Problem — ranking three lamp configurations

Problem

Three lamps are connected to the same fixed-voltage $V$ battery: A: one lamp of resistance $R$; B: two lamps of resistance $R$ in series; C: two lamps of resistance $R$ in parallel. Rank the total power dissipated from smallest to largest.

Solution

Reasoning. The battery has fixed voltage $V$; the total power dissipated is $P = V^2/R_\text{eq}$, so what matters is the equivalent resistance seen by the battery: less resistance, more power.

Equivalent resistances:

• A: a single lamp, $R_\text{eq} = R$.
• B: two in series, $R_\text{eq} = 2R$.
• C: two in parallel, $R_\text{eq} = R/2$.

Total powers: $P_A = \frac{V^2}{R}, \qquad P_B = \frac{V^2}{2R}, \qquad P_C = \frac{V^2}{R/2} = \frac{2V^2}{R}$

Comparing: $P_B = \tfrac12 (V^2/R) < P_A = V^2/R < P_C = 2\,(V^2/R)$.

$\ev{P_B < P_A < P_C}$

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Topics: Circuiti elettrici Concepts: Potenza Skills: Ragionamento di ranking