A non conducting sphere of radius R is charged uniformly. At what distance from its surface is the electrostatic potential is half the potential at its centre ?

Options

A. R
B. R/2
C. R/3
D. 2R

Explanation

To solve the problem, we need to find the distance from the surface of a uniformly charged non-conducting sphere at which the electrostatic potential is half of the potential at its center.

### Step-by-Step Solution:

1. **Understanding the Potential at the Center**:
The potential \( V_0 \) at the center of a uniformly charged non-conducting sphere of radius \( R \) can be calculated using the formula:
\[ V_0 = \frac{3kQ}{2R} \]
where \( k \) is Coulomb's constant, \( Q \) is the total charge of the sphere, and \( R \) is the radius of the sphere.

2. **Finding the Potential Outside the Sphere**:
For points outside the sphere (at a distance \( r \) from the center where \( r > R \)), the potential \( V \) is given by:
\[ V = \frac{kQ}{r} \]

3. **Setting Up the Equation**:
We need to find the distance \( d \) from the surface of the sphere where the potential \( V \) is half of the potential at the center:
\[ V = \frac{V_0}{2} = \frac{3kQ}{4R} \]
Therefore, we set up the equation:
\[ \frac{kQ}{r} = \frac{3kQ}{4R} \]

4. **Solving for \( r \)**:
Canceling \( kQ \) from both sides (assuming \( kQ \neq 0 \)):
\[ \frac{1}{r} = \frac{3}{4R} \]
Taking the reciprocal gives:
\[ r = \frac{4R}{3} \]

5. **Finding the Distance from the Surface**:
The distance from the surface of the sphere is:
\[ d = r - R = \frac{4R}{3} - R = \frac{4R}{3} - \frac{3R}{3} = \frac{R}{3} \]

### Final Answer:
The distance from the surface of the sphere at which the electrostatic potential is half the potential at its center is:
\[ \frac{R}{3} \]