Earnshaw's Theorem
A charged particle can't be held in a stable equilibrium by electrostatic forces alone.
This is because in free space the Electric Potential satisfies Laplace's equation. And the extrema (local minima, maxima) of Laplace's equation lie only at the boundary.
This is a consequence of electric field having zero divergence in free space and also having zero curl. Thus a stable equlibria which requires forces to point to same direction (a sink), or the forces to spiral towards a point is not possible in free space.
