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Since Maxwell's equations are phenomenological, I'm looking for the actual deep reason why the magnetic field is oriented circularly around a straight conductor.

Could anybody knowledgeable on quantum electrodynamics please elaborate on this subject? Is there an explanation, or does QED itself depend on Maxwell's description, without offering some particular underlying mechanism?

(Alternatively, or complementarily, does general relativity offer some explanation for the circular geometry of the magnetic field lines?)

shredEngineer
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  • Maxwell's equations were originally derived phenomenologically, but are these days theoretically founded in the gauge theory called QED. (Posting this as a comment until someone writes out a nice answer.) – NDewolf Apr 04 '22 at 08:07
  • Thanks, I'd like to know the most important details of this foundation in QED! :) – shredEngineer Apr 04 '22 at 08:54
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    Does this answer your question? How do you go from quantum electrodynamics to Maxwell's equations? – John Rennie Apr 04 '22 at 09:19
  • @JohnRennie Unfortunately not. As far as I am able to understand the equations (being an engineer), B=∇A is assumed a priori. To express it another way: I want to know why the photons of the magnetic field "curl" around the moving electrons. What I don't want is a formal proof that QED somehow reduces to classical EM in the classical limit. – shredEngineer Apr 04 '22 at 09:28
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    QED is a theory to calculate scattering probabilities. It cannot be used to calculate magnetic field lines, or at least not directly. However when we require that QED has a U(1) local gauge symmetry this requires us to introduce a field that turns out to be the electromagnetic four potential used in Maxwell's equations, and this does predict circular magnetic field lines. So reducing QED to classical EM is the answer to your question and the linked question is indeed a duplicate. – John Rennie Apr 04 '22 at 09:35
  • So there is really no fundamental explanation for why the magnetic field is circular? Every argument seems to start and end at Maxwells equations, without answering the "why". The rotation operator ∇× applied to the vector potential seems to do the trick, but nobody knows why? – shredEngineer Apr 04 '22 at 09:51
  • If the answer to my question is that "no, QED does not at all give an explanation", then indeed this question may be closed, as a duplicate if you insist. – shredEngineer Apr 04 '22 at 09:55
  • Well, the "fundamental reason" is that, as far as we know, nature does not admit magnetic monopoles and, hence, magnetic field lines are closed. This translates to the lack of a current in the Bianchi identity for the stress-energy tensor or $B = \nabla\times A$. – NDewolf Apr 04 '22 at 10:15
  • @NDewolf Thanks for pointing me to the Bianchi Identity, I'll attempt to understand some of it. Your argument that the B-field lines should be closed seems like half of the answer. But I was hoping for some explanation like "the photons emitted by the moving electron are forced into a circular orbit because some force acts on them like so and so". – shredEngineer Apr 04 '22 at 10:37
  • Thinking of the magnetic field as a vector is flawed in the first place (because the $\times$ operation itself is only a computational hack). If you see it correctly as a bivector (little swirls perpendicular to the magnetic field not-actually-a-vector) then it makes a lot more sense, since it "billows out from" or is "sent into rotation by" the current (like water around a jet). But there are no circles wrapping around the wire; that's just an artifact of using insufficiently sophisticated math. Asking for a "deeper why" still seems quite vague.... – HTNW Apr 04 '22 at 10:57
  • @HTNW Thank you. I am happy to change my perspective on the magnetic field. I have to admit that I barely ever heard of "bivectors" before. Do you know of any good visual representation of bivectors in a magnetic context, something that even an engineer would understand? :) – shredEngineer Apr 04 '22 at 11:13
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    @shredEngineer The only image I could find after a quick search was this one, though it's a bit sparse for my liking. Hopefully you can still see why it is a more "natural" picture than the typical one. But a bivector should be visually intuitive: it is a oriented (i.e. swirling) plane segment, just like a vector is an oriented (i.e. directional) line segment. – HTNW Apr 04 '22 at 11:18
  • @HTNW Thanks a bunch! So the "modern" way to treat EM is to use bivectors, which are two-dimensional entites defined at every point in space (or even spacetime)? And these bivectors are fundamental "vortices" which are the sources of the magnetic field lines? Also, do bivectors have any relation to the vector potential? – shredEngineer Apr 04 '22 at 11:31
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    these may help you with the bi-vector concept: https://physics.stackexchange.com/questions/160993/why-does-a-magnetic-field-go-anticlockwise-of-the-direction-of-current/161036#161036 and https://physics.stackexchange.com/questions/410714/why-does-a-magnetic-field-curl-around-a-current-carrying-element/410735#410735 – hyportnex Apr 04 '22 at 13:39
  • @hyportnex Thank you! The article mentioned in the 2nd link is quite helpful to my understanding. (https://doi.org/10.1088/0143-0807/22/3/301) – shredEngineer Apr 04 '22 at 14:18
  • John Roche has written maybe a dozen tutorial papers on various EM and mechanical subjects, they are all superbly crafted and worth studying. – hyportnex Apr 04 '22 at 14:30
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    @shredEngineer If you want to learn more about the general subject of bivectors there is a nice YouTube video on geometric algebra here. – John Rennie Apr 04 '22 at 16:06

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