Editing Earnshaw's theorem (section)

=== In electrostatics ===
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 inward, toward that position. If all of the surrounding field lines point toward 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 {{math|'''F'''('''r''')}} deriving from a potential {{math|''U''('''r''')}} will always be divergenceless (satisfy [[Laplace's equation]]):

<math display="block">\nabla \cdot \mathbf{F} = \nabla \cdot (-\nabla U) = -\nabla^2 U = 0.</math>

Therefore, there are no local [[minimum|minima]] or [[maximum|maxima]] of the field potential in free space, only [[saddle point]]s. A stable equilibrium of the particle cannot exist and there must be an instability in some direction. This argument may not be sufficient if all the second derivatives of ''U'' are null''.''<ref>{{cite journal |last=Weinstock |first=Robert |date=1976 |title=On a fallacious proof of Earnshaw's theorem |journal=American Journal of Physics |volume=44 |issue=4 |pages=392–393 |doi=10.1119/1.10449|bibcode=1976AmJPh..44..392W }}</ref>

To be completely rigorous, strictly speaking, the existence of a stable point does not require that all neighbouring force vectors point exactly toward the stable point; the force vectors could spiral in toward the stable point, for example. One method for dealing with this invokes the fact that, in addition to the divergence, the [[curl (mathematics)|curl]] of any electric field in free space is also zero (in the absence of any magnetic currents).