1.32 (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 the equilibrium of the test charge be stable. If a test charge is in
displaced from its position in any direction, then it experiences a restoring force point, where the electric field is zero. All the field lines near the null point are o towards the null point. There is a net inward flux of the electric field through an around the null point. According to Gauss's law, the flux of electric field this which is not enclosing any charge is zero. Hence, the equilibrium of the testable.
(b)
Two charges of the same magnitude and the same sign are placed at a certain dis point of the joining line of the charges is the null point. When a test charged is the line, it experiences a restoring force. If it is displaced normal to the joint net force takes it away from the null point. Hence, the charge is unstable because equilibrium requires restoring force in all directions.
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