10 min readConcept sheetNumericals of 2, 3 and 5 marks, and a 2 or 3-mark question on why homes use parallel wiring

Resistors in series and in parallel

In series the same current flows and Rs = R₁ + R₂ + R₃. In parallel the same potential difference acts and 1/Rp = 1/R₁ + 1/R₂ + 1/R₃.

SeriesRs = R₁ + R₂ + R₃; same current through each

Parallel1/Rp = 1/R₁ + 1/R₂ + 1/R₃; same potential difference across each

Two in parallelRp = R₁R₂/(R₁ + R₂)

Key resultRp is smaller than the smallest resistor in the group

Series vs parallel

SeriesResistors are joined end to end, in one line. The same current flows through each. The potential differences add up: V = V₁ + V₂ + V₃. Rs = R₁ + R₂ + R₃, so the total is more than the largest resistor. If one part breaks, the whole circuit stops.

ParallelResistors are joined between the same two points. Each has the same potential difference. The currents add up: I = I₁ + I₂ + I₃. 1/Rp = 1/R₁ + 1/R₂ + 1/R₃, so the total is less than the smallest resistor. If one branch breaks, the others keep working.

Quick check: n equal resistors R give nR in series and R/n in parallel.

Uses in daily life: why homes are wired in parallel

Homes are wired in parallel. Each appliance gets the full 220 V and draws the current it needs.
Each appliance can have its own switch, so a fan can run while the light is off.
In series, if one appliance fails, all of them stop. The voltage is also shared, so none gets the full 220 V.
Adding appliances in parallel lowers the total resistance, so the main current rises. Too many on one socket can overload the circuit.
A fuse is joined in series with the circuit it protects, so all the current passes through it.
Some festival serial lights have many tiny bulbs in series, so one fused bulb can make a whole string go dark.

Activity: equivalent resistance in series and parallel

Use three resistors, for example 1 Ω, 2 Ω and 3 Ω, a battery, a plug key, an ammeter and a voltmeter.

1
Join the three resistors end to end. Connect them in series with the battery, key and ammeter. Put the voltmeter across all three.
2
Close the key and find V/I. Observation: V/I = R₁ + R₂ + R₃ = 6 Ω.
3
Move the ammeter to different places in the loop. Observation: it shows the same current everywhere.
4
Now join the same resistors in parallel between two points X and Y. Put the voltmeter across XY and the ammeter in the main line.
5
Close the key and find V/I. Observation: I/V = 1/R₁ + 1/R₂ + 1/R₃, and V/I is less than 1 Ω, the smallest resistor.
6
Put the ammeter in each branch in turn. Observation: the branch currents add up to the main current.

How to draw series and parallel circuits

Series: draw a battery, a plug key, an ammeter and three zigzag resistors R₁, R₂, R₃ one after another in a single loop.
Draw the voltmeter across all three resistors (or across one, to show V₁).
Parallel: draw two short vertical lines for points X and Y. Draw R₁, R₂ and R₃ as three separate branches between X and Y.
Join X and Y to the battery through a plug key and an ammeter in the main line.
Draw the voltmeter across X and Y. Label every resistor, the + and − ends of the battery, and the currents I, I₁, I₂, I₃.

Worked example: series and parallel

Problem

(a) Resistors of 2 Ω, 3 Ω and 5 Ω are joined in series to a 12 V battery. Find the total resistance, the current, and the potential difference across each resistor. (b) Resistors of 3 Ω and 6 Ω are joined in parallel to a 6 V battery. Find the total resistance, the main current and the current in each resistor.

Working

(a) Rs = 2 Ω + 3 Ω + 5 Ω = 10 Ω.
I = V/Rs = 12 V ÷ 10 Ω = 1.2 A, the same in each resistor.
V₁ = 1.2 A × 2 Ω = 2.4 V; V₂ = 1.2 A × 3 Ω = 3.6 V; V₃ = 1.2 A × 5 Ω = 6.0 V.
Check: 2.4 V + 3.6 V + 6.0 V = 12 V.
(b) 1/Rp = 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2, so Rp = 2 Ω.
Main current I = V/Rp = 6 V ÷ 2 Ω = 3 A.
I₁ = 6 V ÷ 3 Ω = 2 A; I₂ = 6 V ÷ 6 Ω = 1 A. Check: 2 A + 1 A = 3 A.

Answer

(a) 10 Ω; 1.2 A; 2.4 V, 3.6 V and 6.0 V. (b) 2 Ω; 3 A; 2 A in 3 Ω and 1 A in 6 Ω.

In real life

In your home, the fan, tube light, fridge and TV are all in parallel, which is why switching off the TV does not switch off the fridge. Some Diwali serial lights are in series, so one fused bulb can darken the whole string.

Exam angle

2 marks: “Give two reasons why a series arrangement is not used for domestic circuits.” If one appliance fails, all stop. The voltage is shared, so no appliance gets the full 220 V, and all must take the same current. Appliances cannot be switched on and off separately.

3 marks: “How will you join three resistors of 6 Ω each to get (a) 9 Ω and (b) 4 Ω?” (a) Two in parallel give 3 Ω; add the third in series: 3 + 6 = 9 Ω. (b) Two in series give 12 Ω; put the third in parallel: (12 × 6) ÷ 18 = 4 Ω.

5 marks: “A 4 Ω resistor is joined in series with a parallel pair of 6 Ω and 12 Ω across a 12 V battery. Find the total resistance, the main current and the current in each parallel resistor.” Pair = 72/18 = 4 Ω; total = 8 Ω; I = 12/8 = 1.5 A; PD across pair = 6 V; 1 A in 6 Ω and 0.5 A in 12 Ω.

Remember it

Series: Same current, Sum the resistances. Parallel: same Potential, add the reciprocals, and the parallel answer is always smaller than the smallest.