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Jefimenko's equations 
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#1
Apr2912, 01:09 AM

P: 440

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? 


#2
Apr2912, 02:47 AM

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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 timedependent 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 timevarying electric field causing a timevarying 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 framedependent construct, i.e., through Lorentz transformations they mix with each other as determined by the transformation law of a 2ndrank tensor. 


#3
Apr3012, 12:55 AM

P: 612

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 BiotSavart 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. 


#4
Apr3012, 06:24 AM

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Jefimenko's equations
To me this sounds somehow making the physics less intuitive. Of course, Maxwell's displacement current doesn't belong to the righthand side (sources) of the inhomogeneous Maxwell equations but to the lefthand 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 fourvector current of electric charge).
In relativistically covariant notation (in HeavisideLorentz 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] 


#5
Apr3012, 08:13 AM

P: 612

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. 


#6
Apr3012, 10:27 AM

P: 440

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? 


#7
Apr3012, 11:45 AM

P: 612

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. 


#8
Apr3012, 04:47 PM

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#9
May112, 03:11 AM

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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 fourvector 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 LienardWiechert equations for the radiation from an accelerated charged particle, which is a special case of these more general equations for arbitrary chargecurrent 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 gaugefield 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 fourvector 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. 


#10
May112, 05:54 AM

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#11
May112, 06:04 AM

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#12
May112, 11:42 AM

P: 97

There are subtleties here, involving radiation reaction and the selfenergy of a charged particle, that present difficulties for both the field interpretation and the distantaction interpretation. WheelerFeynman 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 distantaction interpretation of classical electrodynamics... at least as viable as the field interpretation. It sometimes surprises students to learn that there is no fully selfconsistent classical electrodynamics, either with fields or with distantaction, but see the last chapter of Jackson for why our classical theories work as well as they do, in spite of this. 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". 


#13
May212, 01:12 AM

P: 612

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 doublecheck. 


#14
May212, 10:14 AM

P: 97




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