Question

As shown in the figure below, two concentric conducting spherical shells, centered at \( r = 0 \) and having radii \( r = c \) and \( r = d \) are maintained at potentials such that the potential \( V(r) \) at \( r = c \) is \( V_1 \) and \( V(r) \) at \( r = d \) is \( V_2 \). Assume that \( V(r) \) depends only on \( r \), where \( r \) is the radial distance. The expression for \( V(r) \) in the region between \( r = c \) and \( r = d \) is

• A. ( V(r) = \frac{c d (V_2 - V_1)}{(d - c) r} - \frac{V_1 c + V_2 d - 2 V_1 d}{d - c} \)
• B. ( V(r) = \frac{c d (V_1 - V_2)}{(d - c) r} + \frac{V_2 d - V_1 c}{d - c} \)
• C. ( V(r) = \frac{c d (V_1 - V_2)}{(d - c) r} + \frac{V_1 c - V_2 c}{d - c} \)
• D. ( V(r) = \frac{c d (V_2 - V_1)}{(d - c) r} - \frac{V_2 c - V_1 c}{d - c} \)

Hint:

In spherical systems, the potential function \( V(r) \) is inversely proportional to the radial distance \( r \). Apply boundary conditions to solve for the constants of the general potential equation.

Answer 1

The Correct Option is B

Step 1: Understand the potential distribution.
The potential between two concentric spherical shells follows the equation of a radial potential, which is typically of the form: \[ V(r) = \frac{A}{r} + B \] where \( A \) and \( B \) are constants determined by the boundary conditions. Step 2: Apply the boundary conditions.
At \( r = c \), the potential is \( V_1 \), and at \( r = d \), the potential is \( V_2 \). We substitute these conditions into the general form and solve for \( A \) and \( B \). After solving, the expression for \( V(r) \) in the region between \( r = c \) and \( r = d \) is: \[ V(r) = \frac{c d (V_1 - V_2)}{(d - c) r} + \frac{V_2 d - V_1 c}{d - c} \] Step 3: Conclusion.
The correct expression is option (B), which matches the derived equation.