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Concept
Earnshaw's theorem
Summary
Earnshaw's theorem states that a collection of point charges cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic interaction of the charges. This was first proven by British mathematician Samuel Earnshaw in 1842.
It is usually cited in reference to magnetic fields, but was first applied to electrostatic fields.
Earnshaw's theorem applies to classical inverse-square law forces (electric and gravitational) and also to the magnetic forces of permanent magnets, if the magnets are hard (the magnets do not vary in strength with external fields). Earnshaw's theorem forbids magnetic levitation in many common situations.
If the materials are not hard, Braunbeck's extension shows that materials with relative magnetic permeability greater than one (paramagnetism) are further destabilising, but materials with a permeability less than one (diamagnetic materials) permit stable configurations.
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 F(r) deriving from a potential U(r) 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 some direction.
Official source
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