Hi Wu Fan and Qihan,
That is a very good question, because it addresses a feature of magnetic forces which normally does not arise in our experience with small magnets.
The force on an object is related to the change in the energy of a system (not including the kinetic or thermal energy of the object) when the object is moved. We write
F = (change in Energy)/(change in position)
For static fields. The change in position has a direction, and so the force does too (you need some vector algebra with a dot product to express this exactly).
Two small magnets placed together with like poles close to each other feel a repulsive force because of the energy stored in the magnetic field. The energy density in space is proportional to the magnetic field squared, and when the close-by poles are the same, their fields add in more places than they subtract, and so the total energy is higher for this case than when opposite poles are closer, where the field is smaller in more places.
There are two things about the Earthís magnetic field which makes this effect much smaller. For one, the field is very weak at the surface (about a gauss or less). The more important reason is that because the field extends over such a large space and because we on the surface are far away from the center of the Earthís dipole, the Earthís magnetic field strength is very uniform if you look at it over a region of space that is reasonable in size (like the size of the magnet you propose to use).
If you put these two pieces together, you find that the force on a magnet due to the Earthís field is very small -- if you move the magnet from one place to another, its field adds to the Earthís field in almost the same way because the Earthís field is very little different from one place to another, and the total magnetic energy changes by a very very tiny amount. In fact, the total magnetic force on a magnet in a uniform magnetic field is exactly zero, and the forces we normally associate with magnets repelling or attracting are proportional to the rate of change of the field strength with position.
This isnít the end of the story, however, because the magnetic energy of the system depends on which way the magnet is pointing, relative to the Earthís field. If it points along the field, the fields add, for a higher energy. If it points the other way, the fields subtract, for a lower energy, and so the magnet prefers to turn to point in this way. Magnets in uniform fields feel torques which make them turn around if they are not pointing in the right direction, but there is no net force making the magnet want to levitate.
That having been said, if you had a really really big magnet, whose field extended over such a large region that the Earthís field changes noticeably over that region (you might need another Earth-sized bar magnet), then yes, a noticeable force can be produced.
As for actual levitation, that can only happen with materials whose magnetic moment actually points the wrong way, increasing the energy in a magnetic field. These are called diamagnets. Diamagnetism is purely a quantum mechanical effect, with no classical explanation. By far the most intense diamagnets are superconductors. You may have seen superconductors levitating over magnets, or vice-versa. The Earthís magnetic field does not change rapidly enough from place to place to levitate even a superconductor.
Tom (w Mike)
(published on 10/22/2007)