Earnshaw’s theorem (Prob. 3.2) says that you cannot trap a charged particle in an electr...

Earnshaw’s theorem (Prob. 3.2) says that you cannot trap a charged particle in an electrostatic field. Question: Could you trap a neutral (but polarizable) atom in an electrostatic field?

(a) Show that the force on the atom is

(b) The question becomes, therefore: Is it possible for E₂ to have a local maximum (in a charge-free region)? In that case the force would push the atom back to its equilibrium position. Show that the answer is no. [Hint: Use Prob. 3.4(a).]²²

Reference prob 3.4

(a) Show that the average electric field over a spherical surface, due to charges outside the sphere, is the same as the field at the center.

(b) What is the average due to charges inside the sphere?