived at\nthe most important\nconclusion that light is\nan \nwave electromagnetic\nInterestingly,\nMaxwell did not agree\nwith the idea (strongly\nsuggested \nby \nthe\nFaraday\u2019s \nlaws \nof\nelectrolysis) \nthat\nelectricity \nwas\nparticulate in nature Rationalised 2023-24\n203\nElectromagnetic\nWaves\nNow, consider a different surface, which has the same boundary"}, {"Chapter": "1", "sentence_range": "6155-6158", "Text": "From these he arrived at\nthe most important\nconclusion that light is\nan \nwave electromagnetic\nInterestingly,\nMaxwell did not agree\nwith the idea (strongly\nsuggested \nby \nthe\nFaraday\u2019s \nlaws \nof\nelectrolysis) \nthat\nelectricity \nwas\nparticulate in nature Rationalised 2023-24\n203\nElectromagnetic\nWaves\nNow, consider a different surface, which has the same boundary This\nis a pot like surface [Fig"}, {"Chapter": "1", "sentence_range": "6156-6159", "Text": "electromagnetic\nInterestingly,\nMaxwell did not agree\nwith the idea (strongly\nsuggested \nby \nthe\nFaraday\u2019s \nlaws \nof\nelectrolysis) \nthat\nelectricity \nwas\nparticulate in nature Rationalised 2023-24\n203\nElectromagnetic\nWaves\nNow, consider a different surface, which has the same boundary This\nis a pot like surface [Fig 8"}, {"Chapter": "1", "sentence_range": "6157-6160", "Text": "Rationalised 2023-24\n203\nElectromagnetic\nWaves\nNow, consider a different surface, which has the same boundary This\nis a pot like surface [Fig 8 1(b)] which nowhere touches the current, but\nhas its bottom between the capacitor plates; its mouth is the circular\nloop mentioned above"}, {"Chapter": "1", "sentence_range": "6158-6161", "Text": "This\nis a pot like surface [Fig 8 1(b)] which nowhere touches the current, but\nhas its bottom between the capacitor plates; its mouth is the circular\nloop mentioned above Another such surface is shaped like a tiffin box\n(without the lid) [Fig"}, {"Chapter": "1", "sentence_range": "6159-6162", "Text": "8 1(b)] which nowhere touches the current, but\nhas its bottom between the capacitor plates; its mouth is the circular\nloop mentioned above Another such surface is shaped like a tiffin box\n(without the lid) [Fig 8"}, {"Chapter": "1", "sentence_range": "6160-6163", "Text": "1(b)] which nowhere touches the current, but\nhas its bottom between the capacitor plates; its mouth is the circular\nloop mentioned above Another such surface is shaped like a tiffin box\n(without the lid) [Fig 8 1(c)]"}, {"Chapter": "1", "sentence_range": "6161-6164", "Text": "Another such surface is shaped like a tiffin box\n(without the lid) [Fig 8 1(c)] On applying Ampere\u2019s circuital law to such\nsurfaces with the same perimeter, we find that the left hand side of\nEq"}, {"Chapter": "1", "sentence_range": "6162-6165", "Text": "8 1(c)] On applying Ampere\u2019s circuital law to such\nsurfaces with the same perimeter, we find that the left hand side of\nEq (8"}, {"Chapter": "1", "sentence_range": "6163-6166", "Text": "1(c)] On applying Ampere\u2019s circuital law to such\nsurfaces with the same perimeter, we find that the left hand side of\nEq (8 1) has not changed but the right hand side is zero and not m0i,\nsince no current passes through the surface of Fig"}, {"Chapter": "1", "sentence_range": "6164-6167", "Text": "On applying Ampere\u2019s circuital law to such\nsurfaces with the same perimeter, we find that the left hand side of\nEq (8 1) has not changed but the right hand side is zero and not m0i,\nsince no current passes through the surface of Fig 8"}, {"Chapter": "1", "sentence_range": "6165-6168", "Text": "(8 1) has not changed but the right hand side is zero and not m0i,\nsince no current passes through the surface of Fig 8 1(b) and (c)"}, {"Chapter": "1", "sentence_range": "6166-6169", "Text": "1) has not changed but the right hand side is zero and not m0i,\nsince no current passes through the surface of Fig 8 1(b) and (c) So we\nhave a contradiction; calculated one way, there is a magnetic field at a\npoint P; calculated another way, the magnetic field at P is zero"}, {"Chapter": "1", "sentence_range": "6167-6170", "Text": "8 1(b) and (c) So we\nhave a contradiction; calculated one way, there is a magnetic field at a\npoint P; calculated another way, the magnetic field at P is zero Since the contradiction arises from our use of Ampere\u2019s circuital law,\nthis law must be missing something"}, {"Chapter": "1", "sentence_range": "6168-6171", "Text": "1(b) and (c) So we\nhave a contradiction; calculated one way, there is a magnetic field at a\npoint P; calculated another way, the magnetic field at P is zero Since the contradiction arises from our use of Ampere\u2019s circuital law,\nthis law must be missing something The missing term must be such\nthat one gets the same magnetic field at point P,  no matter what surface\nis used"}, {"Chapter": "1", "sentence_range": "6169-6172", "Text": "So we\nhave a contradiction; calculated one way, there is a magnetic field at a\npoint P; calculated another way, the magnetic field at P is zero Since the contradiction arises from our use of Ampere\u2019s circuital law,\nthis law must be missing something The missing term must be such\nthat one gets the same magnetic field at point P,  no matter what surface\nis used We can actually guess the missing term by looking carefully at\nFig"}, {"Chapter": "1", "sentence_range": "6170-6173", "Text": "Since the contradiction arises from our use of Ampere\u2019s circuital law,\nthis law must be missing something The missing term must be such\nthat one gets the same magnetic field at point P,  no matter what surface\nis used We can actually guess the missing term by looking carefully at\nFig 8"}, {"Chapter": "1", "sentence_range": "6171-6174", "Text": "The missing term must be such\nthat one gets the same magnetic field at point P,  no matter what surface\nis used We can actually guess the missing term by looking carefully at\nFig 8 1(c)"}, {"Chapter": "1", "sentence_range": "6172-6175", "Text": "We can actually guess the missing term by looking carefully at\nFig 8 1(c) Is there anything passing through the surface S between the\nplates of the capacitor"}, {"Chapter": "1", "sentence_range": "6173-6176", "Text": "8 1(c) Is there anything passing through the surface S between the\nplates of the capacitor Yes, of course, the electric field"}, {"Chapter": "1", "sentence_range": "6174-6177", "Text": "1(c) Is there anything passing through the surface S between the\nplates of the capacitor Yes, of course, the electric field If the plates of the\ncapacitor have an area A, and a total charge Q, the magnitude of the\nelectric field E between the plates is (Q/A)/e0 (see Eq"}, {"Chapter": "1", "sentence_range": "6175-6178", "Text": "Is there anything passing through the surface S between the\nplates of the capacitor Yes, of course, the electric field If the plates of the\ncapacitor have an area A, and a total charge Q, the magnitude of the\nelectric field E between the plates is (Q/A)/e0 (see Eq 2"}, {"Chapter": "1", "sentence_range": "6176-6179", "Text": "Yes, of course, the electric field If the plates of the\ncapacitor have an area A, and a total charge Q, the magnitude of the\nelectric field E between the plates is (Q/A)/e0 (see Eq 2 41)"}, {"Chapter": "1", "sentence_range": "6177-6180", "Text": "If the plates of the\ncapacitor have an area A, and a total charge Q, the magnitude of the\nelectric field E between the plates is (Q/A)/e0 (see Eq 2 41) The field is\nperpendicular to the surface S of Fig"}, {"Chapter": "1", "sentence_range": "6178-6181", "Text": "2 41) The field is\nperpendicular to the surface S of Fig 8"}, {"Chapter": "1", "sentence_range": "6179-6182", "Text": "41) The field is\nperpendicular to the surface S of Fig 8 1(c)"}, {"Chapter": "1", "sentence_range": "6180-6183", "Text": "The field is\nperpendicular to the surface S of Fig 8 1(c) It has the same magnitude\nover the area A of the capacitor plates, and vanishes outside it"}, {"Chapter": "1", "sentence_range": "6181-6184", "Text": "8 1(c) It has the same magnitude\nover the area A of the capacitor plates, and vanishes outside it So what\nis the electric flux FE through the surface S"}, {"Chapter": "1", "sentence_range": "6182-6185", "Text": "1(c) It has the same magnitude\nover the area A of the capacitor plates, and vanishes outside it So what\nis the electric flux FE through the surface S Using Gauss\u2019s law, it is\nE\n0\n0\n1\n=\n=\nQ\nQ\nA\nAA\n\u03a6\n\u03b5\n=\u03b5\nE\n(8"}, {"Chapter": "1", "sentence_range": "6183-6186", "Text": "It has the same magnitude\nover the area A of the capacitor plates, and vanishes outside it So what\nis the electric flux FE through the surface S Using Gauss\u2019s law, it is\nE\n0\n0\n1\n=\n=\nQ\nQ\nA\nAA\n\u03a6\n\u03b5\n=\u03b5\nE\n(8 3)\nNow if the charge Q on the capacitor plates changes with time, there is a\ncurrent i = (dQ/dt), so that using Eq"}, {"Chapter": "1", "sentence_range": "6184-6187", "Text": "So what\nis the electric flux FE through the surface S Using Gauss\u2019s law, it is\nE\n0\n0\n1\n=\n=\nQ\nQ\nA\nAA\n\u03a6\n\u03b5\n=\u03b5\nE\n(8 3)\nNow if the charge Q on the capacitor plates changes with time, there is a\ncurrent i = (dQ/dt), so that using Eq (8"}, {"Chapter": "1", "sentence_range": "6185-6188", "Text": "Using Gauss\u2019s law, it is\nE\n0\n0\n1\n=\n=\nQ\nQ\nA\nAA\n\u03a6\n\u03b5\n=\u03b5\nE\n(8 3)\nNow if the charge Q on the capacitor plates changes with time, there is a\ncurrent i = (dQ/dt), so that using Eq (8 3), we have\nd\nd\ndd\nd\nd\nt\u03a6E\nt\nQ\ntQ\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\u03b5\uf8f8\uf8f7 =\n\u03b5\n0\n0\n1\nThis implies that for consistency,\n\u03b50\nd\nd\nt\u03a6E\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7  = i\n(8"}, {"Chapter": "1", "sentence_range": "6186-6189", "Text": "3)\nNow if the charge Q on the capacitor plates changes with time, there is a\ncurrent i = (dQ/dt), so that using Eq (8 3), we have\nd\nd\ndd\nd\nd\nt\u03a6E\nt\nQ\ntQ\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\u03b5\uf8f8\uf8f7 =\n\u03b5\n0\n0\n1\nThis implies that for consistency,\n\u03b50\nd\nd\nt\u03a6E\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7  = i\n(8 4)\nThis is the missing term in Ampere\u2019s circuital law"}, {"Chapter": "1", "sentence_range": "6187-6190", "Text": "(8 3), we have\nd\nd\ndd\nd\nd\nt\u03a6E\nt\nQ\ntQ\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\u03b5\uf8f8\uf8f7 =\n\u03b5\n0\n0\n1\nThis implies that for consistency,\n\u03b50\nd\nd\nt\u03a6E\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7  = i\n(8 4)\nThis is the missing term in Ampere\u2019s circuital law If we generalise\nthis law by adding to the total current carried by conductors through\nthe surface, another term which is e0 times the rate of change of electric\nflux through the same surface, the total has the same value of current i\nfor all surfaces"}, {"Chapter": "1", "sentence_range": "6188-6191", "Text": "3), we have\nd\nd\ndd\nd\nd\nt\u03a6E\nt\nQ\ntQ\n=\n\uf8eb\n\uf8ed\uf8ec\n\uf8f6\n\u03b5\uf8f8\uf8f7 =\n\u03b5\n0\n0\n1\nThis implies that for consistency,\n\u03b50\nd\nd\nt\u03a6E\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7  = i\n(8 4)\nThis is the missing term in Ampere\u2019s circuital law If we generalise\nthis law by adding to the total current carried by conductors through\nthe surface, another term which is e0 times the rate of change of electric\nflux through the same surface, the total has the same value of current i\nfor all surfaces If this is done, there is no contradiction in the value of B\nobtained anywhere using the generalised Ampere\u2019s law"}, {"Chapter": "1", "sentence_range": "6189-6192", "Text": "4)\nThis is the missing term in Ampere\u2019s circuital law If we generalise\nthis law by adding to the total current carried by conductors through\nthe surface, another term which is e0 times the rate of change of electric\nflux through the same surface, the total has the same value of current i\nfor all surfaces If this is done, there is no contradiction in the value of B\nobtained anywhere using the generalised Ampere\u2019s law B at the point P\nis non-zero no matter which surface is used for calculating it"}, {"Chapter": "1", "sentence_range": "6190-6193", "Text": "If we generalise\nthis law by adding to the total current carried by conductors through\nthe surface, another term which is e0 times the rate of change of electric\nflux through the same surface, the total has the same value of current i\nfor all surfaces If this is done, there is no contradiction in the value of B\nobtained anywhere using the generalised Ampere\u2019s law B at the point P\nis non-zero no matter which surface is used for calculating it B at a\npoint P outside the plates [Fig"}, {"Chapter": "1", "sentence_range": "6191-6194", "Text": "If this is done, there is no contradiction in the value of B\nobtained anywhere using the generalised Ampere\u2019s law B at the point P\nis non-zero no matter which surface is used for calculating it B at a\npoint P outside the plates [Fig 8"}, {"Chapter": "1", "sentence_range": "6192-6195", "Text": "B at the point P\nis non-zero no matter which surface is used for calculating it B at a\npoint P outside the plates [Fig 8 1(a)] is the same as at a point M just\ninside, as it should be"}, {"Chapter": "1", "sentence_range": "6193-6196", "Text": "B at a\npoint P outside the plates [Fig 8 1(a)] is the same as at a point M just\ninside, as it should be The current carried by conductors due to flow of\ncharges is called conduction current"}, {"Chapter": "1", "sentence_range": "6194-6197", "Text": "8 1(a)] is the same as at a point M just\ninside, as it should be The current carried by conductors due to flow of\ncharges is called conduction current The current, given by Eq"}, {"Chapter": "1", "sentence_range": "6195-6198", "Text": "1(a)] is the same as at a point M just\ninside, as it should be The current carried by conductors due to flow of\ncharges is called conduction current The current, given by Eq (8"}, {"Chapter": "1", "sentence_range": "6196-6199", "Text": "The current carried by conductors due to flow of\ncharges is called conduction current The current, given by Eq (8 4), is a\nnew term, and is due to changing electric field (or electric displacement,\nan old term still used sometimes)"}, {"Chapter": "1", "sentence_range": "6197-6200", "Text": "The current, given by Eq (8 4), is a\nnew term, and is due to changing electric field (or electric displacement,\nan old term still used sometimes) It is, therefore, called displacement\ncurrent or Maxwell\u2019s displacement current"}, {"Chapter": "1", "sentence_range": "6198-6201", "Text": "(8 4), is a\nnew term, and is due to changing electric field (or electric displacement,\nan old term still used sometimes) It is, therefore, called displacement\ncurrent or Maxwell\u2019s displacement current Figure 8"}, {"Chapter": "1", "sentence_range": "6199-6202", "Text": "4), is a\nnew term, and is due to changing electric field (or electric displacement,\nan old term still used sometimes) It is, therefore, called displacement\ncurrent or Maxwell\u2019s displacement current Figure 8 2 shows the electric\nand magnetic fields inside the parallel plate capacitor discussed above"}, {"Chapter": "1", "sentence_range": "6200-6203", "Text": "It is, therefore, called displacement\ncurrent or Maxwell\u2019s displacement current Figure 8 2 shows the electric\nand magnetic fields inside the parallel plate capacitor discussed above The generalisation made by Maxwell then is the following"}, {"Chapter": "1", "sentence_range": "6201-6204", "Text": "Figure 8 2 shows the electric\nand magnetic fields inside the parallel plate capacitor discussed above The generalisation made by Maxwell then is the following The source\nof a magnetic field is not just the conduction electric current due to flowing\nFIGURE 8"}, {"Chapter": "1", "sentence_range": "6202-6205", "Text": "2 shows the electric\nand magnetic fields inside the parallel plate capacitor discussed above The generalisation made by Maxwell then is the following The source\nof a magnetic field is not just the conduction electric current due to flowing\nFIGURE 8 1 A\nparallel plate\ncapacitor C, as part of\na circuit through\nwhich a time\ndependent current\ni (t) flows,  (a) a loop of\nradius r, to determine\nmagnetic field at a\npoint P on the loop;\n(b) a pot-shaped\nsurface passing\nthrough the interior\nbetween the capacitor\nplates with the loop\nshown in (a) as its\nrim; (c) a tiffin-\nshaped surface with\nthe circular loop as\nits rim and a flat\ncircular bottom S\nbetween the capacitor\nplates"}, {"Chapter": "1", "sentence_range": "6203-6206", "Text": "The generalisation made by Maxwell then is the following The source\nof a magnetic field is not just the conduction electric current due to flowing\nFIGURE 8 1 A\nparallel plate\ncapacitor C, as part of\na circuit through\nwhich a time\ndependent current\ni (t) flows,  (a) a loop of\nradius r, to determine\nmagnetic field at a\npoint P on the loop;\n(b) a pot-shaped\nsurface passing\nthrough the interior\nbetween the capacitor\nplates with the loop\nshown in (a) as its\nrim; (c) a tiffin-\nshaped surface with\nthe circular loop as\nits rim and a flat\ncircular bottom S\nbetween the capacitor\nplates The arrows\nshow uniform electric\nfield between the\ncapacitor plates"}, {"Chapter": "1", "sentence_range": "6204-6207", "Text": "The source\nof a magnetic field is not just the conduction electric current due to flowing\nFIGURE 8 1 A\nparallel plate\ncapacitor C, as part of\na circuit through\nwhich a time\ndependent current\ni (t) flows,  (a) a loop of\nradius r, to determine\nmagnetic field at a\npoint P on the loop;\n(b) a pot-shaped\nsurface passing\nthrough the interior\nbetween the capacitor\nplates with the loop\nshown in (a) as its\nrim; (c) a tiffin-\nshaped surface with\nthe circular loop as\nits rim and a flat\ncircular bottom S\nbetween the capacitor\nplates The arrows\nshow uniform electric\nfield between the\ncapacitor plates Rationalised 2023-24\nPhysics\n204\ncharges, but also the time rate of change of electric field"}, {"Chapter": "1", "sentence_range": "6205-6208", "Text": "1 A\nparallel plate\ncapacitor C, as part of\na circuit through\nwhich a time\ndependent current\ni (t) flows,  (a) a loop of\nradius r, to determine\nmagnetic field at a\npoint P on the loop;\n(b) a pot-shaped\nsurface passing\nthrough the interior\nbetween the capacitor\nplates with the loop\nshown in (a) as its\nrim; (c) a tiffin-\nshaped surface with\nthe circular loop as\nits rim and a flat\ncircular bottom S\nbetween the capacitor\nplates The arrows\nshow uniform electric\nfield between the\ncapacitor plates Rationalised 2023-24\nPhysics\n204\ncharges, but also the time rate of change of electric field More\nprecisely, the total current i is the sum of the