Preface

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About *University Physics*

*University Physics* is designed for the two- or three-semester calculus-based physics course. The text has been developed to meet the scope and sequence of most university physics courses and provides a foundation for a career in mathematics, science, or engineering. The book provides an important opportunity for students to learn the core concepts of physics and understand how those concepts apply to their lives and to the world around them.

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About the authors

Thermodynamics

Temperature and Heat

Introduction

Temperature and Thermal Equilibrium

Abstract:
By the end of this section, you will be able to:

* Define temperature and describe it qualitatively
* Explain thermal equilibrium
* Explain the zeroth law of thermodynamics

Temperature

The concept of temperature has evolved from the common concepts of hot and cold. The scientific definition of temperature explains more than our senses of hot and cold. As you may have already learned, many physical quantities are defined solely in terms of how they are observed or measured, that is, they are defined *operationally*. **Temperature** is operationally defined as the quantity of what we measure with a thermometer. As we will see in detail in a later chapter on the kinetic theory of gases, temperature is proportional to the average kinetic energy of translation, a fact that provides a more physical definition. Differences in temperature maintain the transfer of heat, or *heat transfe*r, throughout the universe. **Heat transfer** is the movement of energy from one place or material to another as a result of a difference in temperature. (You will learn more about heat transfer later in this chapter.)

Thermal Equilibrium

An important concept related to temperature is thermal equilibrium. Two objects are in thermal equilibrium if they are in close contact that allows either to gain energy from the other, but nevertheless, no net energy is transferred between them, or, even when not in contact, if no net energy is transferred between them when they are brought into contact. If two objects remain in contact for a long time, they typically come to equilibrium. In other words, two objects in thermal equilibrium do not exchange energy.

Thermometers and Temperature Scales

Abstract:
By the end of this section, you will be able to:

* Describe several different types of thermometers
* Convert temperatures between the Celsius, Fahrenheit, and Kelvin scales

Thermal Expansion

Abstract:
By the end of this section, you will be able to:

* Answer qualitative questions about the effects of thermal expansion
* Solve problems involving thermal expansion, including those involving thermal stress

Thermal Expansion in Two and Three Dimensions

Unconstrained objects expand in all dimensions, as illustrated in [link]. That is, their areas and volumes, as well as their lengths, increase with temperature. Because the proportions stay the same, holes and container volumes also get larger with temperature. If you cut a hole in a metal plate, the remaining material will expand exactly as it would if the piece you removed were still in place. The piece would get bigger, so the hole must get bigger too.

Thermal Stress

If you change the temperature of an object while preventing it from expanding or contracting, the object is subjected to stress that is compressive if the object would expand in the absence of constraint and tensile if it would contract. This stress resulting from temperature changes is known as thermal stress. It can be quite large and can cause damage.

Heat Transfer, Specific Heat, and Calorimetry

Abstract:
By the end of this section, you will be able to:

* Explain phenomena involving heat as a form of energy transfer
* Solve problems involving heat transfer

Internal Energy and Heat

A thermal system has *internal energy*, which is the sum of the microscopic energies of the system. This includes thermal energy, which is associated with the mechanical energies of its molecules and which is proportional to the system’s temperature. As we saw earlier in this chapter, if two objects at different temperatures are brought into contact with each other, energy is transferred from the hotter to the colder object until the bodies reach thermal equilibrium (that is, they are at the same temperature). No work is done by either object because no force acts through a distance (as we discussed in Work and Kinetic Energy). These observations reveal that heat is energy transferred spontaneously due to a temperature difference. [link] shows an example of heat transfer.

Mechanical Equivalent of Heat

It is also possible to change the temperature of a substance by doing work, which transfers energy into or out of a system. This realization helped establish that heat is a form of energy. James Prescott Joule (1818–1889) performed many experiments to establish the mechanical equivalent of heat—*the work needed to produce the same effects as heat transfer*. In the units used for these two quantities, the value for this equivalence is

Temperature Change and Heat Capacity

We have noted that heat transfer often causes temperature change. Experiments show that with no phase change and no work done on or by the system, the transferred heat is typically directly proportional to the change in temperature and to the mass of the system, to a good approximation. (Below we show how to handle situations where the approximation is not valid.) The constant of proportionality depends on the substance and its phase, which may be gas, liquid, or solid. We omit discussion of the fourth phase, plasma, because although it is the most common phase in the universe, it is rare and short-lived on Earth.

Phase Changes

Abstract:
By the end of this section, you will be able to:

* Describe phase transitions and equilibrium between phases
* Solve problems involving latent heat
* Solve calorimetry problems involving phase changes

Phase Diagrams

The phase of a given substance depends on the pressure and temperature. Thus, plots of pressure versus temperature showing the phase in each region provide considerable insight into thermal properties of substances. Such a *pT* graph is called a phase diagram.

Equilibrium

At the melting temperature, the solid and liquid phases are in equilibrium. If heat is added, some of the solid will melt, and if heat is removed, some of the liquid will freeze. The situation is somewhat more complex for liquid-gas equilibrium. Generally, liquid and gas are in equilibrium at any temperature. We call the gas phase a vapor when it exists at a temperature below the critical temperature, as it does for water at $20.0{\rule{0.2em}{0ex}}\text{°}\text{C}$. Liquid in a closed container at a fixed temperature evaporates until the pressure of the gas reaches a certain value, called the vapor pressure, which depends on the gas and the temperature. At this equilibrium, if heat is added, some of the liquid will evaporate, and if heat is removed, some of the gas will condense; molecules either join the liquid or form suspended droplets. If there is not enough liquid for the gas to reach the vapor pressure in the container, all the liquid eventually evaporates.

Phase Change and Latent Heat

So far, we have discussed heat transfers that cause temperature change. However, in a phase transition, heat transfer does not cause any temperature change.

Mechanisms of Heat Transfer

Abstract:
By the end of this section, you will be able to:

* Explain some phenomena that involve conductive, convective, and radiative heat transfer
* Solve problems on the relationships between heat transfer, time, and rate of heat transfer
* Solve problems using the formulas for conduction and radiation

Conduction

As you walk barefoot across the living room carpet in a cold house and then step onto the kitchen tile floor, your feet feel colder on the tile. This result is intriguing, since the carpet and tile floor are both at the same temperature. The different sensation is explained by the different rates of heat transfer: The heat loss is faster for skin in contact with the tiles than with the carpet, so the sensation of cold is more intense.

Convection

In convection, thermal energy is carried by the large-scale flow of matter. It can be divided into two types. In *forced convection*, the flow is driven by fans, pumps, and the like. A simple example is a fan that blows air past you in hot surroundings and cools you by replacing the air heated by your body with cooler air. A more complicated example is the cooling system of a typical car, in which a pump moves coolant through the radiator and engine to cool the engine and a fan blows air to cool the radiator.

Radiation

You can feel the heat transfer from the Sun. The space between Earth and the Sun is largely empty, so the Sun warms us without any possibility of heat transfer by convection or conduction. Similarly, you can sometimes tell that the oven is hot without touching its door or looking inside—it may just warm you as you walk by. In these examples, heat is transferred by radiation ([link]). That is, the hot body emits electromagnetic waves that are absorbed by the skin. No medium is required for electromagnetic waves to propagate. Different names are used for electromagnetic waves of different wavelengths: radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays.

The Kinetic Theory of Gases

Introduction

Molecular Model of an Ideal Gas

Abstract:
By the end of this section, you will be able to:

* Apply the ideal gas law to situations involving the pressure, volume, temperature, and the number of molecules of a gas
* Use the unit of moles in relation to numbers of molecules, and molecular and macroscopic masses
* Explain the ideal gas law in terms of moles rather than numbers of molecules
* Apply the van der Waals gas law to situations where the ideal gas law is inadequate

The Gas Laws

In the previous chapter, we saw one consequence of the large intermolecular spacing in gases: Gases are easily compressed. [link] shows that gases have larger coefficients of volume expansion than either solids or liquids. These large coefficients mean that gases expand and contract very rapidly with temperature changes. We also saw (in the section on thermal expansion) that most gases expand at the same rate or have the same coefficient of volume expansion, $\beta$. This raises a question: Why do all gases act in nearly the same way, when all the various liquids and solids have widely varying expansion rates?

Moles and Avogadro’s Number

It is often convenient to measure the amount of substance with a unit on a more human scale than molecules. The SI unit for this purpose was developed by the Italian scientist Amedeo Avogadro (1776–1856). (He worked from the hypothesis that equal volumes of gas at equal pressure and temperature contain equal numbers of molecules, independent of the type of gas. As mentioned above, this hypothesis has been confirmed when the ideal gas approximation applies.) A mole (abbreviated mol) is defined as the amount of any substance that contains as many molecules as there are atoms. (Technically, we should say “formula units,” not “molecules,” but this distinction is irrelevant for our purposes.) The number of molecules in one mole is called Avogadro’s number $\left({N}_{\text{A}}\right),$ and the value of Avogadro’s number is now known to be

The Ideal Gas Law Restated using Moles

A very common expression of the ideal gas law uses the number of moles in a sample, *n*, rather than the number of molecules, *N*. We start from the ideal gas law,

The Van der Waals Equation of State

We have repeatedly noted that the ideal gas law is an approximation. How can it be improved upon? The van der Waals equation of state (named after the Dutch physicist Johannes van der Waals, 1837−1923) improves it by taking into account two factors. First, the attractive forces between molecules, which are stronger at higher density and reduce the pressure, are taken into account by adding to the pressure a term equal to the square of the molar density multiplied by a positive coefficient *a*. Second, the volume of the molecules is represented by a positive constant *b*, which can be thought of as the volume of a mole of molecules. This is subtracted from the total volume to give the remaining volume that the molecules can move in. The constants *a* and *b* are determined experimentally for each gas. The resulting equation is

*pV* Diagrams

We can examine aspects of the behavior of a substance by plotting a *pV* diagram, which is a graph of pressure versus volume. When the substance behaves like an ideal gas, the ideal gas law $pV=nRT$ describes the relationship between its pressure and volume. On a *pV* diagram, it’s common to plot an *isotherm*, which is a curve showing *p* as a function of *V* with the number of molecules and the temperature fixed. Then, for an ideal gas, $pV=\text{constant}\text{.}$ For example, the volume of the gas decreases as the pressure increases. The resulting graph is a hyperbola.

Pressure, Temperature, and RMS Speed

Abstract:
By the end of this section, you will be able to:

* Explain the relations between microscopic and macroscopic quantities in a gas
* Solve problems involving mixtures of gases
* Solve problems involving the distance and time between a gas molecule’s collisions

Vapor Pressure, Partial Pressure, and Dalton’s Law

The pressure a gas would create if it occupied the total volume available is called the gas’s partial pressure. If two or more gases are mixed, they will come to thermal equilibrium as a result of collisions between molecules; the process is analogous to heat conduction as described in the chapter on temperature and heat. As we have seen from kinetic theory, when the gases have the same temperature, their molecules have the same average kinetic energy. Thus, each gas obeys the ideal gas law separately and exerts the same pressure on the walls of a container that it would if it were alone. Therefore, in a mixture of gases, *the total pressure is the sum of partial pressures of the component gases*, assuming ideal gas behavior and no chemical reactions between the components. This law is known as Dalton’s law of partial pressures, after the English scientist John Dalton (1766–1844) who proposed it. Dalton’s law is consistent with the fact that pressures add according to Pascal’s principle.

