Text: Molecular Magnetism and Levitation. (The Frog Which Learned to Fly) A little frog (alive !) and a water ball levitate inside a Ø40mm vertical bore of a Bitter solenoid in a magnetic field of about 16 Tesla at the Nijmegen High Field Magnet laboratory. The image of a high-temperature superconductor levitating above a magnet in fog of liquid nitrogen can hardly surprise anyone these days - it has become common knowledge that superconductors are ideal diamagnetics and magnetic field must expel them. On the other hand, the enclosed photographs of water and a frog hovering inside a magnet (not on board are spacecraft) are somewhat counterintuitive and will probably take many people (even physicists) by surprise. This is the first observation of magnetic levitation of living organisms as well as the first images of diamagnetics levitated in a normal, room-temperature environment (if we disregard the tale about Flying Coffin of Mohammed as such evidence, of course). In fact, it is possible to levitate magnetically every material and every living creature on the earth due to the always present molecular magnetism. The molecular magnetism is very weak (millions times weaker than ferromagnetism) and usually remains unnoticed in everyday life, thereby producing the wrong impression that materials around us are mainly nonmagnetic. But they are all magnetic. It is just that magnetic fields required to levitate all these "nonmagnetic" materials have to be approximately 100 times larger than for the case of, say, superconductors. Whether an object will or will not levitate in a magnetic field B is defined by the balance between the magnetic force F = MB and gravity mg = V g where is the material density, V is the volume and g = 9.8m/s2. The magnetic moment M = (/ µ0)VB so that F = (/µ0)BVB = (/2µ0)VB2. Therefore, the vertical field gradient B2 required for levitation has to be larger than 2µ0 g/. Molecular susceptibilities are typically 10-5 for diamagnetics and 10-3 for paramagnetic materials and, since is most often a few g/cm3, their magnetic levitation requiresfield gradients ~1000 and 10 T2/m, respectively. Taking l = 10cm as a typical size of high-field magnets and B2 ~ B2/l as an estimate, we find that fields of the order of 1 and 10T are sufficient to cause levitation of para- and diamagnetics. This result should not come as a surprise because, as we know, magnetic fields of less than 0.1T can levitate a superconductor (= -1) and, from the formulas above, the magnetic force increases as B2. The water and the frog are but two examples of magnetic levitation. We have observed plenty of other materials floating in magnetic field - from simple metals (Bi and Sb), liquids (propanol, acetone and liquid nitrogen) and various polymers to everyday things such as various plants and living creatures (frogs and fish). We hope that our photographs will help many - particularly, non-physicists - to appreciate the importance of magnetism in the world around us. For instance, it is not always necessary to organize a space mission to study the effects of microgravity- some experiments, e.g. plants or crystal growth, can be performed inside a magnet instead. Importantly, the ability to levitate does not depend on the amount of material involved, V, and high-field magnets can be made to accommodate large objects, animals or even man. In the case of living organisms, no adverse effects of strong static magnetic fields are known - after all, our frog levitated in fields comparable to those used in commercial in-vivo imaging systems (currently up to 10T). The small frog looked comfortable inside the magnet and, afterwards, happily joined its fellow frogs in a biology department. A.K. Geim, J.C. Maan. The work is featured in: Physics World, April 1997, p. 28 A.K. Geim et al, Molecular Magnetism and Levitation, in Proceeding of European Low Gravity Association (ELGRA), Biannual Meeting, Paris, 17 March 1997. Is Magnetic Levitation Possible? 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 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 it towards the point of equilibrium on some small sphere around the point. However, by Gauss' theorem, | F(x).dS = | divF dV 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) 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 "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 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 anti-parallel magnetic field and it levitates. ref: Berry, Proc Roy Soc London 452, 1207-1220 (1996).

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