Question

For the arrangement of the potentiometer shown in the figure, the balance point is obtained at a distance $75\text{cm}$ from $A$, when the key $k$ is open. The second balance point is obtained at $60\text{cm}$ from $A$, when the key $k$ is closed . Find the internal resistance of the battery $E_1$ (in $\Omega )$ is

Answer 1

Let $\lambda$ be resistance per unit length of wire $AB$. When $k$ is opened, $I\left(\lambda x_1\right)=E_1....\left(1\right)$
If $k$ is closed,
$I\lambda x_2=E_1-ir.....\left(2\right)$
From secondary circuit
$i=\frac{E_1}{R+r}.....\left(3\right)$
$\frac{x_1}{x_2}=\frac{E_1}{E_1-ir}$
using $3$, we get
$\frac{x_1}{x_2}=\frac{R+r}{R}$
$\Rightarrow r=(\frac{x_1}{x_2}-1)R=(\frac{0.75}{0.60}-1)6$
$r=1.5\Omega$