SPI Physics · Electricity and Magnetism

Series and Parallel Circuits

Resistors in series share one current and their resistances add; resistors in parallel share one voltage and the reciprocals of their resistances add.

All Electricity and Magnetism notes

Why this matters

Four questions in the bank, two EASY and two MODERATE: what stays the same in a parallel circuit, the smallest resistance from equal resistors, a wire cut in two and joined in parallel, and the voltage across one resistor in a small network. Each needs the right rule for each group, so know which quantity is shared.

Concept 1 of 2: Adding resistors in series and in parallel

In series the resistors sit one after another on a single path, so the same current flows through all of them and the battery's voltage is shared out. In parallel each resistor is its own path between the same two points, so each gets the full voltage and the current splits.

Definition

Series: same current through each; voltages add up to the battery voltage; $R_s = R_1 + R_2 + R_3$ . The total is bigger than the biggest resistor.
Parallel: same voltage across each; currents add up to the total current; $\dfrac{1}{R_p} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3}$ . The total is smaller than the smallest resistor.
For two in parallel: $R_p = \dfrac{R_1 R_2}{R_1 + R_2}$ (product over sum).
Houses are wired in parallel: every appliance gets the full mains voltage and can be switched on or off by itself.
In a network, find each group's resistance first. A branch joined straight across the battery has the full battery voltage across it.

Series and parallel

R_s = R_1 + R_2 + R_3 \qquad \frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \qquad R_p = \frac{R_1 R_2}{R_1 + R_2}

Worked example

A 3

\Omega

and a 6

\Omega

resistor are joined in parallel across a 6 V battery. Find the combined resistance and the current through each.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Electricity and Magnetism · Series and Parallel Circuits

Which one among the following statements is true about parallel circuit?

Adding resistances that are in parallel

Plain adding is only for series. A 3

\Omega

and a 6

\Omega

resistor in parallel give 2

\Omega

, not 9

\Omega

: a parallel total is always smaller than the smallest resistor.

Swapping what is shared

In series the CURRENT is the same and the voltages differ; in parallel the VOLTAGE is the same and the currents differ.

Splitting the battery voltage between parallel branches

Two branches joined straight across a battery each have the full battery voltage. Share the voltage only between resistors that are in series inside one branch.

Drill 1 more on adding resistors in series and in parallel

Concept 2 of 2: Equal resistors and cutting a wire

With identical resistors the formulas collapse into one multiplication or division. And cutting a wire into pieces makes each piece a smaller resistor, because resistance is proportional to length.

Definition

n equal resistors R in series: $nR$ . This is the largest total you can make.
n equal resistors R in parallel: $\dfrac{R}{n}$ . This is the smallest total you can make.
A wire of resistance R cut into n equal pieces: each piece is $\dfrac{R}{n}$ . Joined in parallel they give $\dfrac{R/n}{n} = \dfrac{R}{n^2}$ .

n equal resistors

R_s = nR \qquad R_p = \frac{R}{n} \qquad \text{cut into } n \text{ and joined in parallel: } \frac{R}{n^2}

Worked example

You have four resistors of 2

\Omega

each. Find the largest and the smallest resistance you can make with all four.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Electricity and Magnetism · Series and Parallel Circuits

A wire that has resistance R is cut into two equal pieces. The two parts are joined in parallel. What is the resistance of the combination?

Forgetting that cutting changes each piece

Each half of a cut wire has half the resistance. Joining two halves of a 10

\Omega

wire in parallel gives

\frac{5}{2} = 2.5\,\Omega

, not

\frac{10}{2} = 5\,\Omega

Picking series for the minimum

Series always gives the largest total and parallel the smallest. 'Minimum resistance' from equal resistors means all of them in parallel.

Drill 1 more on equal resistors and cutting a wire

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Adding resistors in series and in parallel
Series and parallel
$R_s = R_1 + R_2 + R_3 \qquad \frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \qquad R_p = \frac{R_1 R_2}{R_1 + R_2}$
Equal resistors and cutting a wire
n equal resistors
$R_s = nR \qquad R_p = \frac{R}{n} \qquad \text{cut into } n \text{ and joined in parallel: } \frac{R}{n^2}$

Watch out for (5)

Adding resistances that are in parallel
→ Adding resistors in series and in parallel
Swapping what is shared
→ Adding resistors in series and in parallel
Splitting the battery voltage between parallel branches
→ Adding resistors in series and in parallel
Forgetting that cutting changes each piece
→ Equal resistors and cutting a wire
Picking series for the minimum
→ Equal resistors and cutting a wire

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