Question

What is Earnshaw's theorem?

Answer 1

Earnshaw's theorem says that a group of electric charges cannot be maintained only by an electric field. This is derived from the fact the the divergence of the gradient of electric potential is zero. What that means is that for electric potential, there are solutions to it where there are extrema. Yet on top of this, the above does not allow for minima or maxima, so the only solutions left for the above are saddle points, which are not stable forms for electric charges to be settled.

In other words, the first line (lol). In truth, Earnshaw's theorem, to be fully appreciated, needs an electrodynamics background.