Mean Free Path and Mean Free Time

We now consider collisions explicitly. The usual first step (which is all we’ll take) is to calculate the mean free path, $\lambda ,$ the average distance a molecule travels between collisions with other molecules, and the *mean free time* $\tau$, the average time between the collisions of a molecule. If we assume all the molecules are spheres with a radius *r*, then a molecule will collide with another if their centers are within a distance 2*r* of each other. For a given particle, we say that the area of a circle with that radius, $4\pi {r}^{2}$, is the “cross-section” for collisions. As the particle moves, it traces a cylinder with that cross-sectional area. The mean free path is the length $\lambda$ such that the expected number of other molecules in a cylinder of length $\lambda$ and cross-section $4\pi {r}^{2}$ is 1. If we temporarily ignore the motion of the molecules other than the one we’re looking at, the expected number is the number density of molecules, *N*/*V*, times the volume, and the volume is $4\pi {r}^{2}\lambda$, so we have $\left(N\text{/}V\right)4\pi {r}^{2}\lambda =1,$ or

Heat Capacity and Equipartition of Energy

Abstract:
By the end of this section, you will be able to:

* Solve problems involving heat transfer to and from ideal monatomic gases whose volumes are held constant
* Solve similar problems for non-monatomic ideal gases based on the number of degrees of freedom of a molecule
* Estimate the heat capacities of metals using a model based on degrees of freedom

Heat Capacity of an Ideal Monatomic Gas at Constant Volume

We define the *molar heat capacity at constant volume* ${C}_{V}$ as

Degrees of Freedom

In the previous section, we found that $\frac{1}{2}m\stackrel{\text{–}}{{v}^{2}}=\frac{3}{2}{k}_{\text{B}}T$ and $\stackrel{\text{–}}{{v}^{2}}=3\stackrel{\text{–}}{{v}_{x}^{2}}$, from which it follows that $\frac{1}{2}m\stackrel{\text{–}}{{v}_{x}^{2}}=\frac{1}{2}{k}_{\text{B}}T$. The same equation holds for $\stackrel{\text{–}}{{v}_{y}^{2}}$ and for $\stackrel{\text{–}}{{v}_{z}^{2}}$. Thus, we can look at our energy of $\frac{3}{2}{k}_{\text{B}}T$ as the sum of contributions of $\frac{1}{2}{k}_{\text{B}}T$ from each of the three dimensions of translational motion. Shifting to the gas as a whole, we see that the 3 in the formula ${C}_{V}=\frac{3}{2}R$ also reflects those three dimensions. We define a degree of freedom as an independent possible motion of a molecule, such as each of the three dimensions of translation. Then, letting *d* represent the number of degrees of freedom, the molar heat capacity at constant volume of a monatomic ideal gas is ${C}_{V}=\frac{d}{2}R,$ where $d=3$.

Molar Heat Capacity of Solid Elements

The idea of equipartition leads to an estimate of the molar heat capacity of solid elements at ordinary temperatures. We can model the atoms of a solid as attached to neighboring atoms by springs ([link]).

Distribution of Molecular Speeds

Abstract:
By the end of this section, you will be able to:

* Describe the distribution of molecular speeds in an ideal gas
* Find the average and most probable molecular speeds in an ideal gas

The Maxwell-Boltzmann Distribution

The motion of molecules in a gas is random in magnitude and direction for individual molecules, but a gas of many molecules has a predictable distribution of molecular speeds. This predictable distribution of molecular speeds is known as the Maxwell-Boltzmann distribution, after its originators, who calculated it based on kinetic theory, and it has since been confirmed experimentally ([link]).

The First Law of Thermodynamics

Introduction

Thermodynamic Systems

Abstract:
By the end of this section, you will be able to:

* Define a thermodynamic system, its boundary, and its surroundings
* Explain the roles of all the components involved in thermodynamics
* Define thermal equilibrium and thermodynamic temperature
* Link an equation of state to a system

Work, Heat, and Internal Energy

Abstract:
By the end of this section, you will be able to:

* Describe the work done by a system, heat transfer between objects, and internal energy change of a system
* Calculate the work, heat transfer, and internal energy change in a simple process

Work Done by a System

A force created from any source can do work by moving an object through a displacement. Then how does a thermodynamic system do work? [link] shows a gas confined to a cylinder that has a movable piston at one end. If the gas expands against the piston, it exerts a force through a distance and does work on the piston. If the piston compresses the gas as it is moved inward, work is also done—in this case, on the gas. The work associated with such volume changes can be determined as follows: Let the gas pressure on the piston face be *p*. Then the force on the piston due to the gas is *pA*, where *A* is the area of the face. When the piston is pushed outward an infinitesimal distance *dx*, the magnitude of the work done by the gas is

Internal Energy

The internal energy ${E}_{\text{int}}$ of a thermodynamic system is, by definition, the sum of the mechanical energies of all the molecules or entities in the system. If the kinetic and potential energies of molecule *i* are ${K}_{i}$ and ${U}_{i},$ respectively, then the internal energy of the system is the average of the total mechanical energy of all the entities:

First Law of Thermodynamics

Abstract:
By the end of this section, you will be able to:

* State the first law of thermodynamics and explain how it is applied
* Explain how heat transfer, work done, and internal energy change are related in any thermodynamic process

Thermodynamic Processes

Abstract:
By the end of this section, you will be able to:

* Define a thermodynamic process
* Distinguish between quasi-static and non-quasi-static processes
* Calculate physical quantities, such as the heat transferred, work done, and internal energy change for isothermal, adiabatic, and cyclical thermodynamic processes

Quasi-static and Non-quasi-static Processes

A quasi-static process refers to an idealized or imagined process where the change in state is made infinitesimally slowly so that at each instant, the system can be assumed to be at a thermodynamic equilibrium with itself and with the environment. For instance, imagine heating 1 kg of water from a temperature $20{\rule{0.2em}{0ex}}\text{°}\text{C}$ to $21{\rule{0.2em}{0ex}}\text{°}\text{C}$ at a constant pressure of 1 atmosphere. To heat the water very slowly, we may imagine placing the container with water in a large bath that can be slowly heated such that the temperature of the bath can rise infinitesimally slowly from $20{\rule{0.2em}{0ex}}\text{°}\text{C}$ to $21{\rule{0.2em}{0ex}}\text{°}\text{C}$. If we put 1 kg of water at $20{\rule{0.2em}{0ex}}\text{°}\text{C}$ directly into a bath at $21{\rule{0.2em}{0ex}}\text{°}\text{C}$, the temperature of the water will rise rapidly to $21{\rule{0.2em}{0ex}}\text{°}\text{C}$ in a non-quasi-static way.

Isothermal Processes

An isothermal process is a change in the state of the system at a constant temperature. This process is accomplished by keeping the system in thermal equilibrium with a large heat bath during the process. Recall that a heat bath is an idealized “infinitely” large system whose temperature does not change. In practice, the temperature of a finite bath is controlled by either adding or removing a finite amount of energy as the case may be.

Adiabatic Processes

In an adiabatic process, the system is insulated from its environment so that although the state of the system changes, no heat is allowed to enter or leave the system, as seen in [link]. An adiabatic process can be conducted either quasi-statically or non-quasi-statically. When a system expands adiabatically, it must do work against the outside world, and therefore its energy goes down, which is reflected in the lowering of the temperature of the system. An adiabatic expansion leads to a lowering of temperature, and an adiabatic compression leads to an increase of temperature. We discuss adiabatic expansion again in Adiabatic Processes for an ideal Gas.

Cyclic Processes

We say that a system goes through a cyclic process if the state of the system at the end is same as the state at the beginning. Therefore, state properties such as temperature, pressure, volume, and internal energy of the system do not change over a complete cycle:

Heat Capacities of an Ideal Gas

Abstract:
By the end of this section, you will be able to:

* Define heat capacity of an ideal gas for a specific process
* Calculate the specific heat of an ideal gas for either an isobaric or isochoric process
* Explain the difference between the heat capacities of an ideal gas and a real gas
* Estimate the change in specific heat of a gas over temperature ranges

Adiabatic Processes for an Ideal Gas

Abstract:
By the end of this section, you will be able to:

* Define adiabatic expansion of an ideal gas
* Demonstrate the qualitative difference between adiabatic and isothermal expansions

The Second Law of Thermodynamics

Introduction

Reversible and Irreversible Processes

Abstract:
By the end of this section, you will be able to:

* Define reversible and irreversible processes
* State the second law of thermodynamics via an irreversible process

Heat Engines

Abstract:
By the end of this section, you will be able to:

* Describe the function and components of a heat engine
* Explain the efficiency of an engine
* Calculate the efficiency of an engine for a given cycle of an ideal gas

Refrigerators and Heat Pumps

Abstract:
By the end of this section, you will be able to:

* Describe a refrigerator and a heat pump and list their differences
* Calculate the performance coefficients of simple refrigerators and heat pumps

Statements of the Second Law of Thermodynamics

Abstract:
By the end of this section, you will be able to:

* Contrast the second law of thermodynamics statements according to Kelvin and Clausius formulations
* Interpret the second of thermodynamics via irreversibility

The Carnot Cycle

Abstract:
* Describe the Carnot cycle with the roles of all four processes involved
* Outline the Carnot principle and its implications
* Demonstrate the equivalence of the Carnot principle and the second law of thermodynamics

Entropy

Abstract:
By the end of this section you will be able to:

* Describe the meaning of entropy
* Calculate the change of entropy for some simple processes

Entropy on a Microscopic Scale

Abstract:
By the end of this section, you will be able to:

* Interpret the meaning of entropy at a microscopic scale
* Calculate a change in entropy for an irreversible process of a system and contrast with the change in entropy of the universe
* Explain the third law of thermodynamics

Electricity and Magnetism

Electric Charges and Fields

Introduction

Electric Charge

Abstract:
By the end of this section, you will be able to:

* Describe the concept of electric charge
* Explain qualitatively the force electric charge creates

Discoveries

You have probably experienced the phenomenon of static electricity: When you first take clothes out of a dryer, many (not all) of them tend to stick together; for some fabrics, they can be very difficult to separate. Another example occurs if you take a woolen sweater off quickly—you can feel (and hear) the static electricity pulling on your clothes, and perhaps even your hair. If you comb your hair on a dry day and then put the comb close to a thin stream of water coming out of a faucet, you will find that the water stream bends toward (is attracted to) the comb ([link]).

Properties of Electric Charge

In addition to the existence of two types of charge, several other properties of charge have been discovered.

The Source of Charges: The Structure of the Atom

Once it became clear that all matter was composed of particles that came to be called atoms, it also quickly became clear that the constituents of the atom included both positively charged particles and negatively charged particles. The next question was, what are the physical properties of those electrically charged particles?

A Note on Terminology

As noted previously, electric charge is a property that an object can have. This is similar to how an object can have a property that we call mass, a property that we call density, a property that we call temperature, and so on. Technically, we should always say something like, “Suppose we have a particle that carries a charge of $3{\rule{0.2em}{0ex}}\mu \text{C}.$” However, it is very common to say instead, “Suppose we have a $3\text{-}\mu \text{C}$ charge.” Similarly, we often say something like, “Six charges are located at the vertices of a regular hexagon.” A charge is not a particle; rather, it is a *property* of a particle. Nevertheless, this terminology is extremely common (and is frequently used in this book, as it is everywhere else). So, keep in the back of your mind what we really mean when we refer to a “charge.”

Conductors, Insulators, and Charging by Induction

Abstract:
By the end of this section, you will be able to:

* Explain what a conductor is
* Explain what an insulator is
* List the differences and similarities between conductors and insulators
* Describe the process of charging by induction

Conductors and Insulators

As discussed in the previous section, electrons surround the tiny nucleus in the form of a (comparatively) vast cloud of negative charge. However, this cloud does have a definite structure to it. Let’s consider an atom of the most commonly used conductor, copper.

