The length of a potentiometer wire is $l$. $A$ cell of $\mathrm{emf}$ $E$ is balanced at a length $l/3$ from the positive end of the wire. If the length of the wire is increased by $l/2$. At what distance will the same cell give a balance point.
Diffcult
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Let $x$ be the desired length.
Potential gradient in the first case $=\frac{E_{0}}{l}$
$E=\left(\frac{l}{3}\right) \cdot\left(\frac{E_{0}}{l}\right)=\frac{E_{0}}{3}….(\mathrm{i})$
Potential gradient in the second case.
$=\frac{E_{0}}{3 l / 2}=\frac{2 E_{0}}{3 l}$
$\therefore E=(x) \frac{2 E_{0}}{3 l}$$...(II)$
From equations $(i)$ and $(ii),$ we get
$\frac{E_{0}}{3}=\left(\frac{2 E_{0}}{3 l}\right) x$
$\Rightarrow x=\frac{1}{2}$
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