Jefimenko's equations

dipole

So I recently have been introduced to the retarded solutions of Maxwell's equations, which are referred to as Jefimenko's equations:

http://en.wikipedia.org/wiki/Jefimenko%27s_equations

And I'm curious as to how to interpret these equations (if you want to see a simple derivation of these equations see Griffiths). Unlike in Maxwell's equations, these forms do not seems to couple the magnetic and electric fields - they appear as independent fields.

Does anyone know more about this? Can electromagnetic waves exist in the retarded solutions of maxwell equations?

vanhees71

I never understood, why these equations are named after Jefimenko since they are known for over a century. Usually they are derived from the retarded potentials in Lorenz gauge, which is the most natural way to derive them from a relativistic point of view. Of course, the Jefimenko equations are advantageous in the sense that they are gauge invariant, i.e., as soon as one assumes retarded boundary conditions, one uniquely gets these solutions for the electromagnetic field, while of course the potentials are gauge dependent and not necessarily explicitly retarded. There are nice papers on this by Jackson and other autors in Am. J. Phys. If needed, I can look for these references.

Then, it is an oldfashioned view that there is an electric and a magnetic field and that for time-dependent fields one would cause the other. Since Minkowski it should be clear that there is only one entity called the electromagnetic field, which is a massless vector field, represented by either a four vector (the four potential) or the antisymmetric Faraday tensor (antisymmetric tensor). This particularly means that there is no clear interpretation of the time-varying electric field causing a time-varying magnetic field and vice versa. This becomes immediately clear from the fact that the distinction between electric and magnetic components of the Faraday tensor is a frame-dependent construct, i.e., through Lorentz transformations they mix with each other as determined by the transformation law of a 2nd-rank tensor.

PhilDSP

Jackson's text book has a concise derivation of the equations that is quite different from the derivation Jefimenko used. Jackson characterizes the equations as "Jefimenko's generalization of the Coulomb and Biot-Savart Laws".

Jackson also shows how they are related (but definitely not the same) to a pair of equations that can be used to provide a similar result. But the related equations are specialized in that they apply to point charges only. Jackson attributes the electric field equation to Feynman and the magnetic field equation to Heaviside.

Jefimenko has developed an entire discipline based on the insights his equations and approach provide. Among many other things, Jefimenko shows that Maxwell's displacement current might not be physical and provides a substitute concept which I think he calls Electrokinetic field. He gives an alternative interpretation of Faraday's moving magnet experiment in which electromotive force is generated by electric current, not the magnetic field. His equations are not extremely easy to work with, so Maxwell's displacement current may still be highly useful in practice. But the concepts Jefimenko develops may be simpler and easy to grasp.

The basic interpretation of his equations is that both dynamic electric and magnetic fields are generated by electric current.

vanhees71

To me this sounds somehow making the physics less intuitive. Of course, Maxwell's displacement current doesn't belong to the right-hand side (sources) of the inhomogeneous Maxwell equations but to the left-hand side, since it's part of the four dimensional divergence of the Faraday tensor in the relativistically covariant description. Thus, it's clear that the true sources of the em. field are the charge density and current density (or relativistically covariant the four-vector current of electric charge).

In relativistically covariant notation (in Heaviside-Lorentz units) the Maxwell equations read:

[tex]\partial_{\mu} ({}^{\dagger} F)^{\mu \nu}=0, \quad \partial_{\mu} F^{\mu \nu}=\frac{1}{c} j_{\nu}.[/tex]

Here the Hodge dual of the Faraday tensor is defined as

[tex]({}^{\dagger}F)^{\mu \nu}=\frac{1}{2}\epsilon^{\mu \nu \rho \sigma} F_{\rho \sigma}.[/tex]

PhilDSP

Good points both of you. In the Maxwell I've read I don't recall any indications that Maxwell expected displacement current to act as a source generating electromagnetic fields, probably because in his theory it *is* the electromagnetic field (the dynamic part at least) and the response to moving sources.

Of course Jefimenko gives us yet another tool or set of tools. So we can choose as we need them.

dipole

Interesting discussion so far, but my original question was regarding electromagnetic waves - if we take Jefimenko's equations, then how can we couple the E and B fields such that they propagate together as a wave?

