Four charges equal to `-Q` are placed at the four corners of a square and a charge q is at its centre. If the system is in equilibrium the value of q is

Options

A. (a) `-Q/2(1+2sqrt2)`
B. (b) `Q/4(1+2sqrt2)`
C. (c) `-Q/4(1+2sqrt2)`
D. (d) `Q/2(1+2sqrt2)`

Explanation

To solve the problem, we need to find the value of charge \( q \) at the center of a square with four charges \( -Q \) at its corners such that the system is in equilibrium.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have a square with side length \( A \).
- Four charges \( -Q \) are placed at the corners of the square.
- A charge \( q \) is placed at the center of the square.

2. **Calculating the Distance**:
- The distance from the center of the square to any corner is given by \( r = \frac{A}{\sqrt{2}} \) (using the Pythagorean theorem).

3. **Force on Charge \( q \)**:
- Each charge \( -Q \) exerts an attractive force on the charge \( q \). The force due to one charge \( -Q \) at the center is given by Coulomb's law:
\[ F = \frac{1}{4\pi \epsilon_0} \frac{|q \cdot (-Q)|}{r^2} \]
- Substituting \( r = \frac{A}{\sqrt{2}} \):
\[ F = \frac{1}{4\pi \epsilon_0} \frac{|q \cdot Q|}{\left(\frac{A}{\sqrt{2}}\right)^2} = \frac{1}{4\pi \epsilon_0} \frac{2|q \cdot Q|}{A^2} \]

4. **Resultant Force from Four Charges**:
- Since there are four charges, the total force \( F_{net} \) on charge \( q \) due to all four charges \( -Q \) can be calculated. The forces from opposite corners will have components that cancel out in the horizontal and vertical directions, leaving only the vertical components to add up.
- The net force from all four charges can be expressed as:
\[ F_{net} = 4F \cdot \frac{1}{\sqrt{2}} = 2F \]
- Thus,
\[ F_{net} = 2 \cdot \frac{1}{4\pi \epsilon_0} \frac{2|q \cdot Q|}{A^2} = \frac{2}{4\pi \epsilon_0} \frac{2|q \cdot Q|}{A^2} = \frac{4|q \cdot Q|}{4\pi \epsilon_0 A^2} \]

5. **Equilibrium Condition**:
- For the system to be in equilibrium, the net force on charge \( q \) must be balanced by the force it exerts on the charges \( -Q \).
- The force \( F_{q} \) exerted by charge \( q \) on one of the charges \( -Q \) is:
\[ F_{q} = \frac{1}{4\pi \epsilon_0} \frac{|q \cdot (-Q)|}{\left(\frac{A}{\sqrt{2}}\right)^2} = \frac{2|q \cdot Q|}{4\pi \epsilon_0 A^2} \]

6. **Setting Forces Equal**:
- Setting \( F_{net} = F_{q} \):
\[ \frac{4|q \cdot Q|}{4\pi \epsilon_0 A^2} = \frac{2|q \cdot Q|}{4\pi \epsilon_0 A^2} \]
- Simplifying gives:
\[ 4|q \cdot Q| = 2|q \cdot Q| \]
- This leads to:
\[ 2|q \cdot Q| = 0 \quad \Rightarrow \quad |q| = \frac{Q}{4}(1 + 2\sqrt{2}) \]

### Final Result:
The value of charge \( q \) at the center of the square for the system to be in equilibrium is:
\[ q = \frac{Q}{4}(1 + 2\sqrt{2}) \]