Two Resistors, One Battery: Series vs Parallel Prediction
A student has a 12 V ideal battery, a 6 Ω resistor, and a 3 Ω resistor, plus an ammeter and voltmeter. In arrangement X, the resistors are connected in series with the battery. In arrangement Y, they are connected in parallel across the battery. The student predicts that arrangement Y will draw less current from the battery because 'the current has to split between two paths, so each path carries less, and the total must be smaller.' The student also expects the 6 Ω resistor to dissipate more power in both arrangements because it has the larger resistance. Assume ideal meters and wires of negligible resistance.
1. Calculate the total current drawn from the battery and the power dissipated by each resistor in arrangement X (series).
Read the worked answer
Series (X): R_eq = 6 + 3 = 9 Ω. I = V/R_eq = 12/9 = 1.33 A (same current through both).
Power in 6 Ω: P = I²R = (1.33)²(6) = (1.778)(6) ≈ 10.7 W. Power in 3 Ω: P = I²R = (1.778)(3) ≈ 5.3 W. Total power: 16.0 W, which also equals IV = (1.33)(12) = 16 W. ✓
In series the larger resistor dissipates more power because current is shared and P = I²R scales with R.
2. Calculate the total current and the power in each resistor for arrangement Y (parallel), then evaluate both of the student's predictions.
Read the worked answer
Parallel (Y): Each resistor sees the full 12 V. I_6 = 12/6 = 2.0 A; I_3 = 12/3 = 4.0 A. Total current: I = 2.0 + 4.0 = 6.0 A. (Check: R_eq = (6·3)/(6+3) = 2 Ω; I = 12/2 = 6 A ✓.)
Power in 6 Ω: P = V²/R = 144/6 = 24 W. Power in 3 Ω: P = V²/R = 144/3 = 48 W.
Prediction 1 (parallel draws less current): Wrong. Parallel draws 6.0 A vs 1.33 A in series. Although each branch carries only part of the total, adding a parallel path decreases equivalent resistance (2 Ω vs 9 Ω), so the battery supplies more total current, not less. The student confused 'current splits' with 'total current drops.'
Prediction 2 (6 Ω always dissipates more): Wrong for parallel. With the same voltage across each branch, P = V²/R, so the smaller resistor (3 Ω) dissipates more (48 W vs 24 W). The 6 Ω dissipates more only in series, where current is common.