Four point charges each +q is placed on the circumference of a circle of diameter `2 d` in such a way that they form a square. The potential at the centre is

Options

A. `0`
B. `(4kq)/d`
C. `(4d)/q`
D. `q/(4d)`

Answer

To find the potential at the center of a circle where four point charges, each of charge +q, are placed at the corners of a square inscribed in the circle, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have four charges, each of magnitude +q, placed at the corners of a square. The square is inscribed in a circle with a diameter of 2d.
- The radius of the circle (and the distance from the center to each charge) is \( r = d \).

2. **Formula for Electric Potential**:
- The electric potential \( V \) due to a point charge \( Q \) at a distance \( r \) is given by the formula:
\[ V = k \frac{Q}{r} \]
where \( k \) is Coulomb's constant.

3. **Calculating the Total Potential at the Center**:
- Since potential is a scalar quantity, we can simply add the potentials due to each charge at the center.
- The distance from each charge to the center is \( d \). Therefore, the potential due to one charge at the center is:
\[ V_1 = k \frac{q}{d} \]
- Since there are four identical charges, the total potential \( V_0 \) at the center is:
\[ V_0 = V_1 + V_1 + V_1 + V_1 = 4 \left( k \frac{q}{d} \right) \]
- Thus, we can write:
\[ V_0 = \frac{4kq}{d} \]

4. **Final Answer**:
- The potential at the center of the circle is:
\[ V_0 = \frac{4kq}{d} \]