An infinite sheet of charge has surface charge density `sigma` . The separation between two equipotential surfaces, whose potential differ by V is:

Options

A. `(sigma V)/( 2 in _(0))`
B. `(2 in_(0) V)/( sigma)`
C. `( in_(0)V)/(sigma)`
D. `(2 sigma in_(0))/(V)`

Explanation

To solve the problem of finding the separation between two equipotential surfaces with a potential difference \( V \) due to an infinite sheet of charge with surface charge density \( \sigma \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Electric Field due to an Infinite Sheet of Charge**:
The electric field \( E \) produced by an infinite sheet of charge with surface charge density \( \sigma \) is given by the formula:
\[ E = \frac{\sigma}{2 \epsilon_0} \]
This electric field is uniform and acts perpendicular to the sheet.

2. **Relate Electric Field to Potential Difference**:
The relationship between electric field \( E \), potential difference \( V \), and separation \( D \) between two equipotential surfaces is given by:
\[ V = E \cdot D \]
Rearranging this equation gives us:
\[ D = \frac{V}{E} \]

3. **Substitute the Expression for Electric Field**:
Substitute the expression for \( E \) into the equation for \( D \):
\[ D = \frac{V}{\frac{\sigma}{2 \epsilon_0}} = \frac{V \cdot 2 \epsilon_0}{\sigma} \]

4. **Final Expression for Separation**:
Thus, the separation \( D \) between the two equipotential surfaces is:
\[ D = \frac{2 \epsilon_0 V}{\sigma} \]

### Conclusion:
The separation between the two equipotential surfaces whose potential differs by \( V \) is given by:
\[ D = \frac{2 \epsilon_0 V}{\sigma} \]