Two heaters A and B have power rating of 1kW and 2kW, 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 [2024]

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Solution

Q.
Two heaters A and B have power rating of 1kW and 2kW, 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 [2024]

1 1 : 1

2 2 : 9

3 1 : 2

4 2 : 3

Ans.
(2)

Given that, $P_{A} = 1$ $kW, P_{B} = 2$ $kW$

For the series combination of two heaters A and B of same voltage, equivalent resistance is,

$R_{s} = R_{A} + R_{B}$ ...(i)

Power across both heaters, $P_{A} = \frac{V^{2}}{R_{A}}, P_{B} = \frac{V^{2}}{R_{B}}$

$∴$ $\frac{V^{2}}{P} = \frac{V^{2}}{P_{A}} + \frac{V^{2}}{P_{B}}$ (Using (i))

$∴$ $\frac{1}{P_{s}} = \frac{1}{P_{A}} + \frac{1}{P_{B}} ∴ P_{s} = \frac{P_{A} \cdot P_{B}}{P_{A} + P_{B}} ∴ P_{s} = \frac{1 \times 2}{1 + 2} = \frac{2}{3}$ $kW$

For the parallel combination of the two heaters,

$\frac{1}{R_{p}} = \frac{1}{R_{A}} + \frac{1}{R_{B}};$ $\frac{P_{P}}{V^{2}} = \frac{P_{A}}{V^{2}} + \frac{P_{B}}{V^{2}}$

$∴ P_{P} = P_{A} + P_{B} = 3$ $kW$

So the ratio of power output in series to power output in parallel is $= \frac{2}{3} \times \frac{1}{3} = \frac{2}{9}$

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