(a) Consider an arbitrary electrostatic field configuration. A small test charge is placed at a null point (i.e, where `vec(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.

Explanation

(a) If a charge is in stable equilibrium at a null point, then on displacing it slightly away from this point in any direction the particle will experience restoring force. For electric force to act on the test charge towards the null point, the electric lines in the region nearby the null point must be directed inwards towards this null point. This means that there will be a net flux linked with the surface enclosing the null point. For the net flux, there must be a net charge enclosed within the Gaussian surface, but there is no such charge at the null point. Hence it is not possible. So the particle can never be in the state of stable equilibrium at null point in all directions.
(b) If we consider two positive charges kept at some distance from each other, then at the midpoint of the line joining them the electric field intensity will be zero. But if you displace the test charge along the perpendicular bisector, then the positive charges will further repel the test charge away.