Explanation
Informally, the case of a point charge in an arbitrary static electric field is a simple consequence of Gauss's law. For a particle to be in a stable equilibrium, small perturbations ("pushes") on the particle in any direction should not break the equilibrium; the particle should "fall back" to its previous position. This means that the force field lines around the particle's equilibrium position should all point inwards, towards that position. If all of the surrounding field lines point towards the equilibrium point, then the divergence of the field at that point must be negative (i.e. that point acts as a sink). However, Gauss's Law says that the divergence of any possible electric force field is zero in free space. In mathematical notation, an electrical force deriving from a potential will always be divergenceless (satisfy Laplace's equation):
Therefore, there are no local minima or maxima of the field potential in free space, only saddle points. A stable equilibrium of the particle cannot exist and there must be an instability in at least one direction.
To be completely rigorous, strictly speaking, the existence of a stable point does not require that all neighboring force vectors point exactly toward the stable point; the force vectors could spiral in towards the stable point, for example. One method for dealing with this invokes the fact that, in addition to the divergence, the curl of any electric field in free space is also zero (in the absence of any magnetic currents).
This theorem also states that there is no possible static configuration of ferromagnets which can stably levitate an object against gravity, even when the magnetic forces are stronger than the gravitational forces.
Earnshaw's theorem has even been proven for the general case of extended bodies, and this is so even if they are flexible and conducting, provided they are not diamagnetic, as diamagnetism constitutes a (small) repulsive force, but no attraction.
There are, however, several exceptions to the rule's assumptions which allow magnetic levitation.
Read more about this topic: Earnshaw's Theorem
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