Two cells, having the same $e.m.f.$ are connected in series through an external resistance $R.$ Cells have internal resistances $r_1$ and $r_2\,\, (r_1 > r_2)$ respectively. When the circuit is closed, the potential difference across the first cell is zero. The value of $R$ is
AIPMT 2006,JEE MAIN 2022, Medium
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(b) Let the voltage across any one cell is $V$, then
$V = E - ir = E - {r_1}\,\left( {\frac{{2E}}{{{r_1} + {r_2} + R}}} \right)$
But $V = 0$
$==>$ $E - \frac{{2E{r_1}}}{{{r_1} + {r_2} + R}} = 0$
$==>$ ${r_1} + {r_2} + R = 2{r_1}$
$==>$ $R = {r_1} - {r_2}$
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