Earnshaw's theorem says that you cannot trap a charged particle in an electrostatic field. Could you trap a neutral but polarizable atom in an electrostatic field?
a) Show that the force on the atom is:
F=1/2a Gradient E^2
b) The question becomes: is it possible for E2 to have a local maximum in a charge free region? In that case the force would push the atoms back to its equilibrium position. Show that the answer is no.