Problem Statement: Four charges equal to -Q are placed at the 4 corners of a square and a charge q is placed at the centre. If the system is in equilibrium, what is the value of q?

Solution:

Understanding the Problem: To solve this problem, we need to understand the concept of electric forces and the conditions for equilibrium.

Conditions for Equilibrium: The conditions for equilibrium in an electrostatic system are:

1. The net force on each charge is zero.
2. The net torque on each charged object is zero.

Calculating the Net Force:

- The charge at the center of the square experiences a repulsive force from each of the four corner charges.
- The force due to each corner charge is given by Coulomb’s law: F = k(Qq/r^2), where k is the Coulomb constant, Q is the magnitude of the corner charge, q is the magnitude of the center charge, and r is the distance between them.
- Since the corner charges are located at the corners of a square, the distance between the center and each corner is r = d/√2, where d is the length of the side of the square.
- The net force on the center charge is the vector sum of the four individual forces.

Calculating the Net Torque:

- The torque on the center charge due to each of the corner charges is given by τ = r x F, where r is the vector from the center charge to the corner charge, and F is the force on the center charge due to the corner charge.
- Since the corner charges are located at the corners of a square, the direction of the torque due to opposite corner charges cancel out.
- The net torque on the center charge is the sum of the torques due to the adjacent corner charges.

Equating the Net Force and Net Torque to Zero:

- In order for the system to be in equilibrium, the net force and net torque on the center charge must be zero.
- We can equate the net force and net torque to zero to get two equations in two unknowns (q and d).
- Solving these equations, we get the value of q.

Final Answer: The value of q is q = √2Q.

The answer is Q/4 (2√2 + 1) only.. take one corner and Mark the direction of all the resultant forces on it, considering q to be positive for maintaining stable equilibrium. now find the resultant force due to the three corners equating it with the force due to charge at centre.