Was Maxwell wrong, do his equations in fact not imply the existence of electromagnetic waves?

PhilDSP

Hi Dipole,

The wave equations for the E and B fields have the exact same form or kernel. They are entirely independent of each other (neither contains any source term of the other). That fact alone ought to indicate that Jefimenko's findings are appropriate.

The Maxwell equations or his theory or electromagnetic waves are not at all held suspect according to Jefimenko's work. On the contrary they are all necessary. It may be worth considering that the potentials were primal in Maxwell's theory and they provide the coupling to either the E or B fields separately. It was Heaviside who pushed to remove the potentials from the standard representation of the Maxwell equations and that was probably to the benefit of everyone who first wanted to understand the equations, but it does leave out the underlying structure.

Russell E

Not quite accurate. Jackson says "These two results, sometimes known as Jefimenko's generalizations of the Coulomb and Biot-Savart laws, were popularized in this author's text [referring to J's 1966 text on EM]." The phrase "sometimes known as" and the word "popularized" are both significant qualifiers.

Hmmm... those equations are just the solution of Maxwell's equations for a given distribution of electric charges and currents. The "approach" goes back to the Lorenz, Weber, Lienard-Wiechert tradition of retarded action at a distance, which is just another way of looking at the same things. These equations don't represent any new physics.

Again, hmmm.... He can assign new names to things, but there is no new physics in those equations.

That isn't a new or novel insight, and it doesn't emerge uniquely from those equations. Again, this is the basic retarded distant action tradition of Weber et al.

vanhees71

This is precisely the wrong impression, Jefimenko's somewhat strange interpretation of his equations seem to imply, and that's why I don't like this hype about them.

What's written as "|Jefimenko's equation" is nothing else than the solutions of Maxwell's equations for retarded boundary conditions, i.e., for the situation that you have a given charge and current distribution (better a given four-vector electric current) and you look for the electromagnetic waves emitted from the corresponding motion of charged particles. This solution has been known long before Jefimenko. I'll have to look the history up somewhere. For sure they where known at the end of the 19th century in form of the Lienard-Wiechert equations for the radiation from an accelerated charged particle, which is a special case of these more general equations for arbitrary charge-current distributions.

Also it is clear that electric and magnetc fields are no independent quantities. They are the 6 components of the Faraday tensor [itex]F_{\mu \nu}[/itex] with respect to a given frame of reference. Doing a Lorentz boost to another frame mixes the "old" electric and magnetic components to the corresponding "new" components. There is no physically sensible objective distinction into electric and magnetic field but only the one and only electromagnetic field, represented by the Faraday tensor.

Also the retarded solutions ("Jefimenko's equations") do not show that electric and magnetic fields are independent quantities. This becomes clear from the fact that electromagnetism is a gauge-field theory and there's the continuity equation as a consistency or integrability constraint,

[tex]\partial_t \rho + \vec{\nabla} \cdot \vec{j}=\partial_{\mu} j^{\mu}=0,[/tex]

of Maxwell's equations. Only for four-vector currents fulling the retarded solutions are really solutions of Maxwell's equations and thus only such currents and the corresponding retarded Faraday tensor describe a physically meaningful situation.

I consider Jackson's Classical Electrodynamics as the far better textbook on electromagnetism than Jefimenko's. A very good book is also Schwinger's Classical Electrodynamics and the best one is volume 2 and volume 8 of Landau and Lifhitz's Course on Theoretical Physics.

PhilDSP

Jackson probably should have placed a footnote reference on the term "in this author's text". Notice that "(Jefimenko)" appears immediately after it. I'd interpret that to refer to Jefimenko's 1966 book on EM which is listed in the bibliography. Jackson didn't include his own material in the bibliography so we would have to track down the various publications of his text book to see where Jackson's first publication of the material occurred.

True, in one sense at least. But Jefimenko brings further clarity to the question of what causes what and does not rest in complacency on his initial findings. He pursues that question and others very actively and deeply to arrive at at least new perspectives on older matters. He even has proposed experiments which potentially show different results from how the behavior of EM differs from common expectations.

