# Why are the magnetic moment and the angular moment related?

Why are the magnetic moment and the angular moment related? I've always read everywhere that they are related but found nowhere a satisfactory explanation of the cause

• – joshphysics May 25 '13 at 4:41

Let's first look at the classical situation. A charged particle moving round a circular loop had an angular momentum and because it is also a current, it produces a magnetic moment. Therefore, it can be considered to be a magnetic dipole with a moment $\vec{m}$. The magnetic moment and the angular momentum are proportional to each other with the constant of proportionality called the gyromagnetic ratio.

Going to the quantum world, some particles are observed to have an intrinsic magnetic moment the way they can have a mass or charge. We can define a quantity $\vec{S}$, the intrinsic angular momentum, from $\vec{m}$ using an appropriate gyromagnetic ratio. It is experimentally confirmed that we need $\vec{S}$ for angular momentum conservation. That is, the orbital angular momentum $\vec{L}$ alone is not conserved but the total angular momentum $\vec{J} = \vec{L} + \vec{S}$ is.

• Yeah, I know that, and I also see the classic case. But I would like to know why something with angular momentum (be it intrinsic or orbital) there must be a magnetic moment associated. Take the example with an electron with l=0, only having spin. How do you justificate the magnetic moment? What is the exact reason? – Yossarian May 25 '13 at 22:34

You can understand, it through Einstein-de Hass effect or other way Barnett effect;

In classical mechanics, when an object of mass $m$ moves circularly, it gives rise to angular momentum. Similarly, when a charge particle moves circularly it gives rise to magnetic moment. So charge particles are not massless, they do have some mass, their orbital motion results into this angular momentum.

As per above effect means that: You have a ferromagnetic substance hanged by a thin fibre. Now, you magnetise the ferromagnetic rod, it starts rotating other way, because of conservation of angular momentum arising from the moving mass charges inside the rod.

Magnetic moment, angular momentum, and charge are related, because the magnetic field is how the electromagnetic interaction carries angular momentum.

If there were an intrinsic relationship between magnetic moment and angular momentum, you would expect the neutrino to have a magnetic moment. The current PDG reports an upper limit $\mu_\nu < 29\times10^{-12}\,\mu_B$ from experiments with reactor neutrinos, quite different from the electron's magnetic moment $\mu_e \approx 2\mu_B$. Note that an electron-type neutrino will spend part of its time as an $e^-$-$W^+$ loop (and similar for neutrinos with contributions from the other flavors), which will give it some miniscule magnetic moment whose predicted value I don't know. However at tree level the neutrino's coupling to the photon is exactly zero, which means it carries angular momentum without magnetic moment.

• this looks nice. Could you please elaborate more on what you mean by "the magnetic field is how the electromagnetic interaction carries angular momentum"? – Yossarian Mar 16 '15 at 7:36
• Classically, you must introduce angular momentum to create a magnetic field (and computing the angular momentum stored in a magnetic field is a familiar E&M problem). Quantum-mechanically, charge couples to photons which contain both electric and magnetic components; the photon carries spin $\hbar$, which plays a major role in which electronic transitions in atoms are allowed and which are forbidden. – rob Mar 17 '15 at 6:14

The way I understand it, a charged particle, which has an angular momentum, will have a magnetic moment associated with it. This could be because an angular momentum is associated with some kind of rotation. For a charged particle, this rotation could be thought to constitute a current loop. And a current loop can always be associated with a magnetic moment at its centre. Thus charge + rotation (angular momentum) --> magnetic moment. The moment comes out to be proportional to the angular momentum.

For the quantum case, the angular momentum is probably linked to some rotation we don't understand yet, which could be giving rise to the magnetic moment...

In some sense, this is just a longer version of Amey Joshi's answer, but hope it helps. (Though I AM a year late! :P)

• actually you are two years late! XD thanks for your answer but i don't feel like it answers my question. I do get the heuristic point that a rotating charge is a current which generates a magnetic field, but this is a very weak argument, since first, the world is quantum and this viewpoint is classical, and second, if we have an electron without angular momentum (that is only with spin) this argument also fails since there is no current there. – Yossarian Mar 12 '15 at 15:02