A point charge Q is placed inside a conducting spherical shell of inner radius 3R and outer radius 5R at a distance R from the centre of the shell. The electric potential at the centre of the shell will be

Options

A. A) `(1)/(4piepsilon_(0)).(Q)/(R)`
B. B) `(1)/(4piepsilon_(0)).(5Q)/(6R)`
C. C) `(1)/(4piepsilon_(0)).(13Q)/(15Q)`
D. D) `(1)/(4piepsilon_(0)).(7Q)/(9Q)`

Answer

To find the electric potential at the center of a conducting spherical shell with a point charge \( Q \) placed inside it, we can follow these steps:

### Step 1: Understand the Configuration
We have a conducting spherical shell with:
- Inner radius = \( 3R \)
- Outer radius = \( 5R \)
- A point charge \( Q \) is placed at a distance \( R \) from the center of the shell.

### Step 2: Electric Potential Due to the Point Charge
The electric potential \( V_1 \) due to a point charge \( Q \) at a distance \( r \) from the charge is given by the formula:
\[ V_1 = \frac{KQ}{r} \]
where \( K = \frac{1}{4\pi \epsilon_0} \).

In this case, the distance from the point charge to the center of the shell is \( 2R \) (since the charge is at a distance \( R \) from the center of the shell which has an inner radius of \( 3R \)). Thus, we need to calculate the potential at the center due to the point charge:
\[ V_1 = \frac{KQ}{2R} \]

### Step 3: Electric Potential Due to the Inner Surface of the Shell
Since the shell is conducting, the inner surface will have an induced charge due to the point charge \( Q \). The potential \( V_2 \) at the center due to the inner surface (radius \( 3R \)) is:
\[ V_2 = \frac{KQ_{\text{induced}}}{3R} \]
For a conducting shell, the induced charge on the inner surface will be \( -Q \) (to maintain neutrality), thus:
\[ V_2 = \frac{-KQ}{3R} \]

### Step 4: Electric Potential Due to the Outer Surface of the Shell
The outer surface of the shell (radius \( 5R \)) will have a charge of \( +Q \) (to maintain neutrality). The potential \( V_3 \) at the center due to the outer surface is:
\[ V_3 = \frac{KQ}{5R} \]

### Step 5: Total Electric Potential at the Center
The total electric potential \( V \) at the center of the shell is the sum of the potentials due to the point charge, the inner surface, and the outer surface:
\[ V = V_1 + V_2 + V_3 \]
Substituting the values we calculated:
\[ V = \frac{KQ}{2R} + \left(\frac{-KQ}{3R}\right) + \frac{KQ}{5R} \]

### Step 6: Simplifying the Expression
To simplify, we can find a common denominator, which is \( 30R \):
\[ V = \frac{15KQ}{30R} - \frac{10KQ}{30R} + \frac{6KQ}{30R} \]
Combining these terms:
\[ V = \frac{(15 - 10 + 6)KQ}{30R} = \frac{11KQ}{30R} \]

### Step 7: Final Result
Thus, the electric potential at the center of the shell is:
\[ V = \frac{11KQ}{30R} \]