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Idea of Temperature

A cell of EMF...

Question

A cell of EMF $E$ and resistance $r$ supplies currents for the same time $t$ through external resistances $R_{1}$ and $R_{2}$ respectively. If the heat produced in both cases is the same then the internal resistance is given by

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

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

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

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r = R_{1} + R_{2}

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Solution

The correct option is C $r = \sqrt{R_{1} R_{2}}$
Heat produced $= i^{2} R T$
Given $i_{1}^{2} R_{1} T = i_{2}^{2} R_{2} T$
$\Rightarrow \frac{E^{2} R_{1}}{(R_{1} + r)^{2}} = \frac{R_{2} E^{2}}{(R_{2} + r)^{2}}$ $E = i_{1} (R_{1} + r) = i_{2} (R_{2} + r)$
$\Rightarrow R_{1} R_{2}^{2} + 2 R_{1} R_{2} r + r^{2} R_{1} = R_{2} . R_{1}^{2} + 2 R_{2} R_{1} R_{2} r + r^{2} R_{2}$
$r^{2} (R_{1} - R_{2}) = R_{1} R_{2} (R_{1} - R_{2})$
$r^{2} = R_{1} R_{2}$
$r = \sqrt{R_{1} R_{2}}$

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