Jefimenko injected a bit of renewed vitality and an alternate way of looking at the same old issues. That's worth something, isn't it? Not everyone is comfortable with the idea that we know as much about EM behavior as we ever will. And apparently the lessons of Weber, Lorentz, Lienard and Wiechert have never really been learned by the vast majority of 20th century students of electromagnetism.

PhilDSP

I probably agree with you there. Jackson presents a very broad scope of findings in many, many different areas. One that apparently attempts to survey most every significant technical observation in the past one hundred years. Jackson's text book may be the desert island type of book for most of us.

Russell E

There are two different ways of looking at electrodynamics, one based on the field interpretation of Faraday, Maxwell, Hertz, et al, and the other based on the retarded distant action interpretation of Weber, Lienard-Weichert, Tetrode-Fokker, et al. It was once thought that the phenomena of electromagnetic waves was unique to the field interpretation, so when Hertz discovered behavior that was interpreted as evidence of electromagnetic waves it was taken as a sign by many people that Maxwell was right and the action-at-a-distance theories were wrong. However, in later years it was realized that exactly the same phenomena that was interpreted as wave action also emerges from the action-at-a-distance interpretation - with suitable retarded action and boundary conditions. Ultimately all the observable effects are interactions between electric charges. (This of course is a hypothesis... it's conceivable in Maxwell's interpretation that there could be electromagnetic waves in the universe that never originated with any accelerating charges - but we have never observed any such thing.)

There are subtleties here, involving radiation reaction and the self-energy of a charged particle, that present difficulties for both the field interpretation and the distant-action interpretation. Wheeler-Feynman gave an account of radiation reaction within the context of distant action by making use of both the advanced and the retarded solutions, with an ideal absorber in the future. So, at least nominally, there is a viable distant-action interpretation of classical electrodynamics... at least as viable as the field interpretation.

It sometimes surprises students to learn that there is no fully self-consistent classical electrodynamics, either with fields or with distant-action, but see the last chapter of Jackson for why our classical theories work as well as they do, in spite of this.

Yes, I interpret it that way too.

Again, there is no new physics in writing down solutions to Maxwell's equations, so he should not be expecting experimental results to differ from what Maxwell's equations predict. You don't exactly say that he does... instead you say he predicts results that differ from "common expectations". Well, that's an odd thing to say. The common expectation is that experimental results will agree with Maxwell's equations, so if Jefimenko expects something different, his expectations are in conflict with Maxwell's equations - but he also claims to accept the validity of Maxwell's equations - so what you've described seems somewhat confused.

The Wikipedia article on Jefimenko suffers from the same confusion. It says it is commonly believed that a varying E field causes B, and a varying B field causes E, but it gives no reference that supports the claim that this is commonly believed (the one reference it cites actually says something very different), so it's really just a straw man - like your "common expectations".

PhilDSP

Sorry for the ambiguity. Jefimenko has a number of books that focus on different subjects, some of which he develops theory that extends to points which begin to be controversial. What I said was a rough generalization about this work. As far as I know he has made no claims that conflict with the Maxwell equations but he does entertain an alternate scenario for how relativity arises and what its observable effects are. But discussing that is probably off-topic both in this thread and in this forum.

One good reference for the Wikipedia article might be Feynman's basic E & M lecture for undergraduate students - the one included in the "Best of Feynman" audio recordings package. As I recall, he clearly says that a changing magnetic field produces an electric field and vice versa. I'll try to listen to that again to double-check.

Russell E

Indeed, that's crackpottery.

One has to be careful about the informal usage of words like "produces" and "causes". We can trace the flow of energy (for example), but ultimately it isn't possible to decide if the world is causal or not - we observe only associations. For example, Oersted knew how to make a changing electric field, and he found that when he did this, a magnetic field appears. Likewise Faraday knew a way of producing a changing magnetic field, and when he did this, an electric field appears. So one can say they "produced" one thing by doing the other thing, but obviously these kinds of associations can always be viewed in the opposite direction, or as simply associations, or better as different manifestations of the same phenomena. It's silly to make a big deal out of this, and accuse people of harboring fundamental misconceptions, just for using words like "produced" or "caused" when they really meant "is associated with" or some such.