Re: Where is the polarity of a donut shaped magnet?

The polarity of any magnet depends on how it is created. The magnetic field of a material is just the sum of the magnetic fields from all its individual atoms. Magnetized materials have been formed with a large percentage of their atoms polarized in the same direction.

You could take any chunk of polarizable material, apply a large enough external field to align the atoms, and end up with a magnet. In this case, the polarity will depend on the field you applied more than the shape of the magnet. For a uniform applied field, one end of your magnet will act as the north pole, and one end the south pole. Long exposure to the roughly- uniform field of the earth is what has magnetized "natural" magnets, i.e. lodestones.

I suspect that your question was motivated by standard bar and horseshoe magnets. Most people are familiar with the shape of the magnetic field around a bar magnet, from the classic experiment where iron filings line up along magnetic field lines. It is actually more accurate to picture those lines as closed loops. The lines you detect with the iron filings are just the part of the loop that is outside the magnet. All those outside lines continue inside the magnet, forming a kind of circuit.

To form a horseshoe magnet, you bend an already-magnetized bar, bringing the north and south poles of the magnet closer together. This concentrates the external field in the space just around the two poles. The iron filing experiment would show field lines in tight arcs from one pole face to the other. Inside, the field lines still close into loops that pass through the bent magnet. Away from the poles, the field is much weaker than it was before we bent the magnet.

If you want to really concentrate the field, you can bend the poles such that they face one another. Then nearly all of the magnetic field lines are concentrated in straight lines just between the poles, with very little "fringe" field looping out beyond the edges. This magnet configuration is often used in physics experiments to generate small volumes of uniform straight-line magnetic field. This can be used to bend a beam of charged particles, for example. The particles only experience the field when they are between the poles, but can travel in straight lines before and after because there is no field.

Now, take those two pole faces and make them touch. This is, I think, the donut configuration about which you're puzzling. From this discussion, I hope you can see that now all the magnetic field lines - loops, really - are contained entirely inside the magnet. There is essentially no outside field from the magnet anywhere, including in the "hole." By completing the "circuit," you have "shorted out" your magnetic field!

I hope this answers your question.