Suppose we have a horseshoe magnet. Now we bend it in such a way that it becomes donut shaped and poles remain in contact with each other. In this situation, what will happen to the magnet? Will it still behave as a magnet? Where would be its poles? What will happen to the domains inside the magnet?
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Maxwell's equation $\nabla\cdot\vec{B}=0$ tells us that there are no magnetic monopoles (sources of magnetic "charge"). As a result, magnetic field lines must always form closed loops. Closed loops don't really have a beginning or an end ("poles"), but just a sense of direction (clockwise or counter-clockwise). The poles of a magnet just help to figure out which sense the loops emerging from the magnet posses.
Therefore, if you take a U-shaped magnet and bend it into a donut shape, you are just making another loop. Nothing happens to the domains inside the magnet, and the magnet remains a magnet.
Actually, these types of magnets are used to produce toroidal fields. They have important uses in electronics.
N. Charron
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