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ScienceWeek
EXPERIMENTAL PHYSICS: ON DIAMAGNETIC LEVITATION
The following points are made by Ronald E. Pelrine (American Scientist 2004 92:428):
1) The ability of magnets to exert forces on one another without touching intrigues many people. It is a short step from pondering this curious phenomenon to wondering whether the force from one magnet could be used to levitate another, seemingly in defiance of gravity. Unfortunately, the answer is no: A magnetic field can be arranged so that at some position it just balances the gravitational force on a small magnet, but any disturbance to the levitated magnet, no matter how tiny, causes it to crash. This inherent lack of stability is summed up in a statement of physical law known as Earnshaw's Theorem, first elucidated in 1842. It is a direct consequence of Maxwell's fundamental equations describing electricity and magnetism.
2) But Mastery of Maxwell's equations is not needed to understand Earnshaw's Theorem. One needs merely to know that the behavior of a magnet can be described in terms of something called the "magnetic potential", which is analogous to more familiar forms of potential energy (stored energy). Consider a marble placed on an undulating surface. The marble will roll in the direction that decreases its potential energy most rapidly, becoming free of any force only where the potential attains a local minimum -- the flat bottom of a depression. Similarly, a levitated magnet would be stable only if it could be situated at a local minimum of the magnetic potential. But Maxwell's equations dictate that the magnetic potential at a point in space must be the average of the potential at surrounding positions. The magnetic potential thus cannot attain a local minimum anywhere in free space: Some nearby points will always have lower magnetic energy, while others will have higher energy.
3) Faced with the obvious implications of Earnshaw's Theorem, investigators have looked for other ways to levitate. The most common tactic is to use time-varying magnetic fields, to which Earnshaw's Theorem does not apply. Active-feedback levitation, for example, uses sensors to measure the position of a levitated object, adjusting the applied field in just the right way to keep things suspended. This approach has been used for decades in active magnetic bearings and experimental "maglev" trains. Although workable, such systems have considerable drawbacks: They consume power and are relatively complex, which means that they are expensive and can be prone to failure. But it turns out that there is a way to levitate a magnet without such complications. To understand how this feat can be carried out more simply, one needs at least a rudimentary understanding of the different types of magnetic materials.
4) Magnetic materials come in three flavors: ferromagnetic, paramagnetic and diamagnetic. Ferromagnetic materials, such as iron, can often be permanently magnetized, allowing objects made of them, for example, to stick to refrigerator doors indefinitely. Paramagnetic substances, such as the mineral biotite, become magnetized only while they are exposed to an external magnetic field. They are attracted to permanent magnets and thus do not help in the quest for stable, passive levitation. Diamagnetic substances act differently. They repel permanent magnets, and in this way make such levitations rather easy.
5) A very simple model of the atom helps explain why diamagnetic materials act in this way. Consider an electron in orbit around the nucleus of an atom of diamagnetic material. Being a charge in motion, this electron generates a magnetic field that is just like that of a tiny current-carrying loop of wire. In the absence of an external magnetic field, this orbiting electron and its many neighbors generate randomly aligned fields, which cancel one another, so the material does not generate an overall field of its own. But when subjected to an external field (say, one from an approaching permanent magnet), these electrons speed up or slow down so as to oppose the change in the field inside their orbits. (This is just the atomic-scale version of a rule of electricity and magnetism called Lenz's Law.) The net effect is an induced magnetization that opposes the applied field, causing a repulsive force.
6) One can exploit this force to levitate permanent magnets above fixed diamagnetic materials. Or one can reverse things and levitate diamagnetic materials above one or more stationary magnets. The German physicist Werner Braunbeck demonstrated such diamagnetic levitation for the first time in 1939 when he floated some strongly diamagnetic materials (bismuth and graphite) using a fixed electromagnet. The stable levitation of small permanent magnets above superconductors, a familiar sight in recent years, is just another form of diamagnetic levitation: Superconductors are not only perfectly conductive, they are highly diamagnetic.(1-5)
References (abridged):
1. Braunbeck, W. 1939. Free suspension of bodies in electric and magnetic fields. Zeitschrift für Physik 112:753-763
2. Earnshaw, S. 1842. On the nature of the molecular forces which regulate the constitution of the luminferous ether. Transactions of the Cambridge Philosophical Society 7:97-112
3. Geim, A.K., M. D. Simon, M. I. Boamfa and L. O. Heflinger. 1999. Magnet levitation at your fingertips. Nature 400:323-324
4. Pelrine, R. 1995. Magnetic Field Levitation. U.S. Patent 5,396,136
5. Pelrine, R. 1997. Room temperature, open-Loop levitation of microdevices using diamagnetic materials. In Micromechanics and MEMS: Classic and Seminal Papers to 1990, ed. W. Trimmer. New York: IEEE Press, pp. 320-323
American Scientist http://www.americanscientist.org
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Related Material:
MAGNETIC LEVITATION OF ORDINARY OBJECTS
Notes by ScienceWeek:
In this context, "lifting" is distinguished from "levitation", with levitation referring to stable floating in an applied magnetic field.
The following points are made by Andrey Geim (Physics Today September 1998):
1) All materials can be lifted by magnetic fields that are currently standard. Due to the readjustment of electron orbits in a magnetic field, all objects exhibit diamagnetism, which determines the lowest possible limit of their magnetic response. Fields of approximately 10 *tesla are sufficient to lift practically any substance.
2) Magnetic fields strong enough to lift diamagnetic materials became available during the mid 20th century, and superconductors were first levitated in 1947. It took 50 years to rediscover the levitation of conventional room-temperature diamagnetic materials. In 1991, Beaugnon and Tournier magnetically lifted water and a number of organic substances. Other researchers soon levitated liquid hydrogen, helium, and frog eggs. The author's research group at the University of Nijmegen (NL) has levitated practically everything at hand, "from pieces of cheese and pizza to living creatures including frogs and a mouse." (The article includes a photograph of a levitated live frog in the bore of a 20 T magnet, the frog reported to exhibit no adverse effects from exposure to the magnetic field.) The magnetic fields used in these experiments have been available for decades.
3) In contrast to diamagnetic substances, paramagnetic substances cannot levitate. Only diamagnetic substances can flaunt *Earnshaw's theorem, which states that no stationary object made of charges, magnets, and masses can be held in space by any fixed combination of electric, magnetic, and gravitational forces. Diamagnetism involves electron motion around nuclei, and thus is not a fixed configuration as required by the theorem.
4) A diamagnetic substance can levitate only close to an inflection point of the vertical component of the magnetic field. This is a purely geometric condition independent of the field strength.
5) The author suggests an example of the exploitation of the diamagnetic force: the direction of growth of germinating seeds, which ordinarily depends on gravity, can in the absence of gravity (e.g., in a space ship) be determined by a small permanent magnet (O.A. Kuznetsov and K.H. Hasenstein, Planta 1996 198:87).
Physics Today http://www.physicstoday.org
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Notes by ScienceWeek:
tesla: International System unit of magnetic flux density. 1 tesla = 1 weber per square centimeter.
Earnshaw's theorem: The classic statement of this theorem is that a charge cannot be held in stable equilibrium in an electric field under the influence of electric forces alone. The theorem as given in the text by Geim is a recent reformulation by Michael Berry (M.V. Berry and A.K. Geim, Eur. J. Phys. 1997 18:307)
ScienceWeek http://scienceweek.com
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