Question

Two electric heaters have power ratings \( P_1 \) and \( P_2 \), at voltage \( V \). They are connected in series to a DC source of voltage \( V \). Find the power consumed by the combination. Will they consume the same power if connected in parallel across the same source?

Hint:

In a series connection, the total power is given by the harmonic mean of the two power ratings. In a parallel connection, the total power is simply the sum of the individual powers.

Answer 1

Power Consumption in Series and Parallel Connections:
(i) Power Consumption in Series:
- The resistance of each heater is: \[ R_1 = \frac{V^2}{P_1}, \quad R_2 = \frac{V^2}{P_2} \] - The total resistance in series is: \[ R_{\text{eq}} = R_1 + R_2 = \frac{V^2}{P_1} + \frac{V^2}{P_2} \] - The total current in the circuit is: \[ I = \frac{V}{R_{\text{eq}}} \] - The power consumed in series is: \[ P_{\text{series}} = I^2 R_{\text{eq}} \] \[ P_{\text{series}} = \frac{V^2}{R_{\text{eq}}} = \frac{V^2}{\frac{V^2}{P_1} + \frac{V^2}{P_2}} \] \[ P_{\text{series}} = \frac{P_1 P_2}{P_1 + P_2} \] (ii) Power Consumption in Parallel:
- In parallel, the power consumed by each heater remains the same as their original ratings: \[ P_{\text{parallel}} = P_1 + P_2 \] Thus, the heaters consume different power in series and the same power in parallel.