In a meter bridge, the wire of length $1\, m$ has a nonuniform cross-section such that, the variation $\frac{{dR}}{{d\ell }}$ of its resistance $R$ with length $\ell $ is $\frac{{dR}}{{d\ell }} \propto \frac{1}{{\sqrt \ell }}$ 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 $AP$ ? ................ $m$
JEE MAIN 2019, Diffcult
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$\frac{\mathrm{d} \mathrm{R}}{\mathrm{d} \ell}=\frac{\mathrm{k}}{\sqrt{\ell}} \quad \mathrm{k}=\mathrm{constant}$
$\int_{0}^{R} d R=k \int_{0}^{1} \frac{d \ell}{\sqrt{\ell}}$
$\mathrm{R}=2 \mathrm{k}$ resistance of wire $\mathrm{AB}$
Again, $\int_{0}^{R / 2} \mathrm{d} \mathrm{R}=\mathrm{k} \int_{0}^{L} \frac{\mathrm{d} \ell}{\sqrt{\ell}} \quad \mathrm{L} \rightarrow$ Length $\mathrm{AP}$
$\frac{\mathrm{R}}{2}=\mathrm{k} 2 \mathrm{L}^{1 / 2} \quad ; \quad \mathrm{k}=\mathrm{k} 2 \mathrm{L}^{1 / 2}$
$\Rightarrow \quad \mathrm{L}=\frac{1}{4}\, \mathrm{m}=0.25 \,\mathrm{m}$
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