Four charges of same magnitude q are placed at four corners of a square of side a . The value electric potential at the centre of the square will be (where k = `(1)/(4 pi epsilon_(0))` )

Options

A. `(4 kq)/(a ) `
B. `4 sqrt(2) (kq)/(a ) `
C. `(4kq)/(sqrt(2a))`
D. `(kq)/(a sqrt(2))`

Answer

To find the electric potential at the center of a square with four charges of the same magnitude \( q \) placed at its corners, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a square with side length \( a \).
- There are four charges, each of magnitude \( q \), located at the corners of the square.

2. **Finding the Distance from the Charges to the Center**:
- The center of the square is equidistant from all four corners.
- The distance \( r \) from any corner of the square to the center can be calculated using the Pythagorean theorem. Since the center divides the square into two equal halves, the distance is given by:
\[ r = \frac{a}{\sqrt{2}} \]

3. **Calculating the Electric Potential Due to One Charge**:
- The electric potential \( V \) due to a single point charge \( q \) at a distance \( r \) is given by the formula:
\[ V = k \frac{q}{r} \]
- Substituting the value of \( r \):
\[ V = k \frac{q}{\frac{a}{\sqrt{2}}} = k \frac{q \sqrt{2}}{a} \]

4. **Calculating the Total Electric Potential at the Center**:
- Since the potential is a scalar quantity, the total potential \( V_{total} \) at the center due to all four charges is simply the sum of the potentials due to each charge:
\[ V_{total} = 4 \times V = 4 \left( k \frac{q \sqrt{2}}{a} \right) \]
- Therefore, we have:
\[ V_{total} = \frac{4 k q \sqrt{2}}{a} \]

### Final Answer:
The electric potential at the center of the square is:
\[ V = \frac{4 k q \sqrt{2}}{a} \]