From the Lorentz force equation we know that the magnitude of the force between a current carrying wire and a magnet field is defined as $F = ILB$. This implies that there must be a current across the wire in order for there to be a force. Now let's take a look at ferromagnetic materials such as iron for example. Let's assume that I have an iron nail which when it gets close enough to a magnet it gets attracted by the magnetic field of the magnet. How is this force (attraction) possible if there isn't any electric current across the nail?

The only conclusion I've come to is that as soon as the iron nail (or any other ferromagnetic material) gets close enough to the magnet there is a change in magnetic flux. From Lenz's law we know that a change in flux induces an EMF: $V = -N\frac{d\phi }{dt}$ If we divide that voltage by the resistance of the iron nail (which is almost 0) we should get the current across the nail which then satisfies the Lorentz force equation. Is this assumption correct? I've found one drawback from my own conclusion though, if this were true then copper should also get attracted and it doesn't.

If someone could explain how this attraction force works and if there is a mathematical expression for it I'd greatly appreciate it.

  • $\begingroup$ excellent question, most of the litterature focuses on explaining magnetic force upon a moving charge but then that give us no clue about magnet to magnet or magnet to ferromagnetic forces. I believe the answer lies in atomic current, something about electrons rotating around the atom preferentially in a way or another. But I eagerly await for the experts answers $\endgroup$ Sep 19 '18 at 18:34
  • $\begingroup$ This article is highly relevant here en.wikipedia.org/wiki/Magnetic_domain $\endgroup$ Sep 19 '18 at 18:59

How is this force (attraction) possible if there isn't any electric current across the nail?

There indeed is no free electric current in the nail, but there is a magnetization current due to non-uniformity of magnetization in the nail. Much of it is on the surface, where magnetization jumps to zero (outside the nail) and some of it is even inside, due to jumps in magnetization across magnetic domains. It is a bound electric current, it isn't free electric current, so it is not possible to measure it with ammeter, and there are no usual Ohmic losses associated with it. This bound current is due to microscopic bound motion of microscopic charges that the nail consists of. The only apparent manifestation of this bound current is magnetic interaction.

Alternatively, one can use simpler although less powerful theory of magnetic poles, where the magnet and also the nail are just ensembles of magnetic poles that interact according to Coulomb's law for magnets. In this theory, there is no need to introduce the magnetization current, the ILB formula does not apply, but we can still calculate the magnetic forces.

  • $\begingroup$ Wikipedia says that for neutron "the magnetic moment of the neutron can be viewed as resulting from the vector sum of the three quark magnetic moments, plus the orbital magnetic moments caused by the movement of the three charged quarks within the neutron." Then at least some part of magnetization of a ferromagnetic material is not associated with any movement of electric charges - ? $\endgroup$ Sep 19 '18 at 20:44
  • $\begingroup$ Magnetic moment of a particle such as neutron can be viewed either as due to microscopic motion of electric charge, or as independent quantity not connected to motion of electric charge. But it is hard to prove that such a motion is not there. It would be easy for particle we knew is a point, but we have no way to ascertain that. Neutron has some internal structure and charge distribution and magnetic moment may well be due to motion of electric charge. $\endgroup$ Sep 19 '18 at 23:44

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