Earnshaw’s Theorem

Ѳ V = 0 in any region of space containing no charge (V= electric potential). In other words ¶²V/¶x² + ¶²V/¶y² + ¶²V/¶z² = 0. Therefore V cannot have a relative minimum anywhere in the interior of the region, because the first derivatives are zero at a minimum, but the 2^nd derivatives are all positive, so their sum cannot be zero. The consequence is that you cannot have stable equilibrium of a charge in the neighborhood of other charges unless you use other force(s), and it cannot be a constant force like gravity. This is known as Earnshaw’s theorem. Why do you need a minimum V? So that if the charge moves a little, there will be a force pushing it toward equilibrium. It is similar to a marble in a bowl- the gravitational counterpart.

The same applies to magnetism. The magnetic scalar potential f corresponds to electric potential, and Ѳf = 0, so you cannot get a magnet to hover in equilibrium. (Yes you can, just wait.) A ring magnet on a pencil will hover above another magnet, but not without the pencil. With superconductivity, this changes. A magnet will hover above a superconducting surface, because currents in the superconductor produce a magnetic field to cancel the field of the magnet (the Meisner effect) and cause a force on the magnet equal to its weight. Another way to levitate things is dynamic levitation. Provide feedback so that a magnetic field or fields adjust themselves, as in maglev trains.

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