Resistance in series and parallel
Reduce the parallel group first, then add in series: 4 Ω and 12 Ω in parallel make 3 Ω, plus 3 Ω in series gives 6 Ω, so a 12 V cell drives 2 A.
Circuit builder
Tap a resistor to choose it, then change its value or add another in series or in parallel with it. The glowing dots are the current: faster where more flows. Brighter resistors turn more power into heat.
| R | I | V | Power | |
|---|---|---|---|---|
| R₁ | 4 Ω | 1.5 A | 6 V | 9 W |
| R₂ | 12 Ω | 0.5 A | 6 V | 3 W |
| R₃ | 3 Ω | 2 A | 6 V | 12 W |
Step by step
- R₁ and R₂ in parallel: R = R₁R₂ / (R₁ + R₂) = 4 × 12 / (4 + 12) = 3 Ω
- The groups are in series: Req = 3 + 3 = 6 Ω
- Current from the cell: I = V / Req = 12 / 6 = 2 A
- Across the parallel group V = IR = 2 × 3 = 6 V, the same for every branch. It splits R₁: 6 / 4 = 1.5 A, R₂: 6 / 12 = 0.5 A.
- Drops in series: R₃: 2 × 3 = 6 V.
- Power: P = EI = 12 × 2 = 24 W.
Why it works
- Series: one path, so the same current flows through each and the voltages add. Rs = R₁ + R₂ + R₃.
- Parallel: the same voltage is across each and the currents add. 1/Rp = 1/R₁ + 1/R₂ + 1/R₃.
- In a mixed circuit, find a group that is purely in series or purely in parallel and replace it by its equivalent.
- Repeat until one resistor is left. That is the equivalent resistance.
- Then work back out: the total current, the voltage across each group, and the current in each branch.
Series and parallel: quick answers
How do you solve a series and parallel combination?
Replace the innermost group first. Here 4 Ω ∥ 12 Ω = 4 × 12 / 16 = 3 Ω, then 3 + 3 = 6 Ω. The 12 V cell drives 2 A, which splits 1.5 A and 0.5 A in the parallel pair.
What is the difference between series and parallel?
In series the current is the same and the voltages add; the total is bigger than any one. In parallel the voltage is the same and the currents add; the total is smaller than any one.
What is the ratio of series to parallel resistance for n equal resistors?
n equal resistors R give nR in series and R/n in parallel, so the ratio is n². Two equal resistors give a ratio of 4.