$A$ Wheatstone's bridge is balanced with a resistance of $625\, \Omega$ in the third arm, where $P, Q$ and $S$ are in the $1^{st}, 2^{nd}$ and $4^{th}$ arm respectively. If $P$ and $Q$ are interchanged, the resistance in the third arm has to be increased by $51\,\Omega$ to secure balance. The unknown resistance in the fourth arm is ............. $\Omega$
Diffcult
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The balancing condition for Wheatstone's bridge is, $\frac{P}{Q}=\frac{S}{R}$
$\frac{P}{Q}=\frac{625}{R}$$.....................(1)$
When $\mathrm{P}$ and $\mathrm{Q}$ are interchanged we get,
$\frac{Q}{P}=\frac{676}{R}$
$\frac{P}{Q}=\frac{R}{676}$$.....................(2)$
Equating both the equations,
$\frac{P}{Q}=\frac{625}{R}=\frac{R}{676}$
$R^{2}=\sqrt{422500}$
$R=650 \Omega$
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