Charging by Induction

Let’s examine in more detail what happens in a conductor when an electrically charged object is brought close to it. As mentioned, the conduction electrons in the conductor are able to move with nearly complete freedom. As a result, when a charged insulator (such as a positively charged glass rod) is brought close to the conductor, the (total) charge on the insulator exerts an electric force on the conduction electrons. Since the rod is positively charged, the conduction electrons (which themselves are negatively charged) are attracted, flowing toward the insulator to the near side of the conductor ([link]).

Coulomb's Law

Abstract:
By the end of this section, you will be able to:

* Describe the electric force, both qualitatively and quantitatively
* Calculate the force that charges exert on each other
* Determine the direction of the electric force for different source charges
* Correctly describe and apply the superposition principle for multiple source charges

Multiple Source Charges

The analysis that we have done for two particles can be extended to an arbitrary number of particles; we simply repeat the analysis, two charges at a time. Specifically, we ask the question: Given *N* charges (which we refer to as source charge), what is the net electric force that they exert on some other point charge (which we call the test charge)? Note that we use these terms because we can think of the test charge being used to test the strength of the force provided by the source charges.

Electric Field

Abstract:
By the end of this section, you will be able to:

* Explain the purpose of the electric field concept
* Describe the properties of the electric field
* Calculate the field of a collection of source charges of either sign

Defining a Field

Suppose we have *N* source charges ${q}_{1},{q}_{2},{q}_{3}\text{,…},{q}_{N}$, applying *N* electrostatic forces on a test charge *Q*, at displacements ${\stackrel{\to }{\text{r}}}_{1},{\stackrel{\to }{\text{r}}}_{2},{\stackrel{\to }{\text{r}}}_{3}\text{,…},{\stackrel{\to }{\text{r}}}_{N}$ from the source charges. The net force on *Q* is (see [link])

The Meaning of “Field”

Recall from your studies of gravity that the word “field” in this context has a precise meaning. A field, in physics, is a physical quantity whose value depends on (is a function of) position, relative to the source of the field. In the case of the electric field, [link] shows that the value of $\stackrel{\to }{\text{E}}$ (both the magnitude and the direction) depends on where in space the point *P* is located, with ${\stackrel{\to }{\text{r}}}_{i}$ measured from the locations of the source charges ${q}_{i}$.

Superposition

Yet another experimental fact about the field is that it obeys the superposition principle. In this context, that means that we can (in principle) calculate the total electric field of many source charges by calculating the electric field of only ${q}_{1}$ at position *P*, then calculate the field of ${q}_{2}$ at *P*, while—and this is the crucial idea—ignoring the field of, and indeed even the existence of, ${q}_{1}.$ We can repeat this process, calculating the field of each individual source charge, independently of the existence of any of the other charges. The total electric field, then, is the vector sum of all these fields. That, in essence, is what [link] says.

The Direction of the Field

[link] enables us to determine the magnitude of the electric field, but we need the direction also. We use the convention that the direction of any electric field vector is the same as the direction of the electric force vector that the field would apply to a positive test charge placed in that field. Such a charge would be repelled by positive source charges (the force on it would point away from the positive source charge) but attracted to negative charges (the force points toward the negative source).

Calculating Electric Fields of Charge Distributions

Abstract:
By the end of this section, you will be able to:

* Explain what a continuous source charge distribution is and how it is related to the concept of quantization of charge
* Describe line charges, surface charges, and volume charges
* Calculate the field of a continuous source charge distribution of either sign

Electric Field Lines

Abstract:
By the end of this section, you will be able to:

* Explain the purpose of an electric field diagram
* Describe the relationship between a vector diagram and a field line diagram
* Explain the rules for creating a field diagram and why these rules make physical sense
* Sketch the field of an arbitrary source charge

Electric Dipoles

Abstract:
By the end of this section, you will be able to:

* Describe a permanent dipole
* Describe an induced dipole
* Define and calculate an electric dipole moment
* Explain the physical meaning of the dipole moment

Rotation of a Dipole due to an Electric Field

For now, we deal with only the simplest case: The external field is uniform in space. Suppose we have the situation depicted in [link], where we denote the distance between the charges as the vector $\stackrel{\to }{\text{d}},$ pointing from the negative charge to the positive charge. The forces on the two charges are equal and opposite, so there is no net force on the dipole. However, there is a torque about the midpoint between the two charges:

Induced Dipoles

Neutral atoms are, by definition, electrically neutral; they have equal amounts of positive and negative charge. Furthermore, since they are spherically symmetrical, they do not have a “built-in” dipole moment the way most asymmetrical molecules do. They obtain one, however, when placed in an external electric field, because the external field causes oppositely directed forces on the positive nucleus of the atom versus the negative electrons that surround the nucleus. The result is a new charge distribution of the atom, and therefore, an induced dipole moment ([link]).

Gauss's Law

Introduction

Electric Flux

Abstract:
By the end of this section, you will be able to:

* Define the concept of flux
* Describe electric flux
* Calculate electric flux for a given situation

Area Vector

For discussing the flux of a vector field, it is helpful to introduce an area vector $\stackrel{\to }{\text{A}}.$ This allows us to write the last equation in a more compact form. What should the magnitude of the area vector be? What should the direction of the area vector be? What are the implications of how you answer the previous question?

Electric Flux

Now that we have defined the area vector of a surface, we can define the electric flux of a uniform electric field through a flat area as the scalar product of the electric field and the area vector, as defined in Products of Vectors:

Explaining Gauss’s Law

Abstract:
By the end of this section, you will be able to:

* State Gauss’s law
* Explain the conditions under which Gauss’s law may be used
* Apply Gauss’s law in appropriate systems

Electric Field Lines Picture

An alternative way to see why the flux through a closed spherical surface is independent of the radius of the surface is to look at the electric field lines. Note that every field line from *q* that pierces the surface at radius ${R}_{1}$ also pierces the surface at ${R}_{2}$ ([link]).

Statement of Gauss’s Law

Gauss’s law generalizes this result to the case of any number of charges and any location of the charges in the space inside the closed surface. According to Gauss’s law, the flux of the electric field $\stackrel{\to }{\text{E}}$ through any closed surface, also called a Gaussian surface, is equal to the net charge enclosed $\left({q}_{\text{enc}}\right)$ divided by the permittivity of free space $\left({\epsilon }_{0}\right)$:

Applying Gauss’s Law

Abstract:
By the end of this section, you will be able to:

* Explain what spherical, cylindrical, and planar symmetry are
* Recognize whether or not a given system possesses one of these symmetries
* Apply Gauss’s law to determine the electric field of a system with one of these symmetries

Charge Distribution with Spherical Symmetry

A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if you rotate the system, it doesn’t look different. For instance, if a sphere of radius *R* is uniformly charged with charge density ${\rho }_{0}$ then the distribution has spherical symmetry ([link](a)). On the other hand, if a sphere of radius *R* is charged so that the top half of the sphere has uniform charge density ${\rho }_{1}$ and the bottom half has a uniform charge density ${\rho }_{2}\ne {\rho }_{1},$ then the sphere does not have spherical symmetry because the charge density depends on the direction ([link](b)). Thus, it is not the shape of the object but rather the shape of the charge distribution that determines whether or not a system has spherical symmetry.

Charge Distribution with Cylindrical Symmetry

A charge distribution has cylindrical symmetry if the charge density depends only upon the distance *r* from the axis of a cylinder and must not vary along the axis or with direction about the axis. In other words, if your system varies if you rotate it around the axis, or shift it along the axis, you do not have cylindrical symmetry.

Charge Distribution with Planar Symmetry

A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges.

Conductors in Electrostatic Equilibrium

Abstract:
By the end of this section, you will be able to:

* Describe the electric field within a conductor at equilibrium
* Describe the electric field immediately outside the surface of a charged conductor at equilibrium
* Explain why if the field is not as described in the first two objectives, the conductor is not at equilibrium

The Electric Field inside a Conductor Vanishes

If an electric field is present inside a conductor, it exerts forces on the free electrons (also called conduction electrons), which are electrons in the material that are not bound to an atom. These free electrons then accelerate. However, moving charges by definition means nonstatic conditions, contrary to our assumption. Therefore, when electrostatic equilibrium is reached, the charge is distributed in such a way that the electric field inside the conductor vanishes.

Charge on a Conductor

An interesting property of a conductor in static equilibrium is that extra charges on the conductor end up on the outer surface of the conductor, regardless of where they originate. [link] illustrates a system in which we bring an external positive charge inside the cavity of a metal and then touch it to the inside surface. Initially, the inside surface of the cavity is negatively charged and the outside surface of the conductor is positively charged. When we touch the inside surface of the cavity, the induced charge is neutralized, leaving the outside surface and the whole metal charged with a net positive charge.

The Electric Field at the Surface of a Conductor

If the electric field had a component parallel to the surface of a conductor, free charges on the surface would move, a situation contrary to the assumption of electrostatic equilibrium. Therefore, the electric field is always perpendicular to the surface of a conductor.

Electric Potential

Introduction

Electric Potential Energy

Abstract:
By the end of this section, you will be able to:

* Define the work done by an electric force
* Define electric potential energy
* Apply work and potential energy in systems with electric charges

Electric Potential and Potential Difference

Abstract:
By the end of this section, you will be able to:

* Define electric potential, voltage, and potential difference
* Define the electron-volt
* Calculate electric potential and potential difference from potential energy and electric field
* Describe systems in which the electron-volt is a useful unit
* Apply conservation of energy to electric systems

The Electron-Volt

The energy per electron is very small in macroscopic situations like that in the previous example—a tiny fraction of a joule. But on a submicroscopic scale, such energy per particle (electron, proton, or ion) can be of great importance. For example, even a tiny fraction of a joule can be great enough for these particles to destroy organic molecules and harm living tissue. The particle may do its damage by direct collision, or it may create harmful X-rays, which can also inflict damage. It is useful to have an energy unit related to submicroscopic effects.

Conservation of Energy

The total energy of a system is conserved if there is no net addition (or subtraction) due to work or heat transfer. For conservative forces, such as the electrostatic force, conservation of energy states that mechanical energy is a constant.

Voltage and Electric Field

So far, we have explored the relationship between voltage and energy. Now we want to explore the relationship between voltage and electric field. We will start with the general case for a non-uniform $\stackrel{\to }{\text{E}}$ field. Recall that our general formula for the potential energy of a test charge *q* at point *P* relative to reference point *R* is

Calculations of Electric Potential

Abstract:
By the end of this section, you will be able to:

* Calculate the potential due to a point charge
* Calculate the potential of a system of multiple point charges
* Describe an electric dipole
* Define dipole moment
* Calculate the potential of a continuous charge distribution

Systems of Multiple Point Charges

Just as the electric field obeys a superposition principle, so does the electric potential. Consider a system consisting of *N* charges ${q}_{1},{q}_{2},\text{…},{q}_{N}.$ What is the net electric potential *V* at a space point *P* from these charges? Each of these charges is a source charge that produces its own electric potential at point *P*, independent of whatever other changes may be doing. Let ${V}_{1},{V}_{2},\text{…},{V}_{N}$ be the electric potentials at *P* produced by the charges ${q}_{1},{q}_{2},\text{…},{q}_{N},$ respectively. Then, the net electric potential ${V}_{P}$ at that point is equal to the sum of these individual electric potentials. You can easily show this by calculating the potential energy of a test charge when you bring the test charge from the reference point at infinity to point *P*:

The Electric Dipole

An electric dipole is a system of two equal but opposite charges a fixed distance apart. This system is used to model many real-world systems, including atomic and molecular interactions. One of these systems is the water molecule, under certain circumstances. These circumstances are met inside a microwave oven, where electric fields with alternating directions make the water molecules change orientation. This vibration is the same as heat at the molecular level.

