The length of a potentiometer wire is $\ell $. A cell of emf $E$ is balanced at a length $\ell /3$ from the positive end of the wire. If the length of the wire is increased by $\ell /2$ at what distance will the same cell give a balanced point
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$\mathrm{E}=\mathrm{x} \ell_{1}=\frac{\mathrm{V}}{\ell} \ell_{1}=\frac{\mathrm{V}}{\ell}\left(\frac{\ell}{3}\right)$ ..........$(1)$
$\ell=$ length of wire
again ${\rm{E}} = {{\rm{x}}^\prime }{\ell ^\prime } = \frac{{\rm{V}}}{{3\ell /2}}{\ell ^\prime }$ .......$(2)$
Divided eq. $(1)$ by $(2)$
$\ell^{\prime}=\ell / 2$
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