How does the geometry of magnetic bodies influence torque?

[swissgirl1999 Refer]

TL;DR

About the symmetry of magnetism, its unstable repulsion and magnetic torque, an experimental approach.

If I take two rectangular neodymium magnet like these (N35, 20 × 10 × 10 mm ),

map the faces 1 and 2, and on the body 3, 4, 5 6, and test all 64 possible combinations (switching 1 and 2 and rotating 3, 4, 5 and 6 ) , what would be the outcome (attraction-repulsion ratio) ? Why?

If in that configuration the outcome is repulsion, then rotating the magnet , changing face 1 and 2, will it continue to repel, or will it attract? Why?

Finally, if in that configuration the outcome is repulsion , then rotating the magnet through its body (3, 4,5 or 6, tilting it sideways) will keep repelling, or will it attract ? Why?

I’ve been experimenting with different shaped magnets coming to the conclusion that the intensity of torque is directly related with the three-dimensional degree of freedom, so my question is: where in electromagnetism literature could someone find the response to this tabletop experiment?

I mean, given that all the factors necessary to make those predictions (i.e. , material , shape, size of the magnets, etc.) are known, which equations and calculations are used for that?

 

[Ibix]

I think the things in the image are actually plastic cases containing magnets, rather than just magnets. Your results will depend on how the magnets are aligned compared to the case walls and whether they have freedom to move within the cases. I suspect you will need to find these things out (or extract the actual magnets) before you can get interpretable results.

Googling "calculate force between magnets" will give you some answers about the maths. Even the Wikipedia page has the basics. The formulae given there are for pointlike magnets. If that's not accurate enough for you, you'll likely need to do some integration, probably numerically.

 

[Dale]

@Ibix gave the dipole-dipole formulas. Try them out.

However, if you find that the equations do not work, that does not imply that your experiment has violated textbook electromagnetism. It just means that your magnets are not simple dipoles.

 

[Ibix]

It just means that your magnets are not simple dipoles.

Indeed. I suspect the finite size of the magnets is relevant here. That's when the integration comes in.

It's worth noting that even with simple dipoles you can get moderately complex behaviour under rotation if the magnets aren't aligned with the rotation axis. And you'll get different-each-time behaviour if they have some freedom to move in their cases.

With a bit of care you could use iron filings (or more modern alternatives, which are probably less messy) to see if the field from the magnets is more or less axisymmetric. And then shake the case vigorously and repeat the filings experiment to see if the magnets are loose.

Pro tip: do not allow iron filings to come into contact with a magnet, or you will have a hairy magnet forevermore.