Potential difference beween centre and surface of the sphere of radius R and uniorm volume charge density `rho` within it will be

Options

A. `(rhoR^2)/(6epsilon_0)`
B. `(rhoR^2)/(4epsilon_0)`
C. `(rhoR^2)/(3epsilon_0)`
D. `(rhoR^2)/(2epsilon_0)`

Explanation

To find the potential difference between the center and the surface of a sphere with radius \( R \) and uniform volume charge density \( \rho \), we can follow these steps:

### Step 1: Calculate the total charge \( Q \) inside the sphere
The total charge \( Q \) within a sphere of radius \( R \) with uniform volume charge density \( \rho \) can be calculated using the formula:
\[ Q = \rho \cdot V \]
where \( V \) is the volume of the sphere given by:
\[ V = \frac{4}{3} \pi R^3 \]
Thus, we have:
\[ Q = \rho \cdot \frac{4}{3} \pi R^3 \]

### Step 2: Calculate the electric potential \( V_c \) at the center of the sphere
The electric potential \( V_c \) at the center of a uniformly charged sphere can be calculated using the formula:
\[ V_c = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Q}{R} \]
Substituting the expression for \( Q \):
\[ V_c = \frac{1}{4 \pi \epsilon_0} \cdot \frac{\rho \cdot \frac{4}{3} \pi R^3}{R} \]
This simplifies to:
\[ V_c = \frac{\rho}{3 \epsilon_0} R \]

### Step 3: Calculate the electric potential \( V_s \) at the surface of the sphere
The electric potential \( V_s \) at the surface of the sphere (radius \( R \)) is given by:
\[ V_s = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Q}{R} \]
Again substituting for \( Q \):
\[ V_s = \frac{1}{4 \pi \epsilon_0} \cdot \frac{\rho \cdot \frac{4}{3} \pi R^3}{R} \]
This simplifies to:
\[ V_s = \frac{\rho}{3 \epsilon_0} R \]

### Step 4: Calculate the potential difference \( V_{diff} \)
The potential difference \( V_{diff} \) between the center and the surface of the sphere is given by:
\[ V_{diff} = V_c - V_s \]
Substituting the values of \( V_c \) and \( V_s \):
\[ V_{diff} = \left(\frac{\rho}{3 \epsilon_0} R\right) - \left(\frac{\rho}{3 \epsilon_0} R\right) = 0 \]

### Conclusion
The potential difference between the center and the surface of the sphere is:
\[ V_{diff} = 0 \]