Potential of Continuous Charge Distributions

We have been working with point charges a great deal, but what about continuous charge distributions? Recall from [link] that

Determining Field from Potential

Abstract:
By the end of this section, you will be able to:

* Explain how to calculate the electric field in a system from the given potential
* Calculate the electric field in a given direction from a given potential
* Calculate the electric field throughout space from a given potential

Equipotential Surfaces and Conductors

Abstract:
By the end of this section, you will be able to:

* Define equipotential surfaces and equipotential lines
* Explain the relationship between equipotential lines and electric field lines
* Map equipotential lines for one or two point charges
* Describe the potential of a conductor
* Compare and contrast equipotential lines and elevation lines on topographic maps

Distribution of Charges on Conductors

In [link] with a point charge, we found that the equipotential surfaces were in the form of spheres, with the point charge at the center. Given that a conducting sphere in electrostatic equilibrium is a spherical equipotential surface, we should expect that we could replace one of the surfaces in [link] with a conducting sphere and have an identical solution outside the sphere. Inside will be rather different, however.

Applications of Electrostatics

Abstract:
By the end of this section, you will be able to:

* Describe some of the many practical applications of electrostatics, including several printing technologies
* Relate these applications to Newton’s second law and the electric force

The Van de Graaff Generator

Van de Graaff generator**s** (or Van de Graaffs) are not only spectacular devices used to demonstrate high voltage due to static electricity—they are also used for serious research. The first was built by Robert Van de Graaff in 1931 (based on original suggestions by Lord Kelvin) for use in nuclear physics research. [link] shows a schematic of a large research version. Van de Graaffs use both smooth and pointed surfaces, and conductors and insulators to generate large static charges and, hence, large voltages.

Xerography

Most copy machines use an electrostatic process called xerography—a word coined from the Greek words *xeros* for dry and *graphos* for writing. The heart of the process is shown in simplified form in [link].

Laser Printers

Laser printers use the xerographic process to make high-quality images on paper, employing a laser to produce an image on the photoconducting drum as shown in [link]. In its most common application, the laser printer receives output from a computer, and it can achieve high-quality output because of the precision with which laser light can be controlled. Many laser printers do significant information processing, such as making sophisticated letters or fonts, and in the past may have contained a computer more powerful than the one giving them the raw data to be printed.

Ink Jet Printers and Electrostatic Painting

The ink jet printer, commonly used to print computer-generated text and graphics, also employs electrostatics. A nozzle makes a fine spray of tiny ink droplets, which are then given an electrostatic charge ([link]).

Smoke Precipitators and Electrostatic Air Cleaning

Another important application of electrostatics is found in air cleaners, both large and small. The electrostatic part of the process places excess (usually positive) charge on smoke, dust, pollen, and other particles in the air and then passes the air through an oppositely charged grid that attracts and retains the charged particles ([link])

Capacitance

Introduction

Capacitors and Capacitance

Abstract:
By the end of this section, you will be able to:

* Explain the concepts of a capacitor and its capacitance
* Describe how to evaluate the capacitance of a system of conductors

Calculation of Capacitance

We can calculate the capacitance of a pair of conductors with the standard approach that follows.

Parallel-Plate Capacitor

The parallel-plate capacitor ([link]) has two identical conducting plates, each having a surface area *A*, separated by a distance *d*. When a voltage *V* is applied to the capacitor, it stores a charge *Q*, as shown. We can see how its capacitance may depend on *A* and *d* by considering characteristics of the Coulomb force. We know that force between the charges increases with charge values and decreases with the distance between them. We should expect that the bigger the plates are, the more charge they can store. Thus, *C* should be greater for a larger value of *A*. Similarly, the closer the plates are together, the greater the attraction of the opposite charges on them. Therefore, *C* should be greater for a smaller *d*.

Spherical Capacitor

A spherical capacitor is another set of conductors whose capacitance can be easily determined ([link]). It consists of two concentric conducting spherical shells of radii ${R}_{1}$ (inner shell) and ${R}_{2}$ (outer shell). The shells are given equal and opposite charges $+Q$ and $\text{−}Q$, respectively. From symmetry, the electrical field between the shells is directed radially outward. We can obtain the magnitude of the field by applying Gauss’s law over a spherical Gaussian surface of radius *r* concentric with the shells. The enclosed charge is $+Q$; therefore we have

Cylindrical Capacitor

A cylindrical capacitor consists of two concentric, conducting cylinders ([link]). The inner cylinder, of radius ${R}_{1}$, may either be a shell or be completely solid. The outer cylinder is a shell of inner radius ${R}_{2}$. We assume that the length of each cylinder is *l* and that the excess charges $+Q$ and $\text{−}Q$ reside on the inner and outer cylinders, respectively.

Capacitors in Series and in Parallel

Abstract:
By the end of this section, you will be able to:

* Explain how to determine the equivalent capacitance of capacitors in series and in parallel combinations
* Compute the potential difference across the plates and the charge on the plates for a capacitor in a network and determine the net capacitance of a network of capacitors

The Series Combination of Capacitors

[link] illustrates a series combination of three capacitors, arranged in a row within the circuit. As for any capacitor, the capacitance of the combination is related to the charge and voltage by using [link]. When this series combination is connected to a battery with voltage *V*, each of the capacitors acquires an identical charge *Q*. To explain, first note that the charge on the plate connected to the positive terminal of the battery is $+Q$ and the charge on the plate connected to the negative terminal is $\text{−}Q$. Charges are then induced on the other plates so that the sum of the charges on all plates, and the sum of charges on any pair of capacitor plates, is zero. However, the potential drop ${V}_{1}=Q\text{/}{C}_{1}$ on one capacitor may be different from the potential drop ${V}_{2}=Q\text{/}{C}_{2}$ on another capacitor, because, generally, the capacitors may have different capacitances. The series combination of two or three capacitors resembles a single capacitor with a smaller capacitance. Generally, any number of capacitors connected in series is equivalent to one capacitor whose capacitance (called the *equivalent capacitance*) is smaller than the smallest of the capacitances in the series combination. Charge on this equivalent capacitor is the same as the charge on any capacitor in a series combination: That is, *all capacitors of a series combination have the same charge*. This occurs due to the conservation of charge in the circuit. When a charge *Q* in a series circuit is removed from a plate of the first capacitor (which we denote as $\text{−}Q$), it must be placed on a plate of the second capacitor (which we denote as $+Q\right),$ and so on.

The Parallel Combination of Capacitors

A parallel combination of three capacitors, with one plate of each capacitor connected to one side of the circuit and the other plate connected to the other side, is illustrated in [link](a). Since the capacitors are connected in parallel, *they all have the same voltage V across their plates*. However, each capacitor in the parallel network may store a different charge. To find the equivalent capacitance ${C}_{\text{P}}$ of the parallel network, we note that the total charge *Q* stored by the network is the sum of all the individual charges:

Energy Stored in a Capacitor

Abstract:
By the end of this section, you will be able to:

* Explain how energy is stored in a capacitor
* Use energy relations to determine the energy stored in a capacitor network

Capacitor with a Dielectric

Abstract:
By the end of this section, you will be able to:

* Describe the effects a dielectric in a capacitor has on capacitance and other properties
* Calculate the capacitance of a capacitor containing a dielectric

Molecular Model of a Dielectric

Abstract:
By the end of this section, you will be able to:

* Explain the polarization of a dielectric in a uniform electrical field
* Describe the effect of a polarized dielectric on the electrical field between capacitor plates
* Explain dielectric breakdown

Current and Resistance

Introduction

Electrical Current

Abstract:
By the end of this section, you will be able to:

* Describe an electrical current
* Define the unit of electrical current
* Explain the direction of current flow

Defining Current and the Ampere

Electrical current is defined to be the rate at which charge flows. When there is a large current present, such as that used to run a refrigerator, a large amount of charge moves through the wire in a small amount of time. If the current is small, such as that used to operate a handheld calculator, a small amount of charge moves through the circuit over a long period of time.

Current in a Circuit

In the previous paragraphs, we defined the current as the charge that flows through a cross-sectional area per unit time. In order for charge to flow through an appliance, such as the headlight shown in [link], there must be a complete path (or circuit) from the positive terminal to the negative terminal. Consider a simple circuit of a car battery, a switch, a headlight lamp, and wires that provide a current path between the components. In order for the lamp to light, there must be a complete path for current flow. In other words, a charge must be able to leave the positive terminal of the battery, travel through the component, and back to the negative terminal of the battery. The switch is there to control the circuit. Part (a) of the figure shows the simple circuit of a car battery, a switch, a conducting path, and a headlight lamp. Also shown is the schematic of the circuit [part (b)]. A schematic is a graphical representation of a circuit and is very useful in visualizing the main features of a circuit. Schematics use standardized symbols to represent the components in a circuits and solid lines to represent the wires connecting the components. The battery is shown as a series of long and short lines, representing the historic voltaic pile. The lamp is shown as a circle with a loop inside, representing the filament of an incandescent bulb. The switch is shown as two points with a conducting bar to connect the two points and the wires connecting the components are shown as solid lines. The schematic in part (c) shows the direction of current flow when the switch is closed.

Model of Conduction in Metals

Abstract:
By the end of this section, you will be able to:

* Define the drift velocity of charges moving through a metal
* Define the vector current density
* Describe the operation of an incandescent lamp

Drift Velocity

Electrical signals move very rapidly. Telephone conversations carried by currents in wires cover large distances without noticeable delays. Lights come on as soon as a light switch is moved to the ‘on’ position. Most electrical signals carried by currents travel at speeds on the order of ${10}^{8}{\rule{0.2em}{0ex}}\text{m/s}$, a significant fraction of the speed of light. Interestingly, the individual charges that make up the current move much slower on average, typically drifting at speeds on the order of ${10}^{-4}\text{m/s}$. How do we reconcile these two speeds, and what does it tell us about standard conductors?

Current Density

Although it is often convenient to attach a negative or positive sign to indicate the overall direction of motion of the charges, current is a scalar quantity, $I=\frac{dQ}{dt}$. It is often necessary to discuss the details of the motion of the charge, instead of discussing the overall motion of the charges. In such cases, it is necessary to discuss the current density, $\stackrel{\to }{J}$, a vector quantity. The current density is the flow of charge through an infinitesimal area, divided by the area. The current density must take into account the local magnitude and direction of the charge flow, which varies from point to point. The unit of current density is ampere per meter squared, and the direction is defined as the direction of net flow of positive charges through the area.

Resistivity and Resistance

Abstract:
By the end of this section, you will be able to:

* Differentiate between resistance and resistivity
* Define the term conductivity
* Describe the electrical component known as a resistor
* State the relationship between resistance of a resistor and its length, cross-sectional area, and resistivity
* State the relationship between resistivity and temperature

Resistivity

When a voltage is applied to a conductor, an electrical field $\stackrel{\to }{E}$ is created, and charges in the conductor feel a force due to the electrical field. The current density $\stackrel{\to }{J}$ that results depends on the electrical field and the properties of the material. This dependence can be very complex. In some materials, including metals at a given temperature, the current density is approximately proportional to the electrical field. In these cases, the current density can be modeled as

Temperature Dependence of Resistivity

Looking back at [link], you will see a column labeled “Temperature Coefficient.” The resistivity of some materials has a strong temperature dependence. In some materials, such as copper, the resistivity increases with increasing temperature. In fact, in most conducting metals, the resistivity increases with increasing temperature. The increasing temperature causes increased vibrations of the atoms in the lattice structure of the metals, which impede the motion of the electrons. In other materials, such as carbon, the resistivity decreases with increasing temperature. In many materials, the dependence is approximately linear and can be modeled using a linear equation:

Resistance

We now consider the resistance of a wire or component. The resistance is a measure of how difficult it is to pass current through a wire or component. Resistance depends on the resistivity. The resistivity is a characteristic of the material used to fabricate a wire or other electrical component, whereas the resistance is a characteristic of the wire or component.

