Origins of magnetic fields from permanent static magnets versus current

[magnetics]

The videos from Veritasium explaining permanent static magnets and electromagnets were quite good I thought…





But they have me a little confused with regard to the origins of magnetic fields generated by an electric current as opposed to a permanent static magnet from say iron.

1. An electric current induces a magnetic field which is explained as a consequence of special relativity. That the magnetic field is just an electric field viewed from a different frame of reference.

2. Whereas the explanation of the origins of a permanent static magnet were not explained in terms of special relativity, but as a quantum mechanical effect... Electrical charge, intrinsic magnetic moments, unpaired electrons, ferromagnetism and domains.

Are the two descriptions simply two ways of expressing moving charged particles or is there something fundamentally different between the source of these two magnetic fields?

Thank you.

 

[Dale]

An electric current induces a magnetic field which is explained as a consequence of special relativity. That the magnetic field is just an electric field viewed from a different frame of reference.

There is one subtlety that most people (including myself) miss the first time they are presented with this argument. It is actually a magnetic force that is just an electric force viewed from another frame. With magnetic forces it is always possible to transform to a frame where the charge is at rest, and therefore the force is purely electric. However, there are electromagnetic fields such that there are no frames where the field is purely electric and there is always some amount of magnetic field.

Whereas the explanation of the origins of a permanent static magnet were not explained in terms of special relativity, but as a quantum mechanical effect.

Modern quantum mechanics is fully relativistic so there is no inherent conflict in the two.

 

[magnetics]

Thanks Dale for clearing up the field/force description.

The question however referred to the field, not the force. I’m trying to understand the similarity or difference in the origins of the two magnetic fields. For instance, in terms of the electrons traveling in the current, do they also have a spin component like the orbital electrons in permanent magnets? Or is the source of the field in a current purely from the movement of the electrons through the conductor?

Thank you.

 

[Dale]

The question however referred to the field, not the force.

I understood that. The question is based on a wrong premise which I was correcting. Magnetic fields are not just electric fields viewed from a different reference as stated in point 1 of your question.

Or is the source of the field in a current purely from the movement of the electrons through the conductor?

The source of magnetic fields in both cases is current density. In the case of an electromagnet it is a classical free current density and in the case of a permanent magnet it is a quantum mechanical version of current density.

 

[vanhees71]

Well, that seems to be a misconception due to the famous (for me rather infamous) textbook by Purcell (vol. 2 of the Berkeley physics course). Relativity does not tell us at all that

An electric current induces a magnetic field which is explained as a consequence of special relativity.

It rather tells us that there is an electromagnetic field, described as an antisymmetric tensor $F_{\mu \nu}$ in Minkowski space (let's not deal with the full general-relativistic treatment yet), which can be split into electric and magnetic field components with respect to (inertial) reference frames.

Concerning the description of permanent magnets it's indeed true that there's no way to understand it without quantum theory. That's true about most things having to do with the description of matter. Of course in many cases there's an effectively classical description derivable from quantum-statistical many-body theory. In the case of macroscopic electrodynamics it's of course wise to use relativistic many-body theory and only than check, where one can get along with simplified non-relativistic approximations, because then one avoids some paradoxes which are just still remnants of the historical development that electromagnetics has been first developed as a non-relativistic theory which thereafter lead to the discovery that the non-relativistic space-time description is only an approximation.

 

[vanhees71]

The source of magnetic fields in both cases is current density. In the case of an electromagnet it is a classical free current density and in the case of a permanent magnet it is a quantum mechanical version of current density.

That's not right either! The source of the magnetic field of a (hard) ferromagnet is not only due to current densities but also due to macroscopic magnetization, i.e., the spontaneous breaking of rotational invariance due to a phase transition where fundamental magnetic moments of electrons are aligned due to exchange effects. Not only charge-current distributions are fundamental in the sense that elementary particles like electrons carry an electric charge but also magnetic moments. Despite carrying one negative elementary charge electrons also carry a magnetic moment due to their spin.

Of course, on the macroscopic level you can formally describe also the magnetization equivalently by current densities, $\vec{j}_{\text{mag}}=\vec{\nabla} \times \vec{M}$. However, that doesn't mean, that these magnetic fields are due to classical "molecular currents". That's clearly contradicted by the fact that the gyro-factor in the Einstein-de Haas experiment is not 1 (as Einstein and de Haas erroreneously first thought to have measured ;-)).

https://en.wikipedia.org/wiki/Einstein–de_Haas_effect

 

[Dale]

source of the magnetic field of a (hard) ferromagnet is not only due to current densities but also due to macroscopic magnetization, i.e., the spontaneous breaking of rotational invariance due to a phase transition where fundamental magnetic moments of electrons

If you write down the probability current density for such an electron is it not non-zero?

