Of flying frogs and levitrons

Diamagnetic objects are repelled by magnetic fields. If the fields are strong enough, this repulsion can balance gravity, and objects levitated in this way can be held in stable equilibrium, apparently violating Earnshaw's theorem. In fact Earnshaw's theorem does not apply to induced magnetism, and it is possible for the total energy (gravitational + magnetic) to possess a minimum. General stability conditions are derived, and it is shown that stable zones always exist on the axis of a field with rotational symmetry, and include the inflection point of the magnitude of the field. For the field inside a solenoid, the zone is calculated in detail; if the solenoid is long, the zone is centred on the top end, and its vertical extent is about half the radius of the solenoid. The theory explains recent experiments by Geim et al, in which a variety of objects (one of which was a living frog) was levitated in a field of about 16 T. Similar ideas explain the stability of a spinning magnet above a magnetized base plate. Stable levitation of paramagnets is impossible.