conduction current\ndenoted by ic, and the displacement current denoted by id (= e0 (dFE/\ndt))"}, {"Chapter": "1", "sentence_range": "6206-6209", "Text": "The arrows\nshow uniform electric\nfield between the\ncapacitor plates Rationalised 2023-24\nPhysics\n204\ncharges, but also the time rate of change of electric field More\nprecisely, the total current i is the sum of the conduction current\ndenoted by ic, and the displacement current denoted by id (= e0 (dFE/\ndt)) So we have\n0\nd\nd\nE\nc\nd\nc\ni\ni\ni\ni\n\u03a6t\n\u03b5\n=\n+\n=\n+\n(8"}, {"Chapter": "1", "sentence_range": "6207-6210", "Text": "Rationalised 2023-24\nPhysics\n204\ncharges, but also the time rate of change of electric field More\nprecisely, the total current i is the sum of the conduction current\ndenoted by ic, and the displacement current denoted by id (= e0 (dFE/\ndt)) So we have\n0\nd\nd\nE\nc\nd\nc\ni\ni\ni\ni\n\u03a6t\n\u03b5\n=\n+\n=\n+\n(8 5)\nIn explicit terms, this means that outside the capacitor plates,\nwe have only conduction current ic = i, and no displacement\ncurrent, i"}, {"Chapter": "1", "sentence_range": "6208-6211", "Text": "More\nprecisely, the total current i is the sum of the conduction current\ndenoted by ic, and the displacement current denoted by id (= e0 (dFE/\ndt)) So we have\n0\nd\nd\nE\nc\nd\nc\ni\ni\ni\ni\n\u03a6t\n\u03b5\n=\n+\n=\n+\n(8 5)\nIn explicit terms, this means that outside the capacitor plates,\nwe have only conduction current ic = i, and no displacement\ncurrent, i e"}, {"Chapter": "1", "sentence_range": "6209-6212", "Text": "So we have\n0\nd\nd\nE\nc\nd\nc\ni\ni\ni\ni\n\u03a6t\n\u03b5\n=\n+\n=\n+\n(8 5)\nIn explicit terms, this means that outside the capacitor plates,\nwe have only conduction current ic = i, and no displacement\ncurrent, i e , id = 0"}, {"Chapter": "1", "sentence_range": "6210-6213", "Text": "5)\nIn explicit terms, this means that outside the capacitor plates,\nwe have only conduction current ic = i, and no displacement\ncurrent, i e , id = 0 On the other hand, inside the capacitor, there is\nno conduction current, i"}, {"Chapter": "1", "sentence_range": "6211-6214", "Text": "e , id = 0 On the other hand, inside the capacitor, there is\nno conduction current, i e"}, {"Chapter": "1", "sentence_range": "6212-6215", "Text": ", id = 0 On the other hand, inside the capacitor, there is\nno conduction current, i e , ic = 0, and there is only displacement\ncurrent, so that id = i"}, {"Chapter": "1", "sentence_range": "6213-6216", "Text": "On the other hand, inside the capacitor, there is\nno conduction current, i e , ic = 0, and there is only displacement\ncurrent, so that id = i The generalised (and correct) Ampere\u2019s circuital law has the same\nform as Eq"}, {"Chapter": "1", "sentence_range": "6214-6217", "Text": "e , ic = 0, and there is only displacement\ncurrent, so that id = i The generalised (and correct) Ampere\u2019s circuital law has the same\nform as Eq (8"}, {"Chapter": "1", "sentence_range": "6215-6218", "Text": ", ic = 0, and there is only displacement\ncurrent, so that id = i The generalised (and correct) Ampere\u2019s circuital law has the same\nform as Eq (8 1), with one difference: \u201cthe total current passing\nthrough any surface of which the closed loop is the perimeter\u201d is\nthe sum of the conduction current and the displacement current"}, {"Chapter": "1", "sentence_range": "6216-6219", "Text": "The generalised (and correct) Ampere\u2019s circuital law has the same\nform as Eq (8 1), with one difference: \u201cthe total current passing\nthrough any surface of which the closed loop is the perimeter\u201d is\nthe sum of the conduction current and the displacement current The generalised law is\nB\n\ufffdgl\nd =\nd\nd\n0\n\u00b5\n\u00b5\n\u03b5\n0\n0\ni\nt\nc\nE\n+\n\u222b\n\u03a6\n(8"}, {"Chapter": "1", "sentence_range": "6217-6220", "Text": "(8 1), with one difference: \u201cthe total current passing\nthrough any surface of which the closed loop is the perimeter\u201d is\nthe sum of the conduction current and the displacement current The generalised law is\nB\n\ufffdgl\nd =\nd\nd\n0\n\u00b5\n\u00b5\n\u03b5\n0\n0\ni\nt\nc\nE\n+\n\u222b\n\u03a6\n(8 6)\nand is known as Ampere-Maxwell law"}, {"Chapter": "1", "sentence_range": "6218-6221", "Text": "1), with one difference: \u201cthe total current passing\nthrough any surface of which the closed loop is the perimeter\u201d is\nthe sum of the conduction current and the displacement current The generalised law is\nB\n\ufffdgl\nd =\nd\nd\n0\n\u00b5\n\u00b5\n\u03b5\n0\n0\ni\nt\nc\nE\n+\n\u222b\n\u03a6\n(8 6)\nand is known as Ampere-Maxwell law In all respects, the displacement current has the same physical\neffects as the conduction current"}, {"Chapter": "1", "sentence_range": "6219-6222", "Text": "The generalised law is\nB\n\ufffdgl\nd =\nd\nd\n0\n\u00b5\n\u00b5\n\u03b5\n0\n0\ni\nt\nc\nE\n+\n\u222b\n\u03a6\n(8 6)\nand is known as Ampere-Maxwell law In all respects, the displacement current has the same physical\neffects as the conduction current In some cases, for example, steady\nelectric fields in a conducting wire, the displacement current may\nbe zero since the electric field E does not change with time"}, {"Chapter": "1", "sentence_range": "6220-6223", "Text": "6)\nand is known as Ampere-Maxwell law In all respects, the displacement current has the same physical\neffects as the conduction current In some cases, for example, steady\nelectric fields in a conducting wire, the displacement current may\nbe zero since the electric field E does not change with time In other\ncases, for example, the charging capacitor above, both conduction\nand displacement currents may be present in different regions of\nspace"}, {"Chapter": "1", "sentence_range": "6221-6224", "Text": "In all respects, the displacement current has the same physical\neffects as the conduction current In some cases, for example, steady\nelectric fields in a conducting wire, the displacement current may\nbe zero since the electric field E does not change with time In other\ncases, for example, the charging capacitor above, both conduction\nand displacement currents may be present in different regions of\nspace In most of the cases, they both may be present in the same\nregion of space, as there exist no perfectly conducting or perfectly\ninsulating medium"}, {"Chapter": "1", "sentence_range": "6222-6225", "Text": "In some cases, for example, steady\nelectric fields in a conducting wire, the displacement current may\nbe zero since the electric field E does not change with time In other\ncases, for example, the charging capacitor above, both conduction\nand displacement currents may be present in different regions of\nspace In most of the cases, they both may be present in the same\nregion of space, as there exist no perfectly conducting or perfectly\ninsulating medium Most interestingly, there may be large regions\nof space where there is no conduction current, but there is only a\ndisplacement current due to time-varying electric fields"}, {"Chapter": "1", "sentence_range": "6223-6226", "Text": "In other\ncases, for example, the charging capacitor above, both conduction\nand displacement currents may be present in different regions of\nspace In most of the cases, they both may be present in the same\nregion of space, as there exist no perfectly conducting or perfectly\ninsulating medium Most interestingly, there may be large regions\nof space where there is no conduction current, but there is only a\ndisplacement current due to time-varying electric fields In such a\nregion, we expect a magnetic field, though there is no (conduction)\ncurrent source nearby"}, {"Chapter": "1", "sentence_range": "6224-6227", "Text": "In most of the cases, they both may be present in the same\nregion of space, as there exist no perfectly conducting or perfectly\ninsulating medium Most interestingly, there may be large regions\nof space where there is no conduction current, but there is only a\ndisplacement current due to time-varying electric fields In such a\nregion, we expect a magnetic field, though there is no (conduction)\ncurrent source nearby The prediction of such a displacement current\ncan be verified experimentally"}, {"Chapter": "1", "sentence_range": "6225-6228", "Text": "Most interestingly, there may be large regions\nof space where there is no conduction current, but there is only a\ndisplacement current due to time-varying electric fields In such a\nregion, we expect a magnetic field, though there is no (conduction)\ncurrent source nearby The prediction of such a displacement current\ncan be verified experimentally For example, a magnetic field (say at point\nM) between the plates of the capacitor in Fig"}, {"Chapter": "1", "sentence_range": "6226-6229", "Text": "In such a\nregion, we expect a magnetic field, though there is no (conduction)\ncurrent source nearby The prediction of such a displacement current\ncan be verified experimentally For example, a magnetic field (say at point\nM) between the plates of the capacitor in Fig 8"}, {"Chapter": "1", "sentence_range": "6227-6230", "Text": "The prediction of such a displacement current\ncan be verified experimentally For example, a magnetic field (say at point\nM) between the plates of the capacitor in Fig 8 2(a) can be measured and\nis seen to be the same as that just outside (at P)"}, {"Chapter": "1", "sentence_range": "6228-6231", "Text": "For example, a magnetic field (say at point\nM) between the plates of the capacitor in Fig 8 2(a) can be measured and\nis seen to be the same as that just outside (at P) The displacement current has (literally) far reaching consequences"}, {"Chapter": "1", "sentence_range": "6229-6232", "Text": "8 2(a) can be measured and\nis seen to be the same as that just outside (at P) The displacement current has (literally) far reaching consequences One thing we immediately notice is that the laws of electricity and\nmagnetism are now more symmetrical*"}, {"Chapter": "1", "sentence_range": "6230-6233", "Text": "2(a) can be measured and\nis seen to be the same as that just outside (at P) The displacement current has (literally) far reaching consequences One thing we immediately notice is that the laws of electricity and\nmagnetism are now more symmetrical* Faraday\u2019s law of induction states\nthat there is an induced emf equal to the rate of change of magnetic flux"}, {"Chapter": "1", "sentence_range": "6231-6234", "Text": "The displacement current has (literally) far reaching consequences One thing we immediately notice is that the laws of electricity and\nmagnetism are now more symmetrical* Faraday\u2019s law of induction states\nthat there is an induced emf equal to the rate of change of magnetic flux Now, since the emf between two points 1 and 2 is the work done per unit\ncharge in taking it from 1 to 2, the existence of an emf implies the existence\nof an electric field"}, {"Chapter": "1", "sentence_range": "6232-6235", "Text": "One thing we immediately notice is that the laws of electricity and\nmagnetism are now more symmetrical* Faraday\u2019s law of induction states\nthat there is an induced emf equal to the rate of change of magnetic flux Now, since the emf between two points 1 and 2 is the work done per unit\ncharge in taking it from 1 to 2, the existence of an emf implies the existence\nof an electric field So, we can rephrase Faraday\u2019s law of electromagnetic\ninduction by saying that a magnetic field, changing with time, gives rise\nto an electric field"}, {"Chapter": "1", "sentence_range": "6233-6236", "Text": "Faraday\u2019s law of induction states\nthat there is an induced emf equal to the rate of change of magnetic flux Now, since the emf between two points 1 and 2 is the work done per unit\ncharge in taking it from 1 to 2, the existence of an emf implies the existence\nof an electric field So, we can rephrase Faraday\u2019s law of electromagnetic\ninduction by saying that a magnetic field, changing with time, gives rise\nto an electric field Then, the fact that an electric field changing with\ntime gives rise to a magnetic field, is the symmetrical counterpart, and is\nFIGURE 8"}, {"Chapter": "1", "sentence_range": "6234-6237", "Text": "Now, since the emf between two points 1 and 2 is the work done per unit\ncharge in taking it from 1 to 2, the existence of an emf implies the existence\nof an electric field So, we can rephrase Faraday\u2019s law of electromagnetic\ninduction by saying that a magnetic field, changing with time, gives rise\nto an electric field Then, the fact that an electric field changing with\ntime gives rise to a magnetic field, is the symmetrical counterpart, and is\nFIGURE 8 2 (a) The\nelectric and magnetic\nfields E and B between\nthe capacitor plates, at\nthe point M"}, {"Chapter": "1", "sentence_range": "6235-6238", "Text": "So, we can rephrase Faraday\u2019s law of electromagnetic\ninduction by saying that a magnetic field, changing with time, gives rise\nto an electric field Then, the fact that an electric field changing with\ntime gives rise to a magnetic field, is the symmetrical counterpart, and is\nFIGURE 8 2 (a) The\nelectric and magnetic\nfields E and B between\nthe capacitor plates, at\nthe point M (b) A cross\nsectional view of Fig"}, {"Chapter": "1", "sentence_range": "6236-6239", "Text": "Then, the fact that an electric field changing with\ntime gives rise to a magnetic field, is the symmetrical counterpart, and is\nFIGURE 8 2 (a) The\nelectric and magnetic\nfields E and B between\nthe capacitor plates, at\nthe point M (b) A cross\nsectional view of Fig (a)"}, {"Chapter": "1", "sentence_range": "6237-6240", "Text": "2 (a) The\nelectric and magnetic\nfields E and B between\nthe capacitor plates, at\nthe point M (b) A cross\nsectional view of Fig (a) *\nThey are still not perfectly symmetrical; there are no known sources of magnetic\nfield (magnetic monopoles) analogous to electric charges which are sources of\nelectric field"}, {"Chapter": "1", "sentence_range": "6238-6241", "Text": "(b) A cross\nsectional view of Fig (a) *\nThey are still not perfectly symmetrical; there are no known sources of magnetic\nfield (magnetic monopoles) analogous to electric charges which are sources of\nelectric field Rationalised 2023-24\n205\nElectromagnetic\nWaves\na consequence of the displacement current being a source of a magnetic\nfield"}, {"Chapter": "1", "sentence_range": "6239-6242", "Text": "(a) *\nThey are still not perfectly symmetrical; there are no known sources of magnetic\nfield (magnetic monopoles) analogous to electric charges which are sources of\nelectric field Rationalised 2023-24\n205\nElectromagnetic\nWaves\na consequence of the displacement current being a source of a magnetic\nfield Thus, time- dependent electric and magnetic fields give rise to each\nother"}, {"Chapter": "1", "sentence_range": "6240-6243", "Text": "*\nThey are still not perfectly symmetrical; there are no known sources of magnetic\nfield (magnetic monopoles) analogous to electric charges which are sources of\nelectric field Rationalised 2023-24\n205\nElectromagnetic\nWaves\na consequence of the displacement current being a source of a magnetic\nfield Thus, time- dependent electric and magnetic fields give rise to each\nother Faraday\u2019s law of electromagnetic induction and Ampere-Maxwell\nlaw give a quantitative expression of this statement, with the current\nbeing the total current, as in Eq"}, {"Chapter": "1", "sentence_range": "6241-6244", "Text": "Rationalised 2023-24\n205\nElectromagnetic\nWaves\na consequence of the displacement current being a source of a magnetic\nfield Thus, time- dependent electric and magnetic fields give rise to each\nother Faraday\u2019s law of electromagnetic induction and Ampere-Maxwell\nlaw give a quantitative expression of this statement, with the current\nbeing the total current, as in Eq (8"}, {"Chapter": "1", "sentence_range": "6242-6245", "Text": "Thus, time- dependent electric and magnetic fields give rise to each\nother Faraday\u2019s law of electromagnetic induction and Ampere-Maxwell\nlaw give a quantitative expression of this statement, with the current\nbeing the total current, as in Eq (8 5)"}, {"Chapter": "1", "sentence_range": "6243-6246", "Text": "Faraday\u2019s law of electromagnetic induction and Ampere-Maxwell\nlaw give a quantitative expression of this statement, with the current\nbeing the total current, as in Eq (8 5) One very important consequence\nof this symmetry is the existence of electromagnetic waves, which we\ndiscuss qualitatively in the next section"}, {"Chapter": "1", "sentence_range": "6244-6247", "Text": "(8 5) One very important consequence\nof this symmetry is the existence of electromagnetic waves, which we\ndiscuss qualitatively in the next section MAXWELL\u2019S EQUATIONS IN VACUUM\n1"}, {"Chapter": "1", "sentence_range": "6245-6248", "Text": "5) One very important consequence\nof this symmetry is the existence of electromagnetic waves, which we\ndiscuss qualitatively in the next section MAXWELL\u2019S EQUATIONS IN VACUUM\n1 \u201cE"}, {"Chapter": "1", "sentence_range": "6246-6249", "Text": "One very important consequence\nof this symmetry is the existence of electromagnetic waves, which we\ndiscuss qualitatively in the next section MAXWELL\u2019S EQUATIONS IN VACUUM\n1 \u201cE dA = Q/\u27120\n(Gauss\u2019s Law for electricity)\n2"}, {"Chapter": "1", "sentence_range": "6247-6250", "Text": "MAXWELL\u2019S EQUATIONS IN VACUUM\n1 \u201cE dA = Q/\u27120\n(Gauss\u2019s Law for electricity)\n2 \u201cB"}, {"Chapter": "1", "sentence_range": "6248-6251", "Text": "\u201cE dA = Q/\u27120\n(Gauss\u2019s Law for electricity)\n2 \u201cB dA = 0\n(Gauss\u2019s Law for magnetism)\n3"}, {"Chapter": "1", "sentence_range": "6249-6252", "Text": "dA = Q/\u27120\n(Gauss\u2019s Law for electricity)\n2 \u201cB dA = 0\n(Gauss\u2019s Law for magnetism)\n3 \u201cE"}, {"Chapter": "1", "sentence_range": "6250-6253", "Text": "\u201cB dA = 0\n(Gauss\u2019s Law for magnetism)\n3 \u201cE dl = \u2013d\nd\nt\u03a6B\nl=\n(Faraday\u2019s Law)\n4"}, {"Chapter": "1", "sentence_range": "6251-6254", "Text": "dA = 0\n(Gauss\u2019s Law for magnetism)\n3 \u201cE dl = \u2013d\nd\nt\u03a6B\nl=\n(Faraday\u2019s Law)\n4 \u201cB"}, {"Chapter": "1", "sentence_range": "6252-6255", "Text": "\u201cE dl = \u2013d\nd\nt\u03a6B\nl=\n(Faraday\u2019s Law)\n4 \u201cB dl ==\nd\nd\n0\n\u00b5\n\u00b5 \u03b5\n0\n0\ni\nt\nc\nE\n+\n\u03a6\n(Ampere \u2013 Maxwell Law)\n8"}, {"Chapter": "1", "sentence_range": "6253-6256", "Text": "dl = \u2013d\nd\nt\u03a6B\nl=\n(Faraday\u2019s Law)\n4 \u201cB dl ==\nd\nd\n0\n\u00b5\n\u00b5 \u03b5\n0\n0\ni\nt\nc\nE\n+\n\u03a6\n(Ampere \u2013 Maxwell Law)\n8 3  ELECTROMAGNETIC WAVES\n8"}, {"Chapter": "1", "sentence_range": "6254-6257", "Text": "\u201cB dl ==\nd\nd\n0\n\u00b5\n\u00b5 \u03b5\n0\n0\ni\nt\nc\nE\n+\n\u03a6\n(Ampere \u2013 Maxwell Law)\n8 3  ELECTROMAGNETIC WAVES\n8 