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If a magnetic material is exposed to a magnetic field, and all of the the magnetic dipoles align in the same direction, then the bound currents, while in the inside the material, cancel each other out, on the boundary of the material, there is nothing to cancel them out, meaning we could interpret this so individual tiny loops of bound current constitute one large loop of current on the outside of the object.

So what about domains then? Are they not the same way? Could a magnetic domain, with all of it's loops of bound current, be interpreted as one large loop of current around the domain? Forming some sort of "domain current", even though this would be cancelled out in the end by all of the adjacent domains anyway?

Also, are there any other forms of current that can be found within magnetic materials? Apart from free and bound currents?

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Yes, the field produced by the alignment of atomic dipoles within a domain is the same as the one that would be produced by a current flowing around the the boundary of the domain.
As far as I know, the only other currents associate with a permanent magnet are those associated with individual atomic dipoles.

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