In an experiment to measure the internal resistance of a cell by potentiometer, it is found that the balance point is at a length of $2\text{m}$ when the cell is shunted by a $4\mathrm{ }\mathrm{\Omega }$ resistance and at $3\text{m}$ when cell is shunted by a $8\mathrm{ }\mathrm{\Omega }$ resistance. The internal resistance of cell is -

• A

  $12\mathrm{ }\mathrm{\Omega }$
• Correct answer
  B

  $8\mathrm{ }\mathrm{\Omega }$
• C

  $16\mathrm{ }\mathrm{\Omega }$
• D

  $1\mathrm{ }\mathrm{\Omega }$

Solution

$V_A-V_B=\varphi \times l$

$\mathrm{E}-\frac{\mathrm{E}\mathrm{r}}{\mathrm{r}+4}=\mathrm{\varphi }\times 2$ ....(i).

$\mathrm{E}\left(\frac{4}{\mathrm{r}+4}\right)=\mathrm{\varphi }\times 2$ .... (i)

Similarly, $\mathrm{E}\left(\frac{8}{\mathrm{r}+8}\right)=\mathrm{\varphi }\times 3$ ....(ii)

$\frac{\mathrm{e}{\mathrm{q}}_1}{\mathrm{e}{\mathrm{q}}_2}\Rightarrow \frac{\mathrm{r}+8}{2\left(\mathrm{r}+4\right)}=\frac{2}{3}\Rightarrow 3\mathrm{r}+24=4\mathrm{r}+16\Rightarrow \mathrm{r}=\mathrm{8 }\mathrm{\Omega }$

Hence, the correct answer is option: 'B'.