Electrostatics: Equilibrium due to two point charges.

Solution Preview

Please see the attached file.

Need help with this problem: Consider a positive point charge. i) Give an example of how you might place two other point charges so that the net force on the initial charge is zero. ii) If the net force on the charge is zero then this charge is in equilibrium. The equilibrium will be stable if, when the charge is displaced slightly in any direction from its position of equilibrium, there is a force pushing it back toward the equilibrium position. Sketch the necessary electric field for this to be the case. iii) Show, using Gauss' law, that this is not possible for a distribution of surrounding charges.

Solution:
i)
Let charge +Q is under consideration. Consider two point charges q (may be either positive or both negative) placed at point A and B at a distance 2d. Let we put charge +Q at the midpoint M of AB.
The force experienced per unit test ...