Resistors (Series & Parallel)
Add resistors in series or combine reciprocals in parallel — total resistance, with the steps.
2 resistors in series.
Two resistors of 4 Ω and 6 Ω combine differently by wiring. In series they add: 4 + 6 = 10 Ω. In parallel you sum the reciprocals — 1/4 + 1/6 = 5/12 — and invert, giving 4 × 6 ÷ (4 + 6) = 24 ÷ 10 = 2.4 Ω.
Series vs parallel resistance
How resistors combine depends entirely on how they are wired. In a series chain the same current flows through every resistor, so the resistances simply add. In a parallel bank the same voltage sits across every resistor and the currents split, so it is the reciprocals (conductances) that add. Once you have the total, feed it into Ohm’s law to find current or voltage.
Series resistances add directly; parallel reciprocals (1/R) add, then invert for R.
Worked example
Combine a 4 Ω and a 6 Ω resistor in parallel:
- 1 Pick the wiring. Series → add resistances. Parallel → add reciprocals, then invert. Here the two resistors are in parallel.
- 2 Sum the reciprocals. 1/R = 1/4 + 1/6 = 3/12 + 2/12 = 5/12 per ohm.
- 3 Invert to get R. R = 1 ÷ (5/12) = 12/5 = 2.4 Ω.
- 4 Shortcut for two resistors. Product over sum: R = (4 × 6) ÷ (4 + 6) = 24 ÷ 10 = 2.4 Ω — the same answer.
Series vs parallel at a glance
Same components, opposite behaviour. The product-over-sum shortcut applies to exactly two parallel resistors.
| Property | Series | Parallel |
|---|---|---|
| Total resistance | R = R₁ + R₂ + ⋯ | 1/R = 1/R₁ + 1/R₂ + ⋯ |
| Two resistors | R = R₁ + R₂ | R = R₁R₂ ÷ (R₁ + R₂) |
| Shared quantity | Same current through each | Same voltage across each |
| Total vs each | Larger than any single resistor | Smaller than the smallest resistor |
| Example (4 Ω, 6 Ω) | 10 Ω | 2.4 Ω |
Why parallel is always smaller
A parallel total is always less than the smallest resistor in the group. Adding another path in parallel gives current a new route to flow through, which can only increase the total current for a given voltage — and more current at the same voltage means less effective resistance. You are summing conductances (1/R), so the combined conductance can only grow, and its reciprocal can only shrink. With 4 Ω and 6 Ω, the 2.4 Ω result sits below the 4 Ω branch on its own.