What Exactly Is The Physical Content Of Maxwells Equations.

[bhobba]

It doesn't contain anything new, and nothing I haven't seen before, but as people who have been around a while know, I have a bit of a fascination with 'justifications' for Maxwell's equations.

I recently came acrossss the following:
https://www.vttoth.com/CMS/physics-notes/123-the-electromagnetic-field-tensor

We know the theorems by Wigner: the no-interaction theorem and the fact that all fields must be tensors (the Wigner-Eckart Theorem) strongly suggests the examination of a 4-vector field. But the above link would seem to then imply Maxwell's equations as a consequence.

Ultimately, of course, the validity of Maxwell's equations is an experimental matter, but it's hard not to get the feeling that something deep is going on here. Try as I might, though, I have no idea what it is.

Any ideas?

Thanks
Bill

 

[PeterDonis]

I recently came acrossss the following

Have you read Chapters 3 and 4 of Misner, Thorne, & Wheeler? That's where I first encountered the material that's briefly summarized in the post you reference. MTW was published in 1973.

the above link would seem to then imply Maxwell's equations as a consequence.

$F = dA$ implying $dF = 0$ is indeed a geometric property that will be possessed by any 2-form that is the exterior derivative of a 1-form. (Indeed, $d^2 = 0$ is a general identity for differential forms.)

However, that's only two of Maxwell's Equations. The other two, which Toth dismisses as "merely the defining equations of the charge density and current", are in fact the connection between the EM field and its source, which is not a "mere" definition; it's where the actual physics is. (This is a good illustration of how a "part-time physicist" can miss the boat.)

In differential form language, this can be written as $d (*F) = 4 \pi *J$, where the $*$ is the Hodge dual operator and $J$ is the charge-current 4-vector (which because of the way the Hodge dual works in 4D spacetime, makes $* J$ a 3-form, as expected for the exterior derivative of the 2-form $* F$--MTW chapter 4 goes into more detail about all this). Toth's observation that you get automatic conservation of the source (the continuity equation) from this is just another application of $d^2 = 0$--apply $d$ to both sides of $d (* F) = 4 \pi * J$ and you get $d * J = 0$, i.e., charge-current conservation. But to call all this a "mere" definition of charge and current misses the point. Automatic conservation of the source is part of the physics; it's what tells you that you've got the right connection between the field and its source.

it's hard not to get the feeling that something deep is going on here.

There's certainly important physics involved--and similar principles are used in GR to tell us that the Einstein Field Equation is the right one for the connection between spacetime curvature and its source, stress-snergy. (MTW goes into detail about this as well.) If that's "deep", then yes, there is indeed something deep going on.

 

[Sagittarius A-Star]

I recently came acrossss the following:

What also may help for understanding the EM field is to use Geometric Algebra, applied to 4D-spacetime.

The wedge product $\wedge$ (=outer product) of two vectors is a bivector.

The components of a 4-vector are projections on the four axes of the coordinate system (x, y, z, t).
The components of a bivector in 4D-spacetime are projections on the six (oriented) planes of the coordinate system (yz, zx, xy, xt, yt, zt).

The EM field bivector is $F = \nabla \wedge A$

Maxwells's equations:
$\nabla \cdot F = J$ (vector)
$\nabla \wedge F= 0$ (trivector, no magnetic monopoles)

See also

 

[FactChecker]

IMHO, as a casual amateur, geometric algebra (and geometric calculus) goes a long way to demystifying the subject. Unfortunately, it involves a long, tedious path of bookkeeping. It starts with dirt-simple observations/rules of how products of entities of different dimensions give results in other dimensions. From those rules, a lot of bookkeeping gives very satisfying results. The books by Alan Macdonald ("Linear and Geometric Algebra" and "Vector and Geometric Calculus") give a thorough explanation.
The end result, for me, was that there is no mystery in Maxwell's equations, just a lot of bookkeeping from basic geometry.

 

[weirdoguy]

What also may help for understanding the EM field is to use Geometric Algebra, applied to 4D-spacetime.

May or may not. Differential geometry is my specialisation within mathematical physics, and when I tried to study geometric algebra it only gave me headache and different forms of depression. Differential geometry, done properly (not like most physicstis do) is (for me) the way to go. Long live jet bundles!