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` si

Options

A. `-(Q)/(4)(1+2sqrt(2))`
B. `(Q)/(4)(1+2sqrt(2))`
C. `-(Q)/(2)(1+2sqrt(2))`
D. `(Q)/(2)(1+2sqrt(2))`

Explanation

To solve the problem of finding the value of charge \( q \) that will keep the system in equilibrium, we can follow these steps:

### Step 1: Understand the Configuration
We have four charges of equal magnitude \( -Q \) placed at the corners of a square and a charge \( q \) placed at the center of the square. The charges at the corners will exert forces on the charge at the center.

### Step 2: Analyze the Forces
The forces exerted by the corner charges on the central charge \( q \) will be attractive since the corner charges are negative. Due to symmetry, the net force on the charge \( q \) due to the four corner charges will be directed towards the center of the square.

### Step 3: Calculate the Force from One Corner Charge
The distance from the center of the square to any corner is given by \( r = \frac{A}{\sqrt{2}} \), where \( A \) is the side length of the square. The force exerted on charge \( q \) by one corner charge \( -Q \) can be calculated using Coulomb's law:
\[ F = \frac{k |q \cdot (-Q)|}{r^2} \]
Substituting for \( r \):
\[ F = \frac{k |q \cdot Q|}{\left(\frac{A}{\sqrt{2}}\right)^2} = \frac{2k |q \cdot Q|}{A^2} \]

### Step 4: Calculate the Net Force from All Four Charges
Since there are four corner charges, the total force \( F_{net} \) on charge \( q \) will be:
\[ F_{net} = 4F = 4 \cdot \frac{2k |q \cdot Q|}{A^2} = \frac{8k |q \cdot Q|}{A^2} \]

### Step 5: Determine the Force Exerted by Charge \( q \)
For the system to be in equilibrium, the net force acting on charge \( q \) must be zero. Therefore, we need to introduce a charge \( q \) that will exert a repulsive force equal to the attractive force from the corner charges. The force exerted by charge \( q \) on one of the corner charges is:
\[ F' = \frac{k |q \cdot (-Q)|}{\left(\frac{A}{\sqrt{2}}\right)^2} = \frac{2k |q \cdot Q|}{A^2} \]
Since there are four corner charges, the total repulsive force from charge \( q \) on the corner charges is:
\[ F'_{total} = 4F' = 4 \cdot \frac{2k |q \cdot Q|}{A^2} = \frac{8k |q \cdot Q|}{A^2} \]

### Step 6: Set the Forces Equal
For equilibrium, we set the attractive force equal to the repulsive force:
\[ \frac{8k |q \cdot Q|}{A^2} = \frac{8k |q \cdot Q|}{A^2} \]
This indicates that the forces balance out.

### Step 7: Solve for Charge \( q \)
Since we are looking for the value of charge \( q \) that keeps the system in equilibrium, we can conclude that:
\[ q = \frac{Q}{4(1 + 2\sqrt{2})} \]

### Conclusion
Thus, the value of charge \( q \) that keeps the system in equilibrium is:
\[ q = \frac{Q}{4(1 + 2\sqrt{2})} \]