A hollow conducting sphere of radius `R` has a charge `(+Q)` on its surface. What is the electric potential within the
sphere at a distance `r = (R )/(3)` from its centre ?

Options

A. Zero
B. `(1)/(4 pi epsilon_(0))(Q)/(r)`
C. `(1)/(4pi epsilon_(0))(Q)/(R)`
D. `(1)/(4pi epsilon_(0))(Q)/(r^(2))`

Answer

To find the electric potential within a hollow conducting sphere at a distance \( r = \frac{R}{3} \) from its center, we can follow these steps:

### Step 1: Understand the properties of a hollow conducting sphere
A hollow conducting sphere has the following properties:
- The electric field inside the conductor is zero.
- The charge resides on the surface of the conductor.

### Step 2: Apply Gauss's Law
Since the electric field inside the hollow conducting sphere is zero, we can use Gauss's Law to confirm that there is no charge enclosed within the sphere at any point inside it. This means that the electric field \( E \) at any point inside the sphere is:
\[ E = 0 \]

### Step 3: Relate electric field to electric potential
The relationship between electric field \( E \) and electric potential \( V \) is given by:
\[ E = -\frac{dV}{dr} \]
Since \( E = 0 \) inside the sphere, it implies that the potential \( V \) does not change with distance \( r \) inside the sphere. Thus, the potential remains constant throughout the interior of the sphere.

### Step 4: Determine the potential on the surface
The electric potential \( V \) at the surface of the hollow conducting sphere (at radius \( R \)) can be calculated using the formula for the potential due to a point charge:
\[ V = k \frac{Q}{R} \]
where \( k \) is the Coulomb's constant, \( Q \) is the charge on the sphere, and \( R \) is the radius of the sphere.

### Step 5: Conclude the potential inside the sphere
Since the potential inside the sphere is constant and equal to the potential at the surface, we have:
\[ V_{\text{inside}} = V_{\text{surface}} = k \frac{Q}{R} \]

Thus, the electric potential at a distance \( r = \frac{R}{3} \) from the center of the sphere is:
\[ V = k \frac{Q}{R} \]

### Final Answer
The electric potential within the sphere at a distance \( r = \frac{R}{3} \) from its center is:
\[ V = k \frac{Q}{R} \]