3"}, {"Chapter": "1", "sentence_range": "6255-6258", "Text": "dl ==\nd\nd\n0\n\u00b5\n\u00b5 \u03b5\n0\n0\ni\nt\nc\nE\n+\n\u03a6\n(Ampere \u2013 Maxwell Law)\n8 3  ELECTROMAGNETIC WAVES\n8 3 1  Sources of electromagnetic waves\nHow are electromagnetic waves produced"}, {"Chapter": "1", "sentence_range": "6256-6259", "Text": "3  ELECTROMAGNETIC WAVES\n8 3 1  Sources of electromagnetic waves\nHow are electromagnetic waves produced Neither stationary charges\nnor charges in uniform motion (steady currents) can be sources of\nelectromagnetic waves"}, {"Chapter": "1", "sentence_range": "6257-6260", "Text": "3 1  Sources of electromagnetic waves\nHow are electromagnetic waves produced Neither stationary charges\nnor charges in uniform motion (steady currents) can be sources of\nelectromagnetic waves The former produces only electrostatic fields, while\nthe latter produces magnetic fields that, however, do not vary with time"}, {"Chapter": "1", "sentence_range": "6258-6261", "Text": "1  Sources of electromagnetic waves\nHow are electromagnetic waves produced Neither stationary charges\nnor charges in uniform motion (steady currents) can be sources of\nelectromagnetic waves The former produces only electrostatic fields, while\nthe latter produces magnetic fields that, however, do not vary with time It is an important result of Maxwell\u2019s theory that accelerated charges\nradiate electromagnetic waves"}, {"Chapter": "1", "sentence_range": "6259-6262", "Text": "Neither stationary charges\nnor charges in uniform motion (steady currents) can be sources of\nelectromagnetic waves The former produces only electrostatic fields, while\nthe latter produces magnetic fields that, however, do not vary with time It is an important result of Maxwell\u2019s theory that accelerated charges\nradiate electromagnetic waves The proof of this basic result is beyond\nthe scope of this book, but we can accept it on the basis of rough,\nqualitative reasoning"}, {"Chapter": "1", "sentence_range": "6260-6263", "Text": "The former produces only electrostatic fields, while\nthe latter produces magnetic fields that, however, do not vary with time It is an important result of Maxwell\u2019s theory that accelerated charges\nradiate electromagnetic waves The proof of this basic result is beyond\nthe scope of this book, but we can accept it on the basis of rough,\nqualitative reasoning Consider a charge oscillating with some frequency"}, {"Chapter": "1", "sentence_range": "6261-6264", "Text": "It is an important result of Maxwell\u2019s theory that accelerated charges\nradiate electromagnetic waves The proof of this basic result is beyond\nthe scope of this book, but we can accept it on the basis of rough,\nqualitative reasoning Consider a charge oscillating with some frequency (An oscillating charge is an example of accelerating charge"}, {"Chapter": "1", "sentence_range": "6262-6265", "Text": "The proof of this basic result is beyond\nthe scope of this book, but we can accept it on the basis of rough,\nqualitative reasoning Consider a charge oscillating with some frequency (An oscillating charge is an example of accelerating charge )   This\nproduces an oscillating electric field in space, which produces an\noscillating magnetic field, which in turn, is a source of oscillating electric\nfield, and so on"}, {"Chapter": "1", "sentence_range": "6263-6266", "Text": "Consider a charge oscillating with some frequency (An oscillating charge is an example of accelerating charge )   This\nproduces an oscillating electric field in space, which produces an\noscillating magnetic field, which in turn, is a source of oscillating electric\nfield, and so on The oscillating electric and magnetic fields thus\nregenerate each other, so to speak, as the wave propagates through the\nspace"}, {"Chapter": "1", "sentence_range": "6264-6267", "Text": "(An oscillating charge is an example of accelerating charge )   This\nproduces an oscillating electric field in space, which produces an\noscillating magnetic field, which in turn, is a source of oscillating electric\nfield, and so on The oscillating electric and magnetic fields thus\nregenerate each other, so to speak, as the wave propagates through the\nspace The frequency of the electromagnetic wave naturally equals the\nfrequency of oscillation of the charge"}, {"Chapter": "1", "sentence_range": "6265-6268", "Text": ")   This\nproduces an oscillating electric field in space, which produces an\noscillating magnetic field, which in turn, is a source of oscillating electric\nfield, and so on The oscillating electric and magnetic fields thus\nregenerate each other, so to speak, as the wave propagates through the\nspace The frequency of the electromagnetic wave naturally equals the\nfrequency of oscillation of the charge The energy associated with the\npropagating wave comes at the expense of the energy of the source \u2013 the\naccelerated charge"}, {"Chapter": "1", "sentence_range": "6266-6269", "Text": "The oscillating electric and magnetic fields thus\nregenerate each other, so to speak, as the wave propagates through the\nspace The frequency of the electromagnetic wave naturally equals the\nfrequency of oscillation of the charge The energy associated with the\npropagating wave comes at the expense of the energy of the source \u2013 the\naccelerated charge From the preceding discussion, it might appear easy to test the\nprediction that light is an electromagnetic wave"}, {"Chapter": "1", "sentence_range": "6267-6270", "Text": "The frequency of the electromagnetic wave naturally equals the\nfrequency of oscillation of the charge The energy associated with the\npropagating wave comes at the expense of the energy of the source \u2013 the\naccelerated charge From the preceding discussion, it might appear easy to test the\nprediction that light is an electromagnetic wave We might think that all\nwe needed to do was to set up an ac circuit in which the current oscillate\nat the frequency of visible light, say, yellow light"}, {"Chapter": "1", "sentence_range": "6268-6271", "Text": "The energy associated with the\npropagating wave comes at the expense of the energy of the source \u2013 the\naccelerated charge From the preceding discussion, it might appear easy to test the\nprediction that light is an electromagnetic wave We might think that all\nwe needed to do was to set up an ac circuit in which the current oscillate\nat the frequency of visible light, say, yellow light But, alas, that is not\npossible"}, {"Chapter": "1", "sentence_range": "6269-6272", "Text": "From the preceding discussion, it might appear easy to test the\nprediction that light is an electromagnetic wave We might think that all\nwe needed to do was to set up an ac circuit in which the current oscillate\nat the frequency of visible light, say, yellow light But, alas, that is not\npossible The frequency of yellow light is about 6 \u00d7 1014 Hz, while the\nfrequency that we get even with modern electronic circuits is hardly about\n1011 Hz"}, {"Chapter": "1", "sentence_range": "6270-6273", "Text": "We might think that all\nwe needed to do was to set up an ac circuit in which the current oscillate\nat the frequency of visible light, say, yellow light But, alas, that is not\npossible The frequency of yellow light is about 6 \u00d7 1014 Hz, while the\nfrequency that we get even with modern electronic circuits is hardly about\n1011 Hz This is why the experimental demonstration of electromagnetic\nRationalised 2023-24\nPhysics\n206\nwave had to come in the low frequency region (the radio\nwave region), as in the Hertz\u2019s experiment (1887)"}, {"Chapter": "1", "sentence_range": "6271-6274", "Text": "But, alas, that is not\npossible The frequency of yellow light is about 6 \u00d7 1014 Hz, while the\nfrequency that we get even with modern electronic circuits is hardly about\n1011 Hz This is why the experimental demonstration of electromagnetic\nRationalised 2023-24\nPhysics\n206\nwave had to come in the low frequency region (the radio\nwave region), as in the Hertz\u2019s experiment (1887) Hertz\u2019s successful experimental test of Maxwell\u2019s\ntheory created a sensation and sparked off other\nimportant works in this field"}, {"Chapter": "1", "sentence_range": "6272-6275", "Text": "The frequency of yellow light is about 6 \u00d7 1014 Hz, while the\nfrequency that we get even with modern electronic circuits is hardly about\n1011 Hz This is why the experimental demonstration of electromagnetic\nRationalised 2023-24\nPhysics\n206\nwave had to come in the low frequency region (the radio\nwave region), as in the Hertz\u2019s experiment (1887) Hertz\u2019s successful experimental test of Maxwell\u2019s\ntheory created a sensation and sparked off other\nimportant works in this field Two important\nachievements in this connection deserve mention"}, {"Chapter": "1", "sentence_range": "6273-6276", "Text": "This is why the experimental demonstration of electromagnetic\nRationalised 2023-24\nPhysics\n206\nwave had to come in the low frequency region (the radio\nwave region), as in the Hertz\u2019s experiment (1887) Hertz\u2019s successful experimental test of Maxwell\u2019s\ntheory created a sensation and sparked off other\nimportant works in this field Two important\nachievements in this connection deserve mention Seven\nyears after Hertz, Jagdish Chandra Bose, working at\nCalcutta (now Kolkata), succeeded in producing and\nobserving electromagnetic waves of much shorter\nwavelength (25 mm to 5 mm)"}, {"Chapter": "1", "sentence_range": "6274-6277", "Text": "Hertz\u2019s successful experimental test of Maxwell\u2019s\ntheory created a sensation and sparked off other\nimportant works in this field Two important\nachievements in this connection deserve mention Seven\nyears after Hertz, Jagdish Chandra Bose, working at\nCalcutta (now Kolkata), succeeded in producing and\nobserving electromagnetic waves of much shorter\nwavelength (25 mm to 5 mm) His experiment, like that\nof Hertz\u2019s, was confined to the laboratory"}, {"Chapter": "1", "sentence_range": "6275-6278", "Text": "Two important\nachievements in this connection deserve mention Seven\nyears after Hertz, Jagdish Chandra Bose, working at\nCalcutta (now Kolkata), succeeded in producing and\nobserving electromagnetic waves of much shorter\nwavelength (25 mm to 5 mm) His experiment, like that\nof Hertz\u2019s, was confined to the laboratory At around the same time, Guglielmo Marconi in Italy\nfollowed Hertz\u2019s work and succeeded in transmitting\nelectromagnetic waves over distances of many kilometres"}, {"Chapter": "1", "sentence_range": "6276-6279", "Text": "Seven\nyears after Hertz, Jagdish Chandra Bose, working at\nCalcutta (now Kolkata), succeeded in producing and\nobserving electromagnetic waves of much shorter\nwavelength (25 mm to 5 mm) His experiment, like that\nof Hertz\u2019s, was confined to the laboratory At around the same time, Guglielmo Marconi in Italy\nfollowed Hertz\u2019s work and succeeded in transmitting\nelectromagnetic waves over distances of many kilometres Marconi\u2019s experiment marks the beginning of the field of\ncommunication using electromagnetic waves"}, {"Chapter": "1", "sentence_range": "6277-6280", "Text": "His experiment, like that\nof Hertz\u2019s, was confined to the laboratory At around the same time, Guglielmo Marconi in Italy\nfollowed Hertz\u2019s work and succeeded in transmitting\nelectromagnetic waves over distances of many kilometres Marconi\u2019s experiment marks the beginning of the field of\ncommunication using electromagnetic waves 8"}, {"Chapter": "1", "sentence_range": "6278-6281", "Text": "At around the same time, Guglielmo Marconi in Italy\nfollowed Hertz\u2019s work and succeeded in transmitting\nelectromagnetic waves over distances of many kilometres Marconi\u2019s experiment marks the beginning of the field of\ncommunication using electromagnetic waves 8 3"}, {"Chapter": "1", "sentence_range": "6279-6282", "Text": "Marconi\u2019s experiment marks the beginning of the field of\ncommunication using electromagnetic waves 8 3 2  Nature of electromagnetic waves\nIt can be shown from Maxwell\u2019s equations that electric\nand magnetic fields in an electromagnetic wave are\nperpendicular to each other, and to the direction of\npropagation"}, {"Chapter": "1", "sentence_range": "6280-6283", "Text": "8 3 2  Nature of electromagnetic waves\nIt can be shown from Maxwell\u2019s equations that electric\nand magnetic fields in an electromagnetic wave are\nperpendicular to each other, and to the direction of\npropagation It appears reasonable, say from our\ndiscussion of the displacement current"}, {"Chapter": "1", "sentence_range": "6281-6284", "Text": "3 2  Nature of electromagnetic waves\nIt can be shown from Maxwell\u2019s equations that electric\nand magnetic fields in an electromagnetic wave are\nperpendicular to each other, and to the direction of\npropagation It appears reasonable, say from our\ndiscussion of the displacement current Consider\nFig"}, {"Chapter": "1", "sentence_range": "6282-6285", "Text": "2  Nature of electromagnetic waves\nIt can be shown from Maxwell\u2019s equations that electric\nand magnetic fields in an electromagnetic wave are\nperpendicular to each other, and to the direction of\npropagation It appears reasonable, say from our\ndiscussion of the displacement current Consider\nFig 8"}, {"Chapter": "1", "sentence_range": "6283-6286", "Text": "It appears reasonable, say from our\ndiscussion of the displacement current Consider\nFig 8 2"}, {"Chapter": "1", "sentence_range": "6284-6287", "Text": "Consider\nFig 8 2 The electric field inside the plates of the capacitor\nis directed perpendicular to the plates"}, {"Chapter": "1", "sentence_range": "6285-6288", "Text": "8 2 The electric field inside the plates of the capacitor\nis directed perpendicular to the plates The magnetic\nfield this gives rise to via the displacement current is\nalong the perimeter of a circle parallel to the capacitor\nplates"}, {"Chapter": "1", "sentence_range": "6286-6289", "Text": "2 The electric field inside the plates of the capacitor\nis directed perpendicular to the plates The magnetic\nfield this gives rise to via the displacement current is\nalong the perimeter of a circle parallel to the capacitor\nplates So B and E are perpendicular in this case"}, {"Chapter": "1", "sentence_range": "6287-6290", "Text": "The electric field inside the plates of the capacitor\nis directed perpendicular to the plates The magnetic\nfield this gives rise to via the displacement current is\nalong the perimeter of a circle parallel to the capacitor\nplates So B and E are perpendicular in this case This\nis a general feature"}, {"Chapter": "1", "sentence_range": "6288-6291", "Text": "The magnetic\nfield this gives rise to via the displacement current is\nalong the perimeter of a circle parallel to the capacitor\nplates So B and E are perpendicular in this case This\nis a general feature In Fig"}, {"Chapter": "1", "sentence_range": "6289-6292", "Text": "So B and E are perpendicular in this case This\nis a general feature In Fig 8"}, {"Chapter": "1", "sentence_range": "6290-6293", "Text": "This\nis a general feature In Fig 8 3, we show a typical example of a plane\nelectromagnetic wave propagating along the z direction\n(the fields are shown as a function of the z coordinate, at\na given time t)"}, {"Chapter": "1", "sentence_range": "6291-6294", "Text": "In Fig 8 3, we show a typical example of a plane\nelectromagnetic wave propagating along the z direction\n(the fields are shown as a function of the z coordinate, at\na given time t) The electric field Ex is along the x-axis,\nand varies sinusoidally with z, at a given time"}, {"Chapter": "1", "sentence_range": "6292-6295", "Text": "8 3, we show a typical example of a plane\nelectromagnetic wave propagating along the z direction\n(the fields are shown as a function of the z coordinate, at\na given time t) The electric field Ex is along the x-axis,\nand varies sinusoidally with z, at a given time The\nmagnetic field By is along the y-axis, and again varies\nsinusoidally with z"}, {"Chapter": "1", "sentence_range": "6293-6296", "Text": "3, we show a typical example of a plane\nelectromagnetic wave propagating along the z direction\n(the fields are shown as a function of the z coordinate, at\na given time t) The electric field Ex is along the x-axis,\nand varies sinusoidally with z, at a given time The\nmagnetic field By is along the y-axis, and again varies\nsinusoidally with z The electric and magnetic fields Ex\nand By are perpendicular to each\nother, and to the direction z of\npropagation"}, {"Chapter": "1", "sentence_range": "6294-6297", "Text": "The electric field Ex is along the x-axis,\nand varies sinusoidally with z, at a given time The\nmagnetic field By is along the y-axis, and again varies\nsinusoidally with z The electric and magnetic fields Ex\nand By are perpendicular to each\nother, and to the direction z of\npropagation We can write Ex and\nBy as follows:\nEx= E0 sin (kz\u2013wt)\n[8"}, {"Chapter": "1", "sentence_range": "6295-6298", "Text": "The\nmagnetic field By is along the y-axis, and again varies\nsinusoidally with z The electric and magnetic fields Ex\nand By are perpendicular to each\nother, and to the direction z of\npropagation We can write Ex and\nBy as follows:\nEx= E0 sin (kz\u2013wt)\n[8 7(a)]\nBy= B0 sin (kz\u2013wt)\n[8"}, {"Chapter": "1", "sentence_range": "6296-6299", "Text": "The electric and magnetic fields Ex\nand By are perpendicular to each\nother, and to the direction z of\npropagation We can write Ex and\nBy as follows:\nEx= E0 sin (kz\u2013wt)\n[8 7(a)]\nBy= B0 sin (kz\u2013wt)\n[8 7(b)]\nHere k is related to the wave length\nl of the wave by the usual\nequation\n2\nk\n\u03bb\n\u03c0\n=\n(8"}, {"Chapter": "1", "sentence_range": "6297-6300", "Text": "We can write Ex and\nBy as follows:\nEx= E0 sin (kz\u2013wt)\n[8 7(a)]\nBy= B0 sin (kz\u2013wt)\n[8 7(b)]\nHere k is related to the wave length\nl of the wave by the usual\nequation\n2\nk\n\u03bb\n\u03c0\n=\n(8 8)\n EXAMPLE 8"}, {"Chapter": "1", "sentence_range": "6298-6301", "Text": "7(a)]\nBy= B0 sin (kz\u2013wt)\n[8 7(b)]\nHere k is related to the wave length\nl of the wave by the usual\nequation\n2\nk\n\u03bb\n\u03c0\n=\n(8 8)\n EXAMPLE 8 1\nHeinrich Rudolf Hertz\n(1857 \u2013 1894) German\nphysicist who was the\nfirst to broadcast and\nreceive radio waves"}, {"Chapter": "1", "sentence_range": "6299-6302", "Text": "7(b)]\nHere k is related to the wave length\nl of the wave by the usual\nequation\n2\nk\n\u03bb\n\u03c0\n=\n(8 8)\n EXAMPLE 8 1\nHeinrich Rudolf Hertz\n(1857 \u2013 1894) German\nphysicist who was the\nfirst to broadcast and\nreceive radio waves He\nproduced \nelectro-\nmagnetic waves, sent\nthem through space, and\nmeasured their wave-\nlength and speed"}, {"Chapter": "1", "sentence_range": "6300-6303", "Text": "8)\n EXAMPLE 8 