A cell of emf $E$ is connected to a resistance $R_{1}$ for time $t$ and the amount of heat generated in it is $H$ . If the resistance $R_{1}$ is replaced by another resistance $R_{2}$ and is connected to the cell for the same time $t$ , the amount of heat generated in $R_{2}$ is $4 H$ . Then internal resistance of the cell is:

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

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

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

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\sqrt{R_{1} R_{2}} \times \frac{\sqrt{R_{2}} - \sqrt{R_{1}}}{\sqrt{R_{2}} + \sqrt{R_{1}}}

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Solution

The correct option is C $\sqrt{R_{1} R_{2}} \times \frac{\sqrt{R_{2}} - 2 \sqrt{R_{1}}}{2 \sqrt{R_{2}} - \sqrt{R_{1}}}$
Heat generated is given as, $H = I^{2} R \times t$
According to given condition:
$\Rightarrow I_{1}^{2} R_{1} \times t = H$ and $I_{2}^{2} R_{2} \times t = 4 H$
$\Rightarrow \frac{E^{2}}{{(R_{1} + r)}^{2}} R_{1} \times t = H$ and $\frac{E^{2}}{{(R_{2} + r)}^{2}} R_{2} \times t = 4 H$

$\Rightarrow \frac{R_{2}}{{(R_{2} + r)}^{2}} = 4 \frac{R_{1}}{{(R_{1} + r)}^{2}}$

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

On solving, $r = \sqrt{R_{1} R_{2}} \times \frac{\sqrt{R_{2}} - 2 \sqrt{R_{1}}}{2 \sqrt{R_{2}} - \sqrt{R_{1}}}$

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A cell of emf E is connected to a resistance R1 for time t and the amount of heat generated in it is H. If the resistance R1 is replaced by another resistance R2 and is connected to the cell for the same time t, the amount of heat generated in R2 is 4H. Then internal resistance of the cell is: