Question

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 _______

Answer 1

For the balanced Wheatstone bridge
$\frac{ R _1}{ R _2}=\frac{\int\limits_0^{0.5} \frac{\rho dx }{\pi r _x^2}}{\int\limits_{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$