(Original by Philip Gibbs and Andre Geim,1997)
A theorem due to Samuel Earnshaw proves that it is not possible to achieve static levitation using any combination of fixed magnets and electric charges. Static levitation means stable suspension of an object against gravity. There are, however, a few ways of to levitate by getting round the assumptions of the theorem.
The proof of Earnshaw's theorem is simple if you understand some basic vector calculus. The static force as a function of position F(r) acting on any body in vacuum due to gravitation, electrostatic and magnetostatic fields, will always be divergenceless. divF = 0.
At a point of equilibrium the force is zero. If the equilibrium is stable, the force must point in towards the point of equilibrium on some small sphere around the point. However, by Gauss' theorem,
the integral of the radial component of the force over the surface must be equal to the integral of the divergence of the force over the volume inside which is zero. QED!
This theorem even applies to extended bodies which may even be flexible and conducting so long as they are not diamagnetic. They will always be unstable to lateral rigid displacements of the body in some direction about any position of equilibrium. You cannot get round it using any combination of fixed magnets with fixed pendulums or whatever.
ref: Earnshaw, S., On the nature of the molecular forces which regulate the constitution of the luminferous ether., Trans. Camb. Phil. Soc., 7, pp 97-112 (1842)
There are not really exceptions to any theorem but there are ways around it which violate the assumptions. Here are some of them.
Technically any body sitting on a surface is levitated a microscopic distance above it. This is due to electromagnetic intermolecular forces and is not what is really meant by the term "levitation". Because of the small distances, quantum effects are significant but Earnshaw's theorem assumes that only classical physics is relevant.
If you can detect the position of an object in space and feed it into a control system which can vary the strength of electromagnets acting on the object, it is not difficult to keep it levitated. You just have to program the system to weaken the strength of the magnet whenever the object approaches it and strengthen when it moves away. You could even do it with movable permanent magnets. These methods violate the assumption of Earnshaw's theorem that the magnets are fixed. Electromagnetic suspension is one system used in magnetic levitation trains such as the one at Birmingham airport, England. It is also possible to buy gadgets which levitate objects in this way.
It is possible to levitate superconductors and other diamagnetic materials. This is also used in magnetic levitated trains. It has become common place to see the new high temperature superconducting materials levitated in this way. A superconductor is perfectly diamagnetic which means it expels a magnetic field. Other diamagnetic materials are common place and can also be levitated in a magnetic field if it is strong enough. Water droplets and even frogs have been levitated in this way at HFML-FELIX.
A high temperature superconductor in magnetic suspension
Earnshaw's theorem does not apply to diamagnetics as they behave like "anti-magnets": they align ANTI-parallel to magnetic lines while the magnets meant in the theorem always try to align in parallel. In diamagnetics, electrons adjust their trajectories to compensate the influence of the external magnetic field and this results in an induced magnetic field which is directed in the opposite direction. It means that the induced magnetic moment is antiparallel to the external field. Superconductors are diamagnetics with the macroscopic change in trajectories (screening current at the surface). The frog is another example but the electron orbits are changed in every molecule of its body.
refs: Braunbeck, W. Free suspension of bodies in electric and magnetic fields, Zeitschrift für Physik, 112, 11, pp753-763 (1939)
Brandt, Science, Jan 1989
An oscillating magnetic field will induce an alternating current in a conductor and thus generate a levitating force. A similar effect can be achieved with a suitably cut rotating disc. The Oscillating field is a way of making a diamagnetic of a conducting body. Due to a finite resistance, the induced changes in electron trajectories disappear after a short time but you can create a permanent screening current at the surface by applying an oscillating field and conducting bodies behave just like superconducting bodies.
ref: B.V. Jayawant, "Electromagnetic Levitation and Suspension Systems", Publishers: Edward Arnold, London, 1981.
Surprisingly, it is possible to levitate a rotating object with fixed magnets. The levitron is a commercial toy which exploits the effect. The spinning top can levitate delicately above a base with a careful arrangement of magnets so long as its rotation speed and height remains within certain limits. This solution is particularly clever because it only uses permanent magnets. Ceramic materials are used to prevent induced currents which would dissipate the rotational energy.
Actually, the levitron can also be considered as a sort of diamagnetic. By rotation, you stabilise the direction of the magnetic moment in space (magnetic gyroscope). Then you place this magnet with the fixed magnetisation (in contrast to the "fixed magnet") in an anti-parallel magnetic field and it levitates.
ref: Berry, Proc Roy Soc London 452, 1207-1220 (1996).