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Inductors in Circuits

Two cells eac...

Question

Two cells each of same emf $e$ but of internal resistances $r_{1}$ and $r_{2}$ are connected in series through an external resistance $R$ . If the potential difference between the ends of the first cell is zero, what will be the value of $R$ in terms of $r_{1}$ and $r_{2}$ ?

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Solution

Current in the circuit $I = \frac{2 e}{(r_{1} + r_{2}) + R}$
and $e - I r_{1} = 0$
$\Rightarrow e - \frac{2 e r_{1}}{(r_{1} + r_{2}) + R} = 0$
$\Rightarrow \frac{e r_{2} - e r_{1} + e R}{(r_{1} + r_{2}) + R} = 0$
$\Rightarrow r_{2} - r_{1} + R = 0$
or $R = (r_{1} - r_{2})$

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