A potentiometer wire $A B$ having length $L$ and resistance $12 \mathrm{r}$ is joined to a cell $\mathrm{D}$ of emf $\varepsilon$ and internal resistance r. A cell C having emf $\varepsilon / 2$ and internal resistance $3 \mathrm{r}$ is connected. The length $\mathrm{AJ}$ at which the galvanometer as shown in fig. shows no deflection is:
Correct Option: , 3
(3) Let $x$ be the length $A J$ at which galvanometer shows
null deflection current,
$\mathrm{i}=\frac{\varepsilon}{12 \mathrm{r}+\mathrm{r}}=\frac{3}{13 \mathrm{r}}$
or, $i\left(\frac{x}{L} 12 r\right)=\frac{\varepsilon}{2}$
$\Rightarrow \frac{\varepsilon}{13 \mathrm{r}}\left[\frac{\mathrm{x}}{\mathrm{L}} \cdot 12 \mathrm{r}\right]=\frac{\varepsilon}{2}$
or, $x=\frac{13 L}{24}$
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