For the arrangement of the potentiometer shown in the figure, the balance point is obtained at $a$ distance $75\,cm $ from $A$ when the key $k$ is open. The second balance point is obtained at $60\, cm$ from $A$ when the key $k$ is closed. Find the internal resistance (in $ \Omega$) of the battery $E_1$.
Diffcult
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Let $\lambda$ is resistance per unit length of wire $\mathrm{AB}.$ When
$\mathrm{k}$ is opened
$\left| {\left( {\lambda {x_1}} \right) = {E_1}} \right. \ldots ..(1)$
$\mathrm{k}$ is closed
$|\lambda {x_2} = {E_1} - ir \ldots (2)$
$\mathrm{i}=\frac{\mathrm{E}_{1}}{\mathrm{R}+\mathrm{r}} \ldots(3)$
$\Rightarrow r=\left(\frac{x_{1}}{x_{2}}-1\right) R=\left(\frac{0.75}{0.60}-1\right) 24 ; r=6\, \Omega$
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