A room heater is rated 750 W, 220 V. An electric bulb rated 200 W, 220 V is connected in series with this heater. What will be the power consumed by the bulb and the heater respectively, when the supply is at 220 V?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( P_B = 124.8\,\mathrm{W},\ P_H = 33.25\,\mathrm{W} \)
A device rated P at V has a resistance \( R = \frac{V^{2}}{P} \). In series the same current flows through both, and the power in each is \( I^{2}R \).
Resistances
\( \displaystyle R_H = \frac{220^{2}}{750}\,\Omega \approx 64.5\,\Omega \)
\( \displaystyle R_B = \frac{220^{2}}{200}\,\Omega = 242\,\Omega \)
Current in the series circuit
\( \displaystyle I = \frac{220\,\mathrm{V}}{64.5\,\Omega + 242\,\Omega} = \frac{220}{306.5}\,\mathrm{A} \approx 0.718\,\mathrm{A} \)
Powers
\( P_B = I^{2}R_B = (0.718)^{2}\times242\,\mathrm{W} \approx 124.8\,\mathrm{W} \)
\( P_H = I^{2}R_H = (0.718)^{2}\times64.5\,\mathrm{W} \approx 33.25\,\mathrm{W} \)
Shortcut: In series the same current flows, so \( P = I^{2}R \) is larger in the larger resistance. The bulb (242 Ω) has a much larger resistance than the heater (about 64.5 Ω), so \( P_B \gt P_H \). Only Option (1) has this and does not make the two equal.