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Kirchhoff's Voltage Law

A cell develo...

Question

A cell develops the same power across two resistance $R_{1}$ and $R_{2}$ separately. The internal resistance of the cell is-

\sqrt{R_{1} R_{2}}

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\sqrt{R_{1} + R_{2}}

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R_{1} + R_{2}

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

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Solution

The correct option is A $\sqrt{R_{1} R_{2}}$
Power delivered across resistor $R_{1}$ ,

$i_{1} = \frac{E}{R_{1} + r}$

$P_{1} = i_{1}^{2} R_{1} = {(\frac{E}{R_{1} + r})}^{2} R_{1} . . . . . (1)$

Power delivered across resistor $R_{2}$ ,

$i_{2} = \frac{E}{R_{2} + r}$

$P_{2} = i_{2}^{2} R_{2} = {(\frac{E}{R_{2} + r})}^{2} R_{2} . . . . . (2)$

Given, that $P_{1} = P_{2}$

${(\frac{E}{R_{1} + r})}^{2} R_{1} = {(\frac{E}{R_{2} + r})}^{2} R_{2}$

$\sqrt{R_{1}} (R_{2} + r) = \sqrt{R_{2}} (R_{1} + r)$

$(\sqrt{R_{1}} - \sqrt{R_{2}}) r = \sqrt{R_{2}} R_{1} - \sqrt{R_{1}} R_{2}$

$r = \frac{\sqrt{R_{1} R_{2}} (\sqrt{R_{1}} - \sqrt{R_{2}})}{(\sqrt{R_{1}} - \sqrt{R_{2})}}$

$r = \sqrt{R_{1} R_{2}}$

 Hence, $(A)$ is the correct answer.

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A cell develops the same power across two resistance R1 and R2 separately. The internal resistance of the cell is-

A cell develops the same power across two resistance $R_{1}$ and $R_{2}$ separately. The internal resistance of the cell is-