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Sir Andre Geim is famous for two things: 1) winning the Nobel Prize in 2010 for his work on graphene and 2) winning the Ig Noble Prize in 2000 for levitating a live frog in a magnetic field.1 The video above shows the frog levitating in an ultra high magnetic field produced by a Bitter electromagnet.

It's hard to levitate a frog without special equipment, but there are other objects you can find at home or in the classsroom that are easy to levitate.

how does a frog levitate?

It turns out that many materials are 'diamagnetic', meaning that they are repelled by a magnet (in contrast to 'magnetic' materials, which are attracted by a magnet). Although diamagnetism is fairly common, it is a very weak effect, much weaker than the attractive forces of magnetic materials such as iron. For that reason, diamagnetism is rarely ever noticed.

One of the most common diamagnetic materials is water. It's diamagnetism is extremely weak, but it can be observed in very low friction environments. The wonderful video below by Nurdrage shows the repulsive force between water and a strong neodymium magnet.

Since biological materials contain water, they are subject to diamagnetic repulsion. In 1997, Andre Geim and collaborators demonstrated diamagnetic levitation of a frog (which has a particularly high water content) using the repulsive force from a very strong magnet to counteract the force of gravity.

In order to generate a strong enough repulsive force to balance the weight of the frog, Geim's group had to use the extremely high magnetic field generated by a Bitter electromagnet at the Nijmegen High Field Magnet Laboratory in the Netherlands. A Bitter electromagnet, invented by Francis Bitter in 1933, uses conducting plates (instead of the more commonly used conducting wire) to carry the high electric currents and withstand the tremendous heating and stresses associated with high fields. The magnet used in the frog video above generated a magetic field of roughly 16 Teslas (about 10,000 times stronger than the surface field of a typical refrigerator magnet). At that value, the diamagnetic repulsion is just enough to balance the weight of the frog.

the microscopic origin of diamagnetism (notes for geeks)

All materials exhibit diamagnetic tendencies to some extent, which result from the motion of electrons inside atoms. When the diamagnetic material is placed in an external magnetic field, the magnetic forces on the moving electrons cause the atoms to precess, or rotate, around the direction of the external magnetic field. These forces are called Lorentz forces, and the precession is called Larmor precession, named after Joseph Larmor. The Larmor precession of the atom is similar to the precession of a spinning top. A top will wobble around a vertical axis shortly before it falls over, rotating around the direction of the (vertical) gravitational field.

The Larmor precession of an atom moves the atomic electron clouds in circles around the magnetic field, thereby giving rise to electric currents. These currents in turn produce a magnetic field2, which is opposite the direction of the applied magnetic field. In essence, Larmor precession turns the atom into a microscopic electromagnet that is aligned opposite to the external magnetic field. As a result, the atom is repelled from the source of the external field. (For more details on diamagnetism, Larmor precession and associated magnetic fields, try the excellent article3 by Hanno Essén. It assumes an undergraduate background in phyiscs.)

For atoms that have no other significant source of magnetism, the diamagnetic effect is visible as a magnetic repulsion. Generally, this is only true for atoms that have filled electron shells, for which all intrinsic magnetic fields of the electrons cancel.4 For atoms that have unpaired electrons, the intrinsic magnetic field associated with the unpaired electron(s) overwhelms the diamagnetic effects of Larmor precession. These materials are 'paramagnetic' or 'ferromagnetic', rather than diamagnetic. For a more detailed explanation of paramagnetic and ferromagnetic behavior, please see our lesson on Iron and Magnets.

Table 1: Volume magnetic susceptibility for some common diamagnetic materials

material	susceptibility
Pyrolytic Carbon	-40.9
Bismuth	-16.6
Mercury	-2.9
Silver	-2.6
Diamond	-2.1
Lead	-1.8
Graphite	-1.6
Copper	-1.0
Water	-0.9

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Fig. 1: Pyrolitic graphite levitating above an array of neodymium magnets. The magnets are arranged in a checkerboard pattern, with the north poles pointing up for two magnets kitty-corner from each other, and the south poles pointing up for the other two kitty-corner magnets.

levitation at home

It's nearly impossible to produce high enough field strength at home to levitate water (unless you have a really amazing shop in your garage). However, a few other materials are more strongly diamagnetic than water and can be used to perform levitation at home. Table 1 shows the magnetic susceptibility (a measure of the strength of the material's magnetic field) for a number of common diamagnetic materials. The susceptibilities of diamagnetic materials are all negative, indicating they generate a magnetic field opposite the direction of the external field.

Graphite (pencil lead) is one of the most common diamagnetic solids. With strong enough magnets, it is possible to diamagnetically repel graphite against the force of gravity, so that the graphite levitates. The video below gives a demonstration of the diamagnetic effects of pencil leads, including levitation over an array of neodymium magnets. In the video, the magnets are arranged in a checkerboard pattern: the red magnets have their north poles pointing up and the silver magnets have their south poles pointing up. If only one magnet were used, the pencil lead would be repelled, but balance would be unstable. The pencil lead would quickly slide off to one side. By using a checkerboard pattern, we create a stable arrangement of magnetic field. The field is strongest in the middle of each face of a magnet. The field is weakest at the edges where two magnets are touching; at that location, the fields from the two magnets partially cancel.5 This arrangement creates a region of weak field at the boundaries between magnets, and the pencil lead settles into this region of weak field, flanked by regions of stronger field that keep it in place.

