A conducting shell `S_1` having a charge Q is surrounded by an uncharged concentric conducting spherical shell `S_2`. Let the potential difference between `S_1` and that `S_2` be V. If the shell `S_2` is now given a charge `-3Q`, the new potential difference between the same two shells is

Options

A. `V`
B. `2V`
C. `4V`
D. `-2V`

Explanation

To solve the problem, we need to find the potential difference between two concentric conducting shells, \( S_1 \) and \( S_2 \), under two different charge conditions.

### Step-by-Step Solution:

1. **Understand the Initial Setup:**
- We have a conducting shell \( S_1 \) with charge \( Q \).
- Surrounding it is another conducting shell \( S_2 \) which is initially uncharged.

2. **Determine Initial Potential Difference \( V \):**
- The potential \( V_{S_1} \) at shell \( S_1 \) is given by:
\[ V_{S_1} = k \frac{Q}{R_1} \]
- The potential \( V_{S_2} \) at shell \( S_2 \) (since it is uncharged, it has no contribution from its own charge) is:
\[ V_{S_2} = k \frac{Q}{R_2} \]
- The potential difference \( V \) between \( S_1 \) and \( S_2 \) is:
\[ V = V_{S_1} - V_{S_2} = k \frac{Q}{R_1} - k \frac{Q}{R_2} \]
- This simplifies to:
\[ V = kQ \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

3. **Change the Charge on Shell \( S_2 \):**
- Now, \( S_2 \) is given a charge of \( -3Q \).
- The charge on \( S_1 \) remains \( Q \).

4. **Determine New Potential Difference:**
- The new potential \( V_{S_1} \) remains:
\[ V_{S_1} = k \frac{Q}{R_1} \]
- The new potential \( V_{S_2} \) with charge \( -3Q \) is:
\[ V_{S_2} = k \frac{Q}{R_2} + k \frac{-3Q}{R_2} = k \frac{Q - 3Q}{R_2} = k \frac{-2Q}{R_2} \]
- The new potential difference \( V' \) is:
\[ V' = V_{S_1} - V_{S_2} = k \frac{Q}{R_1} - \left( k \frac{-2Q}{R_2} \right) \]
- This simplifies to:
\[ V' = k \frac{Q}{R_1} + k \frac{2Q}{R_2} \]

5. **Relate New Potential Difference to Initial:**
- We can express the new potential difference \( V' \) as:
\[ V' = kQ \left( \frac{1}{R_1} + \frac{2}{R_2} \right) \]
- Since the initial potential difference \( V \) was:
\[ V = kQ \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
- We can see that the new potential difference \( V' \) does not equal \( V \) but rather has additional terms due to the charge on \( S_2 \).

6. **Final Result:**
- The new potential difference \( V' \) can be expressed in terms of \( V \), but it is not equal to \( V \). The potential difference remains dependent on the geometry of the shells and the charges present.

### Conclusion:
The new potential difference between the shells \( S_1 \) and \( S_2 \) when \( S_2 \) is charged with \( -3Q \) is:
\[ V' = kQ \left( \frac{1}{R_1} + \frac{2}{R_2} \right) \]