9 min readConcept sheetPart III (4 marks) numericals; Part IV (7 marks) derivation of equivalent resistance

Resistors in series and in parallel

In series, resistances add: Rs = R₁ + R₂ + R₃. In parallel, reciprocals add: 1/Rp = 1/R₁ + 1/R₂ + 1/R₃.

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

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

Two in parallelRp = R₁R₂/(R₁ + R₂)

n equal resistors RSeries: nR; parallel: R/n

Series vs parallel

Series connectionResistors are joined end to end in one path. The same current flows through each. The battery voltage is shared: V = V₁ + V₂ + V₃. The equivalent resistance is more than the largest one. If one breaks, the whole circuit stops.

Parallel connectionResistors are joined across the same two points, each in its own branch. The same potential difference is across each. The total current divides: I = I₁ + I₂ + I₃. The equivalent resistance is less than the smallest one. If one breaks, the others still work.

Equivalent resistance is the single resistance that could replace the whole group and draw the same current from the battery.

Derivation: resistors in series

Three resistors R₁, R₂ and R₃ are joined in series to a battery of voltage V. The same current I flows through each.

1
By Ohm's law, the potential difference across each is V₁ = IR₁, V₂ = IR₂, V₃ = IR₃.
2
The battery voltage is shared: V = V₁ + V₂ + V₃.
3
So V = IR₁ + IR₂ + IR₃ = I(R₁ + R₂ + R₃).
4
For the equivalent resistance Rs, V = IRs.
5★
So IRs = I(R₁ + R₂ + R₃), giving Rs = R₁ + R₂ + R₃.

★ key step

Derivation: resistors in parallel

Three resistors R₁, R₂ and R₃ are joined in parallel across a battery of voltage V. The total current I divides into I₁, I₂ and I₃.

1
The same V is across each, so I₁ = V/R₁, I₂ = V/R₂, I₃ = V/R₃.
2
The total current is the sum: I = I₁ + I₂ + I₃.
3
So I = V/R₁ + V/R₂ + V/R₃ = V(1/R₁ + 1/R₂ + 1/R₃).
4
For the equivalent resistance Rp, I = V/Rp.
5★
So V/Rp = V(1/R₁ + 1/R₂ + 1/R₃), giving 1/Rp = 1/R₁ + 1/R₂ + 1/R₃.

★ key step

How to draw series and parallel circuits

Series: draw a battery and a key. Then draw R₁, R₂ and R₃ as zig-zags one after another in a single loop, with an ammeter in the same loop.
Parallel: draw two short vertical lines as junctions. Draw R₁, R₂ and R₃ as three separate branches between them, one above the other.
Join the two junctions to the battery, key and ammeter.
Label each resistor, the battery (+ and −), the key and the ammeter. In the parallel circuit, mark I₁, I₂ and I₃ in the branches.

Worked example: series and parallel

Problem

(a) Resistors of 2 Ω, 3 Ω and 5 Ω are joined in series to a 10 V battery. Find the equivalent resistance, the current and the voltage across each. (b) Resistors of 3 Ω and 6 Ω are joined in parallel across a 12 V battery. Find the equivalent resistance and the current in each.

Working

(a) Rs = R₁ + R₂ + R₃ = 2 + 3 + 5 = 10 Ω.
I = V/Rs = 10 V ÷ 10 Ω = 1 A, the same in each resistor.
V₁ = IR₁ = 1 × 2 = 2 V; V₂ = 1 × 3 = 3 V; V₃ = 1 × 5 = 5 V. Check: 2 + 3 + 5 = 10 V.
(b) 1/Rp = 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2, so Rp = 2 Ω.
Total current I = V/Rp = 12 V ÷ 2 Ω = 6 A.
I₁ = V/R₁ = 12 ÷ 3 = 4 A; I₂ = 12 ÷ 6 = 2 A. Check: 4 + 2 = 6 A.

Answer

(a) 10 Ω; 1 A; 2 V, 3 V and 5 V (b) 2 Ω; 4 A through 3 Ω and 2 A through 6 Ω

In real life

At home, fans, lights and the TV are wired in parallel, so each gets the full mains voltage and has its own switch. Older strings of decorative serial lights were wired in series, so one fused bulb put the whole string out.

Exam angle

2 marks: “Why are household appliances connected in parallel?” Each gets the same full voltage, can be switched on and off on its own, and the others keep working if one fails.

4 marks: “Three resistors of 6 Ω each are connected (i) in series (ii) in parallel. Find the equivalent resistance in each case.” (i) Rs = 6 + 6 + 6 = 18 Ω. (ii) 1/Rp = 1/6 + 1/6 + 1/6 = 3/6 = 1/2, so Rp = 2 Ω.

7 marks: “Derive the expression for the equivalent resistance of three resistors connected in parallel.” Same V across each; I = I₁ + I₂ + I₃; put I = V/R for each; cancel V; 1/Rp = 1/R₁ + 1/R₂ + 1/R₃, with a labelled circuit.

Remember it

Series: Same current, Sum of R. Parallel: same Potential difference, reciProcals add.