A point charge q is placed at a distance of r from cebtre of an uncharged conducting sphere of rad `R(lt r)`. The potential at any point on the sphere is

Options

A. a. zero
B. b. `(1)/(4piepsilon_(0))(q)/(r)`
C. c.`(1)/(4pi epsilon_(0))(qR)/(r^(2))`
D. d. `(1)/(4piepsilon_(0))(qr^(2))/(R)`

Answer

To solve the problem, we need to find the potential at any point on the surface of an uncharged conducting sphere when a point charge \( q \) is placed at a distance \( r \) from the center of the sphere, where \( R < r \).

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a point charge \( q \) located at a distance \( r \) from the center of an uncharged conducting sphere of radius \( R \).
- Since \( R < r \), the point charge is outside the conducting sphere.

2. **Properties of Conductors**:
- In electrostatics, the electric field inside a conductor in electrostatic equilibrium is zero.
- The potential throughout the conducting material and on its surface is constant.

3. **Calculating the Potential at the Surface of the Sphere**:
- The potential \( V \) at any point due to a point charge \( q \) at a distance \( r \) is given by the formula:
\[ V = \frac{q}{4 \pi \epsilon_0 r} \]
- Here, \( r \) is the distance from the charge to the point where we are calculating the potential.

4. **Applying the Formula**:
- For any point on the surface of the conducting sphere (which is at a distance \( R \) from the center), the potential \( V \) can be calculated as:
\[ V = \frac{q}{4 \pi \epsilon_0 r} \]
- Since the point charge is outside the sphere, the potential at the surface of the sphere is the same as the potential at the location of the charge \( q \) at distance \( r \).

5. **Conclusion**:
- Therefore, the potential at any point on the surface of the sphere is:
\[ V = \frac{q}{4 \pi \epsilon_0 r} \]

### Final Answer:
The potential at any point on the sphere is \( \frac{q}{4 \pi \epsilon_0 r} \).