(a) Consider an arbitary 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 equillibrium of the test charge is necessarlly unsutable.
(b) Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart.

Explanation

(a) To prove this, suppose the equilibrium of test charge is stabel. When the test charge is displaced from the null point in any direction, it must experience a restoring force towards the null point. This indicates that there is net inward flux of electric field through a closed surface around the null point. But, according to Gauss's law, the electric flux through a surface not enclosing any charge must be zero. Hence the equllibrium cannot be stable.
(b) The middle point of the line joining two like charges is a null point. When we displace a test charge slightly along the line, a restoring force tends to bring the test charge back to the center. But when we displace the test charge normal to the line, the net force on the test charge takes it further it further away from the null point. Hence the equilibrium cannot be stable.