Question

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

Consider an arbitrary electrostatic field configuration. A small test charge is placed at a null point (i.e., where $E=0$) of the configuration. 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.

Answer 1

1) Let us assume that the test charge is placed at the null point (i.e., where $E=0$) is in stable equilibrium. If a test charge is in stable equilibrium and displaced from its position in any direction, then it experiences a restoring force towards a null point.

Thus, all the field lines near the null point are directed inward and towards the null point and there is a net inward flux of electric field through a closed surface around the null point.

But according to Gauss's law, the net electric flux through a closed surface is equal to total charge enclosed by the surface divided by${ϵ}_0$. Here, since there is no charge within the surface the net electric flux must be zero.

This means there is no restoring force acting on our test charge that will bring it to its original position. But this is a contradiction to our assumption. Hence, the equilibrium of the test charge is necessarily unstable.

2) Now if we place two charges of the same magnitude and same sign at a certain distance apart and the null point is the mid-point of the line joining two charges. A restoring force is experienced by the charges when a test charge is displaced along the line. If it is displaced along the normal to the line joining two charges, then the net force takes it away from the null point. Hence, the charge is unstable.