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

Answer

To solve the problem of finding the potential at any point on the surface of an uncharged conducting sphere with a point charge \( q \) placed at a distance \( r \) from its center (where \( R < r \)), we can follow these steps:

### Step 1: Understand the Configuration
We have a point charge \( q \) located outside an uncharged conducting sphere of radius \( R \). The distance from the center of the sphere to the point charge is \( r \), and it is given that \( r > R \).

### Step 2: Identify the Induced Charges
When the point charge \( q \) is placed near the conducting sphere, it induces charges on the surface of the sphere. Specifically, a negative charge will be induced on the side of the sphere closest to the point charge, and a positive charge will be induced on the opposite side. The total charge on the sphere remains zero since it was initially uncharged.

### Step 3: Electric Field Inside the Conductor
Inside a conductor in electrostatic equilibrium, the electric field is zero. This means that the induced charges rearrange themselves in such a way that they cancel out any electric field within the conducting material.

### Step 4: Potential Inside the Conductor
Since the electric field inside the conductor is zero, the potential throughout the conductor (including at any point on the surface) is constant. This constant potential is equal to the potential at the surface of the sphere.

### Step 5: Calculate the Potential at the Surface
To find the potential at the surface of the sphere, we can calculate the potential due to the point charge \( q \) at a distance \( r \) (the distance from the point charge to the center of the sphere). The formula for the electric potential \( V \) due to a point charge is given by:

\[ V = \frac{kq}{r} \]

where \( k \) is the Coulomb's constant.

### Step 6: Conclusion
The potential at any point on the surface of the conducting sphere is equal to the potential due to the point charge \( q \) at the distance \( r \):

\[ V_{\text{sphere}} = \frac{kq}{r} \]

### Summary
Thus, the potential at any point on the surface of the sphere is given by:

\[ V = \frac{kq}{r} \]