A cell sends a current through a resistance $R$ for time $t$. Now the same cell sends current through another resistance $r$ for the same time. If same amount of heat is developed in both the resistance, then the internal resistance of cell is
Medium
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(d)
$H_1=\frac{E^2 R}{\left(R+r_1\right)^2}$
$H_2=\frac{E^2 R}{\left(r+r_1\right)^2}$
$\frac{R}{\left(R+r_1\right)^2}=\frac{r}{\left(r+r_1\right)^2}$
$\frac{r+r_1}{R+r_1}=\frac{\sqrt{r}}{\sqrt{R}}$
$\sqrt{R} r_1+r \sqrt{R}=\sqrt{r} r_1+R \sqrt{r}$
$r_1(\sqrt{R}-\sqrt{r})=\sqrt{R r}(\sqrt{R}-\sqrt{r})$
$r_1=\sqrt{R r}$
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