JEE Main · 2009 · 12 Apr 2009 · Offline · Electrostatics

Label the square so that the two charges $q$ sit at adjacent corners and the two charges $Q$ at the opposite diagonal corners. On one $Q$ the two neighbouring $q$ charges, each a distance $d$ away, give forces whose resultant has magnitude $\sqrt{2}\frac{kQq}{d^2}$ directed along the diagonal, while the other $Q$ across the diagonal at distance $\sqrt{2}d$ gives $\frac{k{Q}^2}{2d^2}$ along the same line. For the net force on $Q$ to vanish these must oppose and balance, which requires $q$ to be negative and leads to $\frac{Q}{q}=-2\sqrt{2}$.