Ohm's Law

Abstract:
By the end of this section, you will be able to:

* Describe Ohm’s law
* Recognize when Ohm’s law applies and when it does not

Description of Ohm’s Law

The current that flows through most substances is directly proportional to the voltage *V* applied to it. The German physicist Georg Simon Ohm (1787–1854) was the first to demonstrate experimentally that the current in a metal wire is *directly proportional to the voltage applied*:

Ohm’s Experiment

In a paper published in 1827, Georg Ohm described an experiment in which he measured voltage across and current through various simple electrical circuits containing various lengths of wire. A similar experiment is shown in [link]. This experiment is used to observe the current through a resistor that results from an applied voltage. In this simple circuit, a resistor is connected in series with a battery. The voltage is measured with a voltmeter, which must be placed across the resistor (in parallel with the resistor). The current is measured with an ammeter, which must be in line with the resistor (in series with the resistor).

Electrical Energy and Power

Abstract:
By the end of this section, you will be able to:

* Express electrical power in terms of the voltage and the current
* Describe the power dissipated by a resistor in an electric circuit
* Calculate the energy efficiency and cost effectiveness of appliances and equipment

Power in Electric Circuits

Power is associated by many people with electricity. Power transmission lines might come to mind. We also think of light bulbs in terms of their power ratings in watts. What is the expression for electric power?

The Cost of Electricity

The more electric appliances you use and the longer they are left on, the higher your electric bill. This familiar fact is based on the relationship between energy and power. You pay for the energy used. Since $P=\frac{dE}{dt}$, we see that

Comparison of Incandescent Light Bulbs, CFL Light Bulbs, and LEDs

The energy savings can be significant when replacing an incandescent light bulb or a CFL light bulb with an LED. Light bulbs are rated by the amount of power that the bulb consumes, and the amount of light output is measured in lumens. The lumen (lm) is the SI -derived unit of luminous flux and is a measure of the total quantity of visible light emitted by a source. A 60-W incandescent light bulb can be replaced with a 13- to 15-W CFL bulb or a 6- to 8-W LED, all three of which have a light output of approximately 800 lm. A table of light output for some commonly used light bulbs appears in [link].

Summary of Relationships

In this chapter, we have discussed relationships between voltage, current, resistance, and power. [link] shows a summary of the relationships between these measurable quantities for ohmic devices. (Recall that ohmic devices follow Ohm’s law $V=IR$.) For example, if you need to calculate the power, use the pink section, which shows that $P=VI$, $P=\frac{{V}^{2}}{R}$, and $P={I}^{2}R$.

Superconductors

Abstract:
By the end of this section, you will be able to:

* Describe the phenomenon of superconductivity
* List applications of superconductivity

The Resistance of Mercury

In 1911, Heike Kamerlingh Onnes of Leiden University, a Dutch physicist, was looking at the temperature dependence of the resistance of the element mercury. He cooled the sample of mercury and noticed the familiar behavior of a linear dependence of resistance on temperature; as the temperature decreased, the resistance decreased. Kamerlingh Onnes continued to cool the sample of mercury, using liquid helium. As the temperature approached $4.2{\rule{0.2em}{0ex}}\text{K}\left(-269.2{\rule{0.2em}{0ex}}\text{°}\text{C}\right)$, the resistance abruptly went to zero ([link]). This temperature is known as the critical temperature ${T}_{\text{c}}$ for mercury. The sample of mercury entered into a phase where the resistance was absolutely zero. This phenomenon is known as superconductivity. (*Note:* If you connect the leads of a three-digit ohmmeter across a conductor, the reading commonly shows up as $0.00{\rule{0.2em}{0ex}}\text{Ω}$. The resistance of the conductor is not actually zero, it is less than $0.01{\rule{0.2em}{0ex}}\text{Ω}$.) There are various methods to measure very small resistances, such as the four-point method, but an ohmmeter is not an acceptable method to use for testing resistance in superconductivity.

Other Superconducting Materials

As research continued, several other materials were found to enter a superconducting phase, when the temperature reached near absolute zero. In 1941, an alloy of niobium-nitride was found that could become superconducting at ${T}_{\text{c}}=16{\rule{0.2em}{0ex}}\text{K}\left(-257{\rule{0.2em}{0ex}}\text{°}\text{C}\right)$ and in 1953, vanadium-silicon was found to become superconductive at ${T}_{\text{c}}=17.5{\rule{0.2em}{0ex}}\text{K}\left(-255.7{\rule{0.2em}{0ex}}\text{°}\text{C}\right).$ The temperatures for the transition into superconductivity were slowly creeping higher. Strangely, many materials that make good conductors, such as copper, silver, and gold, do not exhibit superconductivity. Imagine the energy savings if transmission lines for electric power-generating stations could be made to be superconducting at temperatures near room temperature! A resistance of zero ohms means no ${I}^{2}R$ losses and a great boost to reducing energy consumption. The problem is that ${T}_{\text{c}}=17.5{\rule{0.2em}{0ex}}\text{K}$ is still very cold and in the range of liquid helium temperatures. At this temperature, it is not cost effective to transmit electrical energy because of the cooling requirements.

Introduction to BCS Theory

Type I superconductors, along with some Type II superconductors can be modeled using the BCS theory, proposed by John Bardeen, Leon Cooper, and Robert Schrieffer. Although the theory is beyond the scope of this chapter, a short summary of the theory is provided here. (More detail is provided in Condensed Matter Physics.) The theory considers pairs of electrons and how they are coupled together through lattice-vibration interactions. Through the interactions with the crystalline lattice, electrons near the Fermi energy level feel a small attractive force and form pairs (Cooper pairs), and the coupling is known as a phonon interaction. Single electrons are fermions, which are particles that obey the Pauli exclusion principle. The Pauli exclusion principle in quantum mechanics states that two identical fermions (particles with half-integer spin) cannot occupy the same quantum state simultaneously. Each electron has four quantum numbers $\left(n,l,{m}_{l},{m}_{s}\right)$. The principal quantum number (*n*) describes the energy of the electron, the orbital angular momentum quantum number (*l*) indicates the most probable distance from the nucleus, the magnetic quantum number $\left({m}_{l}\right)$ describes the energy levels in the subshell, and the electron spin quantum number $\left({m}_{s}\right)$ describes the orientation of the spin of the electron, either up or down. As the material enters a superconducting state, pairs of electrons act more like bosons, which can condense into the same energy level and need not obey the Pauli exclusion principle. The electron pairs have a slightly lower energy and leave an energy gap above them on the order of 0.001 eV. This energy gap inhibits collision interactions that lead to ordinary resistivity. When the material is below the critical temperature, the thermal energy is less than the band gap and the material exhibits zero resistivity.

Applications of Superconductors

Superconductors can be used to make superconducting magnets. These magnets are 10 times stronger than the strongest electromagnets. These magnets are currently in use in magnetic resonance imaging (MRI), which produces high-quality images of the body interior without dangerous radiation.

Direct-Current Circuits

Introduction

Electromotive Force

Abstract:
By the end of the section, you will be able to:
* Describe the electromotive force (emf) and the internal resistance of a battery
* Explain the basic operation of a battery

Introduction to Electromotive Force

Voltage has many sources, a few of which are shown in [link]. All such devices create a potential difference and can supply current if connected to a circuit. A special type of potential difference is known as electromotive force (emf). The emf is not a force at all, but the term ‘electromotive force’ is used for historical reasons. It was coined by Alessandro Volta in the 1800s, when he invented the first battery, also known as the voltaic pile. Because the electromotive force is not a force, it is common to refer to these sources simply as sources of emf (pronounced as the letters “ee-em-eff”), instead of sources of electromotive force.

The Origin of Battery Potential

The combination of chemicals and the makeup of the terminals in a battery determine its emf. The lead acid battery used in cars and other vehicles is one of the most common combinations of chemicals. [link] shows a single cell (one of six) of this battery. The cathode (positive) terminal of the cell is connected to a lead oxide plate, whereas the anode (negative) terminal is connected to a lead plate. Both plates are immersed in sulfuric acid, the electrolyte for the system.

Internal Resistance and Terminal Voltage

The amount of resistance to the flow of current within the voltage source is called the internal resistance. The internal resistance *r* of a battery can behave in complex ways. It generally increases as a battery is depleted, due to the oxidation of the plates or the reduction of the acidity of the electrolyte. However, internal resistance may also depend on the magnitude and direction of the current through a voltage source, its temperature, and even its history. The internal resistance of rechargeable nickel-cadmium cells, for example, depends on how many times and how deeply they have been depleted. A simple model for a battery consists of an idealized emf source $\epsilon$ and an internal resistance *r* ([link]).

Battery Testers

Battery testers, such as those in [link], use small load resistors to intentionally draw current to determine whether the terminal potential drops below an acceptable level. Although it is difficult to measure the internal resistance of a battery, battery testers can provide a measurement of the internal resistance of the battery. If internal resistance is high, the battery is weak, as evidenced by its low terminal voltage.

Resistors in Series and Parallel

Abstract:
By the end of this section, you will be able to:

* Define the term equivalent resistance
* Calculate the equivalent resistance of resistors connected in series
* Calculate the equivalent resistance of resistors connected in parallel

Resistors in Series

Resistors are said to be in series whenever the current flows through the resistors sequentially. Consider [link], which shows three resistors in series with an applied voltage equal to ${V}_{ab}.$ Since there is only one path for the charges to flow through, the current is the same through each resistor. The equivalent resistance of a set of resistors in a series connection is equal to the algebraic sum of the individual resistances.

Resistors in Parallel

[link] shows resistors in parallel, wired to a voltage source. Resistors are in parallel when one end of all the resistors are connected by a continuous wire of negligible resistance and the other end of all the resistors are also connected to one another through a continuous wire of negligible resistance. The potential drop across each resistor is the same. Current through each resistor can be found using Ohm’s law $I=V\text{/}R,$ where the voltage is constant across each resistor. For example, an automobile’s headlights, radio, and other systems are wired in parallel, so that each subsystem utilizes the full voltage of the source and can operate completely independently. The same is true of the wiring in your house or any building.

Combinations of Series and Parallel

More complex connections of resistors are often just combinations of series and parallel connections. Such combinations are common, especially when wire resistance is considered. In that case, wire resistance is in series with other resistances that are in parallel.

Practical Implications

One implication of this last example is that resistance in wires reduces the current and power delivered to a resistor. If wire resistance is relatively large, as in a worn (or a very long) extension cord, then this loss can be significant. If a large current is drawn, the *IR* drop in the wires can also be significant and may become apparent from the heat generated in the cord.

Kirchhoff's Rules

Abstract:
By the end of this section, you will be able to:

* State Kirchhoff’s junction rule
* State Kirchhoff’s loop rule
* Analyze complex circuits using Kirchhoff’s rules

Kirchhoff’s First Rule

Kirchhoff’s first rule (the junction rule) applies to the charge entering and leaving a junction ([link]). As stated earlier, a junction, or node, is a connection of three or more wires. Current is the flow of charge, and charge is conserved; thus, whatever charge flows into the junction must flow out.

