let q be the charge placed at vertices of square of side 'd' meters each what value of charge should be placed at the cetre of a square to get equilibrium condition at any one corner of a square
To determine the value of the charge that should be placed at the center of a square to achieve equilibrium at any one corner, we need to consider the forces acting on the charges.
Let's assume the charge at each corner of the square is q, and we want the equilibrium condition at one corner, labeled C.
To achieve equilibrium, the net electrostatic force acting on any charge at the corner C must be zero. This means that the sum of the forces from the charges at the other corners and the center of the square must be zero.
Let's consider the forces acting on the charge at corner C:
1. Forces from the other charges at the corners: The distance between the charges at the corners and the charge at C is d (side length of the square). Since all the corner charges are identical, the forces exerted on C by the three other corner charges are equal in magnitude but opposite in direction. Thus, the net force from the corner charges on C is zero.
2. Force from the charge at the center: The distance between the center charge and the charge at C is √2 times the side length of the square (d√2). The electrostatic force between two charges is given by Coulomb's Law, which states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
Therefore, the force exerted on charge C by the charge at the center is given by:
Force = (k * q * q) / (d√2)^2,
where k is the electrostatic constant.
To achieve equilibrium, this force must be zero. Since k and d are constants, we can set the force equal to zero and solve for q:
(k * q * q) / (d√2)^2 = 0.
Since the product of k and q^2 is zero, this implies that either k = 0 (which is not the case) or q = 0.
Hence, for the equilibrium condition at any one corner of the square, the charge placed at the center must be zero, i.e., no charge should be placed at the center of the square.