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I'm trying to find a way to construct a bigger magnet from smaller ones. It seems like there is a way to be able to do it at least as far as I've read. I'm having trouble understanding how that would work if it is possible though. Say I have two square magnets and I want to make a rectangle magnet, if you put them side by side so that North is touching North and South is touching South. This would create a magnetic field with a split in the middle of the magnet wouldn't it? This would produce a different magnetic field than just making a new larger magnet, right?

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Basically, you can add the separate magnetic fields of both magnets together. Far away from the magnets, that will look the same as a single large magnet, because the origins (the locations of the magnets) are so close together. Close to the magnets, things are indeed different.

If you model the magnet as a magnetic dipole, you could use a formula like this one and do some numerical calculations by adding two identical fields shifted by a distance $\Delta r$.

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Whenever you bring two magnets in proximity applying force this means, that the energy you apply is stored in the magnetic field of the magnet: More field lines are pushed to the outside and the magnet becomes stronger. As the size of the magnet grows, the force density may grow to an smaller amount.

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There is a good post about this from K&J Magnetics, Doubled Forces. This post shows that combining magnets can have a big effect on their pulling power but there is also a point where adding other magnets doesn't provide that much more useful force.

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