A potentiometer wire of length $L$ and a resistance $r$ are connected in series with a battery of e.m.f. $E_0$ and a resistance $r_1$. An unknown e.m.f. $E$ is balanced at a length $l$ of the potentiometer wire. The e.m.f. $E$ will be given by
AIPMT 2015, Medium
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The current through the potentiometer wire is
$I=\frac{E_{0}}{\left(r+r_{1}\right)}$
and the potential difference across the wire is
$V=I r=\frac{E_{0} r}{\left(r+r_{1}\right)}$
The potential gradient along the potentiometer wire is
$k=\frac{V}{L}=\frac{E_{0} r}{\left(r+r_{1}\right) L}$
As the unknown e.m.f. $E$ is balanced against length $l$ of the potentiometer wire,
$\therefore \quad E=k l=\frac{E_{0} r}{\left(r+r_{1}\right)} \frac{l}{L}$
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