Resistance in parallel
In parallel the reciprocals add: 1/Rp = 1/R₁ + 1/R₂ + 1/R₃. For 5, 10 and 30 Ω, Rp = 3 Ω, so a 12 V cell draws 4 A, split 2.4 A, 1.2 A, 0.4 A.
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Equivalent resistance3 Ωbetween A and B
Current4 Afrom the cell
Power48 WP = VI
| R | I | V | Power | |
|---|---|---|---|---|
| R₁ | 5 Ω | 2.4 A | 12 V | 28.8 W |
| R₂ | 10 Ω | 1.2 A | 12 V | 14.4 W |
| R₃ | 30 Ω | 0.4 A | 12 V | 4.8 W |
Step by step
- R₁, R₂ and R₃ in parallel: 1/R = 1/5 + 1/10 + 1/30 = 1/3, so R = 3 Ω
- So Req = 3 Ω
- Current from the cell: I = V / Req = 12 / 3 = 4 A
- Across the parallel group V = IR = 4 × 3 = 12 V, the same for every branch. It splits R₁: 12 / 5 = 2.4 A, R₂: 12 / 10 = 1.2 A, R₃: 12 / 30 = 0.4 A.
- Power: P = EI = 12 × 4 = 48 W.
Why it works
- In parallel every resistor joins the same two points, so each has the full voltage V across it.
- The current splits: I = I₁ + I₂ + I₃, with I₁ = V/R₁, I₂ = V/R₂ and I₃ = V/R₃.
- For the whole group I = V/Rp, so V/Rp = V/R₁ + V/R₂ + V/R₃.
- Cancel V: 1/Rp = 1/R₁ + 1/R₂ + 1/R₃. For two resistors this is Rp = R₁R₂ / (R₁ + R₂).
- Rp is always smaller than the smallest resistor in the group.
Resistance in parallel: quick answers
Why is the resistance in parallel less than the smallest resistor?
Each resistor added in parallel opens another path for the current, so more current flows for the same voltage. More current at the same voltage means less resistance.
What is the equivalent of 5, 10 and 30 Ω in parallel?
1/Rp = 1/5 + 1/10 + 1/30 = 1/3, so Rp = 3 Ω. On 12 V the cell gives 4 A: 2.4 A, 1.2 A, 0.4 A through the three.
Why are household appliances connected in parallel?
Each gets the full mains voltage, each can be switched on and off on its own, and one failing does not stop the others.