1\nHeinrich Rudolf Hertz\n(1857 \u2013 1894) German\nphysicist who was the\nfirst to broadcast and\nreceive radio waves He\nproduced \nelectro-\nmagnetic waves, sent\nthem through space, and\nmeasured their wave-\nlength and speed He\nshowed that the nature\nof \ntheir \nvibration,\nreflection and refraction\nwas the same as that of\nlight and heat waves,\nestablishing \ntheir\nidentity for the first time"}, {"Chapter": "1", "sentence_range": "6301-6304", "Text": "1\nHeinrich Rudolf Hertz\n(1857 \u2013 1894) German\nphysicist who was the\nfirst to broadcast and\nreceive radio waves He\nproduced \nelectro-\nmagnetic waves, sent\nthem through space, and\nmeasured their wave-\nlength and speed He\nshowed that the nature\nof \ntheir \nvibration,\nreflection and refraction\nwas the same as that of\nlight and heat waves,\nestablishing \ntheir\nidentity for the first time He \nalso \npioneered\nresearch on discharge of\nelectricity through gases,\nand \ndiscovered \nthe\nphotoelectric effect"}, {"Chapter": "1", "sentence_range": "6302-6305", "Text": "He\nproduced \nelectro-\nmagnetic waves, sent\nthem through space, and\nmeasured their wave-\nlength and speed He\nshowed that the nature\nof \ntheir \nvibration,\nreflection and refraction\nwas the same as that of\nlight and heat waves,\nestablishing \ntheir\nidentity for the first time He \nalso \npioneered\nresearch on discharge of\nelectricity through gases,\nand \ndiscovered \nthe\nphotoelectric effect HEINRICH RUDOLF HERTZ (1857\u20131894)\nFIGURE 8"}, {"Chapter": "1", "sentence_range": "6303-6306", "Text": "He\nshowed that the nature\nof \ntheir \nvibration,\nreflection and refraction\nwas the same as that of\nlight and heat waves,\nestablishing \ntheir\nidentity for the first time He \nalso \npioneered\nresearch on discharge of\nelectricity through gases,\nand \ndiscovered \nthe\nphotoelectric effect HEINRICH RUDOLF HERTZ (1857\u20131894)\nFIGURE 8 3 A linearly polarised electromagnetic wave,\npropagating in the z-direction with the oscillating electric field E\nalong the x-direction and the oscillating magnetic field B along\nthe y-direction"}, {"Chapter": "1", "sentence_range": "6304-6307", "Text": "He \nalso \npioneered\nresearch on discharge of\nelectricity through gases,\nand \ndiscovered \nthe\nphotoelectric effect HEINRICH RUDOLF HERTZ (1857\u20131894)\nFIGURE 8 3 A linearly polarised electromagnetic wave,\npropagating in the z-direction with the oscillating electric field E\nalong the x-direction and the oscillating magnetic field B along\nthe y-direction Rationalised 2023-24\n207\nElectromagnetic\nWaves\nand w   is the angular frequency"}, {"Chapter": "1", "sentence_range": "6305-6308", "Text": "HEINRICH RUDOLF HERTZ (1857\u20131894)\nFIGURE 8 3 A linearly polarised electromagnetic wave,\npropagating in the z-direction with the oscillating electric field E\nalong the x-direction and the oscillating magnetic field B along\nthe y-direction Rationalised 2023-24\n207\nElectromagnetic\nWaves\nand w   is the angular frequency k is the magnitude of the wave vector (or\npropagation vector) k and its direction describes the direction of\npropagation of the wave"}, {"Chapter": "1", "sentence_range": "6306-6309", "Text": "3 A linearly polarised electromagnetic wave,\npropagating in the z-direction with the oscillating electric field E\nalong the x-direction and the oscillating magnetic field B along\nthe y-direction Rationalised 2023-24\n207\nElectromagnetic\nWaves\nand w   is the angular frequency k is the magnitude of the wave vector (or\npropagation vector) k and its direction describes the direction of\npropagation of the wave The speed of propagation of the wave is (w/k)"}, {"Chapter": "1", "sentence_range": "6307-6310", "Text": "Rationalised 2023-24\n207\nElectromagnetic\nWaves\nand w   is the angular frequency k is the magnitude of the wave vector (or\npropagation vector) k and its direction describes the direction of\npropagation of the wave The speed of propagation of the wave is (w/k) Using Eqs"}, {"Chapter": "1", "sentence_range": "6308-6311", "Text": "k is the magnitude of the wave vector (or\npropagation vector) k and its direction describes the direction of\npropagation of the wave The speed of propagation of the wave is (w/k) Using Eqs [8"}, {"Chapter": "1", "sentence_range": "6309-6312", "Text": "The speed of propagation of the wave is (w/k) Using Eqs [8 7(a) and (b)] for Ex and By  and Maxwell\u2019s equations, one\nfinds that\nw = ck, where, c = 1/\n0\n\u00b5 \u03b50\n[8"}, {"Chapter": "1", "sentence_range": "6310-6313", "Text": "Using Eqs [8 7(a) and (b)] for Ex and By  and Maxwell\u2019s equations, one\nfinds that\nw = ck, where, c = 1/\n0\n\u00b5 \u03b50\n[8 9(a)]\nThe relation w = ck is the standard one for waves (see for example,\nSection 15"}, {"Chapter": "1", "sentence_range": "6311-6314", "Text": "[8 7(a) and (b)] for Ex and By  and Maxwell\u2019s equations, one\nfinds that\nw = ck, where, c = 1/\n0\n\u00b5 \u03b50\n[8 9(a)]\nThe relation w = ck is the standard one for waves (see for example,\nSection 15 4 of class XI Physics textbook)"}, {"Chapter": "1", "sentence_range": "6312-6315", "Text": "7(a) and (b)] for Ex and By  and Maxwell\u2019s equations, one\nfinds that\nw = ck, where, c = 1/\n0\n\u00b5 \u03b50\n[8 9(a)]\nThe relation w = ck is the standard one for waves (see for example,\nSection 15 4 of class XI Physics textbook) This relation is often written\nin terms of frequency, n (=w/2p) and wavelength, l (=2p/k) as\n2\n2\n\u03c0\u03bd\n\u03bb\n=\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nc\n\u03c0\n   or\nnl = c\n[8"}, {"Chapter": "1", "sentence_range": "6313-6316", "Text": "9(a)]\nThe relation w = ck is the standard one for waves (see for example,\nSection 15 4 of class XI Physics textbook) This relation is often written\nin terms of frequency, n (=w/2p) and wavelength, l (=2p/k) as\n2\n2\n\u03c0\u03bd\n\u03bb\n=\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nc\n\u03c0\n   or\nnl = c\n[8 9(b)]\nIt is also seen from Maxwell\u2019s equations that the magnitude of the\nelectric and the magnetic fields in an electromagnetic wave are related as\nB0 = (E0/c)\n(8"}, {"Chapter": "1", "sentence_range": "6314-6317", "Text": "4 of class XI Physics textbook) This relation is often written\nin terms of frequency, n (=w/2p) and wavelength, l (=2p/k) as\n2\n2\n\u03c0\u03bd\n\u03bb\n=\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nc\n\u03c0\n   or\nnl = c\n[8 9(b)]\nIt is also seen from Maxwell\u2019s equations that the magnitude of the\nelectric and the magnetic fields in an electromagnetic wave are related as\nB0 = (E0/c)\n(8 10)\nWe here make remarks on some features of electromagnetic waves"}, {"Chapter": "1", "sentence_range": "6315-6318", "Text": "This relation is often written\nin terms of frequency, n (=w/2p) and wavelength, l (=2p/k) as\n2\n2\n\u03c0\u03bd\n\u03bb\n=\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\nc\n\u03c0\n   or\nnl = c\n[8 9(b)]\nIt is also seen from Maxwell\u2019s equations that the magnitude of the\nelectric and the magnetic fields in an electromagnetic wave are related as\nB0 = (E0/c)\n(8 10)\nWe here make remarks on some features of electromagnetic waves They are self-sustaining oscillations of electric and magnetic fields in\nfree space, or vacuum"}, {"Chapter": "1", "sentence_range": "6316-6319", "Text": "9(b)]\nIt is also seen from Maxwell\u2019s equations that the magnitude of the\nelectric and the magnetic fields in an electromagnetic wave are related as\nB0 = (E0/c)\n(8 10)\nWe here make remarks on some features of electromagnetic waves They are self-sustaining oscillations of electric and magnetic fields in\nfree space, or vacuum They differ from all the other waves we have\nstudied so far, in respect that no material medium is involved in the\nvibrations of the electric and magnetic fields"}, {"Chapter": "1", "sentence_range": "6317-6320", "Text": "10)\nWe here make remarks on some features of electromagnetic waves They are self-sustaining oscillations of electric and magnetic fields in\nfree space, or vacuum They differ from all the other waves we have\nstudied so far, in respect that no material medium is involved in the\nvibrations of the electric and magnetic fields But what if a material medium is actually there"}, {"Chapter": "1", "sentence_range": "6318-6321", "Text": "They are self-sustaining oscillations of electric and magnetic fields in\nfree space, or vacuum They differ from all the other waves we have\nstudied so far, in respect that no material medium is involved in the\nvibrations of the electric and magnetic fields But what if a material medium is actually there We know that light,\nan electromagnetic wave, does propagate through glass, for example"}, {"Chapter": "1", "sentence_range": "6319-6322", "Text": "They differ from all the other waves we have\nstudied so far, in respect that no material medium is involved in the\nvibrations of the electric and magnetic fields But what if a material medium is actually there We know that light,\nan electromagnetic wave, does propagate through glass, for example We\nhave seen earlier that the total electric and magnetic fields inside a\nmedium are described in terms of a permittivity e and a magnetic\npermeability m (these describe the factors  by which the total fields differ\nfrom the external fields)"}, {"Chapter": "1", "sentence_range": "6320-6323", "Text": "But what if a material medium is actually there We know that light,\nan electromagnetic wave, does propagate through glass, for example We\nhave seen earlier that the total electric and magnetic fields inside a\nmedium are described in terms of a permittivity e and a magnetic\npermeability m (these describe the factors  by which the total fields differ\nfrom the external fields) These replace e0 and m0 in the description to\nelectric and magnetic fields in Maxwell\u2019s equations with the result that in\na material medium of permittivity e and magnetic permeability m, the\nvelocity of light becomes,\n1\nv\n\u00b5\u03b5\n=\n(8"}, {"Chapter": "1", "sentence_range": "6321-6324", "Text": "We know that light,\nan electromagnetic wave, does propagate through glass, for example We\nhave seen earlier that the total electric and magnetic fields inside a\nmedium are described in terms of a permittivity e and a magnetic\npermeability m (these describe the factors  by which the total fields differ\nfrom the external fields) These replace e0 and m0 in the description to\nelectric and magnetic fields in Maxwell\u2019s equations with the result that in\na material medium of permittivity e and magnetic permeability m, the\nvelocity of light becomes,\n1\nv\n\u00b5\u03b5\n=\n(8 11)\nThus, the velocity of light depends on electric and magnetic properties of\nthe medium"}, {"Chapter": "1", "sentence_range": "6322-6325", "Text": "We\nhave seen earlier that the total electric and magnetic fields inside a\nmedium are described in terms of a permittivity e and a magnetic\npermeability m (these describe the factors  by which the total fields differ\nfrom the external fields) These replace e0 and m0 in the description to\nelectric and magnetic fields in Maxwell\u2019s equations with the result that in\na material medium of permittivity e and magnetic permeability m, the\nvelocity of light becomes,\n1\nv\n\u00b5\u03b5\n=\n(8 11)\nThus, the velocity of light depends on electric and magnetic properties of\nthe medium We shall see in the next chapter that the refractive index of\none medium with respect to the other is equal to the ratio of velocities of\nlight in the two media"}, {"Chapter": "1", "sentence_range": "6323-6326", "Text": "These replace e0 and m0 in the description to\nelectric and magnetic fields in Maxwell\u2019s equations with the result that in\na material medium of permittivity e and magnetic permeability m, the\nvelocity of light becomes,\n1\nv\n\u00b5\u03b5\n=\n(8 11)\nThus, the velocity of light depends on electric and magnetic properties of\nthe medium We shall see in the next chapter that the refractive index of\none medium with respect to the other is equal to the ratio of velocities of\nlight in the two media The velocity of electromagnetic waves in free space or vacuum is an\nimportant fundamental constant"}, {"Chapter": "1", "sentence_range": "6324-6327", "Text": "11)\nThus, the velocity of light depends on electric and magnetic properties of\nthe medium We shall see in the next chapter that the refractive index of\none medium with respect to the other is equal to the ratio of velocities of\nlight in the two media The velocity of electromagnetic waves in free space or vacuum is an\nimportant fundamental constant It has been shown by experiments on\nelectromagnetic waves of different wavelengths that this velocity is the\nsame (independent of wavelength) to within a few metres per second, out\nof a value of 3\u00d7108 m/s"}, {"Chapter": "1", "sentence_range": "6325-6328", "Text": "We shall see in the next chapter that the refractive index of\none medium with respect to the other is equal to the ratio of velocities of\nlight in the two media The velocity of electromagnetic waves in free space or vacuum is an\nimportant fundamental constant It has been shown by experiments on\nelectromagnetic waves of different wavelengths that this velocity is the\nsame (independent of wavelength) to within a few metres per second, out\nof a value of 3\u00d7108 m/s The constancy of the velocity of em waves in\nvacuum is so strongly supported by experiments and the actual value is\nso well known now that this is used to define a standard of length"}, {"Chapter": "1", "sentence_range": "6326-6329", "Text": "The velocity of electromagnetic waves in free space or vacuum is an\nimportant fundamental constant It has been shown by experiments on\nelectromagnetic waves of different wavelengths that this velocity is the\nsame (independent of wavelength) to within a few metres per second, out\nof a value of 3\u00d7108 m/s The constancy of the velocity of em waves in\nvacuum is so strongly supported by experiments and the actual value is\nso well known now that this is used to define a standard of length The great technological importance of electromagnetic waves stems\nfrom their capability to carry energy from one place to another"}, {"Chapter": "1", "sentence_range": "6327-6330", "Text": "It has been shown by experiments on\nelectromagnetic waves of different wavelengths that this velocity is the\nsame (independent of wavelength) to within a few metres per second, out\nof a value of 3\u00d7108 m/s The constancy of the velocity of em waves in\nvacuum is so strongly supported by experiments and the actual value is\nso well known now that this is used to define a standard of length The great technological importance of electromagnetic waves stems\nfrom their capability to carry energy from one place to another The\nradio and TV signals from broadcasting stations carry energy"}, {"Chapter": "1", "sentence_range": "6328-6331", "Text": "The constancy of the velocity of em waves in\nvacuum is so strongly supported by experiments and the actual value is\nso well known now that this is used to define a standard of length The great technological importance of electromagnetic waves stems\nfrom their capability to carry energy from one place to another The\nradio and TV signals from broadcasting stations carry energy Light\ncarries energy from the sun to the earth, thus making life possible on\nthe earth"}, {"Chapter": "1", "sentence_range": "6329-6332", "Text": "The great technological importance of electromagnetic waves stems\nfrom their capability to carry energy from one place to another The\nradio and TV signals from broadcasting stations carry energy Light\ncarries energy from the sun to the earth, thus making life possible on\nthe earth Rationalised 2023-24\nPhysics\n208\n EXAMPLE 8"}, {"Chapter": "1", "sentence_range": "6330-6333", "Text": "The\nradio and TV signals from broadcasting stations carry energy Light\ncarries energy from the sun to the earth, thus making life possible on\nthe earth Rationalised 2023-24\nPhysics\n208\n EXAMPLE 8 2\n EXAMPLE 8"}, {"Chapter": "1", "sentence_range": "6331-6334", "Text": "Light\ncarries energy from the sun to the earth, thus making life possible on\nthe earth Rationalised 2023-24\nPhysics\n208\n EXAMPLE 8 2\n EXAMPLE 8 1\nExample 8"}, {"Chapter": "1", "sentence_range": "6332-6335", "Text": "Rationalised 2023-24\nPhysics\n208\n EXAMPLE 8 2\n EXAMPLE 8 1\nExample 8 1 A plane electromagnetic wave of frequency\n25 MHz travels in free space along the x-direction"}, {"Chapter": "1", "sentence_range": "6333-6336", "Text": "2\n EXAMPLE 8 1\nExample 8 1 A plane electromagnetic wave of frequency\n25 MHz travels in free space along the x-direction At a particular\npoint in space and time, E = 6"}, {"Chapter": "1", "sentence_range": "6334-6337", "Text": "1\nExample 8 1 A plane electromagnetic wave of frequency\n25 MHz travels in free space along the x-direction At a particular\npoint in space and time, E = 6 3 \u02c6j  V/m"}, {"Chapter": "1", "sentence_range": "6335-6338", "Text": "1 A plane electromagnetic wave of frequency\n25 MHz travels in free space along the x-direction At a particular\npoint in space and time, E = 6 3 \u02c6j  V/m What is B at this point"}, {"Chapter": "1", "sentence_range": "6336-6339", "Text": "At a particular\npoint in space and time, E = 6 3 \u02c6j  V/m What is B at this point Solution Using Eq"}, {"Chapter": "1", "sentence_range": "6337-6340", "Text": "3 \u02c6j  V/m What is B at this point Solution Using Eq (8"}, {"Chapter": "1", "sentence_range": "6338-6341", "Text": "What is B at this point Solution Using Eq (8 10), the magnitude of B is\n\u20138\n8\n6"}, {"Chapter": "1", "sentence_range": "6339-6342", "Text": "Solution Using Eq (8 10), the magnitude of B is\n\u20138\n8\n6 3 V/m\n2"}, {"Chapter": "1", "sentence_range": "6340-6343", "Text": "(8 10), the magnitude of B is\n\u20138\n8\n6 3 V/m\n2 1 10\nT\n3 10 m/s\nE\nB\nc\n=\n=\n=\n\u00d7\n\u00d7\nTo find the direction, we note that E is along y-direction and the\nwave propagates along x-axis"}, {"Chapter": "1", "sentence_range": "6341-6344", "Text": "10), the magnitude of B is\n\u20138\n8\n6 3 V/m\n2 1 10\nT\n3 10 m/s\nE\nB\nc\n=\n=\n=\n\u00d7\n\u00d7\nTo find the direction, we note that E is along y-direction and the\nwave propagates along x-axis Therefore, B should be in a direction\nperpendicular to both x- and y-axes"}, {"Chapter": "1", "sentence_range": "6342-6345", "Text": "3 V/m\n2 1 10\nT\n3 10 m/s\nE\nB\nc\n=\n=\n=\n\u00d7\n\u00d7\nTo find the direction, we note that E is along y-direction and the\nwave propagates along x-axis Therefore, B should be in a direction\nperpendicular to both x- and y-axes Using vector algebra, E \u00d7 B should\nbe along  x-direction"}, {"Chapter": "1", "sentence_range": "6343-6346", "Text": "1 10\nT\n3 10 m/s\nE\nB\nc\n=\n=\n=\n\u00d7\n\u00d7\nTo