When setting the pencil lead on top of the magnets, make sure you lay the pencil lead along the join line between magnets. If you push it too far to one side or the other, it may fall off.

A slightly stronger effect can be seen (figure 1) with pyrolytic carbon (sometimes called pyrolytic graphite), which is available from some specialty shops. The folks at http://sci-toys.com have written up a detailed description of levitation techniques with pyrolitic carbon at http://sci-toys.com/scitoys/scitoys/magnets/pyrolytic_graphite.html. They also sell pyrolytic carbon in their store. Other suppliers are provided in the equipment list at the top of this page.

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Video showing several tricks with diamagnetic pencil leads, including levitation above an array of neodymium magnets.

other activites with diamagnets

spin a pencil lead on a watch glass
Place a watch glass, or some other dome shaped object, upside down on a table. Balance a pencil lead across the top so that it is free to rotate in any direction. Now bring a strong magnet up to one end of the pencil lead and watch it spin away. 

chase a pencil lead across a mirror
Lay a mirror or sheet of glass on a table and place a pencil lead on top. Bring a strong magnet near the pencil lead and watch it roll away. Have a race to see who can chase a pencil lead across the finish line first.

levitate a magnet using diamagnets
Another levitation trick involves using diamagnets to stabilize the attractive pull between two strong magnets. In principle, one could levitate a small magnet by holding a large magnet above it at precisely the right distance so that the attractive magnetic force exactly cancels the pull of gravity. Unfortunately, the arrangement is not stable. If the small magnet moves downward even a tiny amount, the magnetic attraction decreases and gravity takes over, pulling the magnet farther down. The magnet continues to accelerate downward until it eventually hits the ground. If the small magnet moves up, the magnetic attraction overwhelms gravity and pulls the magnet further upwards until it hits the large magnet.

The system can be stabilized using pieces of diamagnetic material above and below the small magnet. When the magnet strays too far in one direction or the other, it moves close to the diamagnet, which repels it back in the direction it came from. The folks at scitoys have written detailed instructions for constructing the levitation device using homemade Bismuth disks. A video of a similar device is available at https://www.youtube.com/watch?v=jOaBnJpIRzM. A technical paper6 on the subject was written by (you guessed it) Andre Geim etal.

questions to ponder

• Why doesn't it matter which way the magnet is pointed?
  The atoms in a diamagnet generate electric currents that always oppose an external magnetic field. In an external magnetic field, each atom becomes a microscopic electromagnet. If you hold the north pole of a permanent magnet near a diamagnetic material, the atoms in the material will align with their north poles facing the permanent magnet, thereby repelling from the permanent magnet. If you hold the south pole of a permanent magnet near a diamagnetic material, the atoms in the material will align with their south poles facing the permanent magnet. If you lay the diamagnetic material down on an array of north and south magnetic poles, the portions of the diamagnet above the north poles will form micro magnets with north poles downward, and the portions above the south poles will form micro magnets with their south poles downwards. No matter which type of magnet pole is nearby, the diamagnet forms a magnet to repel it.
   
• Can I levitate a magnet above a diamagnet?
  Trying to levitate a single magnet above a single diamagnet won't work because the configuration is not stable. The magnet would slide off to one side. (See the discussion above about using a checkerboard array of magnets.) In principle, we might be able to levitate a checkerboard array of magnets above a square mesh of diamagnets that line up with the join lines of the checkerboard array. However, in practice the magnetic array will be too heavy for the magnetic repulsion to support its weight. The best way to levitate a magnet is to use another  strong magnet to provide additional lift. This idea is behind the activity "levitate a magnet using diamagnets" above.
   
• Are there other directions on a checkerboard array of magnets that are stable?
  Yes. You can experiment to find them. Lying the pencil lead at an angle that crosses many intersections of four magnets may be stable even when the lead is not always on a join line. Try it out.
   
• Why does pyrolitic graphite levitate better than pencil graphite?
  A graphite crystal is more diamagnetic in some directions than in others. A pure graphite crystal forms in layers of hexagonal (honeycomb) arrays. The crystal is most strongly diamagnetic in the direction perpendicular to the layers. In pencil lead, microscopic crystals of graphite are randomly oriented. The observed diamagnetism is therefore an average of the diamagnetism available in different directions. In pyrolytic graphite, the graphite crystals form macroscopic layered sheets, all aligned in (more or less) the same direction. In this case, the diamagnetism observed perpendicular to the sheets is substantially higher than average. It may levitate as much as three times higher than pencil lead.

• 1. M V Berry and A K Geim. "Of flying frogs and levitrons." European Journal of Physics 18.4 (1997): 307. http://iopscience.iop.org/0143-0807/18/4/012/
• 2. According to Faraday's law, an electric current produces a magnetic field. See our lesson on Basic Electromagnets for examples.
• 3. Hanno Essén. "Magnetic Fields, Rotating Atoms and the Origin of Diamagnetism." Physica Scripta 40 (1989): 761-767.
• 4. For atoms with unfilled shells, the magnetic fields due to the intrinsic quantum spin of the electrons overwhelm any diamagnetic effect. For atoms with filled shells, the intrinsic magnetic fields of the electrons tend to cancel each other in pairs, with the magnetic field of one electron pointing opposite to the magnetic field of its partner.
• 5. Strictly, the vertical components of the fields cancel, while the horizontal components remain.
• 6. A. K. Geim, M. D. Simon, M. I. Boamfa, L. O. Heflinger. "Magnetic levitation at your fingertips." Nature 400.6742 (1999): 323.