First slide

Electrical power

Question

A cell of constant e.m.f. is first connected to a resistance R₁ and then connected to a resistance R₂. If power delivered in both cases is same, then the internal resistance of the cell is

Moderate

A

$\sqrt{R_{1} R_{2}}$
B

$\sqrt{\frac{R_{1}}{R_{2}}}$
C

$\frac{R_{1} - R_{2}}{2}$
D

$\frac{R_{1} + R_{2}}{2}$

Solution

Power dissipated $= i^{2} R = {(\frac{E}{R + r})}^{2} R$
$\begin{array}{l} ∴ {(\frac{E}{R_{1} + r})}^{2} R_{1} = {(\frac{E}{R_{2} + r})}^{2} R_{2} \\ \Rightarrow R_{1} (R_{2}^{2} + r^{2} + 2 R_{2} r) = R_{2} (R_{1}^{2} + r^{2} + 2 R_{1} r) \\ \Rightarrow R_{2}^{2} R_{1} + R_{1} r^{2} + 2 R_{1} R_{2} r = R_{1}^{2} R_{2} + R_{2} r^{2} + 2 R_{1} R_{2} r \\ \Rightarrow (R_{1} - R_{2}) r^{2} = (R_{1} - R_{2}) R_{1} R_{2} \\ \Rightarrow r = \sqrt{R_{1} R_{2}} \end{array}$

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