Kirchhoff’s Second Rule

Kirchhoff’s second rule (the loop rule) applies to potential differences. The loop rule is stated in terms of potential *V* rather than potential energy, but the two are related since $U=qV.$ In a closed loop, whatever energy is supplied by a voltage source, the energy must be transferred into other forms by the devices in the loop, since there are no other ways in which energy can be transferred into or out of the circuit. Kirchhoff’s loop rule states that the algebraic sum of potential differences, including voltage supplied by the voltage sources and resistive elements, in any loop must be equal to zero. For example, consider a simple loop with no junctions, as in [link].

Applying Kirchhoff’s Rules

By applying Kirchhoff’s rules, we generate a set of linear equations that allow us to find the unknown values in circuits. These may be currents, voltages, or resistances. Each time a rule is applied, it produces an equation. If there are as many independent equations as unknowns, then the problem can be solved.

Multiple Voltage Sources

Many devices require more than one battery. Multiple voltage sources, such as batteries, can be connected in series configurations, parallel configurations, or a combination of the two.

Solar Cell Arrays

Another example dealing with multiple voltage sources is that of combinations of solar cells—wired in both series and parallel combinations to yield a desired voltage and current. Photovoltaic generation, which is the conversion of sunlight directly into electricity, is based upon the photoelectric effect. The photoelectric effect is beyond the scope of this chapter and is covered in Photons and Matter Waves, but in general, photons hitting the surface of a solar cell create an electric current in the cell.

Electrical Measuring Instruments

Abstract:
By the end of this section, you will be able to:

* Describe how to connect a voltmeter in a circuit to measure voltage
* Describe how to connect an ammeter in a circuit to measure current
* Describe the use of an ohmmeter

DC Voltmeters and Ammeters

Whereas voltmeter**s** measure voltage, ammeter**s** measure current. Some of the meters in automobile dashboards, digital cameras, cell phones, and tuner-amplifiers are actually voltmeters or ammeters ([link]). The internal construction of the simplest of these meters and how they are connected to the system they monitor give further insight into applications of series and parallel connections.

Measuring Current with an Ammeter

To measure the current through a device or component, the ammeter is placed in series with the device or component. A series connection is used because objects in series have the same current passing through them. (See [link], where the ammeter is represented by the symbol A.)

Measuring Voltage with a Voltmeter

A voltmeter is connected in parallel with whatever device it is measuring. A parallel connection is used because objects in parallel experience the same potential difference. (See [link], where the voltmeter is represented by the symbol V.)

Analog and Digital Meters

You may encounter two types of meters in the physics lab: analog and digital. The term ‘analog’ refers to signals or information represented by a continuously variable physical quantity, such as voltage or current. An analog meter uses a galvanometer, which is essentially a coil of wire with a small resistance, in a magnetic field, with a pointer attached that points to a scale. Current flows through the coil, causing the coil to rotate. To use the galvanometer as an ammeter, a small resistance is placed in parallel with the coil. For a voltmeter, a large resistance is placed in series with the coil. A digital meter uses a component called an analog-to-digital (A to D) converter and expresses the current or voltage as a series of the digits 0 and 1, which are used to run a digital display. Most analog meters have been replaced by digital meters.

Ohmmeters

An ohmmeter is an instrument used to measure the resistance of a component or device. The operation of the ohmmeter is based on Ohm’s law. Traditional ohmmeters contained an internal voltage source (such as a battery) that would be connected across the component to be tested, producing a current through the component. A galvanometer was then used to measure the current and the resistance was deduced using Ohm’s law. Modern digital meters use a constant current source to pass current through the component, and the voltage difference across the component is measured. In either case, the resistance is measured using Ohm’s law $\left(R=V\text{/}I\right),$ where the voltage is known and the current is measured, or the current is known and the voltage is measured.

RC Circuits

Abstract:
By the end of this section, you will be able to:

* Describe the charging process of a capacitor
* Describe the discharging process of a capacitor
* List some applications of RC circuits

Circuits with Resistance and Capacitance

An *RC* circuit is a circuit containing resistance and capacitance. As presented in Capacitance, the capacitor is an electrical component that stores electric charge, storing energy in an electric field.

Charging a Capacitor

We can use Kirchhoff’s loop rule to understand the charging of the capacitor. This results in the equation $\epsilon -{V}_{R}-{V}_{c}=0.$ This equation can be used to model the charge as a function of time as the capacitor charges. Capacitance is defined as $C=q\text{/}V,$ so the voltage across the capacitor is ${V}_{C}=\frac{q}{C}$. Using Ohm’s law, the potential drop across the resistor is ${V}_{R}=IR$, and the current is defined as $I=dq\text{/}dt.$

Discharging a Capacitor

When the switch in [link](a) is moved to position *B*, the circuit reduces to the circuit in part (c), and the charged capacitor is allowed to discharge through the resistor. A graph of the charge on the capacitor as a function of time is shown in [link](a). Using Kirchhoff’s loop rule to analyze the circuit as the capacitor discharges results in the equation $\text{−}{V}_{R}-{V}_{c}=0$, which simplifies to $IR+\frac{q}{C}=0$. Using the definition of current$\frac{dq}{dt}R=-\frac{q}{C}$ and integrating the loop equation yields an equation for the charge on the capacitor as a function of time:

Household Wiring and Electrical Safety

Abstract:
By the end of this section, you will be able to:

* List the basic concepts involved in house wiring
* Define the terms thermal hazard and shock hazard
* Describe the effects of electrical shock on human physiology and their relationship to the amount of current through the body
* Explain the function of fuses and circuit breakers

Thermal Hazards

Electric power causes undesired heating effects whenever electric energy is converted into thermal energy at a rate faster than it can be safely dissipated. A classic example of this is the short circuit, a low-resistance path between terminals of a voltage source. An example of a short circuit is shown in [link]. A toaster is plugged into a common household electrical outlet. Insulation on wires leading to an appliance has worn through, allowing the two wires to come into contact, or “short.” As a result, thermal energy can quickly raise the temperature of surrounding materials, melting the insulation and perhaps causing a fire.

Shock Hazards

Electric shock is the physiological reaction or injury caused by an external electric current passing through the body. The effect of an electric shock can be negative or positive. When a current with a magnitude above 300 mA passes through the heart, death may occur. Most electrical shock fatalities occur because a current causes ventricular fibrillation, a massively irregular and often fatal, beating of the heart. On the other hand, a heart attack victim, whose heart is in fibrillation, can be saved by an electric shock from a defibrillator.

Electrical Safety: Systems and Devices

[link](a) shows the schematic for a simple ac circuit with no safety features. This is not how power is distributed in practice. Modern household and industrial wiring requires the three-wire system, shown schematically in part (b), which has several safety features, with live, neutral, and ground wires. First is the familiar circuit breaker (or fuse) to prevent thermal overload. Second is a protective case around the appliance, such as a toaster or refrigerator. The case’s safety feature is that it prevents a person from touching exposed wires and coming into electrical contact with the circuit, helping prevent shocks.

Magnetic Forces and Fields

Introduction

Magnetism and Its Historical Discoveries

Abstract:
By the end of this section, you will be able to:

* Explain attraction and repulsion by magnets
* Describe the historical and contemporary applications of magnetism

Brief History of Magnetism

Magnets are commonly found in everyday objects, such as toys, hangers, elevators, doorbells, and computer devices. Experimentation on these magnets shows that all magnets have two poles: One is labeled north (N) and the other is labeled south (S). Magnetic poles repel if they are alike (both N or both S), they attract if they are opposite (one N and the other S), and both poles of a magnet attract unmagnetized pieces of iron. An important point to note here is that you cannot isolate an individual magnetic pole. Every piece of a magnet, no matter how small, which contains a north pole must also contain a south pole.

Contemporary Applications of Magnetism

Today, magnetism plays many important roles in our lives. Physicists’ understanding of magnetism has enabled the development of technologies that affect both individuals and society. The electronic tablet in your purse or backpack, for example, wouldn’t have been possible without the applications of magnetism and electricity on a small scale ([link]). Weak changes in a magnetic field in a thin film of iron and chromium were discovered to bring about much larger changes in resistance, called giant magnetoresistance. Information can then be recorded magnetically based on the direction in which the iron layer is magnetized. As a result of the discovery of giant magnetoresistance and its applications to digital storage, the 2007 Nobel Prize in Physics was awarded to Albert Fert from France and Peter Grunberg from Germany.

Magnetic Fields and Lines

Abstract:
By the end of this section, you will be able to:

* Define the magnetic field based on a moving charge experiencing a force
* Apply the right-hand rule to determine the direction of a magnetic force based on the motion of a charge in a magnetic field
* Sketch magnetic field lines to understand which way the magnetic field points and how strong it is in a region of space

Defining the Magnetic Field

A magnetic field is defined by the force that a charged particle experiences moving in this field, after we account for the gravitational and any additional electric forces possible on the charge. The magnitude of this force is proportional to the amount of charge *q*, the speed of the charged particle *v*, and the magnitude of the applied magnetic field. The direction of this force is perpendicular to both the direction of the moving charged particle and the direction of the applied magnetic field. Based on these observations, we define the magnetic field strength *B* based on the magnetic force $\stackrel{\to }{F}$ on a charge *q* moving at velocity $\stackrel{\to }{v}$ as the cross product of the velocity and magnetic field, that is,

Representing Magnetic Fields

The representation of magnetic fields by magnetic field lines is very useful in visualizing the strength and direction of the magnetic field. As shown in [link], each of these lines forms a closed loop, even if not shown by the constraints of the space available for the figure. The field lines emerge from the north pole (N), loop around to the south pole (S), and continue through the bar magnet back to the north pole.

Motion of a Charged Particle in a Magnetic Field

Abstract:
By the end of this section, you will be able to:

* Explain how a charged particle in an external magnetic field undergoes circular motion
* Describe how to determine the radius of the circular motion of a charged particle in a magnetic field

Magnetic Force on a Current-Carrying Conductor

Abstract:
By the end of this section, you will be able to:

* Determine the direction in which a current-carrying wire experiences a force in an external magnetic field
* Calculate the force on a current-carrying wire in an external magnetic field

Magnetic Fields Produced by Electrical Currents

When discussing historical discoveries in magnetism, we mentioned Oersted’s finding that a wire carrying an electrical current caused a nearby compass to deflect. A connection was established that electrical currents produce magnetic fields. (This connection between electricity and magnetism is discussed in more detail in Sources of Magnetic Fields.)

Calculating the Magnetic Force

Electric current is an ordered movement of charge. A current-carrying wire in a magnetic field must therefore experience a force due to the field. To investigate this force, let’s consider the infinitesimal section of wire as shown in [link]. The length and cross-sectional area of the section are *dl* and *A*, respectively, so its volume is $V=A·dl.$ The wire is formed from material that contains *n* charge carriers per unit volume, so the number of charge carriers in the section is $nA·dl.$ If the charge carriers move with drift velocity ${\stackrel{\to }{v}}_{\text{d}},$ the current *I* in the wire is (from Current and Resistance)

Force and Torque on a Current Loop

Abstract:
By the end of this section, you will be able to:

* Evaluate the net force on a current loop in an external magnetic field
* Evaluate the net torque on a current loop in an external magnetic field
* Define the magnetic dipole moment of a current loop

The Hall Effect

Abstract:
By the end of this section, you will be able to:

* Explain a scenario where the magnetic and electric fields are crossed and their forces balance each other as a charged particle moves through a velocity selector
* Compare how charge carriers move in a conductive material and explain how this relates to the Hall effect

Applications of Magnetic Forces and Fields

Abstract:
By the end of this section, you will be able to:

* Explain how a mass spectrometer works to separate charges
* Explain how a cyclotron works

Mass Spectrometer

The mass spectrometer is a device that separates ions according to their charge-to-mass ratios. One particular version, the Bainbridge mass spectrometer, is illustrated in [link]. Ions produced at a source are first sent through a velocity selector, where the magnetic force is equally balanced with the electric force. These ions all emerge with the same speed $v=E\text{/}B$ since any ion with a different velocity is deflected preferentially by either the electric or magnetic force, and ultimately blocked from the next stage. They then enter a uniform magnetic field ${B}_{0}$ where they travel in a circular path whose radius *R* is given by [link]. The radius is measured by a particle detector located as shown in the figure.

