Question

Two heaters A and B have power rating of 1 kW and 2 kW, respectively. Those two are first connected in series and then in parallel to a fixed power source. The ratio of power outputs for these two cases is :

• A. 1 : 1
• B. 2 : 9
• C. 1 : 2
• D. 2 : 3

Answer 1

The Correct Option is B

This problem requires determining the power output ratio when two heaters, rated at 1 kW (Heater A) and 2 kW (Heater B), are connected first in series and then in parallel to a constant voltage source.

In a series configuration, the total power \( P_s \) is calculated using:

\[ P_s = \frac{V^2}{{R_A + R_B}} \]

The resistances \( R_A \) and \( R_B \) are derived from their power ratings and the source voltage \( V \) using \( P = \frac{V^2}{R} \), yielding:

\[ V^2 = 1 kW \times R_A = 2 kW \times R_B \]

This gives:

\[ R_A = \frac{V^2}{1 \text{ kW}},\ R_B = \frac{V^2}{2 \text{ kW}} \]

For the series connection:

\[ P_s = \frac{V^2}{{\frac{V^2}{1} + \frac{V^2}{2}}} = \frac{V^2}{\frac{3V^2}{2}} = \frac{2}{3} \text{ kW} \]

For the parallel connection, the total power \( P_p \) is the sum of individual powers:

\[ P_p = P_A + P_B = 1 \text{ kW} + 2 \text{ kW} = 3 \text{ kW} \]

The ratio of power outputs is then:

\[ \text{Ratio} = \frac{P_s}{P_p} = \frac{\frac{2}{3}}{3} = \frac{2}{9} \]

The resulting power output ratio is 2:9.