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

Explanation

To find the electric potential at the center of the cavity of a conducting sphere with a point charge placed inside it, 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 non-concentric.
- A point charge \( q_1 \) is placed at a distance \( r \) from the center of the cavity.

2. **Applying Gauss's Law**:
- Since the sphere is conducting, the electric field inside the conductor itself is zero.
- The charge \( q_1 \) will induce a charge of \( -q_1 \) on the inner surface of the cavity.
- Consequently, a charge of \( +q_1 \) will appear on the outer surface of the conducting sphere.

3. **Electric Potential Calculation**:
- 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.

4. **Potential at the Center of the Cavity**:
- The total electric potential at the center of the cavity \( O \) can be calculated as the sum of the potentials due to:
- The point charge \( q_1 \)
- The induced charge on the inner surface of the cavity \( -q_1 \)
- The charge on the outer surface of the conductor \( +q_1 \)

5. **Calculating Each Contribution**:
- The potential due to the point charge \( q_1 \) at the center of the cavity:
\[ V_{q_1} = k \frac{q_1}{d_1} \]
where \( d_1 \) is the distance from \( q_1 \) to the center of the cavity.
- The potential due to the induced charge \( -q_1 \) on the inner surface:
\[ V_{-q_1} = -k \frac{q_1}{R_1} \]
- The potential due to the outer surface charge \( +q_1 \):
\[ V_{+q_1} = k \frac{q_1}{R_2} \]

6. **Total Electric Potential**:
- The total electric potential at the center of the cavity is:
\[ V_{total} = V_{q_1} + V_{-q_1} + V_{+q_1} \]
- Substituting the contributions:
\[ V_{total} = k \frac{q_1}{d_1} - k \frac{q_1}{R_1} + k \frac{q_1}{R_2} \]

7. **Final Expression**:
- Thus, the electric potential at the center of the cavity is:
\[ V = k q_1 \left( \frac{1}{d_1} - \frac{1}{R_1} + \frac{1}{R_2} \right) \]

### Conclusion:
The electric potential at the center of the cavity is given by the expression derived above.