Cyclotron

The cyclotron was developed by E.O. Lawrence to accelerate charged particles (usually protons, deuterons, or alpha-particles) to large kinetic energies. These particles are then used for nuclear-collision experiments to produce radioactive isotopes. A cyclotron is illustrated in [link]. The particles move between two flat, semi-cylindrical metallic containers D1 and D2, called dees. The dees are enclosed in a larger metal container, and the apparatus is placed between the poles of an electromagnet that provides a uniform magnetic field. Air is removed from the large container so that the particles neither lose energy nor are deflected because of collisions with air molecules. The dees are connected to a high-frequency voltage source that provides an alternating electric field in the small region between them. Because the dees are made of metal, their interiors are shielded from the electric field.

Sources of Magnetic Fields

Introduction

The Biot-Savart Law

Abstract:
By the end of this section, you will be able to:

* Explain how to derive a magnetic field from an arbitrary current in a line segment
* Calculate magnetic field from the Biot-Savart law in specific geometries, such as a current in a line and a current in a circular arc

Magnetic Field Due to a Thin Straight Wire

Abstract:
By the end of this section, you will be able to:

* Explain how the Biot-Savart law is used to determine the magnetic field due to a thin, straight wire.
* Determine the dependence of the magnetic field from a thin, straight wire based on the distance from it and the current flowing in the wire.
* Sketch the magnetic field created from a thin, straight wire by using the second right-hand rule.

Magnetic Force between Two Parallel Currents

Abstract:
By the end of this section, you will be able to:

* Explain how parallel wires carrying currents can attract or repel each other
* Define the ampere and describe how it is related to current-carrying wires
* Calculate the force of attraction or repulsion between two current-carrying wires

Magnetic Field of a Current Loop

Abstract:
By the end of this section, you will be able to:

* Explain how the Biot-Savart law is used to determine the magnetic field due to a current in a loop of wire at a point along a line perpendicular to the plane of the loop.
* Determine the magnetic field of an arc of current.

Ampère’s Law

Abstract:
By the end of this section, you will be able to:

* Explain how Ampère’s law relates the magnetic field produced by a current to the value of the current
* Calculate the magnetic field from a long straight wire, either thin or thick, by Ampère’s law

Solenoids and Toroids

Abstract:
By the end of this section, you will be able to:

* Establish a relationship for how the magnetic field of a solenoid varies with distance and current by using both the Biot-Savart law and Ampère’s law
* Establish a relationship for how the magnetic field of a toroid varies with distance and current by using Ampère’s law

Solenoids

A long wire wound in the form of a helical coil is known as a solenoid. Solenoids are commonly used in experimental research requiring magnetic fields. A solenoid is generally easy to wind, and near its center, its magnetic field is quite uniform and directly proportional to the current in the wire.

Toroids

A toroid is a donut-shaped coil closely wound with one continuous wire, as illustrated in part (a) of [link]. If the toroid has *N* windings and the current in the wire is *I*, what is the magnetic field both inside and outside the toroid?

Magnetism in Matter

Abstract:
By the end of this section, you will be able to:

* Classify magnetic materials as paramagnetic, diamagnetic, or ferromagnetic, based on their response to a magnetic field
* Sketch how magnetic dipoles align with the magnetic field in each type of substance
* Define hysteresis and magnetic susceptibility, which determines the type of magnetic material

Paramagnetic Materials

For simplicity, we assume our sample is a long, cylindrical piece that completely fills the interior of a long, tightly wound solenoid. When there is no current in the solenoid, the magnetic dipoles in the sample are randomly oriented and produce no net magnetic field. With a solenoid current, the magnetic field due to the solenoid exerts a torque on the dipoles that tends to align them with the field. In competition with the aligning torque are thermal collisions that tend to randomize the orientations of the dipoles. The relative importance of these two competing processes can be estimated by comparing the energies involved. From [link], the energy difference between a magnetic dipole aligned with and against a magnetic field is ${U}_{B}=2\mu B.$ If $\mu =9.3{\rule{0.2em}{0ex}}×{\rule{0.2em}{0ex}}{10}^{-24}\text{A}·{\text{m}}^{2}$ (the value of atomic hydrogen) and *B* = 1.0 T, then

Diamagnetic Materials

A magnetic field always induces a magnetic dipole in an atom. This induced dipole points opposite to the applied field, so its magnetic field is also directed opposite to the applied field. In paramagnetic and ferromagnetic materials, the induced magnetic dipole is masked by much stronger permanent magnetic dipoles of the atoms. However, in diamagnetic materials, whose atoms have no permanent magnetic dipole moments, the effect of the induced dipole is observable.

Ferromagnetic Materials

Common magnets are made of a ferromagnetic material such as iron or one of its alloys. Experiments reveal that a ferromagnetic material consists of tiny regions known as magnetic domains. Their volumes typically range from ${10}^{-12}$ to ${10}^{-8}{\text{m}}^{3},$ and they contain about ${10}^{17}$ to ${10}^{21}$ atoms. Within a domain, the magnetic dipoles are rigidly aligned in the same direction by coupling among the atoms. This coupling, which is due to quantum mechanical effects, is so strong that even thermal agitation at room temperature cannot break it. The result is that each domain has a net dipole moment. Some materials have weaker coupling and are ferromagnetic only at lower temperatures.

Electromagnetic Induction

Introduction

Faraday’s Law

Abstract:
By the end of this section, you will be able to:

* Determine the magnetic flux through a surface, knowing the strength of the magnetic field, the surface area, and the angle between the normal to the surface and the magnetic field
* Use Faraday’s law to determine the magnitude of induced emf in a closed loop due to changing magnetic flux through the loop

Lenz's Law

Abstract:
By the end of this section, you will be able to:

* Use Lenz’s law to determine the direction of induced emf whenever a magnetic flux changes
* Use Faraday’s law with Lenz’s law to determine the induced emf in a coil and in a solenoid

Motional Emf

Abstract:
By the end of this section, you will be able to:

* Determine the magnitude of an induced emf in a wire moving at a constant speed through a magnetic field
* Discuss examples that use motional emf, such as a rail gun and a tethered satellite

Induced Electric Fields

Abstract:
By the end of this section, you will be able to:

* Connect the relationship between an induced emf from Faraday’s law to an electric field, thereby showing that a changing magnetic flux creates an electric field
* Solve for the electric field based on a changing magnetic flux in time

Eddy Currents

Abstract:
By the end of this section, you will be able to:

* Explain how eddy currents are created in metals
* Describe situations where eddy currents are beneficial and where they are not helpful

Magnetic Damping

Eddy currents can produce significant drag, called magnetic damping, on the motion involved. Consider the apparatus shown in [link], which swings a pendulum bob between the poles of a strong magnet. (This is another favorite physics demonstration.) If the bob is metal, significant drag acts on the bob as it enters and leaves the field, quickly damping the motion. If, however, the bob is a slotted metal plate, as shown in part (b) of the figure, the magnet produces a much smaller effect. There is no discernible effect on a bob made of an insulator. Why does drag occur in both directions, and are there any uses for magnetic drag?

Applications of Magnetic Damping

One use of magnetic damping is found in sensitive laboratory balances. To have maximum sensitivity and accuracy, the balance must be as friction-free as possible. But if it is friction-free, then it will oscillate for a very long time. Magnetic damping is a simple and ideal solution. With magnetic damping, drag is proportional to speed and becomes zero at zero velocity. Thus, the oscillations are quickly damped, after which the damping force disappears, allowing the balance to be very sensitive ([link]). In most balances, magnetic damping is accomplished with a conducting disc that rotates in a fixed field.

Electric Generators and Back Emf

Abstract:
By the end of this section, you will be able to:

* Explain how an electric generator works
* Determine the induced emf in a loop at any time interval, rotating at a constant rate in a magnetic field
* Show that rotating coils have an induced emf; in motors this is called back emf because it opposes the emf input to the motor

Electric Generators

**Electric generators** induce an emf by rotating a coil in a magnetic field, as briefly discussed in Motional Emf. We now explore generators in more detail. Consider the following example.

Back Emf

Generators convert mechanical energy into electrical energy, whereas motors convert electrical energy into mechanical energy. Thus, it is not surprising that motors and generators have the same general construction. A motor works by sending a current through a loop of wire located in a magnetic field. As a result, the magnetic field exerts torque on the loop. This rotates a shaft, thereby extracting mechanical work out of the electrical current sent in initially. (Refer to Force and Torque on a Current Loop for a discussion on motors that will help you understand more about them before proceeding.)

Applications of Electromagnetic Induction

Abstract:
By the end of this section, you will be able to:

* Explain how computer hard drives and graphic tablets operate using magnetic induction
* Explain how hybrid/electric vehicles and transcranial magnetic stimulation use magnetic induction to their advantage

Inductance

Introduction

Mutual Inductance

Abstract:
By the end of this section, you will be able to:

* Correlate two nearby circuits that carry time-varying currents with the emf induced in each circuit
* Describe examples in which mutual inductance may or may not be desirable

Self-Inductance and Inductors

Abstract:
By the end of this section, you will be able to:

* Correlate the rate of change of current to the induced emf created by that current in the same circuit
* Derive the self-inductance for a cylindrical solenoid
* Derive the self-inductance for a rectangular toroid

Inductors

[link] shows some of the magnetic field lines due to the current in a circular loop of wire. If the current is constant, the magnetic flux through the loop is also constant. However, if the current *I* were to vary with time—say, immediately after switch S is closed—then the magnetic flux ${\text{Φ}}_{\text{m}}$ would correspondingly change. Then Faraday’s law tells us that an emf $\epsilon$ would be induced in the circuit, where

Cylindrical Solenoid

Consider a long, cylindrical solenoid with length *l*, cross-sectional area *A*, and *N* turns of wire. We assume that the length of the solenoid is so much larger than its diameter that we can take the magnetic field to be $B={\mu }_{0}nI$ throughout the interior of the solenoid, that is, we ignore end effects in the solenoid. With a current *I* flowing through the coils, the magnetic field produced within the solenoid is

Rectangular Toroid

A toroid with a rectangular cross-section is shown in [link]. The inner and outer radii of the toroid are ${R}_{1}{\rule{0.2em}{0ex}}\text{and}{\rule{0.2em}{0ex}}{R}_{2},{\rule{0.2em}{0ex}}\text{and}{\rule{0.2em}{0ex}}h$ is the height of the toroid. Applying Ampère’s law in the same manner as we did in [link] for a toroid with a circular cross-section, we find the magnetic field inside a rectangular toroid is also given by

Energy in a Magnetic Field

Abstract:
By the end of this section, you will be able to:

* Explain how energy can be stored in a magnetic field
* Derive the equation for energy stored in a coaxial cable given the magnetic energy density

RL Circuits

Abstract:
By the end of this section, you will be able to:

* Analyze circuits that have an inductor and resistor in series
* Describe how current and voltage exponentially grow or decay based on the initial conditions

Oscillations in an LC Circuit

Abstract:
By the end of this section, you will be able to:

