Question

Two resistances $R_1=X\Omega$ and $R_2=1\Omega$ are connected to a wire $AB$ of uniform resistivity, as shown in the figure. The radius of the wire varies linearly along its axis from $0.2mm$ at $A$ to $1mm$ at $B$. A galvanometer (G) connected to the center of the wire, $50cm$ 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

Answer: 5

For the balanced Wheatstone bridge
$\frac{R_1}{R_2}=\frac{\underset{0}{\overset{0.5}{\int }}\frac{\rho dx}{\pi r_x^2}}{\underset{0.5}{\overset{1}{\int }}\frac{\rho dx}{\pi r_x^2}}$
$\frac{R_1}{R_2}=\frac{+{\left[\frac{1}{r_x}\right]}_0^{0.5}}{+{\left[\frac{1}{r_x}\right]}_{0.5}^1}$
$\therefore R_1=5R_2=5\Omega$