Can Electric Forces Alone Give Stable Equilibrium? In Chapter 21, several examples were given of calculating the force exerted on a point charge by other point charges in its surroundings.
(a) Consider a positive point charge +q. Give an example of how you would place two other point charges of your choosing so that the net force on charge +q will be zero.
(b) If the net force on charge +q is zero, then that charge is in equilibrium. The equilibrium will be stable if, when the charge +q is displaced slightly in any direction from its position of equilibrium, the net force on the charge pushes it back toward the equilibrium position. For this to be the case, what must the direction of the electric field E be due to the other charges at points surrounding the equilibrium position of +q?
(c) Imagine that the charge +q is moved very faraway, and imagine a small Gaussian surface centered on the position where +q was in equilibrium. By applying Gauss's law to this surface, show that it is impossible to satisfy the condition for stability described in part (b) in other words; a charge +q cannot be held in stable equilibrium by electrostatic forces alone. This result is known as Earnshaw's theorem.
(d) Parts (a)-(c) referred to the equilibrium of a positive point charge +q. Prove that Earnshaw's theorem also applies to a negative point charge -q.