Electrodynamics from Jefimenko's equations

[Dale]

@vanhees71 are you familiar with anyone who teaches electrodynamics starting from Jefimenko's equations? What would you think about such an approach?

I haven't thought it through carefully yet, but I wonder if it would be easier for students to grasp. The connection to relativity would be a simple matter of explaining how the four current transforms.

 

[vanhees71]

I'm not so sure. Usually you take the four-potential either in Lorenz or Coulomb gauge. In the Lorenz gauge, using Cartesian coordinates, your four components are decoupled and obey the wave equation. Then you choose the retarded solution for these waves. In Coulomb gauge, using Cartesian coordinates, you get a Poisson equation for $\Phi=A^0$ and decoupled wave equations for $\vec{A}$ but with a modified current density. Again using the retarded solution for the latter, you get in both cases the same result for the electromagnetic field (i.e., Jefimenko's equations), as it must be, because Lorenz and Coulomb gauge potentials are just mapped to each other by a gauge transformation.

On the other hand, you can choose any other not so common gauge. I can't find it at the moment, but there was once a very illuminating article in Am. J. Phys. where the author showed that you can define gauges, where part of the potential propagator at either speed you like (in Lorenz gauge all components are retarded with $c$ as the "signal velocity", in Coulomb gauge the scalar potential (temporal part of the four-potential) is intantaneous), all leading to the correct Jefimenko solutions for the field components.

Of course, there's no principle way to argue for the one or the other gauge (except simplifications to find proper solutions for a given problem) how the four-potential propagate, because they always contain unphysical degrees of freedom, which precisely cancel when calculating the electromagnetic field from the potentials. For the electromagnetic field components however, you can argue that it must be retarded with $c$ being the phase velocity of all field components, because these are observable fields. So I think, it's indeed pretty straight forward to derive Jefimenko's equations directly from Maxwell's equations. I'll try to do this later today.

 

[Dale]

Yes I have seen derivations of Jefimenko's equations from Maxwells equations. But I wonder if the reverse were true. Is it possible to derive Maxwell's equations from Jefimenko's.

Regarding the gauge, I was thinking of just sticking with the Lorenz gauge in order to make the later introduction of relativity more natural.

 

[vanhees71]

That's a good question too. I think it should be possible to derive Maxwell's equations from Jefimenko's. Of course you have to assume that strict local charge conservatios, i.e., the continuity equation
$$\partial_t \rho + \vec{\nabla} \cdot \vec{j}=0$$
holds for the electromagnetic charge-current densities.

 

[Dale]

I had forgotten about that. The continuity equation can be derived from Maxwells equations, but I don't think that it can be derived from Jefimenkos equations. So I think it would have to be introduced separately.

 

[vanhees71]

The continuity equation is forced by gauge invariance, i.e., it must hold as an "integrability constraint", independent of the dynamics of the charges. Thus, any model, where the dynamics of the charges is incompatible with charge conservation leads to a contradiction with the em. field equations.

Thus, if you consider the approximation, where the charge-current density is given as an external four-vector field you must assume that the continuity equation holds. Otherwise Jefimenko's equation do not give correct solutions of the Maxwell equations.

 

[Dale]

The continuity equation is forced by gauge invariance

I did not know that. Thanks.