displayx.jpg (32358 bytes)Magnetic Toys
The physics display case in the East Bridge hallway at Caltech


The Levitated Magnet
   It has been shown [1] that it is impossible to levitate, or trap, a magnet using only simple static magnetic fields (just as it is impossible to trap an ion using static electric fields).  Two items in the display case, the Levitated Magnet and the Levitron, demonstrate some tricks to get around this fundamental result.

   The operation of the Levitated Magnet is fairly simple.   The small magnet is held up against gravity by the support magnet, which is oriented to attract the small magnet.  While the support magnet alone (without the graphite blocks) is sufficient to lift the magnet against gravity, in this case the small magnet will not be stable at it's equilibrium position.  In particular, the support is unstable in the vertical direction -- if the small magnet is perturbed upward the support magnet will attract it more strongly, and it will fly upward.  And if the small magnet is perturbed downward the support magnet will attract it less strongly, and it will fall down.  On the other hand, the support magnet alone does provide stable trapping in the horizontal directions -- if the small magnet is perturbed horizontally, the support magnet will pull it back.

levitated.jpg (15859 bytes)   The graphite blocks provide the needed stability in the vertical direction, without changing the stable trapping in the horizontal directions.   Graphite is strongly diamagnetic, so it tends to repel magnetic dipoles.   Placing graphite blocks above and below the small magnet thus can stabilize the levitated magnet.  Calculating the forces is a good problem in magnetostatics, and is left to the reader [2].

   The drive coil simply produces an oscillating magnetic field that causes the levitated magnet to swing back and forth.  Without the drive coil the display is a bit static, and it's hard to see that the small magnet is in fact freely levitated.

   It appears that the Levitated Magnet, in the configuration shown here, was invented by Martin Simon at UCLA, although many others have certainly worked on similar devices in the past.  This device works well only if the levitated magnet is a very strong rare-earth magnet.  Further information can be found at Martin Simon's home page.   A somewhat more difficult demonstration involving diamagnetic levitation is the stable trapping of a live frog, which was recently accomplished at the High Field Magnet Laboratory in Amsterdam.


The Perpetual Top
top.jpg (12725 bytes)   The point of the perpetual top is simply that it continues spinning forever, and the challenge is to understand the driving mechanism.  The top is made of plastic, and contains embedded in it a small permanent magnet, oriented perpendicular to the spin axis of the top.  The base contains a transistor and a coil with two windings, the assembly being driven by a 9-volt power supply.  A schematic of the electrical circuit is shown in the figure.
   As one pole of the magnet (say the south pole) approaches the coil, a current in induced in winding A, in such a direction as to make the base of the transistor (an NPN) go positive.  That makes the emitter-collector current flow through winding B, in the opposite direction to A.  The current through B is larger than that of A (due to the amplification of the transistor), so by Lenz's law the magnet pole will be attracted to the coil.
circuit.jpg (9198 bytes)    Note that the induction from A to B is regenerative, which causes the current pulses in B to be larger.  The circuit may be recognized as a Hartley oscillator circuit.  In this case it does not spontaneously oscillate, partly because the Q factor is quite low, and partly because the transistor is not biased to favor oscillation.
   The beauty of the system is that energy is fed into the spinning top regardless of how the top is spinning.  The south pole is pulled toward the coil as it approaches, and as it moves away from the coil there is no force, since the transistor is then nonconducting.  Alternately, the north pole feels no force as it approaches the coil, but is pushed away as it recedes.  There is no preference to clockwise or counter-clockwise rotation of the top.  Also the circuit draws no power (other than a small leakage current in the transistor) when the top is not spinning.


levitron.jpg (19660 bytes)The Amazing Levitron
   The working of the Levitron is fairly subtle, and has been the subject of a number of research papers.  Thus it is difficult to explain using just words.  The basic levitation force comes from a magnetic repulsion between the top, which is a disk magnet with its poles on the faces of the disk, and the magnetic base.  This force counters gravity, and holds the top up. 
   The top has to spin in order to keep in from flipping over.  When the top is spinning, the magnetic torque acts gyroscopically and the axis does not overturn but rotates about the (nearly vertical) direction of the magnetic field.  Explaining the horizontal stability of the Levitron top is the difficult part, for which it is best to consult the links below, or the research papers.  The basic idea is that the magnetic field from the base is not exactly vertical, but becomes tilted off-axis.   The tilt adds vectorally to the vertical field, so that the total field amplitude increases off-axis.  The top precesses around the field, and thus adiabatically follows the local field.  Because of this the energy of the dipole in the field is
   Eqn1.gif (1526 bytes)
which increases off-axis.  Thus the energy increases off-axis, and the top is stable.   If the top spins too rapidly, the dipole axis of the top will not follow the field, and the Levitron will not be stable horizontally.  Put another way, a top held strictly vertical by gyroscopic action cannot have a position of global stability.

