A thin spherical conducting shell of radius R has a charge q. Another charge Q is placed at the centre of the shell. The electrostatic potential at a point P a distance `R/2` from the centre of the shell is

Options

A. (a) `(2Q)/(4piepsilon_0R)`
B. (b) `(2Q)/(4piepsilon_0R)-(2q)/(4piepsilon_0R)`
C. (c) `(2Q)/(4piepsilon_0R)+(q)/(4piepsilon_0R)`
D. (d) `((q+Q)^2)/(4piepsilon_0R)`

Explanation

To find the electrostatic potential at point P, which is at a distance \( \frac{R}{2} \) from the center of a thin spherical conducting shell of radius \( R \) with charge \( q \) and another charge \( Q \) placed at the center, we can follow these steps:

### Step 1: Understand the System
- We have a thin spherical conducting shell of radius \( R \) with charge \( q \).
- There is a charge \( Q \) placed at the center of this shell.
- We need to find the electrostatic potential at point P, which is located at a distance \( \frac{R}{2} \) from the center.

**Hint:** Recall that the electrostatic potential inside a conductor is constant and equal to the potential on its surface.

### Step 2: Calculate the Potential Due to Charge \( Q \)
- The potential \( V_Q \) due to the charge \( Q \) at a distance \( r \) from it is given by the formula:
\[ V_Q = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r} \]
- Since point P is at a distance \( \frac{R}{2} \) from the center, we substitute \( r = \frac{R}{2} \):
\[ V_Q = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{\frac{R}{2}} = \frac{2Q}{4\pi\epsilon_0 R} \]

**Hint:** Remember that the potential due to a point charge decreases with distance.

### Step 3: Calculate the Potential Due to Charge \( q \) on the Shell
- The conducting shell has a charge \( q \) distributed uniformly on its surface. The potential \( V_q \) at any point inside the shell (including point P) is constant and equal to the potential at the surface of the shell.
- The potential at the surface of the shell (at radius \( R \)) is given by:
\[ V_q = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{R} \]

**Hint:** The potential inside a conducting shell is the same as that at its surface.

### Step 4: Calculate the Total Potential at Point P
- The total potential \( V_P \) at point P is the sum of the potentials due to both charges:
\[ V_P = V_Q + V_q \]
- Substituting the values we calculated:
\[ V_P = \frac{2Q}{4\pi\epsilon_0 R} + \frac{q}{4\pi\epsilon_0 R} \]
- Combine the terms:
\[ V_P = \frac{2Q + q}{4\pi\epsilon_0 R} \]

**Hint:** When adding potentials, make sure to combine like terms properly.

### Final Answer
The electrostatic potential at point P, which is at a distance \( \frac{R}{2} \) from the center of the shell, is:
\[ V_P = \frac{2Q + q}{4\pi\epsilon_0 R} \]