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I have read this Wikipedia page, but I can't understand it.

In this picture from that page, in the formula, there is just one flux density, but what about the flux density of the second magnet? What if the magnets aren't identical?

Also, why do we have both area and radius in the formula? Aren't the two terms proportional? And also, why there are both $\mu_0$ and $B_0$? There is $\mu_0$ in the $B_0$ formula too!

Another thing that I can't get, is that what's the difference between a cylindrical bar magnet and a cylindrical magnet? How are these separated into two different classes with different formulas? Even the latter needs elliptical integrals and magnetization which are very hard.

enter image description here

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A measurement can be done with a dynamometer. For calculations one needs finite-element software and the magnetic properties (hysteresis loop) of the magnets.

There are approximations. The ones that you quote for large distances are not often very relevant. There are formulas for holding power of a magnet against a weak magnet.

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    $\begingroup$ What are those formulas? Your last sentence $\endgroup$
    – Snack Exchange
    Commented Oct 7, 2020 at 19:47
  • $\begingroup$ Holding power is proportional to area of contact. Also depends on magnetic flux density but that is pretty much generally one or two tesla. $\endgroup$
    – user137289
    Commented Oct 7, 2020 at 20:40
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The magnetic field at any point near a permanent magnet is determined by the vector sum of the of each of the fields from all of the atomic dipoles within the magnet. Within another nearby magnet, each atomic dipole is subject to a torque which tends to align it with the field from magnet one, and a net force which depends of the gradient of the field from magnet one (at its location). A reasonable approximation for the net force and torque acting on each magnet may achieved by treating each magnet as a current carrying solenoid of the same size, shape, and dipole moment. Either of these would require a massive numeric simulation (for each relative position). For a rough approximation (which improves with distance) treat each magnet like an electric dipole. Find the “pole strength” and treat a “north pole” like a positive point charge.

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