Kirchhoff’s laws

AS Level Physics · D.C. circuits. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

Two laws let you solve any circuit. Kirchhoff's first law is about junctions: the total current entering a junction equals the total current leaving it. Charge cannot pile up or disappear, so this is conservation of charge.

Kirchhoff's second law is about loops: around any closed loop, the sum of the e.m.f.s equals the sum of the p.d.s. Each coulomb gains energy in the sources and gives exactly that much to the components, so this is conservation of energy.

The resistor formulas come from these laws. In series the current is the same and the p.d.s add. In parallel the p.d. is the same and the currents add.

Rules to remember

Common mistake

In the parallel formula students stop at 1/R. Remember to turn the fraction over at the end: if 1/R = 1/4 Ω−1, then R = 4 Ω.

Worked example

A 12 V supply of negligible internal resistance is connected to a 4.0 Ω resistor in series with a parallel pair of 12 Ω and 6.0 Ω. What is the current in the 6.0 Ω resistor?

  1. Parallel pair: 1/R = 1/12 + 1/6.0 = 3/12, so R = 4.0 Ω.
  2. Total resistance = 4.0 + 4.0 = 8.0 Ω, so supply current = 12 ÷ 8.0 = 1.5 A.
  3. p.d. across the parallel pair = 1.5 × 4.0 = 6.0 V.
  4. Current in the 6.0 Ω resistor = 6.0 ÷ 6.0 = 1.0 A (the other 0.50 A is in the 12 Ω resistor).

Answer: The current in the 6.0 Ω resistor is 1.0 A.

Practice questions

Try each one, then open the answer.

1. Three resistors, each of 12 Ω, are connected in parallel. What is their combined resistance?

  1. A
    0.25 Ω
  2. B
    4.0 Ω
  3. C
    12 Ω
  4. D
    36 Ω
Show answer

Answer: B. 1/R = 1/12 + 1/12 + 1/12 = 3/12, so R = 4.0 Ω. 0.25 is 1/R, not R.

2. Two resistors of 10 Ω and 30 Ω are in parallel across a supply. What fraction of the total current flows through the 10 Ω resistor?

  1. A
    1/4
  2. B
    1/3
  3. C
    3/4
  4. D
    1/2
Show answer

Answer: C. Same p.d. across both; current is inversely proportional to resistance. I10 : I30 = 3 : 1, so 3/4 flows through the 10 Ω resistor.

3. Resistors of 3.0 Ω and 6.0 Ω are connected in parallel, and this combination is in series with a 4.0 Ω resistor. What is the total resistance?

  1. A
    2.0 Ω
  2. B
    6.0 Ω
  3. C
    9.0 Ω
  4. D
    13 Ω
Show answer

Answer: B. Parallel pair: 1/R = 1/3 + 1/6 = 1/2, so 2.0 Ω. In series with 4.0 Ω: 6.0 Ω.

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