find the direction, we note that E is along y-direction and the\nwave propagates along x-axis Therefore, B should be in a direction\nperpendicular to both x- and y-axes Using vector algebra, E \u00d7 B should\nbe along  x-direction Since, (+ \u02c6j ) \u00d7 (+ \u02c6k ) = \u02c6i , B is along the z-direction"}, {"Chapter": "1", "sentence_range": "6344-6347", "Text": "Therefore, B should be in a direction\nperpendicular to both x- and y-axes Using vector algebra, E \u00d7 B should\nbe along  x-direction Since, (+ \u02c6j ) \u00d7 (+ \u02c6k ) = \u02c6i , B is along the z-direction Thus,\nB = 2"}, {"Chapter": "1", "sentence_range": "6345-6348", "Text": "Using vector algebra, E \u00d7 B should\nbe along  x-direction Since, (+ \u02c6j ) \u00d7 (+ \u02c6k ) = \u02c6i , B is along the z-direction Thus,\nB = 2 1 \u00d7 10\u20138 \u02c6k T\nExample 8"}, {"Chapter": "1", "sentence_range": "6346-6349", "Text": "Since, (+ \u02c6j ) \u00d7 (+ \u02c6k ) = \u02c6i , B is along the z-direction Thus,\nB = 2 1 \u00d7 10\u20138 \u02c6k T\nExample 8 2 The magnetic field in a plane electromagnetic wave is\ngiven by By = (2 \u00d7 10\u20137) T sin (0"}, {"Chapter": "1", "sentence_range": "6347-6350", "Text": "Thus,\nB = 2 1 \u00d7 10\u20138 \u02c6k T\nExample 8 2 The magnetic field in a plane electromagnetic wave is\ngiven by By = (2 \u00d7 10\u20137) T sin (0 5\u00d7103x+1"}, {"Chapter": "1", "sentence_range": "6348-6351", "Text": "1 \u00d7 10\u20138 \u02c6k T\nExample 8 2 The magnetic field in a plane electromagnetic wave is\ngiven by By = (2 \u00d7 10\u20137) T sin (0 5\u00d7103x+1 5\u00d71011t)"}, {"Chapter": "1", "sentence_range": "6349-6352", "Text": "2 The magnetic field in a plane electromagnetic wave is\ngiven by By = (2 \u00d7 10\u20137) T sin (0 5\u00d7103x+1 5\u00d71011t) (a) What is the wavelength and frequency of the wave"}, {"Chapter": "1", "sentence_range": "6350-6353", "Text": "5\u00d7103x+1 5\u00d71011t) (a) What is the wavelength and frequency of the wave (b) Write an expression for the electric field"}, {"Chapter": "1", "sentence_range": "6351-6354", "Text": "5\u00d71011t) (a) What is the wavelength and frequency of the wave (b) Write an expression for the electric field Solution\n(a) Comparing the given equation with\nBy = B0 sin 2\u03c0\nx\u03bb\n+Tt\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nWe get, \n3\n2\n0"}, {"Chapter": "1", "sentence_range": "6352-6355", "Text": "(a) What is the wavelength and frequency of the wave (b) Write an expression for the electric field Solution\n(a) Comparing the given equation with\nBy = B0 sin 2\u03c0\nx\u03bb\n+Tt\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nWe get, \n3\n2\n0 5\n\u03c010\n\u03bb =\n\u00d7\n m = 1"}, {"Chapter": "1", "sentence_range": "6353-6356", "Text": "(b) Write an expression for the electric field Solution\n(a) Comparing the given equation with\nBy = B0 sin 2\u03c0\nx\u03bb\n+Tt\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nWe get, \n3\n2\n0 5\n\u03c010\n\u03bb =\n\u00d7\n m = 1 26 cm,\nand\n  \n(\n11)\n1\n1"}, {"Chapter": "1", "sentence_range": "6354-6357", "Text": "Solution\n(a) Comparing the given equation with\nBy = B0 sin 2\u03c0\nx\u03bb\n+Tt\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nWe get, \n3\n2\n0 5\n\u03c010\n\u03bb =\n\u00d7\n m = 1 26 cm,\nand\n  \n(\n11)\n1\n1 5\n10\n/2\n23"}, {"Chapter": "1", "sentence_range": "6355-6358", "Text": "5\n\u03c010\n\u03bb =\n\u00d7\n m = 1 26 cm,\nand\n  \n(\n11)\n1\n1 5\n10\n/2\n23 9 GHz\nT\n=\u03bd\n=\n\u00d7\n\u03c0 =\n(b) E0 = B0c = 2\u00d710\u20137 T \u00d7 3 \u00d7 108 m/s = 6 \u00d7 101 V/m\nThe electric field component is perpendicular to the direction of\npropagation and the direction of magnetic field"}, {"Chapter": "1", "sentence_range": "6356-6359", "Text": "26 cm,\nand\n  \n(\n11)\n1\n1 5\n10\n/2\n23 9 GHz\nT\n=\u03bd\n=\n\u00d7\n\u03c0 =\n(b) E0 = B0c = 2\u00d710\u20137 T \u00d7 3 \u00d7 108 m/s = 6 \u00d7 101 V/m\nThe electric field component is perpendicular to the direction of\npropagation and the direction of magnetic field Therefore, the\nelectric field component along the z-axis is obtained as\nEz = 60 sin (0"}, {"Chapter": "1", "sentence_range": "6357-6360", "Text": "5\n10\n/2\n23 9 GHz\nT\n=\u03bd\n=\n\u00d7\n\u03c0 =\n(b) E0 = B0c = 2\u00d710\u20137 T \u00d7 3 \u00d7 108 m/s = 6 \u00d7 101 V/m\nThe electric field component is perpendicular to the direction of\npropagation and the direction of magnetic field Therefore, the\nelectric field component along the z-axis is obtained as\nEz = 60 sin (0 5 \u00d7 103x + 1"}, {"Chapter": "1", "sentence_range": "6358-6361", "Text": "9 GHz\nT\n=\u03bd\n=\n\u00d7\n\u03c0 =\n(b) E0 = B0c = 2\u00d710\u20137 T \u00d7 3 \u00d7 108 m/s = 6 \u00d7 101 V/m\nThe electric field component is perpendicular to the direction of\npropagation and the direction of magnetic field Therefore, the\nelectric field component along the z-axis is obtained as\nEz = 60 sin (0 5 \u00d7 103x + 1 5 \u00d7 1011 t) V/m\n8"}, {"Chapter": "1", "sentence_range": "6359-6362", "Text": "Therefore, the\nelectric field component along the z-axis is obtained as\nEz = 60 sin (0 5 \u00d7 103x + 1 5 \u00d7 1011 t) V/m\n8 4  ELECTROMAGNETIC SPECTRUM\nAt the time Maxwell predicted the existence of electromagnetic waves, the\nonly familiar electromagnetic waves were the visible light waves"}, {"Chapter": "1", "sentence_range": "6360-6363", "Text": "5 \u00d7 103x + 1 5 \u00d7 1011 t) V/m\n8 4  ELECTROMAGNETIC SPECTRUM\nAt the time Maxwell predicted the existence of electromagnetic waves, the\nonly familiar electromagnetic waves were the visible light waves The existence\nof ultraviolet and infrared waves was barely established"}, {"Chapter": "1", "sentence_range": "6361-6364", "Text": "5 \u00d7 1011 t) V/m\n8 4  ELECTROMAGNETIC SPECTRUM\nAt the time Maxwell predicted the existence of electromagnetic waves, the\nonly familiar electromagnetic waves were the visible light waves The existence\nof ultraviolet and infrared waves was barely established By the end of the\nnineteenth century, X-rays and gamma rays had also been discovered"}, {"Chapter": "1", "sentence_range": "6362-6365", "Text": "4  ELECTROMAGNETIC SPECTRUM\nAt the time Maxwell predicted the existence of electromagnetic waves, the\nonly familiar electromagnetic waves were the visible light waves The existence\nof ultraviolet and infrared waves was barely established By the end of the\nnineteenth century, X-rays and gamma rays had also been discovered We\nnow know that, electromagnetic waves include visible light waves, X-rays,\ngamma rays, radio waves, microwaves, ultraviolet and infrared waves"}, {"Chapter": "1", "sentence_range": "6363-6366", "Text": "The existence\nof ultraviolet and infrared waves was barely established By the end of the\nnineteenth century, X-rays and gamma rays had also been discovered We\nnow know that, electromagnetic waves include visible light waves, X-rays,\ngamma rays, radio waves, microwaves, ultraviolet and infrared waves The\nclassification of em waves according to frequency is the electromagnetic\nspectrum (Fig"}, {"Chapter": "1", "sentence_range": "6364-6367", "Text": "By the end of the\nnineteenth century, X-rays and gamma rays had also been discovered We\nnow know that, electromagnetic waves include visible light waves, X-rays,\ngamma rays, radio waves, microwaves, ultraviolet and infrared waves The\nclassification of em waves according to frequency is the electromagnetic\nspectrum (Fig 8"}, {"Chapter": "1", "sentence_range": "6365-6368", "Text": "We\nnow know that, electromagnetic waves include visible light waves, X-rays,\ngamma rays, radio waves, microwaves, ultraviolet and infrared waves The\nclassification of em waves according to frequency is the electromagnetic\nspectrum (Fig 8 5)"}, {"Chapter": "1", "sentence_range": "6366-6369", "Text": "The\nclassification of em waves according to frequency is the electromagnetic\nspectrum (Fig 8 5) There is no sharp division between one kind of wave\nand the next"}, {"Chapter": "1", "sentence_range": "6367-6370", "Text": "8 5) There is no sharp division between one kind of wave\nand the next The classification is based roughly on how the waves are\nproduced and/or detected"}, {"Chapter": "1", "sentence_range": "6368-6371", "Text": "5) There is no sharp division between one kind of wave\nand the next The classification is based roughly on how the waves are\nproduced and/or detected We briefly describe these different types of electromagnetic waves, in\norder of decreasing wavelengths"}, {"Chapter": "1", "sentence_range": "6369-6372", "Text": "There is no sharp division between one kind of wave\nand the next The classification is based roughly on how the waves are\nproduced and/or detected We briefly describe these different types of electromagnetic waves, in\norder of decreasing wavelengths Electromagnetic spectrum\nhttp://www"}, {"Chapter": "1", "sentence_range": "6370-6373", "Text": "The classification is based roughly on how the waves are\nproduced and/or detected We briefly describe these different types of electromagnetic waves, in\norder of decreasing wavelengths Electromagnetic spectrum\nhttp://www fnal"}, {"Chapter": "1", "sentence_range": "6371-6374", "Text": "We briefly describe these different types of electromagnetic waves, in\norder of decreasing wavelengths Electromagnetic spectrum\nhttp://www fnal gov/pub/inquiring/more/light\nhttp://imagine"}, {"Chapter": "1", "sentence_range": "6372-6375", "Text": "Electromagnetic spectrum\nhttp://www fnal gov/pub/inquiring/more/light\nhttp://imagine gsfc"}, {"Chapter": "1", "sentence_range": "6373-6376", "Text": "fnal gov/pub/inquiring/more/light\nhttp://imagine gsfc nasa"}, {"Chapter": "1", "sentence_range": "6374-6377", "Text": "gov/pub/inquiring/more/light\nhttp://imagine gsfc nasa gov/docs/science/\nRationalised 2023-24\n209\nElectromagnetic\nWaves\nFIGURE 8"}, {"Chapter": "1", "sentence_range": "6375-6378", "Text": "gsfc nasa gov/docs/science/\nRationalised 2023-24\n209\nElectromagnetic\nWaves\nFIGURE 8 5 The electromagnetic spectrum, with common names for various\npart of it"}, {"Chapter": "1", "sentence_range": "6376-6379", "Text": "nasa gov/docs/science/\nRationalised 2023-24\n209\nElectromagnetic\nWaves\nFIGURE 8 5 The electromagnetic spectrum, with common names for various\npart of it The various regions do not have sharply defined boundaries"}, {"Chapter": "1", "sentence_range": "6377-6380", "Text": "gov/docs/science/\nRationalised 2023-24\n209\nElectromagnetic\nWaves\nFIGURE 8 5 The electromagnetic spectrum, with common names for various\npart of it The various regions do not have sharply defined boundaries 8"}, {"Chapter": "1", "sentence_range": "6378-6381", "Text": "5 The electromagnetic spectrum, with common names for various\npart of it The various regions do not have sharply defined boundaries 8 4"}, {"Chapter": "1", "sentence_range": "6379-6382", "Text": "The various regions do not have sharply defined boundaries 8 4 1  Radio waves\nRadio waves are produced by the accelerated motion of charges in conducting\nwires"}, {"Chapter": "1", "sentence_range": "6380-6383", "Text": "8 4 1  Radio waves\nRadio waves are produced by the accelerated motion of charges in conducting\nwires They are used in radio and television communication systems"}, {"Chapter": "1", "sentence_range": "6381-6384", "Text": "4 1  Radio waves\nRadio waves are produced by the accelerated motion of charges in conducting\nwires They are used in radio and television communication systems They\nare generally in the frequency range from 500 kHz to about 1000 MHz"}, {"Chapter": "1", "sentence_range": "6382-6385", "Text": "1  Radio waves\nRadio waves are produced by the accelerated motion of charges in conducting\nwires They are used in radio and television communication systems They\nare generally in the frequency range from 500 kHz to about 1000 MHz The AM (amplitude modulated) band is from 530 kHz to 1710 kHz"}, {"Chapter": "1", "sentence_range": "6383-6386", "Text": "They are used in radio and television communication systems They\nare generally in the frequency range from 500 kHz to about 1000 MHz The AM (amplitude modulated) band is from 530 kHz to 1710 kHz Higher\nfrequencies upto 54 MHz are used for short wave bands"}, {"Chapter": "1", "sentence_range": "6384-6387", "Text": "They\nare generally in the frequency range from 500 kHz to about 1000 MHz The AM (amplitude modulated) band is from 530 kHz to 1710 kHz Higher\nfrequencies upto 54 MHz are used for short wave bands TV waves range\nfrom 54 MHz to 890 MHz"}, {"Chapter": "1", "sentence_range": "6385-6388", "Text": "The AM (amplitude modulated) band is from 530 kHz to 1710 kHz Higher\nfrequencies upto 54 MHz are used for short wave bands TV waves range\nfrom 54 MHz to 890 MHz The FM (frequency modulated) radio band\nextends from 88 MHz to 108 MHz"}, {"Chapter": "1", "sentence_range": "6386-6389", "Text": "Higher\nfrequencies upto 54 MHz are used for short wave bands TV waves range\nfrom 54 MHz to 890 MHz The FM (frequency modulated) radio band\nextends from 88 MHz to 108 MHz Cellular phones use radio waves to\ntransmit voice communication in the ultrahigh frequency (UHF) band"}, {"Chapter": "1", "sentence_range": "6387-6390", "Text": "TV waves range\nfrom 54 MHz to 890 MHz The FM (frequency modulated) radio band\nextends from 88 MHz to 108 MHz Cellular phones use radio waves to\ntransmit voice communication in the ultrahigh frequency (UHF) band How\nthese waves are transmitted and received is described in Chapter 15"}, {"Chapter": "1", "sentence_range": "6388-6391", "Text": "The FM (frequency modulated) radio band\nextends from 88 MHz to 108 MHz Cellular phones use radio waves to\ntransmit voice communication in the ultrahigh frequency (UHF) band How\nthese waves are transmitted and received is described in Chapter 15 8"}, {"Chapter": "1", "sentence_range": "6389-6392", "Text": "Cellular phones use radio waves to\ntransmit voice communication in the ultrahigh frequency (UHF) band How\nthese waves are transmitted and received is described in Chapter 15 8 4"}, {"Chapter": "1", "sentence_range": "6390-6393", "Text": "How\nthese waves are transmitted and received is described in Chapter 15 8 4 2  Microwaves\nMicrowaves (short-wavelength radio waves), with frequencies in the\ngigahertz (GHz) range, are produced by special vacuum tubes (called\nklystrons, magnetrons and Gunn diodes)"}, {"Chapter": "1", "sentence_range": "6391-6394", "Text": "8 4 2  Microwaves\nMicrowaves (short-wavelength radio waves), with frequencies in the\ngigahertz (GHz) range, are produced by special vacuum tubes (called\nklystrons, magnetrons and Gunn diodes) Due to their short wavelengths,\nthey are suitable for the radar systems used in aircraft navigation"}, {"Chapter": "1", "sentence_range": "6392-6395", "Text": "4 2  Microwaves\nMicrowaves (short-wavelength radio waves), with frequencies in the\ngigahertz (GHz) range, are produced by special vacuum tubes (called\nklystrons, magnetrons and Gunn diodes) Due to their short wavelengths,\nthey are suitable for the radar systems used in aircraft navigation Radar\nalso provides the basis for the speed guns used to time fast balls, tennis-\nserves, and automobiles"}, {"Chapter": "1", "sentence_range": "6393-6396", "Text": "2  Microwaves\nMicrowaves (short-wavelength radio waves), with frequencies in the\ngigahertz (GHz) range, are produced by special vacuum tubes (called\nklystrons, magnetrons and Gunn diodes) Due to their short wavelengths,\nthey are suitable for the radar systems used in aircraft navigation Radar\nalso provides the basis for the speed guns used to time fast balls, tennis-\nserves, and automobiles Microwave ovens are an interesting domestic\napplication of these waves"}, {"Chapter": "1", "sentence_range": "6394-6397", "Text": "Due to their short wavelengths,\nthey are suitable for the radar systems used in aircraft navigation Radar\nalso provides the basis for the speed guns used to time fast balls, tennis-\nserves, and automobiles Microwave ovens are an interesting domestic\napplication of these waves In such ovens, the frequency of the microwaves\nis selected to match the resonant frequency of water molecules so that\nenergy from the waves is transferred efficiently to the kinetic energy of\nthe molecules"}, {"Chapter": "1", "sentence_range": "6395-6398", "Text": "Radar\nalso provides the basis for the speed guns used to time fast balls, tennis-\nserves, and automobiles Microwave ovens are an interesting domestic\napplication of these waves In such ovens, the frequency of the microwaves\nis selected to match the resonant frequency of water molecules so that\nenergy from the waves is transferred efficiently to the kinetic energy of\nthe molecules This raises the temperature of any food containing water"}, {"Chapter": "1", "sentence_range": "6396-6399", "Text": "Microwave ovens are an interesting domestic\napplication of these waves In such ovens, the frequency of the microwaves\nis selected to match the resonant frequency of water molecules so that\nenergy from the waves is transferred efficiently to the kinetic energy of\nthe molecules This raises the temperature of any food containing water Rationalised 2023-24\nPhysics\n210\n8"}, {"Chapter": "1", "sentence_range": "6397-6400", "Text": "In such ovens, the frequency of the microwaves\nis selected to match the resonant frequency of water molecules so that\nenergy from the waves is transferred efficiently to the kinetic energy of\nthe molecules This raises the temperature of any food containing water Rationalised 2023-24\nPhysics\n210\n8 4"}, {"Chapter": "1", "sentence_range": "6398-6401", "Text": "This raises the temperature of any food containing water Rationalised 2023-24\nPhysics\n210\n8 4 3 Infrared waves\nInfrared waves are produced by hot bodies and molecules"}, {"Chapter": "1", "sentence_range": "6399-6402", "Text": "Rationalised 2023-24\nPhysics\n210\n8 4 3 Infrared waves\nInfrared waves are produced by hot bodies and molecules This band\nlies adjacent to the low-frequency or long-wave length end of the visible\nspectrum"}, {"Chapter": "1", "sentence_range": "6400-6403", "Text": "4 3 Infrared waves\nInfrared waves are produced by hot bodies and molecules This band\nlies adjacent to the low-frequency or long-wave length end of the visible\nspectrum Infrared waves are sometimes referred to as heat waves"}, {"Chapter": "1", "sentence_range": "6401-6404", "Text": "3 Infrared waves\nInfrared waves are produced by hot bodies and molecules This band\nlies adjacent to the low-frequency or long-wave length end of the visible\nspectrum Infrared waves are sometimes referred to as heat waves This\nis because water molecules present in most materials readily absorb\ninfrared waves (many other molecules, for example, CO2, NH3, also absorb\ninfrared waves)"}, {"Chapter": "1", "sentence_range": "6402-6405", "Text": "This band\nlies