Abstract
Earnshaw’s Theorem, preventing stable levitation using permanent magnets, and its circumvention with diamagnetic materials is introduced in a qualitative manner. A simple home-built levitation set-up using graphite is described.
Keywords
diamagnetism, bismuth, graphite, Earnshaw’s Theorem
Introduction
It is always intriguing to see objects floating in midair without any visible means of support. There are numerous magicians who can cause objects, including members of the audience, to "levitate", and many myths and legends (Icarus and flying carpets, to name but two) deal with flying or floating people. Having many times tried to place one magnet in just the right place to float on top of another (without success), I was very interested to find on the Internet1, 2 details of how to cause a magnet to float stably without any electricity or liquid air. I therefore set out to try and replicate the experiment myself.
In fact, it is theoretically impossible to combine any static arrangement of gravitational and magnetic fields to cause a magnet to float stably in one place3 (Earnshaw’s Theorem). The problem lies in the well-known inverse square law that relates the strength of a field to the distance from its source. Think of a very simple arrangement of a magnet being attracted to a magnet directly above it and pulled down by the Earth’s gravity. The net force on it must be zero for it to stay there; otherwise it would accelerate either up or down. However, if it moves up by a tiny fraction of a centimetre, the force on it will become unbalanced, since it has moved slightly closer to the magnet. This net upward force will make it accelerate upwards, with a larger and larger force acting on it, until it crashes into the magnet itself. Similarly, if it moves down slightly it will have a smaller upward force acting on it, and will continue to fall. It is therefore in an unstable equilibrium, and in practice it is impossible to make a magnet hover in this way.
It is possible to get around this problem, though, by positioning a plate of a diamagnetic material either side of the floating magnet. Diamagnetism is a phenomenon that affects all materials, although normally it is masked by the attractive effects of ferromagnetism (being strongly attracted to a magnet, like iron) and paramagnetism (being weakly attracted to a magnet); it is simply that the material repels all magnetic poles, either north or south. It is strongest in superconductors, which is why they can float above magnets; however, even in bismuth, the element for which the effect is strongest, the repulsion is only about 0.0019% of the magnetic field strength4. While not enough to float a magnet by itself, this is sufficient to stabilise the equilibrium position in the former arrangement of magnets, and thus allow a magnet to float.
Method
The first approach used two plates made from bismuth, a dense silvery metal, very much like lead except that it is non-toxic; it is also the heaviest element with no radioisotopes, consisting entirely of the non-radioactive 209Bi. This explains its use in non-toxic shotgun shot, from where I obtained it; it is also used in medicines. Since its melting point is only 271°C, I was able to cast it easily into two plates which were then machined smooth on the lathe.
The top biasing magnet was a large ferrite ring magnet obtained from a broken magnetron. I tried a variety of small but very powerful NdFeB magnets from earphone speakers, brushless DC motors and head moving mechanisms from computer hard disk drives for the levitated magnet. This approach was unsuccessful; evidently the 3% tin present in "bismuth" shot5 reduces its diamagnetic properties substantially.
Therefore I tried using two graphite plates, sawn off a large lump purchased as a mineral specimen, as the diamagnets. Another change was the use of a set-up which allowed me to adjust the distance between the plates continuously with a lab jack, instead of the multiples of shims I had been using before. After experimentation using the same magnets as before, I was able to float a 5mm wide by 1mm thick NdFeB ring magnet between the two plates, as can be seen in the photographs.
Encouraged by this, I built a wooden stand to mount the magnets and plates on out of 18mm MDF. The biasing magnet is suspended by its own attraction from a washer attached to a piece of threaded rod, which allows its position to be varied continuously. One graphite plate rests on the base of the apparatus, while the other is supported on a copper plate resting on two non-magnetic nuts on non-magnetic bolts, thus allowing its height (and, if necessary, angle) to be varied continuously. The copper plate serves to damp oscillations, to which this apparatus is very prone due to the pendulum nature of the biasing magnet. This seems to be satisfactory, and also allows levitation using only the lower plate, though the height at which the magnet floats is lesser.
Pictures
A close-up of the two plates, showing the magnet between them floating just above the lower one.
Acknowledgements
I am extremely grateful to Winchester College, specifically Dr C S McCaw for providing the graphite, Mr A C Sinclair and Mr G J Penney for helping me to construct the levitation stand; also Dr Cooper for the loan of the digital camera.
References
