Question

Two resistances R₁ = X Ω and R₂ = 1 Ω 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 0.2 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

Correct Answer: 5

The expression for \( R_{AO} \) is given by:

\( R_{AO} = \int_0^{1/2} \frac{\rho \, dx}{\pi (r + 4 \pi x)^2} = \frac{2 \rho}{3 \pi^2} \)

The expression for \( R_{OB} \) is:

\( R_{OB} = \int_{1/2}^1 \frac{\rho \, dx}{\pi (r + 4 \pi x)^2} = \frac{2 \rho}{15 \pi^2} \)

Now, solving for \( x \) with:

\( \left( x \right) \frac{2 \rho}{15 \pi^2} \)

We get the equation:

\( \frac{2 \rho}{3 \pi^2} = \left( 1 \right) \frac{2 \rho}{3 \pi^2} \Rightarrow x = 5 \)