adjacent to the low-frequency or long-wave length end of the visible\nspectrum Infrared waves are sometimes referred to as heat waves This\nis because water molecules present in most materials readily absorb\ninfrared waves (many other molecules, for example, CO2, NH3, also absorb\ninfrared waves) After absorption, their thermal motion increases, that is,\nthey heat up and heat their surroundings"}, {"Chapter": "1", "sentence_range": "6403-6406", "Text": "Infrared waves are sometimes referred to as heat waves This\nis because water molecules present in most materials readily absorb\ninfrared waves (many other molecules, for example, CO2, NH3, also absorb\ninfrared waves) After absorption, their thermal motion increases, that is,\nthey heat up and heat their surroundings Infrared lamps are used in\nphysical therapy"}, {"Chapter": "1", "sentence_range": "6404-6407", "Text": "This\nis because water molecules present in most materials readily absorb\ninfrared waves (many other molecules, for example, CO2, NH3, also absorb\ninfrared waves) After absorption, their thermal motion increases, that is,\nthey heat up and heat their surroundings Infrared lamps are used in\nphysical therapy Infrared radiation also plays an important role in\nmaintaining the earth\u2019s warmth or average temperature through the\ngreenhouse effect"}, {"Chapter": "1", "sentence_range": "6405-6408", "Text": "After absorption, their thermal motion increases, that is,\nthey heat up and heat their surroundings Infrared lamps are used in\nphysical therapy Infrared radiation also plays an important role in\nmaintaining the earth\u2019s warmth or average temperature through the\ngreenhouse effect Incoming visible light (which passes relatively easily\nthrough the atmosphere) is absorbed by the earth\u2019s surface and re-\nradiated as infrared (longer wavelength) radiations"}, {"Chapter": "1", "sentence_range": "6406-6409", "Text": "Infrared lamps are used in\nphysical therapy Infrared radiation also plays an important role in\nmaintaining the earth\u2019s warmth or average temperature through the\ngreenhouse effect Incoming visible light (which passes relatively easily\nthrough the atmosphere) is absorbed by the earth\u2019s surface and re-\nradiated as infrared (longer wavelength) radiations This radiation is\ntrapped by greenhouse gases such as carbon dioxide and water vapour"}, {"Chapter": "1", "sentence_range": "6407-6410", "Text": "Infrared radiation also plays an important role in\nmaintaining the earth\u2019s warmth or average temperature through the\ngreenhouse effect Incoming visible light (which passes relatively easily\nthrough the atmosphere) is absorbed by the earth\u2019s surface and re-\nradiated as infrared (longer wavelength) radiations This radiation is\ntrapped by greenhouse gases such as carbon dioxide and water vapour Infrared detectors are used in Earth satellites, both for military purposes\nand to observe growth of crops"}, {"Chapter": "1", "sentence_range": "6408-6411", "Text": "Incoming visible light (which passes relatively easily\nthrough the atmosphere) is absorbed by the earth\u2019s surface and re-\nradiated as infrared (longer wavelength) radiations This radiation is\ntrapped by greenhouse gases such as carbon dioxide and water vapour Infrared detectors are used in Earth satellites, both for military purposes\nand to observe growth of crops Electronic devices (for example\nsemiconductor light emitting diodes) also   emit infrared and are widely\nused in the remote switches of household electronic systems such as TV\nsets, video recorders and hi-fi systems"}, {"Chapter": "1", "sentence_range": "6409-6412", "Text": "This radiation is\ntrapped by greenhouse gases such as carbon dioxide and water vapour Infrared detectors are used in Earth satellites, both for military purposes\nand to observe growth of crops Electronic devices (for example\nsemiconductor light emitting diodes) also   emit infrared and are widely\nused in the remote switches of household electronic systems such as TV\nsets, video recorders and hi-fi systems 8"}, {"Chapter": "1", "sentence_range": "6410-6413", "Text": "Infrared detectors are used in Earth satellites, both for military purposes\nand to observe growth of crops Electronic devices (for example\nsemiconductor light emitting diodes) also   emit infrared and are widely\nused in the remote switches of household electronic systems such as TV\nsets, video recorders and hi-fi systems 8 4"}, {"Chapter": "1", "sentence_range": "6411-6414", "Text": "Electronic devices (for example\nsemiconductor light emitting diodes) also   emit infrared and are widely\nused in the remote switches of household electronic systems such as TV\nsets, video recorders and hi-fi systems 8 4 4 Visible rays\nIt is the most familiar form of electromagnetic waves"}, {"Chapter": "1", "sentence_range": "6412-6415", "Text": "8 4 4 Visible rays\nIt is the most familiar form of electromagnetic waves It is the part of the\nspectrum that is detected by the human eye"}, {"Chapter": "1", "sentence_range": "6413-6416", "Text": "4 4 Visible rays\nIt is the most familiar form of electromagnetic waves It is the part of the\nspectrum that is detected by the human eye It runs from about\n4 \u00d7 1014 Hz to about 7 \u00d7 1014\n Hz or a wavelength range of about 700 \u2013\n400 nm"}, {"Chapter": "1", "sentence_range": "6414-6417", "Text": "4 Visible rays\nIt is the most familiar form of electromagnetic waves It is the part of the\nspectrum that is detected by the human eye It runs from about\n4 \u00d7 1014 Hz to about 7 \u00d7 1014\n Hz or a wavelength range of about 700 \u2013\n400 nm Visible light emitted or reflected from objects around us provides\nus information about the world"}, {"Chapter": "1", "sentence_range": "6415-6418", "Text": "It is the part of the\nspectrum that is detected by the human eye It runs from about\n4 \u00d7 1014 Hz to about 7 \u00d7 1014\n Hz or a wavelength range of about 700 \u2013\n400 nm Visible light emitted or reflected from objects around us provides\nus information about the world Our eyes are sensitive to this range of\nwavelengths"}, {"Chapter": "1", "sentence_range": "6416-6419", "Text": "It runs from about\n4 \u00d7 1014 Hz to about 7 \u00d7 1014\n Hz or a wavelength range of about 700 \u2013\n400 nm Visible light emitted or reflected from objects around us provides\nus information about the world Our eyes are sensitive to this range of\nwavelengths Different animals are sensitive to different range of\nwavelengths"}, {"Chapter": "1", "sentence_range": "6417-6420", "Text": "Visible light emitted or reflected from objects around us provides\nus information about the world Our eyes are sensitive to this range of\nwavelengths Different animals are sensitive to different range of\nwavelengths For example, snakes can detect infrared waves, and the\n\u2018visible\u2019 range of many insects extends well into the utraviolet"}, {"Chapter": "1", "sentence_range": "6418-6421", "Text": "Our eyes are sensitive to this range of\nwavelengths Different animals are sensitive to different range of\nwavelengths For example, snakes can detect infrared waves, and the\n\u2018visible\u2019 range of many insects extends well into the utraviolet 8"}, {"Chapter": "1", "sentence_range": "6419-6422", "Text": "Different animals are sensitive to different range of\nwavelengths For example, snakes can detect infrared waves, and the\n\u2018visible\u2019 range of many insects extends well into the utraviolet 8 4"}, {"Chapter": "1", "sentence_range": "6420-6423", "Text": "For example, snakes can detect infrared waves, and the\n\u2018visible\u2019 range of many insects extends well into the utraviolet 8 4 5 Ultraviolet rays\nIt covers wavelengths ranging from about 4 \u00d7 10\u20137 m (400 nm) down to\n6 \u00d7 10\u201310m (0"}, {"Chapter": "1", "sentence_range": "6421-6424", "Text": "8 4 5 Ultraviolet rays\nIt covers wavelengths ranging from about 4 \u00d7 10\u20137 m (400 nm) down to\n6 \u00d7 10\u201310m (0 6 nm)"}, {"Chapter": "1", "sentence_range": "6422-6425", "Text": "4 5 Ultraviolet rays\nIt covers wavelengths ranging from about 4 \u00d7 10\u20137 m (400 nm) down to\n6 \u00d7 10\u201310m (0 6 nm) Ultraviolet (UV) radiation is produced by special\nlamps and very hot bodies"}, {"Chapter": "1", "sentence_range": "6423-6426", "Text": "5 Ultraviolet rays\nIt covers wavelengths ranging from about 4 \u00d7 10\u20137 m (400 nm) down to\n6 \u00d7 10\u201310m (0 6 nm) Ultraviolet (UV) radiation is produced by special\nlamps and very hot bodies The sun is an important source of ultraviolet\nlight"}, {"Chapter": "1", "sentence_range": "6424-6427", "Text": "6 nm) Ultraviolet (UV) radiation is produced by special\nlamps and very hot bodies The sun is an important source of ultraviolet\nlight But fortunately, most of it is absorbed in the ozone layer in the\natmosphere at an altitude of about 40 \u2013 50 km"}, {"Chapter": "1", "sentence_range": "6425-6428", "Text": "Ultraviolet (UV) radiation is produced by special\nlamps and very hot bodies The sun is an important source of ultraviolet\nlight But fortunately, most of it is absorbed in the ozone layer in the\natmosphere at an altitude of about 40 \u2013 50 km UV light in large quantities\nhas harmful effects on humans"}, {"Chapter": "1", "sentence_range": "6426-6429", "Text": "The sun is an important source of ultraviolet\nlight But fortunately, most of it is absorbed in the ozone layer in the\natmosphere at an altitude of about 40 \u2013 50 km UV light in large quantities\nhas harmful effects on humans Exposure to UV radiation induces the\nproduction of more melanin, causing tanning of the skin"}, {"Chapter": "1", "sentence_range": "6427-6430", "Text": "But fortunately, most of it is absorbed in the ozone layer in the\natmosphere at an altitude of about 40 \u2013 50 km UV light in large quantities\nhas harmful effects on humans Exposure to UV radiation induces the\nproduction of more melanin, causing tanning of the skin UV radiation is\nabsorbed by ordinary glass"}, {"Chapter": "1", "sentence_range": "6428-6431", "Text": "UV light in large quantities\nhas harmful effects on humans Exposure to UV radiation induces the\nproduction of more melanin, causing tanning of the skin UV radiation is\nabsorbed by ordinary glass Hence, one cannot get tanned or sunburn\nthrough glass windows"}, {"Chapter": "1", "sentence_range": "6429-6432", "Text": "Exposure to UV radiation induces the\nproduction of more melanin, causing tanning of the skin UV radiation is\nabsorbed by ordinary glass Hence, one cannot get tanned or sunburn\nthrough glass windows Welders wear special glass goggles or face masks with glass windows\nto protect their eyes from large amount of UV produced by welding arcs"}, {"Chapter": "1", "sentence_range": "6430-6433", "Text": "UV radiation is\nabsorbed by ordinary glass Hence, one cannot get tanned or sunburn\nthrough glass windows Welders wear special glass goggles or face masks with glass windows\nto protect their eyes from large amount of UV produced by welding arcs Due to its  shorter wavelengths, UV radiations  can be focussed into very\nnarrow beams for high precision applications such as  LASIK (Laser-\nassisted in situ keratomileusis) eye surgery"}, {"Chapter": "1", "sentence_range": "6431-6434", "Text": "Hence, one cannot get tanned or sunburn\nthrough glass windows Welders wear special glass goggles or face masks with glass windows\nto protect their eyes from large amount of UV produced by welding arcs Due to its  shorter wavelengths, UV radiations  can be focussed into very\nnarrow beams for high precision applications such as  LASIK (Laser-\nassisted in situ keratomileusis) eye surgery UV lamps are used to kill\ngerms  in water purifiers"}, {"Chapter": "1", "sentence_range": "6432-6435", "Text": "Welders wear special glass goggles or face masks with glass windows\nto protect their eyes from large amount of UV produced by welding arcs Due to its  shorter wavelengths, UV radiations  can be focussed into very\nnarrow beams for high precision applications such as  LASIK (Laser-\nassisted in situ keratomileusis) eye surgery UV lamps are used to kill\ngerms  in water purifiers Ozone layer in the atmosphere plays a protective role, and hence its\ndepletion by chlorofluorocarbons (CFCs) gas (such as freon) is a matter\nof international concern"}, {"Chapter": "1", "sentence_range": "6433-6436", "Text": "Due to its  shorter wavelengths, UV radiations  can be focussed into very\nnarrow beams for high precision applications such as  LASIK (Laser-\nassisted in situ keratomileusis) eye surgery UV lamps are used to kill\ngerms  in water purifiers Ozone layer in the atmosphere plays a protective role, and hence its\ndepletion by chlorofluorocarbons (CFCs) gas (such as freon) is a matter\nof international concern Rationalised 2023-24\n211\nElectromagnetic\nWaves\n8"}, {"Chapter": "1", "sentence_range": "6434-6437", "Text": "UV lamps are used to kill\ngerms  in water purifiers Ozone layer in the atmosphere plays a protective role, and hence its\ndepletion by chlorofluorocarbons (CFCs) gas (such as freon) is a matter\nof international concern Rationalised 2023-24\n211\nElectromagnetic\nWaves\n8 4"}, {"Chapter": "1", "sentence_range": "6435-6438", "Text": "Ozone layer in the atmosphere plays a protective role, and hence its\ndepletion by chlorofluorocarbons (CFCs) gas (such as freon) is a matter\nof international concern Rationalised 2023-24\n211\nElectromagnetic\nWaves\n8 4 6 X-rays\nBeyond the UV region of the electromagnetic spectrum lies the X-ray\nregion"}, {"Chapter": "1", "sentence_range": "6436-6439", "Text": "Rationalised 2023-24\n211\nElectromagnetic\nWaves\n8 4 6 X-rays\nBeyond the UV region of the electromagnetic spectrum lies the X-ray\nregion We are familiar with X-rays because of its medical applications"}, {"Chapter": "1", "sentence_range": "6437-6440", "Text": "4 6 X-rays\nBeyond the UV region of the electromagnetic spectrum lies the X-ray\nregion We are familiar with X-rays because of its medical applications It\ncovers wavelengths from about 10\u20138 m (10 nm) down to 10\u201313 m\n(10\u20134 nm)"}, {"Chapter": "1", "sentence_range": "6438-6441", "Text": "6 X-rays\nBeyond the UV region of the electromagnetic spectrum lies the X-ray\nregion We are familiar with X-rays because of its medical applications It\ncovers wavelengths from about 10\u20138 m (10 nm) down to 10\u201313 m\n(10\u20134 nm) One common way to generate X-rays is to bombard a metal\ntarget by high energy electrons"}, {"Chapter": "1", "sentence_range": "6439-6442", "Text": "We are familiar with X-rays because of its medical applications It\ncovers wavelengths from about 10\u20138 m (10 nm) down to 10\u201313 m\n(10\u20134 nm) One common way to generate X-rays is to bombard a metal\ntarget by high energy electrons X-rays are used as a diagnostic tool in\nmedicine and as a treatment for certain forms of cancer"}, {"Chapter": "1", "sentence_range": "6440-6443", "Text": "It\ncovers wavelengths from about 10\u20138 m (10 nm) down to 10\u201313 m\n(10\u20134 nm) One common way to generate X-rays is to bombard a metal\ntarget by high energy electrons X-rays are used as a diagnostic tool in\nmedicine and as a treatment for certain forms of cancer Because X-rays\ndamage or destroy living tissues and organisms, care must be taken to\navoid unnecessary or over exposure"}, {"Chapter": "1", "sentence_range": "6441-6444", "Text": "One common way to generate X-rays is to bombard a metal\ntarget by high energy electrons X-rays are used as a diagnostic tool in\nmedicine and as a treatment for certain forms of cancer Because X-rays\ndamage or destroy living tissues and organisms, care must be taken to\navoid unnecessary or over exposure 8"}, {"Chapter": "1", "sentence_range": "6442-6445", "Text": "X-rays are used as a diagnostic tool in\nmedicine and as a treatment for certain forms of cancer Because X-rays\ndamage or destroy living tissues and organisms, care must be taken to\navoid unnecessary or over exposure 8 4"}, {"Chapter": "1", "sentence_range": "6443-6446", "Text": "Because X-rays\ndamage or destroy living tissues and organisms, care must be taken to\navoid unnecessary or over exposure 8 4 7 Gamma rays\nThey lie in the upper frequency range of the electromagnetic spectrum\nand have wavelengths of from about 10\u201310m to less than 10\u201314m"}, {"Chapter": "1", "sentence_range": "6444-6447", "Text": "8 4 7 Gamma rays\nThey lie in the upper frequency range of the electromagnetic spectrum\nand have wavelengths of from about 10\u201310m to less than 10\u201314m This\nhigh frequency radiation is produced in nuclear reactions and\nalso emitted by radioactive nuclei"}, {"Chapter": "1", "sentence_range": "6445-6448", "Text": "4 7 Gamma rays\nThey lie in the upper frequency range of the electromagnetic spectrum\nand have wavelengths of from about 10\u201310m to less than 10\u201314m This\nhigh frequency radiation is produced in nuclear reactions and\nalso emitted by radioactive nuclei They are used in medicine to destroy\ncancer cells"}, {"Chapter": "1", "sentence_range": "6446-6449", "Text": "7 Gamma rays\nThey lie in the upper frequency range of the electromagnetic spectrum\nand have wavelengths of from about 10\u201310m to less than 10\u201314m This\nhigh frequency radiation is produced in nuclear reactions and\nalso emitted by radioactive nuclei They are used in medicine to destroy\ncancer cells Table 8"}, {"Chapter": "1", "sentence_range": "6447-6450", "Text": "This\nhigh frequency radiation is produced in nuclear reactions and\nalso emitted by radioactive nuclei They are used in medicine to destroy\ncancer cells Table 8 1 summarises different types of electromagnetic waves, their\nproduction and detections"}, {"Chapter": "1", "sentence_range": "6448-6451", "Text": "They are used in medicine to destroy\ncancer cells Table 8 1 summarises different types of electromagnetic waves, their\nproduction and detections As mentioned earlier, the demarcation between\ndifferent regions is not sharp and there are overlaps"}, {"Chapter": "1", "sentence_range": "6449-6452", "Text": "Table 8 1 summarises different types of electromagnetic waves, their\nproduction and detections As mentioned earlier, the demarcation between\ndifferent regions is not sharp and there are overlaps TABLE 8"}, {"Chapter": "1", "sentence_range": "6450-6453", "Text": "1 summarises different types of electromagnetic waves, their\nproduction and detections As mentioned earlier, the demarcation between\ndifferent regions is not sharp and there are overlaps TABLE 8 1 DIFFERENT TYPES OF ELECTROMAGNETIC WAVES\nType\nWavelength range\nProduction\nDetection\nRadio\n> 0"}, {"Chapter": "1", "sentence_range": "6451-6454", "Text": "As mentioned earlier, the demarcation between\ndifferent regions is not sharp and there are overlaps TABLE 8 1 DIFFERENT TYPES OF