A conducting sphere of radius `R_(2)` has a spherical cavity of radius `R_(1)` which is non concentric with the sphere. A point charge `q_(1)` is placed at a distance r from the centre of the cavity. For this arrangement, answer the following questions.
Electric potential at the centre of the cavity will be :

Options

A. zero
B. `(kq_(1))/r-(kq_(1))/(R_(1))`
C. `(kq_(1))/r-(kq_(1))/(R_(1))+(Kq_(1))/(R_(2))`
D. None of these

Explanation

To solve the problem regarding the electric potential at the center of the cavity of a conducting sphere with a point charge placed inside, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a conducting sphere of radius \( R_2 \) with a spherical cavity of radius \( R_1 \) that is not concentric with the sphere.
- A point charge \( q_1 \) is placed at a distance \( r \) from the center of the cavity.

2. **Induced Charge on the Inner Surface**:
- When the point charge \( q_1 \) is placed inside the cavity, it induces a charge of \( -q_1 \) on the inner surface of the cavity. This is due to the property of conductors, which states that the electric field inside a conductor in electrostatic equilibrium must be zero.

3. **Charge on the Outer Surface**:
- The total charge of the conducting sphere remains neutral. Therefore, if \( -q_1 \) is induced on the inner surface, a charge of \( +q_1 \) will appear on the outer surface of the conducting sphere.

4. **Electric Potential at the Center of the Cavity**:
- The electric potential \( V \) at a point due to a point charge is given by the formula:
\[ V = k \frac{q}{r} \]
where \( k \) is Coulomb's constant, \( q \) is the charge, and \( r \) is the distance from the charge to the point where the potential is being calculated.

5. **Calculating Contributions to Potential**:
- The potential at the center of the cavity due to the point charge \( q_1 \) located at distance \( r \) from the center of the cavity is:
\[ V_{q_1} = k \frac{q_1}{r} \]
- The potential due to the induced charge \( -q_1 \) on the inner surface of the cavity (which is uniformly distributed) is:
\[ V_{-q_1} = -k \frac{q_1}{R_1} \]
(where \( R_1 \) is the radius of the cavity).
- The potential due to the charge \( +q_1 \) on the outer surface of the conducting sphere at the center of the cavity (which is at a distance \( R_2 \) from the outer surface) is:
\[ V_{+q_1} = k \frac{q_1}{R_2} \]

6. **Total Electric Potential**:
- The total electric potential \( V \) at the center of the cavity is the sum of the potentials due to all the charges:
\[ V = V_{q_1} + V_{-q_1} + V_{+q_1} \]
\[ V = k \frac{q_1}{r} - k \frac{q_1}{R_1} + k \frac{q_1}{R_2} \]

7. **Final Expression**:
- Therefore, the electric potential at the center of the cavity can be expressed as:
\[ V = k q_1 \left( \frac{1}{r} - \frac{1}{R_1} + \frac{1}{R_2} \right) \]