  Another interesting aspect of the Levitron is the Oscillating Field Generator, which keeps the top spinning.  This device produces a roughly horizontal magnetic field that oscillates at about 40 Hz.  This seems to act on the spinning top like an induction motor, keeping the top spinning at about 30 Hz.  This horizontal field also produces a strong oscillating torque on the top, which causes the top to wobble quite a lot as it spins.  The wobble is less without the Oscillating Field Generator, but is still present because of precession and nutation of the top.  All the details of the wobble have yet to be explained in the physics literature.
  The copper block is a new feature we added.  It provides eddy current damping that helps stabilize the top.  The levitron and perpetuator work fine when the block is removed, but then the top rocks back and forth as it spins, and eventually (typically after about a day) the top flops out.  Eddy current damping greatly reduces this motion, and has allowed us to keep the top levitated for months at a time.

There is a lot of additional Levitron information on the web, and here are some useful links:
The Levitron Home Page -- Information on where to buy a Levitron.
A Toy Story -- An article by J. M. McBride explaining the Levitron.
Levitron Physics -- Frequently Asked Questions about the Levitron, along with answers by physicist Michael Berry.  Similar to the explanation above.
Spin-Stabilized Magnetic Levitation -- an article about the Levitron by Martin Simon at UCLA.
Levitron geklont -- Another article, this one in German.
The Hidden History of the Levitron -- The fascinating story of the invention of the Levitron (by a Vermont inventor), and how the idea was then further developed and marketed (without consent of the inventor!).  Both the original and several subsequent patents on the Levitron had the physics all wrong!
The Levitron Story -- Another tale of Levitron history.

Scientific Papers and Preprints related to the Levitron:

  • S. Earnshaw, "On the Nature of the Molecular Forces which regulate the Constitution of the Luminiferous Ether" Trans. Cambridge Philos. Soc., 7, 97-112 (1842) and subsequent explications of this paper in textbooks by Maxwell, Jeans, Page & Adams, Feynman, and others.
  • M. V. Berry "The Levitron: an adiabatic trap for spins" Proc. Roy. Soc. London, 452 1207-1220 (1996).
  • R. Edge, "Levitation using only permanent magnets" Phys. Teach., 33, 252-253 (1995).
  • M. D. Simon, et al. "Spin stabilized magnetic levitation’ Am. J. Phys., 65, 286-282 (1997).
  • T B. Jones, et al., "Simple Theory for the Levitron", J. Appl. Phys., 82, 883-888 (1997).
  • R.F. Gans, et al, "Dynamics of the Levitron", J. Phys. D. 31, 671 (1998).
  • E.H. Brandt, "Levitation in Physics" Science, 243, 349-355 (1989).

It is also interesting to note that the magnetic trapping of neutral atoms, which was an important step in the realization of Bose-Einstein condensation in laser-cooled gases, derives from some of the same principles as the Levitron.

Levitation Timeline for the display case:
September 1, 1999 -- Display case assembled; Levitron started.
October 16, 1999 -- Levitron fell during an earthquake.
November 21, 1999 -- Levitron fell during brief power outage caused by Santa Ana winds.
December 15, 1999 -- Fell; removed 100 mg weight from rotor.
January 8, 2000 -- Fell; put 100 mg weight back.  Also added a larger copper block.
January 31, 2000 -- A black widow spider spun its web around the rotor, causing a fall.
April 19, 2000 -- Levitron fell during brief power outage.
July 31, 2000 -- Fell; removed 100 mg weight from rotor
November 20, 2000 -- Fell; removed 100 mg weight from rotor
December 13, 2000 -- Fell; added 100 mg
June 27, 2001 -- Scheduled power interuption --> fell
July 13, 2001 -- Fell
August 3, 9, 16 -- More falls
October 23, November 1, 26 -- Even more falls
December 14, 2001 -- Replaced whole Levitron with new one
January 15, 2002 -- Fell, but is working much better; Clearly something about the previous unit went bad.
November 25, 2002 -- Fell; couldn't get the beast to work anymore.
January 18, 2003 -- Replaced unit with a new Levitron Omega model, which apparently uses rare earth magnets instead of the ceramic magnets used in previous versions.  This one has a much bigger sweet spot
May 17, 2003 -- Fell; removed red washer (200 mg?)
October 23, 2003 -- Fell because of brief power outage
November 1, 2003 -- Fell again
Longest continuous levitation so far: 314 days (about 10.5 months)


[1] The general idea that stable trapping using simple electrostatic forces is impossible is known as Earnshaw's theorem.  The magnetic version is sometime called Wing's theorem, from the paper "On neutral particle trapping in quasistatic electromagnetic fields," W. H. Wing, Prog. Quant. Elect. 8, 181 (1984).
[2] Some good approaches to tackling magnetostatics problems can be found in Classical Electrodynamics, by J. D. Jackson.

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