Is Magnetic Levitation Possible
(Original by Philip Gibbs and Andre Geim, 18March1997)
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.
Earnshaw's Theorem
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. 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,
/ /
 F(x).dS =  divF dV
/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, S.,
On the nature of the molecular forces which regulate the constitution of the
luminferous ether., Trans. Camb. Phil. Soc., 7, pp 97112 (1842)
Exceptions
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. 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. 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).
Earnshaw's theorem does not apply to diamagnetics as they
behave like "antimagnets": they align ANTIparallel 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,
pp753763 (1939) Brandt, Science, 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
A high temperature superconductor in
magnetic suspension
Rotation: 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 antiparallel magnetic
field and it levitates. ref: Berry, Proc Roy Soc London 452,
12071220 (1996).
A Levitron
The Real Levitation
It is truly fascinating to watch an object freely
hovering in midair and, not surprisingly, levitation has found its way into
myths, science fiction and even politics . Leaving science
fiction aside, physics does know scores of different ways to levitate things.
For instance, a helicopter can be considered as a very impressive levitation
device that uses a stream of air to keep floating. Scientists have also found
many ways to levitate things without any noise or the need for petrol or air,
by using electromagnetic fields. Levitating trains and levitating displays are
but two examples of electromagnetic levitation. However, in all such schemes, a
source of energy (an engine or a battery at least) is always required to keep
an object afloat. Remove the battery and the levitation inevitably stops.
Today's science knows only one way to achieve REAL
levitation, i.e. such that no energy input is required and the levitation can
last forever. The real levitation makes use of
diamagnetism , an
intrinsic property of many materials referring to their ability to expel a
portion, even if a minute one, of an external magnetic field. Electrons in such
materials rearrange their orbits slightly so that they expel the external
field. As a result, diamagnetic materials repel and are repelled by strong
magnetic fields.
Three basic schemes using various aspects of
diamagnetism allow the true levitation:
Superconducting
levitation
Superconductors are ideal diamagnetics and completely expel magnetic field at low temperatures. The
picture shows a superconducting pallet levi tating above a strong magnet. This
levitation does not suffer any stability problems because the magnetic flux is
pinned by defects in superconducting materials. For this fortuitous effect,
superconductors can levitate even below a magnet. Superconducting levitation is
a very well known phenomenon and not discussed further on this site.


Diamagnetic
levitation
An object does not need to be superconducting to levitate. Normal things, even humans, can do it as well, if
placed in a strong magnetic field. Although the majority of ordinary materials,
such as wood or plastic, seem to be nonmagnetic, they, too, expel a very small
portion (0.00001) of an applied magnetic field, i.e. exhibit very weak
diamagnetism. Such materials can be levitated using magnetic fields of about 10
Tesla. For several decades, this levitation possibility had been in oblivion 
even for experts in high magnetic fields  until we levitated a live frog in
1996.


Diamagnetically stabilised
levitation
Low temperatures (such that
air turns liquid) and powerful magnets (such that cooking pans are drawn from a
distance of several meters) are not what one is likely to have at home to be
able to watch the supercon ducting or diamagnetic levitation. Engineers
designing bearings in a motor or a disk drive are also unlikely to have ever
considered levitating devices (magnetic bearings) that would require such
conditions. Now, there is a way  at last  to have miniature levitating
devices that even schoolchildren can make. As the picture shows, the real
levitation is now at our fingertips.


