Under what conditions current passing through a resistance $R$ can be increased by short circuiting the battery of emf $E_2$. The internal resistances of the two batteries are $r_1$ and $r_2$ respectively
Medium
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The current through the circuit before the battery of $\mathrm{emf}$ $\mathrm{E}_{2}$ is short circuited is,
$I_{1}=\frac{E_{1}+E_{2}}{R+r_{1}+r_{2}}$
After short circuiting the battery of $\mathrm{emf}$ $\mathrm{E}_{2}$. current through resistance $\mathrm{R}$ would be,
$I_{2}=\frac{E_{1}}{R+r_{1}}$
Now, $\quad \mathrm{I}_{2}>\mathrm{I}_{1}$
$\therefore $ ${\frac{\mathrm{E}_{1}}{\mathrm{R}+\mathrm{r}_{1}}>\frac{\mathrm{E}_{1}+\mathrm{E}_{2}}{\mathrm{R}+\mathrm{r}_{1}+\mathrm{r}_{2}}} $
${\mathrm{E}_{1} \mathrm{r}_{2}>\mathrm{E}_{2}\left(\mathrm{R}+\mathrm{r}_{1}\right)}$
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