(a) Consider an arbitrary electrostatic field configuration. A small test charge is placed at a null point (⇒, 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) At a null point, the net electrostatic field is zero. If a small test charge is placed at the null point, it experiences no electrostatic force. But if it is displaced from the null point, it tends to move in the direction which is tangential to the field lines ⇒, towards the negative charge and away from the positive charge. So, if the test charge is displaced from the equilibrium position at the null point, it tends to move away from the equilibrium position. Hence, a null point is necessarily unstable.
(b) Let us consider the case of two charges of equal magnitude and opposite sign. The null point occurs at the midpoint of the line joining the two charges. If the test charge is moved away from this null point, it experiences a force towards the negative charge and away from the positive charge. This results in the movement of the charge away from the null point. Hence, the null point is an unstable equilibrium position.
NOTE: A test charge is a positive charge of unit magnitude. A null point or neutral point is a point where magnitude of electric field is zero. A pair of charges of equal magnitude and opposite sign is known as a dipole.
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