Key ideas
Circuit analysis finds the current and voltage in every part of a circuit. Ohm’s law links them for a resistor, and two rules about series and parallel connections do the rest.
- Ohm’s law is V = IR. Power is P = IV = I²R = V² ÷ R, in watts.
- In series the same current flows through every element and the voltages add. Resistances add: R = R₁ + R₂.
- In parallel every branch has the same voltage and the currents add. Conductances add, so 1 ÷ R = 1 ÷ R₁ + 1 ÷ R₂ and the total is always smaller than the smallest branch.
- A resistor and a capacitor charge with time constant τ = RC; after one τ a charging capacitor reaches 63 % of its final voltage.
Key equations
V = IRP = IV = I²R = V² ÷ RR_series = R₁ + R₂ + …1 ÷ R_parallel = 1 ÷ R₁ + 1 ÷ R₂ + …τ = RCV_out = V_in × R₂ ÷ (R₁ + R₂)
Worked example
A 12 V battery drives 6 Ω and 12 Ω resistors in parallel, in series with 2 Ω. The parallel pair is 1 ÷ (1/6 + 1/12) = 4 Ω, so the total is 6 Ω and I = 12 ÷ 6 = 2.0 A. The parallel part has 2.0 × 4 = 8 V across it, so the branch currents are 8 ÷ 6 = 1.33 A and 8 ÷ 12 = 0.67 A, which add to 2.0 A. The power is 12 × 2.0 = 24 W.
Common mistakes
- Adding resistors in parallel as if they were in series.
- Forgetting that current is the same everywhere in a series loop, and voltage is the same across parallel branches.
- Confusing power, in watts, with energy, in joules or kilowatt-hours.
Practice
Test yourself with practice questions on this topic: Electric circuits