Updated 8-September-1997 by PEG

Original by Philip Gibbs and Andre Geim, 18-March-1997

A theorem due to 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 to levitate by getting round the assumptions of the theorem. In case you are wondering, none of these can be used to generate anti-gravity or to fly a craft without wings or jets.

The proof of Earnshaw's theorem is very simple
if you understand some basic vector calculus. The
static force as a function of position
* F(x)* acting
on any body in vacuum due to gravitation,
electrostatic and magnetostatic fields will always
be divergenceless.

/ / |F(x).dS= | divFdV /S /V

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, W., "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.

**Quantum effects:** 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.

**Feedback:** 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 which are 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 (maglev) such
as the one at Birmingham airport, England. It is also
possible to buy gadgets
which levitate objects in this way.

**Diamagnetism:** It is possible to levitate
superconductors and other diamagnetic materials which
magnetise in the opposite sense to a magnetic field in
which they are placed. This is also used in maglev 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
(Meissner-Ochsenfeld effect). 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 a magnetics laboratory in the
Netherlands (Physics World, April 1997). This can only be done
using the strongest magnetic fields which technology has
achieved. The levitated objects sit inside the vertical cylindrical
core of a hollow solenoid.

*ref:* M Berry, A Geim, Eur J Phys 18, p307

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 as iron does (paramagnetics). 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)

E. H. Brandt, "Theory catches up with flying frog", Physics World, 10, p 23, Sept 1997

E. H. Brandt, Science, 243, p349, Jan 1989

**Oscillating Fields:** 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

**Rotation:** Surprisingly, it is possible to
levitate a rotating object with fixed magnets. The
*levitron* is a
commercial toy
which exploitsthe effect, invented by Roy Harrison in
1983. 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).

a levitron

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