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This post Permanent magnet attraction from Maxwell's equations / Lorentz force presents a similar question to mine. I believe it was closed before it was fully answered.

The problem:

In classical electromagnetism (so, putting aside spin).

We model a magnet bar as a current loop and assume a constant external magnetic field perpendicular to the loop surface. We then apply the Lorenz force equation and find that the field applies torque on the current loop but no linear force. So, no pulling or pushing, only rotation.

It seems that the model above should describe the interaction between two bars of magnet when placed south to north. Still, we find that two magnetic bars attract each other.

I suggest the following answer:

Instead of a constant external magnetic field, lets consider the field of a second current loop (this second current loop would model another magnet bar). Now, if we look at the field lines "at and around" the north pole of the current loop we see that they have an orthogonal component facing "north" but also a radial component facing from the axis of the loop outwards.

If we place a current loop A above another current loop B, close to B's north pole, then at every point along the loop A there is a component of the magnetic field facing outwards and orthogonal to the current through A. Appling Lorentz force equation will give us a force perpendicular to loop A surface downwards (towards the loop B).

Does this explain the attraction between two magnets bars for classical electromagnetism?

Qmechanic
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Lior Harari
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2 Answers2

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You are taking the tedious route to the answer.

For two infinitely straight parallel wires carrying current in the same direction, either by Biot-Savart law or Gauß's law, you can find the magnetic field that each wire feels of the other, and then by Lorentz's force law for each electron in it, you can find the famous force on parallel conducting wires, that used to define the SI base unit of Ampère (which is in turn used to define charge Coulomb).

Turning the two infinite straight parallel wires into current loops will produce a similar attraction, and that is more than sufficient to explain the phenomenon you want.

Note that these solutions show that the important physics lies in the fact that the "external" magnetic field is NOT uniform.

While you may derive the attraction between magnets this way, there is no internal self-consistent method to explain magnetism in classical electrodynamics. You can only piecewise model snippets of the behaviour assuming that inconsistencies do not invalidate the results. The relevant no-go theorem goes back to Bohr and is quite important. The only way to properly handle magnetism is in quantum theory.

naturallyInconsistent
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(Too long to be a comment)

You can think of the first magnet creating a given potential and the second one being immersed in said potential. Since magnets behave like dipoles, provided that both bars are close enough to each other, the second magnet will see a uniform magnetic field. This shall introduce a coupling term in the Hamiltonian favoring that both dipoles are aligned, which is why different poles (i.e: North / South) attract and the same poles repel.

If you recall your classical EM lessons, you will surely remember solving the classical exercise of studying the potential energy of a magnetic dipole introduced in a uniform magnetic field and finding its potential energy $V=-\vec\mu \cdot \vec B$ is minimum when the dipole is aligned with the field in the same sense and maximum when it is aligned with the field in the opposite sense. This is the same thing.

Notice I did not talk about spin here, it was all classical. Nonetheless, you might be interested in having a look at Ising-Lenz models for ferromagnetism, they are really instructive and for all that matters you can substitute "particles with spin" by "magnetic dipoles", which are favored energetically if they are aligned with their nearest neighbours.

Lagrangiano
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