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I just read that a magnetic field is generated when electrically charged particles move around. If I understand it correctly, the electric field and the magnetic field are two sides of the same coin, namely the electromagnetic field, and the magnetic field is simply defined as the difference between the actual electromagnetic field and what the electromagnetic field would be if everything was at rest. This difference is generated by strange quantum effects that I dont understand.

However what confuses me in this is, why then do permanent magnets interact with the magnetic field, given that they don't containg electrical currents as in an electric magnet?

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  • $\begingroup$ The difference is not "generated by strange quantum effects" however one of the leading "electrical currents" is in fact a "strange quantum effect": it is that electrons have an intrinsic angular momentum in space, called their "spin". For example in the ground state of the hydrogen atom, those electrons in those $s$-orbitals have zero orbital angular momentum: they aren't "orbiting" in the classical sense. However they have nonzero total angular momentum because they have this intrinsic spin angular momentum. $\endgroup$ – CR Drost Sep 19 '17 at 15:35
  • $\begingroup$ I see. How then is the difference generated? $\endgroup$ – user56834 Sep 20 '17 at 8:37
  • $\begingroup$ Well the world has a curious property that if you accelerate in a certain direction you see clocks tick faster in front of you or slower behind you, in proportion to their distance from you. This turns out to leave invariant a certain distance measure called the "interval" between two events in spacetime. As a result a certain definition of your velocity is as a 4-vector with a fixed length, and if the magnitude is fixed that can only be because the 4-force must always be perpendicular to the 4-velocity. The magnetic field tweaks the 4-force to make this happen. $\endgroup$ – CR Drost Sep 20 '17 at 18:37
  • $\begingroup$ There was a question about “How do I understand Electromagnetism”. Perhaps it’s helpful for you to read this answer. $\endgroup$ – HolgerFiedler Sep 21 '17 at 4:12
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The energy of a magnetic moment $\vec{\mu}$ in a magnetic field $\vec{B}$ is given by $$E = - \vec{\mu} \cdot\vec{B}$$ There are the following contributions to the magnetic moment of an atom:

  • The "rotation" of the electron around the nucleus, which is similar to an electric current. This is called the orbital angular momentum $L$.
  • The spin of the electron $S$. This can be thought of as an internal rotation of the electron around itself.
  • Finally, the nuclear spin $I$. This is analog to the electron spin, but here the nuclei rotate around themselves. However, this is usually small compared to the other two contributions.

So two magnets interact, because they both poses magnetic moments. Therefore, ...

  • the first magnetic moment creates a B-field at the position of the second magnet. Hence, the energy of the second magnet changes to to the upper formula.
  • However, the problem is symmetric. Therefore, the second magnetic moment creates a B-field at the position of the first magnet, as well. Hence, the energy of the first magnet changes to to the upper formula.
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  • $\begingroup$ I see. Thank you. One follow up question: what exactly should I imagine when I think about the "spin" of an electron? Whether I think of an electron as an infinitessimal point or as a cloud, I cannot imagine what it would mean for it to "spin". $\endgroup$ – user56834 Sep 20 '17 at 8:36
  • $\begingroup$ Good question, but I don't have an answer. As far as we know, the electron is a structureless, point-like particle. Nevertheless, when I picture the spin, it's a rotation of a vector (thus length unequal to zero) around an axis. This picture explains, why the spin obtains a minus sign under time reversal symmetry, and it helps to understand the coupling of orbital angular momentum and spin. Although this picture is incomplete due to the extended vector, it help a lot. So whenever you ask yourself the "why" question, answer it with the famous words: "Shut up and calculate!" $\endgroup$ – Semoi Sep 20 '17 at 17:20
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"why then do permanent magnets interact with the magnetic field, given that they don't contain electrical currents as in an electric magnet?"

Why do you think there are no electrical currents? Isn't the electron "orbiting" the nucleus? The currents of the electrons of the atoms create the permanent magnet. You can't use this currents to extract work, but they're there.

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