Four point charges `-Q, -q, 2q` and `2Q` are placed, one at each corner of the square. The relation between `Q` and `q` for which the potential at the centre of the square is zero is

Options

A. `(1)/(q)`
B. `Q =-q`
C. `Q = - (1)/(q)`
D. `Q = q`

Explanation

To solve the problem, we need to find the relation between the charges \( Q \) and \( q \) such that the electric potential at the center of the square formed by the four charges is zero.

### Step-by-Step Solution:

1. **Identify the Charges and Their Positions**:
- Let the charges be placed at the corners of a square with side length \( l \).
- The charges are:
- Charge at (0, 0): \( -Q \)
- Charge at (0, l): \( -q \)
- Charge at (l, 0): \( 2q \)
- Charge at (l, l): \( 2Q \)

2. **Calculate the Distance from Each Charge to the Center**:
- The center of the square is at \( \left( \frac{l}{2}, \frac{l}{2} \right) \).
- The distance from each charge to the center (using the Pythagorean theorem) is:
\[ r = \sqrt{\left( \frac{l}{2} - 0 \right)^2 + \left( \frac{l}{2} - 0 \right)^2} = \sqrt{\frac{l^2}{4} + \frac{l^2}{4}} = \sqrt{\frac{l^2}{2}} = \frac{l}{\sqrt{2}} \]

3. **Write the Expression for Electric Potential at the Center**:
- The electric potential \( V \) at a point due to a point charge is given by:
\[ V = k \frac{q}{r} \]
- Therefore, the total potential \( V_{total} \) at the center due to all four charges is:
\[ V_{total} = k \left( \frac{-Q}{\frac{l}{\sqrt{2}}} + \frac{-q}{\frac{l}{\sqrt{2}}} + \frac{2q}{\frac{l}{\sqrt{2}}} + \frac{2Q}{\frac{l}{\sqrt{2}}} \right) \]
- Simplifying this, we get:
\[ V_{total} = k \frac{\sqrt{2}}{l} \left( -Q - q + 2q + 2Q \right) \]
\[ V_{total} = k \frac{\sqrt{2}}{l} \left( Q + q \right) \]

4. **Set the Total Potential to Zero**:
- For the potential at the center to be zero:
\[ V_{total} = 0 \Rightarrow Q + q = 0 \]
- This implies:
\[ Q = -q \]

### Conclusion:
The relation between \( Q \) and \( q \) for which the potential at the center of the square is zero is:
\[ Q = -q \]