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 :

The system is in equilibrium means the force experienced by each charge is 0. It is clear that charge placed at centre would be in equilibrium for any value of q, so we are considering the equilibrium of charge placed at any corner.FCD+FCA cos450+FCDcos450=0⇒14πε0.(−Q)(−Q)a2+14πε0(−Q)(−Q)(2a)2×12+14πε0(−Q)q(2a/2)2×12=0⇒14πε0Q2a2+14πε0Q22a2.12−14πε02Qqa2×12=0⇒Q+Q22−2q=0⇒22Q+Q−4q=0⇒4q=(22+1)Q⇒q=(22+1)Q4