(a) Consider an arbitrary electrostatic field configuration. A small test charge is placed at a null point (i.e., where E = 0) of the configuration. Show that the equilibrium of the test charge is necessarily unstable. (b) Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart.

(a) Let us consider that the equilibrium is stable. Then if the test charge is displaced in any direction, it will experience a restoring force towards the null point. This means that all the field lines near the null point will be directed towards the null point. That is, there is a net inward flux of electric field through a closed surface around the null point. But according to Gauss’s law, the flux of an electric field through a surface, not enclosing any charge, must be zero. Hence, equilibrium cannot be stable.
(b) The null point is the mid-point of the line joining the two charges. If the test charge is displaced along the line from the null point, there is a restoring force. If the test charge is displaced normally to the line, the net force takes it away from the null point. Hence, the stability of equilibrium needs restoring force in all directions.