6 · Combining series and parallel

The recipe

A mixed circuit looks intimidating at first, but it always breaks down the same way: collapse the parallel section into one equivalent resistor first, then treat the whole thing as a plain series circuit. Here's the full recipe, in order:

Find the combined resistance of the parallel section: 1/R_p = 1/R₁ + 1/R₂
Add that to the series resistor to get the total circuit resistance: R_total = R_s + R_p
Find the total current from the battery: I = EMF / R_total — this is the current everywhere before the junction, including through the series resistor
Find the p.d. across the series resistor: V_s = I × R_s
Find the p.d. across the parallel section: V_p = I × R_p (check: V_s + V_p should equal the EMF)
Find each branch current using V_p, since both branches feel that same p.d.: I₁ = V_p / R₁, I₂ = V_p / R₂

Worked example — the anchor question

A 12 V battery is connected to a 1.0 Ω resistor in series with a parallel combination of a 1.0 Ω and a 2.5 Ω resistor. Find everything.

Step 1 — R_p = (1.0 × 2.5) / (1.0 + 2.5) = 2.5 / 3.5 ≈ 0.71 Ω

Step 2 — R_total = 1.0 + 0.71 = 1.71 Ω

Step 3 — I = 12 / 1.71 ≈ 7.0 A

Step 4 — V_s = 7.0 × 1.0 = 7.0 V

Step 5 — V_p = 7.0 × 0.71 ≈ 5.0 V (check: 7.0 + 5.0 = 12 V ✓)

Step 6 — I₁ = 5.0 / 1.0 = 5.0 A · I₂ = 5.0 / 2.5 = 2.0 A (check: 5.0 + 2.0 = 7.0 A ✓)

And a voltmeter connected straight across the battery terminals would simply read the EMF: 12 V — because it draws no current and taps both terminals directly, no matter what the rest of the circuit is doing.

Why this matters This exact circuit is what the interactive lab is built around. Instead of just reading the six steps above, you can drag the sliders and watch the current split, the voltages settle, and the charge dots visibly spend their energy at each resistor in the right order.