 

[vanhees71]

For a ferromagnet in thermal equilibrium without a current running through, the average velocity of the electrons is 0. So whatever you mean by "probability current density", I'd expect it to be 0 as well, right?

 

[Dale]

the average velocity of the electrons is 0

I know that. That wasn’t the question.

I think that the probability current density is non-zero and that gives rise to the intrinsic magnetization.

 

[Vanadium 50]

Before replying, a) is this A level? And if it is, why are we trying to explain a quantum mechanical effect without quantum mechanics?

 

[Paul Colby]

From a classical view, aren't the fields at a point just the sum of the contribution from each charged particle? Each particle has its own instantaneous rest frame in which the field contributed at each point is purely electric. Now, QM is required to make permanent magnets for the same reason it's needed to explain the stability of solids and atoms.

 

[vanhees71]

In the instantaneous rest frame the particle's em. field has only electric field components if the particle is not accelerated. Otherwise there's a radiation field. The correct view on the issue is not the one put forward by Purcell but that the electromagnetic field is one entity. You cannot reduce it in general to either electric or magnetic fields. These are the components of the field with respect to some reference frame.

QM is needed for the notion of half-integer spin and gyrofactors other than 1.

 

[Dale]

From a classical view, aren't the fields at a point just the sum of the contribution from each charged particle? Each particle has its own instantaneous rest frame in which the field contributed at each point is purely electric.

Superposition doesn’t work across different reference frames.

 

[Paul Colby]

Superposition doesn’t work across different reference frames.

Well, it does in the limited sense that each $(E_n,B_n)$ in one consistent rest frame where the summation is done, is a pure electric field $(E'_n,0)$ obtained by transforming from its individual instantaneous reference frame. One could choose to compute the $(E_n,B_n)$ in this manner AFAIK.

We are assuming no radiation as expected for magnets at rest.

 

[Dale]

What you just described isn’t superposition. I don’t think throwing in the word “limited” changes it.

 

[Paul Colby]

So

$(E,B) = \sum_1^N (E_n,B_n)$

where

$(E_n,B_n) = L_n(E_n',0)$

where, $L_n$ is the $n$-th LT is not considered superposition? Fine, the field is still the sum of contributions which are individually viewed as transformed $E$ fields. Within the constraints of no radiation and ignoring QM which really can't be done for all the reasons stated, is this sum incorrect?

 

[Dale]

So

(E,B)=∑N1(En,Bn)

This is superposition.

where

(En,Bn)=Ln(E′n,0)

This is not.

Both equations are valid, but there is no frame where the field can be seen simply as the Lorentz transform of an E field (unless all the charges are at rest with respect to each other). Thus a B field cannot be seen simply as a relativistic effect of an E field. Not even in classical EM with no radiation, and less so when either of those conditions are considered.

 

[Vanadium 50]

Paul, when you get to the bottom of a hole it is best to stop digging.

There is no frame where the electromagnetic field of an electron is purely electric. 95% of the magnetic field of a magnetized iron domain is from the electrons' spins and only 5% from their orbital angular momentum.

 

[Paul Colby]

It's not a hole as much as asking for clarification. This isn't fair?

 

[Vanadium 50]

You're not asking for clarification. You're making wrong statements.

 

[jartsa]

However, there are electromagnetic fields such that there are no frames where the field is purely electric and there is always some amount of magnetic field.

Is the electromagnetic field around a straight electric wire with a current an example of such field which is not purely electric in any frame?

 

[Dale]

Is the electromagnetic field around a straight electric wire with a current an example of such field which is not purely electric in any frame?

Yes. It turns out that the quantity $E^2-B^2$ (units where c=1) is an invariant of the electromagnetic field tensor. So in the wire’s frame $E=0$ and therefore the invariant is negative. That means that in any frame there must be a B field and that it’s magnitude must be greater than the E field magnitude.

 

[Dale]

You're not asking for clarification. You're making wrong statements.

To be fair, he was asking a series of questions in 11 and 16. It is only 14 where he did not. And there his statements were not so much wrong as mislabeled: you can transform and sum, but only the summation is called superposition.

But let’s try to get back on track.

 

[Vanadium 50]

Fair. But I was mostly responding to 14.

But the relevant point is still valid - magnetism in iron comes from electron spin. Not orbital angular momentum.

 

[bhobba]

But the relevant point is still valid - magnetism in iron comes from electron spin. Not orbital angular momentum.

Indeed. Its one of those effects with its basis in QM. Sometimes you read its caused by currents circulating in domains. Unfortunately its a bit superficial. However someone did a nice insight article using this idea:
https://www.physicsforums.com/insights/permanent-magnets-ferromagnetism-magnetic-surface-currents/

Thanks
Bill