Episode 114: Components in series and parallel

Main aims

• Adding resistors in series increases the effective resistance
• Adding resistors in parallel decreases the effective resistance

Summary

• Demonstration: Combining lamps in series and parallel (10 minutes)
• Discussion: Deriving formulae (20 minutes)
• Worked example: Adding resistors (15 minutes)
• Student experiment: Predicting and measuring resistance (20 minutes)
• Student questions: Practice with the formulae (30 minutes)

Demonstration: Combining lamps in series and parallel
Ask students to recall the equation which defines resistance. (R = V / I where V is the pd across a component and I is the current through it.) Connect 1, 2 and then 3 lamps in series across a supply (constant pd) showing them the reduction in current at the same pd The ratio V/I has increased: adding resistors in series increases overall resistance. You may wish to calculate the resistance each time but beware that, since the temperature of the lamps will be different at different currents you are unlikely to get a simple ratio of resistances.

Repeat the experiment but this time add lamps in parallel. The current increases and the effective (load) resistance decreases. If you calculate the resistance values you should get 1:1/2:1/3 as you add the lamps.

Episode 114-1: Connecting lamps in parallel and in series (Word, 34 KB)

Discussion: Deriving formulae
Now tackle the theory. You are trying to find the single resistor Rtot which will have the same resistance as two or more resistors R1, R2 etc in series. Derive the equations for series and parallel resistance combinations. The starting points are:

Series resistors have the same current but the pds add, so:

Vtot = I Rtot = I R1 + I R2 +.... then divide by Ito get Rtot

Parallel resistors have all have the same pd across them but the currents add, so:

Itot = V / Rtot = V / R1 + V / R2 etc... then divide by V to get the familiar formula.

Beware that the parallel formula is usually quoted as 1 / Rtot so they still have to take the reciprocal to get the value of Rtot. It is unusual (at A-level) for questions to involve more than two resistors in parallel so it is worth pointing out that the parallel formula for two resistors can be rearranged to give:

Rtot = R1 R2 / (R1+R2) or ‘product over sum’. This can save the faint-hearted from reciprocals!

It is also worth pointing out that when n resistors of the same value (R) are connected in parallel the result is an effective resistance R / n. Link this with an n-fold multiplication of current.

Beware that, even at this level, some students will argue that current always ‘takes the path of least resistance’. (Be prepared to discuss what happens in the extreme case when a component is “shorted out” by a piece of wire in parallel. Point out that a small fraction of the overall current still continues to flow through the component. (This case will re-emerge later in potential dividers in episode 118.)

You may need to mention conductance at this point. Conductance G = 1 / R and it is measured in siemens, S. In a parallel combination, conductances add.

Work through examples for two resistors with convenient resistances, e.g. 40 Ω and 60 Ω. These give 100 Ω in series and 24 Ω in parallel. Allow students to perform the calculation for themselves and help out any students who have problems using the reciprocal key on their calculators.

Emphasize that, for resistors in series, the total is always greater than the individual resistances; in parallel, it is less than any of the individual resistances.

Student experiment: Predicting and measuring resistance
Students can measure resistances of different combinations of resistors using an ohm-meter, and compare their answers with calculated values.

You may wish to review the use of the ohm-meter prior to this activity. The range on an ohm-meter can be confusing. Obviously this varies from device to device - make sure they are familiar and competent with the meters you intend them to use; otherwise the points about series and parallel circuits will be lost.

Episode 114-2: Resistors in series and in parallel (Word, 36 KB)

Student questions: Practice with the formulae
Two sets of questions using these formulae.

Kilo- and mega-ohms will catch some out, as will mA.

Episode 114-3: Circuit resistance (Word, 51 KB)

Episode 114-4: Combining resistances (Word, 31 KB)
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}