ELECTROMAGNETIC WAVES\nType\nWavelength range\nProduction\nDetection\nRadio\n> 0 1 m\nRapid  acceleration and\nReceiver\u2019s aerials\ndecelerations of electrons\nin aerials\nMicrowave\n0"}, {"Chapter": "1", "sentence_range": "6452-6455", "Text": "TABLE 8 1 DIFFERENT TYPES OF ELECTROMAGNETIC WAVES\nType\nWavelength range\nProduction\nDetection\nRadio\n> 0 1 m\nRapid  acceleration and\nReceiver\u2019s aerials\ndecelerations of electrons\nin aerials\nMicrowave\n0 1m to 1 mm\nKlystron valve or\nPoint contact diodes\nmagnetron valve\nInfra-red\n1mm to 700 nm\nVibration of atoms\nThermopiles\nand molecules\nBolometer, Infrared\nphotographic film\nLight\n700 nm to 400 nm\nElectrons in atoms emit\nThe eye\nlight when they move from\nPhotocells\none energy level to a\nPhotographic film\nlower energy level\nUltraviolet\n400 nm to 1nm\nInner shell electrons in\nPhotocells\natoms moving from one\nPhotographic film\nenergy level  to a lower level\nX-rays\n1nm to 10\u20133 nm\nX-ray tubes or inner shell\nPhotographic film\nelectrons\nGeiger tubes\nIonisation chamber\nGamma rays\n <10\u20133 nm\nRadioactive decay of the\n-do-\nnucleus\nRationalised 2023-24\nPhysics\n212\nSUMMARY\n1"}, {"Chapter": "1", "sentence_range": "6453-6456", "Text": "1 DIFFERENT TYPES OF ELECTROMAGNETIC WAVES\nType\nWavelength range\nProduction\nDetection\nRadio\n> 0 1 m\nRapid  acceleration and\nReceiver\u2019s aerials\ndecelerations of electrons\nin aerials\nMicrowave\n0 1m to 1 mm\nKlystron valve or\nPoint contact diodes\nmagnetron valve\nInfra-red\n1mm to 700 nm\nVibration of atoms\nThermopiles\nand molecules\nBolometer, Infrared\nphotographic film\nLight\n700 nm to 400 nm\nElectrons in atoms emit\nThe eye\nlight when they move from\nPhotocells\none energy level to a\nPhotographic film\nlower energy level\nUltraviolet\n400 nm to 1nm\nInner shell electrons in\nPhotocells\natoms moving from one\nPhotographic film\nenergy level  to a lower level\nX-rays\n1nm to 10\u20133 nm\nX-ray tubes or inner shell\nPhotographic film\nelectrons\nGeiger tubes\nIonisation chamber\nGamma rays\n <10\u20133 nm\nRadioactive decay of the\n-do-\nnucleus\nRationalised 2023-24\nPhysics\n212\nSUMMARY\n1 Maxwell found an inconsistency in the Ampere\u2019s law and suggested the\nexistence of an additional current, called displacement current, to remove\nthis inconsistency"}, {"Chapter": "1", "sentence_range": "6454-6457", "Text": "1 m\nRapid  acceleration and\nReceiver\u2019s aerials\ndecelerations of electrons\nin aerials\nMicrowave\n0 1m to 1 mm\nKlystron valve or\nPoint contact diodes\nmagnetron valve\nInfra-red\n1mm to 700 nm\nVibration of atoms\nThermopiles\nand molecules\nBolometer, Infrared\nphotographic film\nLight\n700 nm to 400 nm\nElectrons in atoms emit\nThe eye\nlight when they move from\nPhotocells\none energy level to a\nPhotographic film\nlower energy level\nUltraviolet\n400 nm to 1nm\nInner shell electrons in\nPhotocells\natoms moving from one\nPhotographic film\nenergy level  to a lower level\nX-rays\n1nm to 10\u20133 nm\nX-ray tubes or inner shell\nPhotographic film\nelectrons\nGeiger tubes\nIonisation chamber\nGamma rays\n <10\u20133 nm\nRadioactive decay of the\n-do-\nnucleus\nRationalised 2023-24\nPhysics\n212\nSUMMARY\n1 Maxwell found an inconsistency in the Ampere\u2019s law and suggested the\nexistence of an additional current, called displacement current, to remove\nthis inconsistency This displacement current is due to time-varying electric\nfield and is given by\n0\nd\nd\ndi\n\u03a6t\n\u03b5\n\u0395\n=\nand acts as a source of magnetic field in exactly the same way as conduction\ncurrent"}, {"Chapter": "1", "sentence_range": "6455-6458", "Text": "1m to 1 mm\nKlystron valve or\nPoint contact diodes\nmagnetron valve\nInfra-red\n1mm to 700 nm\nVibration of atoms\nThermopiles\nand molecules\nBolometer, Infrared\nphotographic film\nLight\n700 nm to 400 nm\nElectrons in atoms emit\nThe eye\nlight when they move from\nPhotocells\none energy level to a\nPhotographic film\nlower energy level\nUltraviolet\n400 nm to 1nm\nInner shell electrons in\nPhotocells\natoms moving from one\nPhotographic film\nenergy level  to a lower level\nX-rays\n1nm to 10\u20133 nm\nX-ray tubes or inner shell\nPhotographic film\nelectrons\nGeiger tubes\nIonisation chamber\nGamma rays\n <10\u20133 nm\nRadioactive decay of the\n-do-\nnucleus\nRationalised 2023-24\nPhysics\n212\nSUMMARY\n1 Maxwell found an inconsistency in the Ampere\u2019s law and suggested the\nexistence of an additional current, called displacement current, to remove\nthis inconsistency This displacement current is due to time-varying electric\nfield and is given by\n0\nd\nd\ndi\n\u03a6t\n\u03b5\n\u0395\n=\nand acts as a source of magnetic field in exactly the same way as conduction\ncurrent 2"}, {"Chapter": "1", "sentence_range": "6456-6459", "Text": "Maxwell found an inconsistency in the Ampere\u2019s law and suggested the\nexistence of an additional current, called displacement current, to remove\nthis inconsistency This displacement current is due to time-varying electric\nfield and is given by\n0\nd\nd\ndi\n\u03a6t\n\u03b5\n\u0395\n=\nand acts as a source of magnetic field in exactly the same way as conduction\ncurrent 2 An accelerating charge produces electromagnetic waves"}, {"Chapter": "1", "sentence_range": "6457-6460", "Text": "This displacement current is due to time-varying electric\nfield and is given by\n0\nd\nd\ndi\n\u03a6t\n\u03b5\n\u0395\n=\nand acts as a source of magnetic field in exactly the same way as conduction\ncurrent 2 An accelerating charge produces electromagnetic waves An electric charge\noscillating harmonically with frequency n, produces electromagnetic waves\nof the same frequency n"}, {"Chapter": "1", "sentence_range": "6458-6461", "Text": "2 An accelerating charge produces electromagnetic waves An electric charge\noscillating harmonically with frequency n, produces electromagnetic waves\nof the same frequency n An electric dipole is a basic source of\nelectromagnetic waves"}, {"Chapter": "1", "sentence_range": "6459-6462", "Text": "An accelerating charge produces electromagnetic waves An electric charge\noscillating harmonically with frequency n, produces electromagnetic waves\nof the same frequency n An electric dipole is a basic source of\nelectromagnetic waves 3"}, {"Chapter": "1", "sentence_range": "6460-6463", "Text": "An electric charge\noscillating harmonically with frequency n, produces electromagnetic waves\nof the same frequency n An electric dipole is a basic source of\nelectromagnetic waves 3 Electromagnetic waves with wavelength of the order of a few metres were\nfirst produced and detected in the laboratory by Hertz in 1887"}, {"Chapter": "1", "sentence_range": "6461-6464", "Text": "An electric dipole is a basic source of\nelectromagnetic waves 3 Electromagnetic waves with wavelength of the order of a few metres were\nfirst produced and detected in the laboratory by Hertz in 1887 He thus\nverified a basic prediction of Maxwell\u2019s equations"}, {"Chapter": "1", "sentence_range": "6462-6465", "Text": "3 Electromagnetic waves with wavelength of the order of a few metres were\nfirst produced and detected in the laboratory by Hertz in 1887 He thus\nverified a basic prediction of Maxwell\u2019s equations 4"}, {"Chapter": "1", "sentence_range": "6463-6466", "Text": "Electromagnetic waves with wavelength of the order of a few metres were\nfirst produced and detected in the laboratory by Hertz in 1887 He thus\nverified a basic prediction of Maxwell\u2019s equations 4 Electric and magnetic fields oscillate sinusoidally in space and time in an\nelectromagnetic wave"}, {"Chapter": "1", "sentence_range": "6464-6467", "Text": "He thus\nverified a basic prediction of Maxwell\u2019s equations 4 Electric and magnetic fields oscillate sinusoidally in space and time in an\nelectromagnetic wave The oscillating electric and magnetic fields, E and\nB are perpendicular to each other, and to the direction of propagation of\nthe electromagnetic wave"}, {"Chapter": "1", "sentence_range": "6465-6468", "Text": "4 Electric and magnetic fields oscillate sinusoidally in space and time in an\nelectromagnetic wave The oscillating electric and magnetic fields, E and\nB are perpendicular to each other, and to the direction of propagation of\nthe electromagnetic wave For a wave of frequency n, wavelength l,\npropagating along z-direction, we have\nE  = Ex (t) = E0 sin (kz \u2013 w t )\n   = E0 sin    2\n2\n0\n\u03c0\n\u03c0\nz\nt\nE\nz\nTt\n\u03bb\n\u03bd\n\u03bb\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa =\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nsin\nB = By(t) = B0 sin (kz \u2013 w t)\n   = B\nz\nt\nB\nz\nTt\n0\n0\n2\n2\nsin\nsin\n\u03c0\n\u03c0\n\u03bb\n\u03bd\n\u03bb\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa =\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nThey are related by E0/B0 = c"}, {"Chapter": "1", "sentence_range": "6466-6469", "Text": "Electric and magnetic fields oscillate sinusoidally in space and time in an\nelectromagnetic wave The oscillating electric and magnetic fields, E and\nB are perpendicular to each other, and to the direction of propagation of\nthe electromagnetic wave For a wave of frequency n, wavelength l,\npropagating along z-direction, we have\nE  = Ex (t) = E0 sin (kz \u2013 w t )\n   = E0 sin    2\n2\n0\n\u03c0\n\u03c0\nz\nt\nE\nz\nTt\n\u03bb\n\u03bd\n\u03bb\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa =\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nsin\nB = By(t) = B0 sin (kz \u2013 w t)\n   = B\nz\nt\nB\nz\nTt\n0\n0\n2\n2\nsin\nsin\n\u03c0\n\u03c0\n\u03bb\n\u03bd\n\u03bb\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa =\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nThey are related by E0/B0 = c 5"}, {"Chapter": "1", "sentence_range": "6467-6470", "Text": "The oscillating electric and magnetic fields, E and\nB are perpendicular to each other, and to the direction of propagation of\nthe electromagnetic wave For a wave of frequency n, wavelength l,\npropagating along z-direction, we have\nE  = Ex (t) = E0 sin (kz \u2013 w t )\n   = E0 sin    2\n2\n0\n\u03c0\n\u03c0\nz\nt\nE\nz\nTt\n\u03bb\n\u03bd\n\u03bb\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa =\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nsin\nB = By(t) = B0 sin (kz \u2013 w t)\n   = B\nz\nt\nB\nz\nTt\n0\n0\n2\n2\nsin\nsin\n\u03c0\n\u03c0\n\u03bb\n\u03bd\n\u03bb\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa =\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nThey are related by E0/B0 = c 5 The speed c of electromagnetic wave in vacuum is related to m0 and e0 (the\nfree space permeability and permittivity constants) as follows:\n0\n0\n1/\nc\n\u00b5 \u03b5\n="}, {"Chapter": "1", "sentence_range": "6468-6471", "Text": "For a wave of frequency n, wavelength l,\npropagating along z-direction, we have\nE  = Ex (t) = E0 sin (kz \u2013 w t )\n   = E0 sin    2\n2\n0\n\u03c0\n\u03c0\nz\nt\nE\nz\nTt\n\u03bb\n\u03bd\n\u03bb\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa =\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nsin\nB = By(t) = B0 sin (kz \u2013 w t)\n   = B\nz\nt\nB\nz\nTt\n0\n0\n2\n2\nsin\nsin\n\u03c0\n\u03c0\n\u03bb\n\u03bd\n\u03bb\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa =\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nThey are related by E0/B0 = c 5 The speed c of electromagnetic wave in vacuum is related to m0 and e0 (the\nfree space permeability and permittivity constants) as follows:\n0\n0\n1/\nc\n\u00b5 \u03b5\n= The value of c equals the speed of light obtained from\noptical measurements"}, {"Chapter": "1", "sentence_range": "6469-6472", "Text": "5 The speed c of electromagnetic wave in vacuum is related to m0 and e0 (the\nfree space permeability and permittivity constants) as follows:\n0\n0\n1/\nc\n\u00b5 \u03b5\n= The value of c equals the speed of light obtained from\noptical measurements Light is an electromagnetic wave; c is, therefore, also the speed of light"}, {"Chapter": "1", "sentence_range": "6470-6473", "Text": "The speed c of electromagnetic wave in vacuum is related to m0 and e0 (the\nfree space permeability and permittivity constants) as follows:\n0\n0\n1/\nc\n\u00b5 \u03b5\n= The value of c equals the speed of light obtained from\noptical measurements Light is an electromagnetic wave; c is, therefore, also the speed of light Electromagnetic waves other than light also have the same velocity c in\nfree space"}, {"Chapter": "1", "sentence_range": "6471-6474", "Text": "The value of c equals the speed of light obtained from\noptical measurements Light is an electromagnetic wave; c is, therefore, also the speed of light Electromagnetic waves other than light also have the same velocity c in\nfree space The speed of light, or of electromagnetic waves in a material medium is\ngiven by \n1/\nv\n\u00b5 \u03b5\n=\nwhere m is the permeability of the medium and e its permittivity"}, {"Chapter": "1", "sentence_range": "6472-6475", "Text": "Light is an electromagnetic wave; c is, therefore, also the speed of light Electromagnetic waves other than light also have the same velocity c in\nfree space The speed of light, or of electromagnetic waves in a material medium is\ngiven by \n1/\nv\n\u00b5 \u03b5\n=\nwhere m is the permeability of the medium and e its permittivity 6"}, {"Chapter": "1", "sentence_range": "6473-6476", "Text": "Electromagnetic waves other than light also have the same velocity c in\nfree space The speed of light, or of electromagnetic waves in a material medium is\ngiven by \n1/\nv\n\u00b5 \u03b5\n=\nwhere m is the permeability of the medium and e its permittivity 6 The spectrum of electromagnetic waves stretches, in principle, over an\ninfinite range of wavelengths"}, {"Chapter": "1", "sentence_range": "6474-6477", "Text": "The speed of light, or of electromagnetic waves in a material medium is\ngiven by \n1/\nv\n\u00b5 \u03b5\n=\nwhere m is the permeability of the medium and e its permittivity 6 The spectrum of electromagnetic waves stretches, in principle, over an\ninfinite range of wavelengths Different regions are known by different\nnames; g-rays, X-rays, ultraviolet rays, visible rays, infrared rays,\nmicrowaves and radio waves in order of increasing wavelength  from 10\u20132 \u00c5\nor 10\u201312 m to 106 m"}, {"Chapter": "1", "sentence_range": "6475-6478", "Text": "6 The spectrum of electromagnetic waves stretches, in principle, over an\ninfinite range of wavelengths Different regions are known by different\nnames; g-rays, X-rays, ultraviolet rays, visible rays, infrared rays,\nmicrowaves and radio waves in order of increasing wavelength  from 10\u20132 \u00c5\nor 10\u201312 m to 106 m They interact with matter via their electric and magnetic fields which set\nin oscillation charges present in all matter"}, {"Chapter": "1", "sentence_range": "6476-6479", "Text": "The spectrum of electromagnetic waves stretches, in principle, over an\ninfinite range of wavelengths Different regions are known by different\nnames; g-rays, X-rays, ultraviolet rays, visible rays, infrared rays,\nmicrowaves and radio waves in order of increasing wavelength  from 10\u20132 \u00c5\nor 10\u201312 m to 106 m They interact with matter via their electric and magnetic fields which set\nin oscillation charges present in all matter The detailed interaction and\nso the mechanism of absorption, scattering, etc"}, {"Chapter": "1", "sentence_range": "6477-6480", "Text": "Different regions are known by different\nnames; g-rays, X-rays, ultraviolet rays, visible rays, infrared rays,\nmicrowaves and radio waves in order of increasing wavelength  from 10\u20132 \u00c5\nor 10\u201312 m to 106 m They interact with matter via their electric and magnetic fields which set\nin oscillation charges present in all matter The detailed interaction and\nso the mechanism of absorption, scattering, etc , depend on the wavelength\nof the electromagnetic wave, and the nature of the atoms and molecules\nin the medium"}, {"Chapter": "1", "sentence_range": "6478-6481", "Text": "They interact with matter via their electric and magnetic fields which set\nin oscillation charges present in all matter The detailed interaction and\nso the mechanism of absorption, scattering, etc , depend on the wavelength\nof the electromagnetic wave, and the nature of the atoms and molecules\nin the medium Rationalised 2023-24\n213\nElectromagnetic\nWaves\nPOINTS TO PONDER\n1"}, {"Chapter": "1", "sentence_range": "6479-6482", "Text": "The detailed interaction and\nso the mechanism of absorption, scattering, etc , depend on the wavelength\nof the electromagnetic wave, and the nature of the atoms and molecules\nin the medium Rationalised 2023-24\n213\nElectromagnetic\nWaves\nPOINTS TO PONDER\n1 The basic difference between various types of electromagnetic waves\nlies in their wavelengths or frequencies since all of them travel through\nvacuum with the same speed"}, {"Chapter": "1", "sentence_range": "6480-6483", "Text": ", depend on the wavelength\nof the electromagnetic wave, and the nature of the atoms and molecules\nin the medium Rationalised 2023-24\n213\nElectromagnetic\nWaves\nPOINTS TO PONDER\n1 The basic difference between various types of electromagnetic waves\nlies in their wavelengths or frequencies since all of them travel through\nvacuum with the same speed Consequently, the waves differ\nconsiderably in their mode of interaction with matter"}, {"Chapter": "1", "sentence_range": "6481-6484", "Text": "Rationalised 2023-24\n213\nElectromagnetic\nWaves\nPOINTS TO PONDER\n1 The basic difference between various types of electromagnetic waves\nlies in their wavelengths or frequencies since all of them travel through\nvacuum with the same speed Consequently, the waves differ\nconsiderably in their mode of interaction with matter 2"}, {"Chapter": "1", "sentence_range": "6482-6485", "Text": "The basic difference between various types of electromagnetic waves\nlies in their wavelengths or frequencies since all of them travel through\nvacuum with the same speed Consequently, the waves differ\nconsiderably in their mode of interaction with matter 2 Accelerated charged particles radiate electromagnetic waves"}, {"Chapter": "1", "sentence_range": "6483-6486", "Text": "Consequently, the waves differ\nconsiderably in their mode of interaction with matter 2 Accelerated charged particles radiate electromagnetic waves The\nwavelength of the electromagnetic wave is often correlated with the\ncharacteristic size of the system that radiates"}, {"Chapter": "1", "sentence_range": "6484-6487", "Text": "2 Accelerated charged particles radiate electromagnetic waves The\nwavelength of the electromagnetic wave is often correlated with the\ncharacteristic size of the system that radiates Thus, gamma radiation,\nhaving wavelength of 10\u201314 m to 10\u201315 m, typically originate from an\natomic