JEE Main · 2004 · 24 Apr 2004 · Offline · Electrostatics

For the arrangement to balance, the net force on each corner charge must vanish. A corner charge $-Q$ is pushed by the other three corner charges and pulled by the central charge $q$ a distance $\frac{a}{\sqrt{2}}$ away. Summing the three corner contributions and setting the total force to zero fixes the central charge at $q=+\frac{Q}{4}\left(1+2\sqrt{2}\right)$, positive because it must attract the negative corners to hold them in place.