Earnshaw's theorem states that a collection of point charges cannot be maintained in an equilibrium configuration solely by the electrostatic interaction of the charges. This was first stated by Samuel Earnshaw in 1842. It is usually referenced to magnetic fields, but originally applied to electrostatic fields, and, in fact, applies to any classical inverse-square law force or combination of forces (such as magnetic, electric, and gravitational fields).
This follows from Gauss's law. The force acting on an object F(x) (as a function of position) due to a combination of inverse-square law forces (forces deriving from a potential which satisfies Laplace's equation) will always be divergenceless (
·F = 0). What this means is that if the electric (or magnetic, or gravitational) field points inwards towards some point, it will always also point outwards. There are no local minima or maxima of the field in free space, only saddle points.
This theorem 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. There are, however, several exceptions to the rule's assumptions which allow magnetic levitation.
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