* Explain why charge or current oscillates between a capacitor and inductor, respectively, when wired in series
* Describe the relationship between the charge and current oscillating between a capacitor and inductor wired in series

RLC Series Circuits

Abstract:
By the end of this section, you will be able to:

* Determine the angular frequency of oscillation for a resistor, inductor, capacitor $\left(RLC\right)$ series circuit
* Relate the $RLC$ circuit to a damped spring oscillation

Alternating-Current Circuits

Introduction

AC Sources

Abstract:
By the end of this section, you will be able to:

* Explain the differences between direct current (dc) and alternating current (ac)
* Define characteristic features of alternating current and voltage, such as the amplitude or peak and the frequency

Simple AC Circuits

Abstract:
By the end of this section, you will be able to:

* Interpret phasor diagrams and apply them to ac circuits with resistors, capacitors, and inductors
* Define the reactance for a resistor, capacitor, and inductor to help understand how current in the circuit behaves compared to each of these devices

Resistor

First, consider a resistor connected across an ac voltage source. From Kirchhoff’s loop rule, the instantaneous voltage across the resistor of [link](a) is

Capacitor

Now let’s consider a capacitor connected across an ac voltage source. From Kirchhoff’s loop rule, the instantaneous voltage across the capacitor of [link](a) is

Inductor

Lastly, let’s consider an inductor connected to an ac voltage source. From Kirchhoff’s loop rule, the voltage across the inductor *L* of [link](a) is

RLC Series Circuits with AC

Abstract:
By the end of this section, you will be able to:

* Describe how the current varies in a resistor, a capacitor, and an inductor while in series with an ac power source
* Use phasors to understand the phase angle of a resistor, capacitor, and inductor ac circuit and to understand what that phase angle means
* Calculate the impedance of a circuit

Power in an AC Circuit

Abstract:
By the end of this section, you will be able to:

* Describe how average power from an ac circuit can be written in terms of peak current and voltage and of rms current and voltage
* Determine the relationship between the phase angle of the current and voltage and the average power, known as the power factor

Resonance in an AC Circuit

Abstract:
By the end of this section, you will be able to:

* Determine the peak ac resonant angular frequency for a RLC circuit
* Explain the width of the average power versus angular frequency curve and its significance using terms like bandwidth and quality factor

Transformers

Abstract:
By the end of this section, you will be able to:

* Explain why power plants transmit electricity at high voltages and low currents and how they do this
* Develop relationships among current, voltage, and the number of windings in step-up and step-down transformers

Electromagnetic Waves

Introduction

Maxwell’s Equations and Electromagnetic Waves

Abstract:
By the end of this section, you will be able to:

* Explain Maxwell’s correction of Ampère’s law by including the displacement current
* State and apply Maxwell’s equations in integral form
* Describe how the symmetry between changing electric and changing magnetic fields explains Maxwell’s prediction of electromagnetic waves
* Describe how Hertz confirmed Maxwell’s prediction of electromagnetic waves

Maxwell’s Correction to the Laws of Electricity and Magnetism

The four basic laws of electricity and magnetism had been discovered experimentally through the work of physicists such as Oersted, Coulomb, Gauss, and Faraday. Maxwell discovered logical inconsistencies in these earlier results and identified the incompleteness of Ampère’s law as their cause.

Maxwell’s Equations

With the correction for the displacement current, Maxwell’s equations take the form

The Mechanism of Electromagnetic Wave Propagation

To see how the symmetry introduced by Maxwell accounts for the existence of combined electric and magnetic waves that propagate through space, imagine a time-varying magnetic field ${\stackrel{\to }{B}}_{0}\left(t\right)$ produced by the high-frequency alternating current seen in [link]. We represent ${\stackrel{\to }{B}}_{0}\left(t\right)$ in the diagram by one of its field lines. From Faraday’s law, the changing magnetic field through a surface induces a time-varying electric field ${\stackrel{\to }{E}}_{0}\left(t\right)$ at the boundary of that surface. The displacement current source for the electric field, like the Faraday’s law source for the magnetic field, produces only closed loops of field lines, because of the mathematical symmetry involved in the equations for the induced electric and induced magnetic fields. A field line representation of ${\stackrel{\to }{E}}_{0}\left(t\right)$ is shown. In turn, the changing electric field ${\stackrel{\to }{E}}_{0}\left(t\right)$ creates a magnetic field ${\stackrel{\to }{B}}_{1}\left(t\right)$ according to the modified Ampère’s law. This changing field induces ${\stackrel{\to }{E}}_{1}\left(t\right),$ which induces ${\stackrel{\to }{B}}_{2}\left(t\right),$ and so on. We then have a self-continuing process that leads to the creation of time-varying electric and magnetic fields in regions farther and farther away from *O*. This process may be visualized as the propagation of an electromagnetic wave through space.

Hertz’s Observations

The German physicist Heinrich Hertz (1857–1894) was the first to generate and detect certain types of electromagnetic waves in the laboratory. Starting in 1887, he performed a series of experiments that not only confirmed the existence of electromagnetic waves but also verified that they travel at the speed of light.

Plane Electromagnetic Waves

Abstract:
By the end of this section, you will be able to:

* Describe how Maxwell’s equations predict the relative directions of the electric fields and magnetic fields, and the direction of propagation of plane electromagnetic waves
* Explain how Maxwell’s equations predict that the speed of propagation of electromagnetic waves in free space is exactly the speed of light
* Calculate the relative magnitude of the electric and magnetic fields in an electromagnetic plane wave
* Describe how electromagnetic waves are produced and detected

Electromagnetic Waves in One Direction

An electromagnetic wave consists of an electric field, defined as usual in terms of the force per charge on a stationary charge, and a magnetic field, defined in terms of the force per charge on a moving charge. The electromagnetic field is assumed to be a function of only the *x*-coordinate and time. The *y*-component of the electric field is then written as ${E}_{y}\left(x,t\right),$ the *z*-component of the magnetic field as ${B}_{z}\left(x,t\right)$, etc. Because we are assuming free space, there are no free charges or currents, so we can set ${Q}_{\text{in}}=0$ and $I=0$ in Maxwell’s equations.

How the *E* and *B* Fields Are Related

So far, we have seen that the rates of change of different components of the *E* and *B* fields are related, that the electromagnetic wave is transverse, and that the wave propagates at speed *c*. We next show what Maxwell’s equations imply about the ratio of the *E* and *B* field magnitudes and the relative directions of the *E* and *B* fields.

Production and Detection of Electromagnetic Waves

A steady electric current produces a magnetic field that is constant in time and which does not propagate as a wave. Accelerating charges, however, produce electromagnetic waves. An electric charge oscillating up and down, or an alternating current or flow of charge in a conductor, emit radiation at the frequencies of their oscillations. The electromagnetic field of a *dipole antenna* is shown in [link]. The positive and negative charges on the two conductors are made to reverse at the desired frequency by the output of a transmitter as the power source. The continually changing current accelerates charge in the antenna, and this results in an oscillating electric field a distance away from the antenna. The changing electric fields produce changing magnetic fields that in turn produce changing electric fields, which thereby propagate as electromagnetic waves. The frequency of this radiation is the same as the frequency of the ac source that is accelerating the electrons in the antenna. The two conducting elements of the dipole antenna are commonly straight wires. The total length of the two wires is typically about one-half of the desired wavelength (hence, the alternative name *half-wave antenna*), because this allows standing waves to be set up and enhances the effectiveness of the radiation.

Energy Carried by Electromagnetic Waves

Abstract:
By the end of this section, you will be able to:

* Express the time-averaged energy density of electromagnetic waves in terms of their electric and magnetic field amplitudes
* Calculate the Poynting vector and the energy intensity of electromagnetic waves
* Explain how the energy of an electromagnetic wave depends on its amplitude, whereas the energy of a photon is proportional to its frequency

Momentum and Radiation Pressure

Abstract:
By the end of this section, you will be able to:

* Describe the relationship of the radiation pressure and the energy density of an electromagnetic wave
* Explain how the radiation pressure of light, while small, can produce observable astronomical effects

The Electromagnetic Spectrum

Abstract:
By the end of this section, you will be able to:

* Explain how electromagnetic waves are divided into different ranges, depending on wavelength and corresponding frequency
* Describe how electromagnetic waves in different categories are produced
* Describe some of the many practical everyday applications of electromagnetic waves

Radio Waves

The term radio waves refers to electromagnetic radiation with wavelengths greater than about 0.1 m. Radio waves are commonly used for audio communications (i.e., for radios), but the term is used for electromagnetic waves in this range regardless of their application. Radio waves typically result from an alternating current in the wires of a broadcast antenna. They cover a very broad wavelength range and are divided into many subranges, including microwaves, electromagnetic waves used for AM and FM radio, cellular telephones, and TV signals.

Microwaves

**Microwaves** are the highest-frequency electromagnetic waves that can be produced by currents in macroscopic circuits and devices. Microwave frequencies range from about ${10}^{9}\text{Hz}$ to nearly ${10}^{12}\text{Hz}$. Their high frequencies correspond to short wavelengths compared with other radio waves—hence the name “microwave.” Microwaves also occur naturally as the cosmic background radiation left over from the origin of the universe. Along with other ranges of electromagnetic waves, they are part of the radiation that any object above absolute zero emits and absorbs because of thermal agitation, that is, from the thermal motion of its atoms and molecules.

Infrared Radiation

The boundary between the microwave and infrared regions of the electromagnetic spectrum is not well defined (see [link]). **Infrared radiation** is generally produced by thermal motion, and the vibration and rotation of atoms and molecules. Electronic transitions in atoms and molecules can also produce infrared radiation. About half of the solar energy arriving at Earth is in the infrared region, with most of the rest in the visible part of the spectrum. About 23% of the solar energy is absorbed in the atmosphere, about 48% is absorbed at Earth’s surface, and about 29% is reflected back into space.[footnote]

http://earthobservatory.nasa.gov/Features/EnergyBalance/page4.php

Visible Light

Visible light is the narrow segment of the electromagnetic spectrum between about 400 nm and about 750 nm to which the normal human eye responds. Visible light is produced by vibrations and rotations of atoms and molecules, as well as by electronic transitions within atoms and molecules. The receivers or detectors of light largely utilize electronic transitions.

Ultraviolet Radiation

Ultraviolet means “above violet.” The electromagnetic frequencies of ultraviolet radiation **(UV)** extend upward from violet, the highest-frequency visible light. The highest-frequency ultraviolet overlaps with the lowest-frequency X-rays. The wavelengths of ultraviolet extend from 400 nm down to about 10 nm at its highest frequencies. Ultraviolet is produced by atomic and molecular motions and electronic transitions.

X-Rays

X-ray**s** have wavelengths from about ${10}^{\text{−}8}{\text{m to}{\rule{0.2em}{0ex}}10}^{\text{−}12}\text{m}$. They have shorter wavelengths, and higher frequencies, than ultraviolet, so that the energy they transfer at an atomic level is greater. As a result, X-rays have adverse effects on living cells similar to those of ultraviolet radiation, but they are more penetrating. Cancer and genetic defects can be induced by X-rays. Because of their effect on rapidly dividing cells, X-rays can also be used to treat and even cure cancer.

Gamma Rays

Soon after nuclear radioactivity was first detected in 1896, it was found that at least three distinct types of radiation were being emitted, and these were designated as alpha, beta, and gamma rays. The most penetrating nuclear radiation, the gamma ray ($\gamma$ ray), was later found to be an extremely high-frequency electromagnetic wave.

Units

Conversion Factors

Fundamental Constants

Astronomical Data

Mathematical Formulas

Chemistry

The Greek Alphabet