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Edit: My question is different because I am not asking how permanent magnets come about. Rather, I am asking why (taking permanent magnets as a given) they attract or repel each other. I.e. What equation describes the force that causes a permanent magnet to stick to my fridge.

I have been searching online for some qualitative explanation of why permanent magnets attract each other (for example fridge magnets) and can't find anything. The closest I can find is attraction/repulsion of two parallel current carrying wires.

Here's using the same steps to two permanent magnets situated on top of one another:

Lorentz force law says force is proportional to $\vec{v} \times \vec{B}$. Each magnet is composed of many atoms all oriented the same way. All together the electrons orbiting the individual nuclei sum to create a stronger magnetic field (let's say pointing up to the other magnet).

This other magnet is also composed of individual, similarly oriented atoms. Since I know it will attract when the "south" pole of the top magnet is above the "north" pole of the bottom magnet, let's try that situation.

Trying to apply the Lorentz force law, $v$ is going around in a circle on the top magnet. $B$ is coming straight up (from the first magnet). This seems to imply that the force would be perpendicular to the $B$ field, but spinning around in time (as the electron spins around the nucleus).

Where does the force that attracts the two permanent magnets towards one another come from?

Sebastiano
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jbord39
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  • Hello. I have answered the same question here: https://physics.stackexchange.com/q/488813/. Please see if that helps. – Apoorv Jul 12 '19 at 01:02
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    Possible duplicate of How permanent magnets work?Are they working with motion of electrons? – Apoorv Jul 12 '19 at 01:15
  • @ApoorvKhurasia: That answer does not give any indication of why there is a physical force between magnetic poles. It explains why some materials are magnetic (aligned spins). – jbord39 Jul 12 '19 at 19:54
  • @ApoorvKhurasia: Let's just start from the fact that there are permanent magnets. Given that fact (with your excellent answer as an explanation), what makes them have a physical force attracting them together. Is this the Lorentz force? Because as I described it seems that the Lorentz force would always be perpendicular to the "perceived" magnetic force (which would attract the north/south pole together and almost snap them in place). It seems evident that magnetic poles want to align to the fields around them, but why? – jbord39 Jul 12 '19 at 22:45
  • Hey, you have to realize that the feilds are non uniform, so when you sum up all the torque on a wire modeled magnetic dipole in a non uniformed field then you will get some non canceled out force parallel to the magnetic field. In a uniform magnetic field your thinking is correct cuz there is no force outside of the radial direction (towards the center of loop) – Jason Mar 04 '22 at 22:42

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Permanent magnets can't be explained in terms of Maxwell's Equations or Classical Physics in general. As most (if not all) of the phenomena that arrises due to microscopic behaviour, you need Quantum Physics to explain it properly.

Permanent magnetism comes from spin. Spin is an intrinsic property of every particle, like mass. It is quantized, coming in integer multiples of one half. It is one type of angular momentum. The important thing you need to know about spin, in order to answer your question, is that it couples with the Magnetic Field and also 'produces' its own little magnetic field. More precisely, the spin of a particle implies in a magnetic dipole moment.

Just as every fundamental particle (like electrons and protons) has its spin, so do composite particles (like an atom). It happens, though, that the magnetic moment of a single atom is very weak, so you need all the atoms in a lattice to align, so that their fields sum up, in order to get something apreciable and detectable. And that's what happens when you put a non-magnetized piece of iron on a magnetic field. Its atoms align and you get a magnet, simmilarly to when you have lots of tiny magnets togheter.

But, now, why aren't all materials magnetic, you ask?

Because for most atoms the spins of different electrons, togheter with the spin of the nucleus, cancel out. You need a very specific arrangment of atomic structure in order to form crystals that can have its atoms with sobressalent spin and aligned, thus creating a magnet.

Lucas Baldo
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  • I am not asking "how magnets come about". I am asking, "why do magnets attract". I realize that magnetism is a quantum phenomenon (as all other phenomenon at a low enough level). I am wondering about the actual forces that cause a magnet to stick to my fridge. – jbord39 Jul 12 '19 at 19:56
  • As a more specific example, why do the non-magnetized pieces of iron line up with the magnetic field? Is there a force equation that explains this? "The magnetic fields want to align" is a great behavioral description -- but what equation describes this alignment force? – jbord39 Jul 12 '19 at 22:47
  • As I said, 'the spin of a [charged] particle implies in a magnetic dipole moment'. Magnetic dipoles are subject to a torque when in an external magnetic field. If you want equations, the spin couples to the external field in the Hamiltonian. For a lone particle: $\hat{H} = \mu \hat{S_z} B$. The time evolution operator is given by $\hat{U(t)} = e^{-i\hat{H}t}$. There are two possible states for a spin-1/2 particle, and this time evolution will make the particle oscilate between the two, just like a classical particle. I will try to add this to the answer above. – Lucas Baldo Jul 13 '19 at 19:54
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    This is still different from your question. To deal with the full problem you need to take into account that the particles are subject to the field generated by their neighbors. So you need a lattice model, which are very challenging ones (search for Ising model). The general Idea, though, is that the least energetic state of the material is going to be with all spins aligned between themselves and with an external magnetic field. This explains why one magnet forms and how it generates torque on another magnet. To see how they actually atract is a bit harder. – Lucas Baldo Jul 13 '19 at 20:02
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    Near the edge of a magnet the field is straighter, while it bends away the further you go from it (except if you are exactly at the symmetry axis). This means that for two magnets close to each other, the field inside one that is generated by the other is not uniform, and the net result is that the torques at each point tend to, besides aligning them, bring them together. One way of seeing this is noticing this is the least energetic position. – Lucas Baldo Jul 13 '19 at 20:03
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The closest I can find is attraction/repulsion of two parallel current carrying wires

. There you go!

You can model the permanent magnet as made up of lots of little current loops that each form the dipole element of an atom. (Why that’s a good model is a separate question). Together they act like a bigger loop.

The field from that also induces a corresponding magnet in the iron of the refrigerator door. Again, that looks like a current loop.

And those two effective loops are parallel, so they attract.

Bob Jacobsen
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  • Well, almost, since a current loop does not move in a uniform magnetic field because all of the forces (torque really) cancel out. In a non uniform magnetic field then the torques in parallel sum up and results in a more linear force. Thats the thing he was missing. – Jason Mar 04 '22 at 22:37