Overview:
To achieve equilibrium in the system, we need to determine the charge to be placed at the center of the square. We can use the principle of superposition to calculate the net electric field at the center of the square and set it equal to zero. By solving this equation, we can find the required charge.

Given:
- Four identical charges 'q' are placed at the vertices of a square.
- Side of the square is 'a'.

Calculating the Electric Field:
To find the electric field at the center of the square, we consider the contributions from each charge individually.

Charge 1:
- Distance between charge 1 and the center of the square is 'a/√2' (diagonal of the square).
- Electric field at the center due to charge 1 is given by Coulomb's law: E1 = (kq)/(a/√2)^2

Charge 2:
- Distance between charge 2 and the center of the square is also 'a/√2'.
- Electric field at the center due to charge 2 is also given by Coulomb's law: E2 = (kq)/(a/√2)^2

Charge 3:
- Distance between charge 3 and the center of the square is also 'a/√2'.
- Electric field at the center due to charge 3 is also given by Coulomb's law: E3 = (kq)/(a/√2)^2

Charge 4:
- Distance between charge 4 and the center of the square is also 'a/√2'.
- Electric field at the center due to charge 4 is also given by Coulomb's law: E4 = (kq)/(a/√2)^2

Net Electric Field:
The net electric field at the center of the square is the vector sum of the electric fields due to each charge.

- The electric fields E1, E2, E3, and E4 have the same magnitude and direction. They form a square pattern.
- The magnitudes add up: E = E1 + E2 + E3 + E4 = 4E1
- The direction of each electric field is towards the center.

Setting Equilibrium:
To achieve equilibrium, the net electric field at the center of the square must be zero.

- Set E = 0 and solve for q:
4E1 = 0
4(kq)/(a/√2)^2 = 0
4(kq) = 0
q = 0

Conclusion:
To achieve equilibrium in the system, the charge to be placed at the center of the square should be zero (q = 0). This means there should be no charge at the center for the system to remain in equilibrium.