Two resistances $R_1=X \Omega$ and $R_2=1 \Omega$ are connected to a wire $A B$ of uniform resistivity, as shown in the figure. The radius of the wire varies linearly along its axis from $0.2 mm$ at $A$ to $1 mm$ at $B$. A galvanometer ($G$) connected to the center of the wire, $50 cm$ from each end along its axis, shows zero deflection when $A$ and $B$ are connected to a battery. The value of $X$ is. . . . .
IIT 2022, Advanced
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For the balanced Wheatstone bridge
$\frac{ R _1}{ R _2}=\frac{\int_0^{0.5} \frac{\rho d x}{\pi r _x^2}}{\int_{0.5}^1 \frac{\rho dx }{\pi r _x^2}}$
$\frac{ R _1}{ R _2}=\frac{+\left[\frac{1}{ r _x}\right]_0^{0.5}}{+\left[\frac{1}{ r _{ x }}\right]_{0.5}^1}$
$\therefore R _1=5 R _2=5 \Omega$
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