Circuit for the measurement of resistance by potentiometer is shown. The galvanometer is first connected at point $A$ and zero deflection is observed at length $P J = 10\ cm$ . In second case it is connected at point $C$ and zero deflection is observed at a length $30\ cm$ from $P$ . Then the unknown resistance $X$ is
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In potentiometer wire potential difference is directly proportional to length
Let potential drop unit length a potentiometer wire be $\mathrm{K}.$
For zero deflection the current will flow independently in two circles
$\mathrm{IR}=\mathrm{K} \times 10$ ............$(1)$
$\mathrm{IR}+\mathrm{IX}=\mathrm{K} \times 30$ .........$(2)$
$(2)-(1)$
$\Rightarrow \mathrm{IX}-\mathrm{k} \times 20$ ........$(3)$
Divide $(1)$ and $(2)$
$\frac{\mathrm{R}}{\mathrm{X}}=\frac{1}{2} \Rightarrow \mathrm{x}=2 \mathrm{\,R}$
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