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Imagine two magnets on a horizontal surface, with two opposite poles facing each other. If one of them is north, then the other magnet with the south pole facing the north will move towards the north pole. How does the object move? And why doesn't the south pole of the mother magnet pull the north pole of the other magnet?

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I don't quite understand your last sentence. In the case you're describing, the two magnets attract each other and therefore accelerate towards each other. Of course the poles of a macroscopic magnet influence each other on a molecular level, but they arrange themselves in a stable configuration which results in the macroscopic north and south poles of the magnet. Also the sides that are not facing each other influence the whole process, but they only contribute in diminishing the net force that is pulling the magnets together.

If you want to understand this better you should first look at how electric point charges interact with each other and then how this behavior translates to electric/magnetic dipoles.

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