$(a)$ 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.

(N/A) For the equilibrium of a test charge to be stable,the charge must experience a restoring force towards the null point when displaced in any direction. This implies that the electric field lines in the vicinity of the null point must point inwards towards the point. If this were true,there would be a net inward flux of the electric field through a small closed surface surrounding the null point. According to Gauss's law,$\oint E \cdot dA = q_{enclosed} / \epsilon_0$. Since the null point contains no charge $(q_{enclosed} = 0)$,the net flux must be zero. Therefore,it is impossible for the field lines to point inwards in all directions,and the equilibrium is necessarily unstable.
$(b)$ Consider two identical positive charges placed at a distance $2a$ apart. The midpoint is the null point. If a test charge is displaced along the line joining the charges,it experiences a restoring force. However,if it is displaced perpendicular to this line,the components of the electric force from the two charges add up to push the test charge further away from the null point. Since the equilibrium is not stable in all directions,it is unstable.