There are four concentric shells A,B, C and D of radii `a,2a,3a` and `4a` respectively. Shells B and D are given charges `+q` and `-q` respectively. Shell C is now earthed. The potential difference `V_A-V_C` is `k=(1/(4piepsilon_0))`

Options

A. `(kq)/(2a)`
B. `(kq)/(4a)`
C. `(kq)/(3a)`
D. `(kq)/(6a)`

Explanation

To solve the problem, we need to find the potential difference \( V_A - V_C \) given the conditions of the concentric shells. Let's break down the solution step by step.

### Step 1: Understand the configuration
We have four concentric shells A, B, C, and D with radii \( a, 2a, 3a, \) and \( 4a \) respectively. Shell B has a charge of \( +q \) and shell D has a charge of \( -q \). Shell C is earthed, meaning its potential \( V_C = 0 \).

### Step 2: Determine the potential at shell C
Since shell C is earthed, we set its potential \( V_C = 0 \). The potential at any point due to a charged shell can be calculated using the formula:
\[ V = \frac{kQ}{r} \]
where \( k = \frac{1}{4\pi\epsilon_0} \), \( Q \) is the charge, and \( r \) is the distance from the center.

### Step 3: Calculate the potential at shell A
The potential at shell A (\( V_A \)) is influenced by the charges on shells B and D. The contributions to the potential at shell A from these shells can be calculated as follows:

1. **Contribution from shell B** (at \( 2a \)):
- Since shell A is inside shell B, the potential due to shell B at shell A is:
\[ V_B = \frac{k(+q)}{2a} \]

2. **Contribution from shell D** (at \( 4a \)):
- Shell D is outside shell A, so the potential due to shell D at shell A is:
\[ V_D = \frac{k(-q)}{4a} \]

### Step 4: Combine the potentials
Now we can combine the contributions to find the total potential at shell A:
\[ V_A = V_B + V_D = \frac{kq}{2a} - \frac{kq}{4a} \]

### Step 5: Simplify the expression for \( V_A \)
To combine the terms, we need a common denominator:
\[ V_A = \frac{2kq}{4a} - \frac{kq}{4a} = \frac{(2kq - kq)}{4a} = \frac{kq}{4a} \]

### Step 6: Calculate the potential difference \( V_A - V_C \)
Since \( V_C = 0 \):
\[ V_A - V_C = V_A - 0 = V_A = \frac{kq}{4a} \]

### Step 7: Final result
Thus, the potential difference \( V_A - V_C \) is:
\[ V_A - V_C = \frac{kq}{4a} \]

### Summary
The potential difference between shell A and shell C is given by:
\[ V_A - V_C = \frac{kq}{4a} \]