The two rules
- Series: R = R₁ + R₂ + R₃ + … The same current flows through each; the voltage divides.
- Parallel: 1/R = 1/R₁ + 1/R₂ + 1/R₃ + … The same voltage is across each; the current divides.
For just two in parallel, use the shortcut R = R₁R₂ ÷ (R₁ + R₂), "product over sum". For n equal resistors, series gives nR and parallel gives R/n.
The parallel total is always smaller than the smallest resistor. If your answer is bigger, something went wrong.
Capacitors are the other way round
Capacitors in parallel add directly (C = C₁ + C₂ + …), and in series they combine like resistors in parallel (1/C = 1/C₁ + 1/C₂ + …). So to combine capacitors in series, pick "Parallel" above and read the answer in farads.
More circuit formulas are on the physics formula sheet.
Questions students ask
What is the equivalent of 4 Ω and 12 Ω in parallel?
3 Ω, from (4 × 12) ÷ (4 + 12) = 48 ÷ 16.
Why is the parallel resistance smaller?
Each extra branch gives current another path, so the total current goes up for the same voltage, which means lower resistance.
How are household appliances connected?
In parallel, so each gets the full 230 V and can be switched on or off without affecting the others.