It is required to hold four equal point charges each having a charge Q=87(1−22)C in equilibrium at the corners of a square. Find the point charge, in coulomb, that will do this if placed at the center of the square.

Let us consider the equilibrium of charges Q placed at 2. Then for equilibrium .F→2=0→⇒  F2xi^+F2yj^=0→⇒ F2x=0⇒ F21+F24cos⁡45+F25cos⁡45+F23cos⁡90=0 ⇒Q24πε0a2+Q24πε0(2a)212+Qq4πε0a2212+0=0⇒Q+Q22+2q=0⇒Q(22+1)+4q=0 ⇒ q=−Q4(22+1)  Since Q=87(1−22)⇒q=−1487(1−22)(1+22)⇒q=−1487(−7)=2C