Series and parallel circuits

Two conservation laws

• Conservation of charge: at a junction, the total current in equals the total current out.
• Conservation of energy: around any closed loop, the sum of the e.m.f.s equals the sum of the p.d.s.

Series

Same current through each component; the supply p.d. is shared: V = V1 + V2 + ... Since IR = IR1 + IR2 + ..., the total resistance is

R = R1 + R2 + R3 + ...

Parallel

Same p.d. across each branch; the currents add: I = I1 + I2 + ... Since V/R = V/R1 + V/R2 + ..., the total resistance is

1/R = 1/R1 + 1/R2 + 1/R3 + ...

The total is always less than the smallest branch resistance. For two resistors, R = R1R2 / (R1 + R2).

Mixed circuits

Replace each parallel group with its equivalent resistance, then add the series parts. Find the total current, then work back to the p.d. and current in each part.

Worked example

A 6.0 Ω resistor is in series with a parallel pair of 4.0 Ω and 12 Ω, across an 18 V supply. Find the current in the 4.0 Ω resistor.

1. Parallel pair: 4.0 x 12 / (4.0 + 12) = 3.0 Ω; total R = 9.0 Ω
2. Total current = 18 / 9.0 = 2.0 A
3. p.d. across the pair = 2.0 x 3.0 = 6.0 V
4. Current in 4.0 Ω = 6.0 / 4.0

Answer: 1.5 A

Check yourself

1. Find the total resistance of 10 Ω, 22 Ω and 47 Ω in series, in Ω.
   Show answer

   79 Ω. R = 10 + 22 + 47 = 79 Ω.

2. Find the total resistance of 6.0 Ω and 3.0 Ω in parallel, in Ω.
   Show answer

   2 Ω. 1/R = 1/6 + 1/3 = 1/2, so R = 2.0 Ω.

3. Find the total resistance of three 30 Ω resistors in parallel, in Ω.
   Show answer

   10 Ω. 1/R = 3/30, so R = 10 Ω.

4. A current of 2.5 A enters a junction. One branch leaving it carries 0.90 A. Find the current in the only other branch, in A.
   Show answer

   1.6 A. Current in = current out: 2.5 - 0.90 = 1.6 A.

5. A 12 V supply is connected across a 4.0 Ω and an 8.0 Ω resistor in series. Find the p.d. across the 8.0 Ω resistor, in V.
   Show answer

   8 V. I = 12 / 12 = 1.0 A; V = 1.0 x 8.0 = 8.0 V.

6. A 5.0 Ω resistor is in series with two 20 Ω resistors in parallel, across a 30 V supply. Find the current from the supply, in A.
   Show answer

   2 A. Parallel pair = 10 Ω; total = 15 Ω; I = 30 / 15 = 2.0 A.

7. An extra resistor is connected in parallel with an existing one. What happens to the total resistance?
   • It increases
   • It stays the same
   • It decreases
   • It doubles
   Show answer

   It decreases. A new path lets more current flow for the same p.d.

8. The rule that current into a junction equals current out is a consequence of
   • conservation of energy
   • Ohm's law
   • conservation of momentum
   • conservation of charge
   Show answer

   conservation of charge. Charge cannot be created or destroyed at a junction.

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Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics specification (XPH11/YPH11), Issue 3
• Pearson qualifications: International Advanced Levels Physics (2018)

Last updated 7 November 2026. Spotted a mistake? Tell us.