Resistors in series and parallel; solving circuits: WASSCE Physics

Every circuit — from a torch to the wiring of a house — is built from components joined in series, in parallel, or a mixture. Two simple rules let you find the current and voltage anywhere in it.

Learning objectives

By the end of this lesson you should be able to:

find the effective resistance of resistors in series and in parallel
find the current through and p.d. across each resistor in a circuit
solve mixed series–parallel problems
explain why household appliances are connected in parallel

Resistors in series

Three resistors joined end to end in series, and three resistors joined side by side in parallel

Figure 1. Resistors in series and in parallel. WASSCE Tutor diagram.

In series, resistors are joined end to end, so there is only one path:

the same current flows through each
the p.d.s add up to the total p.d.

R = R_1 + R_2 + R_3

The total resistance is larger than any single resistor.

Resistors in parallel

In parallel, resistors are connected side by side between the same two points:

the same p.d. is across each
the currents add up to the total current

\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3}

\begin{aligned}\dfrac{1}{R} &= \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3}\end{aligned}

The total resistance is smaller than the smallest resistor. For two resistors:

R = \dfrac{R_1 R_2}{R_1 + R_2} \quad (\text{"product over sum"})

\begin{gathered}\begin{aligned}R &= \dfrac{R_1 R_2}{R_1 + R_2}\end{aligned} \\[6pt] (\text{"product over sum"})\end{gathered}

Worked example

Find the combined resistance of (a) 4 Ω and 12 Ω in series, (b) 4 Ω and 12 Ω in parallel, (c) three 6 Ω resistors in parallel.

(a) $4 + 12 = 16$ Ω. (b) $\dfrac{4 \times 12}{4 + 12} = 3$ Ω. (c) $\dfrac{6}{3} = 2$ Ω ( $n$ equal resistors in parallel give $R/n$ ).

Solving circuits step by step

Combine resistors into a single effective resistance.
Find the total current from the supply: $I = \dfrac{V}{R_{total}}$ (include internal resistance if given).
Work back: in series, use $V = IR$ for each part; in parallel, the p.d. is shared equally and the current divides.

Worked example

A 6 Ω resistor is connected in series with a parallel pair of 4 Ω and 12 Ω resistors, across a 12 V battery of negligible internal resistance. Find (a) the total resistance, (b) the total current, (c) the p.d. across the parallel pair, (d) the current in the 4 Ω resistor.

(a) Parallel pair: $\dfrac{4 \times 12}{16} = 3$ Ω. Total: $6 + 3 = 9$ Ω.

(b) $I = \dfrac{12}{9} = 1.33$ A.

(d) $I_4 = \dfrac{4.0}{4} = 1.0$ A (and $I_{12} = \dfrac{4.0}{12} = 0.33$ A; together 1.33 A ✓).

Common mistake

In parallel, the current takes more of the easier path: the smaller resistor carries the larger current. And always check that branch currents add up to the total.

Current division (two resistors in parallel)

For a current $I$ splitting between $R_1$ and $R_2$ :

I_1 = I \times \dfrac{R_2}{R_1 + R_2} \qquad I_2 = I \times \dfrac{R_1}{R_1 + R_2}

\begin{gathered}\begin{aligned}I_1 &= I \times \dfrac{R_2}{R_1 + R_2}\end{aligned} \\[6pt] \begin{aligned}I_2 &= I \times \dfrac{R_1}{R_1 + R_2}\end{aligned}\end{gathered}

Why houses are wired in parallel

Each appliance gets the full mains voltage.
Each can be switched on and off independently.
If one appliance or bulb fails, the others keep working.

(Old Christmas-tree lights were wired in series: when one bulb failed, they all went out.)

In Sierra Leone

In a Freetown house, the sockets, the fridge and the lights are all connected in parallel to the 230 V supply. Switching off the fan in one room does not turn off the lights.

Key points

Series: same current; $R = R_1 + R_2 + \ldots$
Parallel: same p.d.; $\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \ldots$ ; two resistors: product over sum.
Solve by combining, finding the total current, then working back.
Household appliances are in parallel.

WASSCE examination focus

WAEC tests this

Mixed series–parallel circuits appear in almost every paper: find the effective resistance, the current in a branch, or the p.d. across a component. Draw the circuit, label each current and p.d., and check that currents at a junction add up. Explain why domestic appliances are connected in parallel.

Quick check

Two 10 Ω resistors are in parallel. What is their combined resistance?
In a series circuit, which quantity is the same for every component?
A 3 A current divides between 2 Ω and 4 Ω resistors in parallel. What current flows in the 2 Ω resistor?

Answers: 1. 5 Ω. 2. The current. 3. $3 \times \dfrac{4}{6} = 2$ A.

Resistors in series and parallel; solving circuits