For charges each equal to q are placed at the corners of a square of side l. The electric potential at the centre of the square is :

Options

A. `1/(4pi epsilon_0) * (4q)/l`
B. `1/(4pi epsilon_0) (4q)/(sqrt2l)`
C. `1/(pi epsilon_0) (sqrt(2q))/l`
D. `1/(pi epsilon_0) (2q)/l`

Answer

To find the electric potential at the center of a square with charges placed at its corners, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Configuration**:
- We have a square with side length \( L \).
- There are four charges, each equal to \( Q \), placed at the corners of the square.

2. **Determine the Distance from the Center to Each Charge**:
- The distance from the center of the square to each corner can be calculated using the Pythagorean theorem.
- The distance \( r \) from the center to any corner is given by:
\[ r = \frac{L}{\sqrt{2}} \]
- This is because the center divides the diagonal of the square into two equal parts, and the diagonal \( d \) of the square is \( d = L\sqrt{2} \).

3. **Calculate the Electric Potential Due to One Charge**:
- The electric potential \( V \) due to a point charge \( Q \) at a distance \( r \) is given by the formula:
\[ V = \frac{kQ}{r} \]
- Substituting \( r = \frac{L}{\sqrt{2}} \):
\[ V = \frac{kQ}{\frac{L}{\sqrt{2}}} = \frac{kQ\sqrt{2}}{L} \]

4. **Calculate the Total Electric Potential at the Center**:
- Since there are four identical charges, the total electric potential \( V_{total} \) at the center is the sum of the potentials due to each charge:
\[ V_{total} = 4 \times \frac{kQ\sqrt{2}}{L} \]
- Therefore:
\[ V_{total} = \frac{4kQ\sqrt{2}}{L} \]

5. **Express in Terms of Constants**:
- The constant \( k \) can be expressed as \( k = \frac{1}{4\pi\epsilon_0} \), where \( \epsilon_0 \) is the permittivity of free space.
- Substituting this into the expression for \( V_{total} \):
\[ V_{total} = \frac{4 \left(\frac{1}{4\pi\epsilon_0}\right) Q \sqrt{2}}{L} \]
- Simplifying gives:
\[ V_{total} = \frac{Q\sqrt{2}}{\pi\epsilon_0 L} \]

### Final Answer:
The electric potential at the center of the square is:
\[ V_{total} = \frac{\sqrt{2} Q}{\pi \epsilon_0 L} \]