nucleus"}, {"Chapter": "1", "sentence_range": "6485-6488", "Text": "Accelerated charged particles radiate electromagnetic waves The\nwavelength of the electromagnetic wave is often correlated with the\ncharacteristic size of the system that radiates Thus, gamma radiation,\nhaving wavelength of 10\u201314 m to 10\u201315 m, typically originate from an\natomic nucleus X-rays are emitted from heavy atoms"}, {"Chapter": "1", "sentence_range": "6486-6489", "Text": "The\nwavelength of the electromagnetic wave is often correlated with the\ncharacteristic size of the system that radiates Thus, gamma radiation,\nhaving wavelength of 10\u201314 m to 10\u201315 m, typically originate from an\natomic nucleus X-rays are emitted from heavy atoms Radio waves\nare produced by accelerating electrons in a circuit"}, {"Chapter": "1", "sentence_range": "6487-6490", "Text": "Thus, gamma radiation,\nhaving wavelength of 10\u201314 m to 10\u201315 m, typically originate from an\natomic nucleus X-rays are emitted from heavy atoms Radio waves\nare produced by accelerating electrons in a circuit A transmitting\nantenna can most efficiently radiate waves having a wavelength of\nabout the same size as the antenna"}, {"Chapter": "1", "sentence_range": "6488-6491", "Text": "X-rays are emitted from heavy atoms Radio waves\nare produced by accelerating electrons in a circuit A transmitting\nantenna can most efficiently radiate waves having a wavelength of\nabout the same size as the antenna Visible radiation emitted by atoms\nis, however, much longer in wavelength than atomic size"}, {"Chapter": "1", "sentence_range": "6489-6492", "Text": "Radio waves\nare produced by accelerating electrons in a circuit A transmitting\nantenna can most efficiently radiate waves having a wavelength of\nabout the same size as the antenna Visible radiation emitted by atoms\nis, however, much longer in wavelength than atomic size 3"}, {"Chapter": "1", "sentence_range": "6490-6493", "Text": "A transmitting\nantenna can most efficiently radiate waves having a wavelength of\nabout the same size as the antenna Visible radiation emitted by atoms\nis, however, much longer in wavelength than atomic size 3 Infrared waves, with frequencies lower than those of visible light,\nvibrate not only the electrons, but entire atoms or molecules of a\nsubstance"}, {"Chapter": "1", "sentence_range": "6491-6494", "Text": "Visible radiation emitted by atoms\nis, however, much longer in wavelength than atomic size 3 Infrared waves, with frequencies lower than those of visible light,\nvibrate not only the electrons, but entire atoms or molecules of a\nsubstance This vibration increases the internal energy and\nconsequently, the temperature of the substance"}, {"Chapter": "1", "sentence_range": "6492-6495", "Text": "3 Infrared waves, with frequencies lower than those of visible light,\nvibrate not only the electrons, but entire atoms or molecules of a\nsubstance This vibration increases the internal energy and\nconsequently, the temperature of the substance This is why infrared\nwaves are often called heat waves"}, {"Chapter": "1", "sentence_range": "6493-6496", "Text": "Infrared waves, with frequencies lower than those of visible light,\nvibrate not only the electrons, but entire atoms or molecules of a\nsubstance This vibration increases the internal energy and\nconsequently, the temperature of the substance This is why infrared\nwaves are often called heat waves 4"}, {"Chapter": "1", "sentence_range": "6494-6497", "Text": "This vibration increases the internal energy and\nconsequently, the temperature of the substance This is why infrared\nwaves are often called heat waves 4 The centre of sensitivity of our eyes coincides with the centre of the\nwavelength distribution of the sun"}, {"Chapter": "1", "sentence_range": "6495-6498", "Text": "This is why infrared\nwaves are often called heat waves 4 The centre of sensitivity of our eyes coincides with the centre of the\nwavelength distribution of the sun It is because humans have evolved\nwith visions most sensitive to the strongest wavelengths from\nthe sun"}, {"Chapter": "1", "sentence_range": "6496-6499", "Text": "4 The centre of sensitivity of our eyes coincides with the centre of the\nwavelength distribution of the sun It is because humans have evolved\nwith visions most sensitive to the strongest wavelengths from\nthe sun EXERCISES\n8"}, {"Chapter": "1", "sentence_range": "6497-6500", "Text": "The centre of sensitivity of our eyes coincides with the centre of the\nwavelength distribution of the sun It is because humans have evolved\nwith visions most sensitive to the strongest wavelengths from\nthe sun EXERCISES\n8 1\nFigure 8"}, {"Chapter": "1", "sentence_range": "6498-6501", "Text": "It is because humans have evolved\nwith visions most sensitive to the strongest wavelengths from\nthe sun EXERCISES\n8 1\nFigure 8 5 shows a capacitor made of two circular plates each of\nradius 12 cm, and separated by 5"}, {"Chapter": "1", "sentence_range": "6499-6502", "Text": "EXERCISES\n8 1\nFigure 8 5 shows a capacitor made of two circular plates each of\nradius 12 cm, and separated by 5 0 cm"}, {"Chapter": "1", "sentence_range": "6500-6503", "Text": "1\nFigure 8 5 shows a capacitor made of two circular plates each of\nradius 12 cm, and separated by 5 0 cm The capacitor is being\ncharged by an external source (not shown in the figure)"}, {"Chapter": "1", "sentence_range": "6501-6504", "Text": "5 shows a capacitor made of two circular plates each of\nradius 12 cm, and separated by 5 0 cm The capacitor is being\ncharged by an external source (not shown in the figure) The\ncharging current is constant and equal to 0"}, {"Chapter": "1", "sentence_range": "6502-6505", "Text": "0 cm The capacitor is being\ncharged by an external source (not shown in the figure) The\ncharging current is constant and equal to 0 15A"}, {"Chapter": "1", "sentence_range": "6503-6506", "Text": "The capacitor is being\ncharged by an external source (not shown in the figure) The\ncharging current is constant and equal to 0 15A (a)\nCalculate the capacitance and the rate of change of potential\ndifference between the plates"}, {"Chapter": "1", "sentence_range": "6504-6507", "Text": "The\ncharging current is constant and equal to 0 15A (a)\nCalculate the capacitance and the rate of change of potential\ndifference between the plates (b)\nObtain the displacement current across the plates"}, {"Chapter": "1", "sentence_range": "6505-6508", "Text": "15A (a)\nCalculate the capacitance and the rate of change of potential\ndifference between the plates (b)\nObtain the displacement current across the plates (c)\nIs Kirchhoff\u2019s first rule (junction rule) valid at each plate of the\ncapacitor"}, {"Chapter": "1", "sentence_range": "6506-6509", "Text": "(a)\nCalculate the capacitance and the rate of change of potential\ndifference between the plates (b)\nObtain the displacement current across the plates (c)\nIs Kirchhoff\u2019s first rule (junction rule) valid at each plate of the\ncapacitor Explain"}, {"Chapter": "1", "sentence_range": "6507-6510", "Text": "(b)\nObtain the displacement current across the plates (c)\nIs Kirchhoff\u2019s first rule (junction rule) valid at each plate of the\ncapacitor Explain FIGURE 8"}, {"Chapter": "1", "sentence_range": "6508-6511", "Text": "(c)\nIs Kirchhoff\u2019s first rule (junction rule) valid at each plate of the\ncapacitor Explain FIGURE 8 5\n8"}, {"Chapter": "1", "sentence_range": "6509-6512", "Text": "Explain FIGURE 8 5\n8 2\nA parallel plate capacitor (Fig"}, {"Chapter": "1", "sentence_range": "6510-6513", "Text": "FIGURE 8 5\n8 2\nA parallel plate capacitor (Fig 8"}, {"Chapter": "1", "sentence_range": "6511-6514", "Text": "5\n8 2\nA parallel plate capacitor (Fig 8 6) made of circular plates each of radius\nR = 6"}, {"Chapter": "1", "sentence_range": "6512-6515", "Text": "2\nA parallel plate capacitor (Fig 8 6) made of circular plates each of radius\nR = 6 0 cm has a capacitance C = 100 pF"}, {"Chapter": "1", "sentence_range": "6513-6516", "Text": "8 6) made of circular plates each of radius\nR = 6 0 cm has a capacitance C = 100 pF The capacitor is connected to\na 230 V ac supply with a (angular) frequency of 300 rad s\u20131"}, {"Chapter": "1", "sentence_range": "6514-6517", "Text": "6) made of circular plates each of radius\nR = 6 0 cm has a capacitance C = 100 pF The capacitor is connected to\na 230 V ac supply with a (angular) frequency of 300 rad s\u20131 Rationalised 2023-24\nPhysics\n214\n(a)\nWhat is the rms value of the conduction current"}, {"Chapter": "1", "sentence_range": "6515-6518", "Text": "0 cm has a capacitance C = 100 pF The capacitor is connected to\na 230 V ac supply with a (angular) frequency of 300 rad s\u20131 Rationalised 2023-24\nPhysics\n214\n(a)\nWhat is the rms value of the conduction current (b)\nIs the conduction current equal to the displacement current"}, {"Chapter": "1", "sentence_range": "6516-6519", "Text": "The capacitor is connected to\na 230 V ac supply with a (angular) frequency of 300 rad s\u20131 Rationalised 2023-24\nPhysics\n214\n(a)\nWhat is the rms value of the conduction current (b)\nIs the conduction current equal to the displacement current (c)\nDetermine the amplitude of B at a point 3"}, {"Chapter": "1", "sentence_range": "6517-6520", "Text": "Rationalised 2023-24\nPhysics\n214\n(a)\nWhat is the rms value of the conduction current (b)\nIs the conduction current equal to the displacement current (c)\nDetermine the amplitude of B at a point 3 0 cm from the axis\nbetween the plates"}, {"Chapter": "1", "sentence_range": "6518-6521", "Text": "(b)\nIs the conduction current equal to the displacement current (c)\nDetermine the amplitude of B at a point 3 0 cm from the axis\nbetween the plates FIGURE 8"}, {"Chapter": "1", "sentence_range": "6519-6522", "Text": "(c)\nDetermine the amplitude of B at a point 3 0 cm from the axis\nbetween the plates FIGURE 8 6\n8"}, {"Chapter": "1", "sentence_range": "6520-6523", "Text": "0 cm from the axis\nbetween the plates FIGURE 8 6\n8 3\nWhat physical quantity is the same for X-rays of wavelength\n10\u201310 m, red light of wavelength 6800 \u00c5 and radiowaves of wavelength\n500m"}, {"Chapter": "1", "sentence_range": "6521-6524", "Text": "FIGURE 8 6\n8 3\nWhat physical quantity is the same for X-rays of wavelength\n10\u201310 m, red light of wavelength 6800 \u00c5 and radiowaves of wavelength\n500m 8"}, {"Chapter": "1", "sentence_range": "6522-6525", "Text": "6\n8 3\nWhat physical quantity is the same for X-rays of wavelength\n10\u201310 m, red light of wavelength 6800 \u00c5 and radiowaves of wavelength\n500m 8 4\nA plane electromagnetic wave travels in vacuum along z-direction"}, {"Chapter": "1", "sentence_range": "6523-6526", "Text": "3\nWhat physical quantity is the same for X-rays of wavelength\n10\u201310 m, red light of wavelength 6800 \u00c5 and radiowaves of wavelength\n500m 8 4\nA plane electromagnetic wave travels in vacuum along z-direction What can you say about the directions of its electric and magnetic\nfield vectors"}, {"Chapter": "1", "sentence_range": "6524-6527", "Text": "8 4\nA plane electromagnetic wave travels in vacuum along z-direction What can you say about the directions of its electric and magnetic\nfield vectors If the frequency of the wave is 30 MHz, what is its\nwavelength"}, {"Chapter": "1", "sentence_range": "6525-6528", "Text": "4\nA plane electromagnetic wave travels in vacuum along z-direction What can you say about the directions of its electric and magnetic\nfield vectors If the frequency of the wave is 30 MHz, what is its\nwavelength 8"}, {"Chapter": "1", "sentence_range": "6526-6529", "Text": "What can you say about the directions of its electric and magnetic\nfield vectors If the frequency of the wave is 30 MHz, what is its\nwavelength 8 5\nA radio can tune in to any station in the 7"}, {"Chapter": "1", "sentence_range": "6527-6530", "Text": "If the frequency of the wave is 30 MHz, what is its\nwavelength 8 5\nA radio can tune in to any station in the 7 5 MHz to 12 MHz band"}, {"Chapter": "1", "sentence_range": "6528-6531", "Text": "8 5\nA radio can tune in to any station in the 7 5 MHz to 12 MHz band What is the corresponding wavelength band"}, {"Chapter": "1", "sentence_range": "6529-6532", "Text": "5\nA radio can tune in to any station in the 7 5 MHz to 12 MHz band What is the corresponding wavelength band 8"}, {"Chapter": "1", "sentence_range": "6530-6533", "Text": "5 MHz to 12 MHz band What is the corresponding wavelength band 8 6\nA charged particle oscillates about its mean equilibrium position\nwith a frequency of 10 9 Hz"}, {"Chapter": "1", "sentence_range": "6531-6534", "Text": "What is the corresponding wavelength band 8 6\nA charged particle oscillates about its mean equilibrium position\nwith a frequency of 10 9 Hz What is the frequency of the\nelectromagnetic waves produced by the oscillator"}, {"Chapter": "1", "sentence_range": "6532-6535", "Text": "8 6\nA charged particle oscillates about its mean equilibrium position\nwith a frequency of 10 9 Hz What is the frequency of the\nelectromagnetic waves produced by the oscillator 8"}, {"Chapter": "1", "sentence_range": "6533-6536", "Text": "6\nA charged particle oscillates about its mean equilibrium position\nwith a frequency of 10 9 Hz What is the frequency of the\nelectromagnetic waves produced by the oscillator 8 7\nThe amplitude of the magnetic field part of a harmonic\nelectromagnetic wave in vacuum is B0 = 510 nT"}, {"Chapter": "1", "sentence_range": "6534-6537", "Text": "What is the frequency of the\nelectromagnetic waves produced by the oscillator 8 7\nThe amplitude of the magnetic field part of a harmonic\nelectromagnetic wave in vacuum is B0 = 510 nT What is the\namplitude of the electric field part of the wave"}, {"Chapter": "1", "sentence_range": "6535-6538", "Text": "8 7\nThe amplitude of the magnetic field part of a harmonic\nelectromagnetic wave in vacuum is B0 = 510 nT What is the\namplitude of the electric field part of the wave 8"}, {"Chapter": "1", "sentence_range": "6536-6539", "Text": "7\nThe amplitude of the magnetic field part of a harmonic\nelectromagnetic wave in vacuum is B0 = 510 nT What is the\namplitude of the electric field part of the wave 8 8\nSuppose that the electric field amplitude of an electromagnetic wave\nis E0 = 120 N/C and that its frequency is n = 50"}, {"Chapter": "1", "sentence_range": "6537-6540", "Text": "What is the\namplitude of the electric field part of the wave 8 8\nSuppose that the electric field amplitude of an electromagnetic wave\nis E0 = 120 N/C and that its frequency is n = 50 0 MHz"}, {"Chapter": "1", "sentence_range": "6538-6541", "Text": "8 8\nSuppose that the electric field amplitude of an electromagnetic wave\nis E0 = 120 N/C and that its frequency is n = 50 0 MHz (a) Determine,\nB0,w, k, and l"}, {"Chapter": "1", "sentence_range": "6539-6542", "Text": "8\nSuppose that the electric field amplitude of an electromagnetic wave\nis E0 = 120 N/C and that its frequency is n = 50 0 MHz (a) Determine,\nB0,w, k, and l (b) Find expressions for E and B"}, {"Chapter": "1", "sentence_range": "6540-6543", "Text": "0 MHz (a) Determine,\nB0,w, k, and l (b) Find expressions for E and B 8"}, {"Chapter": "1", "sentence_range": "6541-6544", "Text": "(a) Determine,\nB0,w, k, and l (b) Find expressions for E and B 8 9\nThe terminology of different parts of the electromagnetic spectrum\nis given in the text"}, {"Chapter": "1", "sentence_range": "6542-6545", "Text": "(b) Find expressions for E and B 8 9\nThe terminology of different parts of the electromagnetic spectrum\nis given in the text Use the formula E = hn (for energy of a quantum\nof radiation: photon) and obtain the photon energy in units of eV for\ndifferent parts of the electromagnetic spectrum"}, {"Chapter": "1", "sentence_range": "6543-6546", "Text": "8 9\nThe terminology of different parts of the electromagnetic spectrum\nis given in the text Use the formula E = hn (for energy of a quantum\nof radiation: photon) and obtain the photon energy in units of eV for\ndifferent parts of the electromagnetic spectrum In what way are\nthe different scales of photon energies that you obtain related to the\nsources of electromagnetic radiation"}, {"Chapter": "1", "sentence_range": "6544-6547", "Text": "9\nThe terminology of different parts of the electromagnetic spectrum\nis given in the text Use the formula E = hn (for energy of a quantum\nof radiation: photon) and obtain the photon energy in units of eV for\ndifferent parts of the electromagnetic spectrum In what way are\nthe different scales of photon energies that you obtain related to the\nsources of electromagnetic radiation 8"}, {"Chapter": "1", "sentence_range": "6545-6548", "Text": "Use the formula E = hn (for energy of a quantum\nof radiation: photon) and obtain the photon energy in units of eV for\ndifferent parts of the electromagnetic spectrum In what way are\nthe different scales of photon energies that you obtain related to the\nsources of electromagnetic radiation 8 10\nIn a plane electromagnetic wave, the electric field oscillates\nsinusoidally at a frequency of 2"}, {"Chapter": "1", "sentence_range": "6546-6549", "Text": "In what way are\nthe different scales of photon energies that you obtain related to the\nsources of electromagnetic radiation 8 10\nIn a plane electromagnetic wave, the electric field oscillates\nsinusoidally at a frequency of 2 0 \u00d7 1010 Hz and amplitude 48 V m\u20131"}, {"Chapter": "1", "sentence_range": "6547-6550", "Text": "8 10\nIn a plane electromagnetic wave, the electric field oscillates\nsinusoidally at a frequency of 2 0 \u00d7 1010 Hz and amplitude 48 V m\u20131 (a)\nWhat is the wavelength of the wave"}, {"Chapter": "1", "sentence_range": "6548-6551", "Text": "10\nIn a plane electromagnetic wave, the electric field oscillates\nsinusoidally at a frequency of 2 0 \u00d7 1010 Hz and amplitude 48 V m\u20131 (a)\nWhat is the wavelength of the wave (b)\nWhat is the amplitude of the oscillating magnetic field"}, {"Chapter": "1", "sentence_range": "6549-6552", "Text": "0 \u00d7 1010 Hz and amplitude 48 V m\u20131 (a)\nWhat is the wavelength of the wave (b)\nWhat is the amplitude of the oscillating magnetic field (c)\nShow that the average energy density of the E field equals the\naverage energy density of the B field"}, {"Chapter": "1", "sentence_range": "6550-6553", "Text": "(a)\nWhat is the wavelength of the wave (b)\nWhat is the amplitude of the oscillating magnetic field (c)\nShow that the average energy density of the E field equals the\naverage energy density of the B field [c = 3 \u00d7 108 m s\u20131"}, {"Chapter": "1", "sentence_range": "6551-6554", "Text": "(b)\nWhat is the amplitude of the oscillating magnetic field (c)\nShow that the average energy density of the E field equals the\naverage energy density of the B field [c = 3 \u00d7 108 m s\u20131 ]\nRationalised 2023-24"}]