In a meter bridge, the wire of length 1m has a non-uniform cross-section such that, the variation $\frac{dR}{d l}$ of its resistance $R$ with length $l$ is $\frac{dR}{d l} \propto \frac{1}{\sqrt{l}}$ . Two equal resistances are connected as shown in the figure. The galvanometer has zero deflection when the jockey is at point P. What is the length AP?

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0.25 m

0.22 m

0.35 m

0.42 m

answer is A.

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Detailed Solution

$\frac{dR}{d ℓ} \propto \frac{1}{\sqrt{ℓ}} \Rightarrow \frac{dR}{dℓ} = k \times \frac{1}{\sqrt{ℓ}} ($ where k is constant)
$\begin{array}{l} dR = k \frac{dℓ}{\sqrt{ℓ}} \\ Let R^{'} = R_{1} = R_{2} \\ \int dR = K \int_{0}^{ℓ} ℓ^{\frac{- 1}{2}} dℓ \\ R_{1} = K \times 2 \sqrt{ℓ} \\ \int dR = K \int_{ℓ}^{1} ℓ^{\frac{- 1}{2}} dℓ \\ R_{2} = K \times 2 (1 - \sqrt{ℓ}) \\ R_{1} = R_{2} \\ K 2 \sqrt{ℓ} = K \times 2 (1 - \sqrt{ℓ}) \\ ℓ = 0.25 m \end{array}$