A potentiometer wire A B having length L
Question:

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:

1. (1) $\frac{11}{12} \mathrm{~L}$

2. (2) $\frac{11}{24} \mathrm{~L}$

3. (3) $\frac{13}{24} \mathrm{~L}$

4. (4) $\frac{5}{12} \mathrm{~L}$

Correct Option: , 3

Solution:

(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, $x=\frac{13 L}{24}$