Talk:Jefimenko's equations

Do the Jefimenko equations need to be supplemented with the continuity equation or the Lorenz gauge condition to be equivalent to Maxwell's laws?

The answer is that the Jefimenko equations as written require the Lorenz gauge condition to be equivalent to Maxwell's laws. All you have to do is try to calculate the divergence of the equation for the electric field and you will find that the continuity equation won't cancel out the terms correctly. —Preceding unsigned comment added by JoeRoser (talk • contribs) 05:39, 24 December 2008 (UTC)

They require other conditions as well, see the JRSpriggs 2006 post at the bottom of this page. -- Army1987 – Deeds, not words. 15:24, 6 January 2009 (UTC)

Retarded versus Advanced time

Would it not be more appropriate to call $t_{r}=t-R/c$ the "advanced" time since it is EARLIER than t? $t+R/c$ would be the retarded time since it is later. Would it not? JRSpriggs 06:02, 25 April 2006 (UTC)

The whole thing is called retardation because the electric and the magnetic fields at a point are not affected by changes in the charge and current density at a remote point immediately, but only at a later time. It is a standard terminology to call the time that needs to be substituted in the integral the retarded time. Yevgeny Kats 11:30, 25 April 2006 (UTC)

Suggestions and questions

• What units are you using? Is it International System of Units or Centimeter gram second system of units?

• Are there similar equations for the vector potential and voltage? If so, what are they and what gauge (such as the Lorenz gauge condition) do they assume?

Assuming the Lorenz gauge condition and Gaussian units...

$\phi ({\vec {r}},t)=\int {\frac {\rho ({\acute {\vec {r}}},t_{r})}{R}}\mathrm {d} ^{3}{\acute {r}}$

${\vec {A}}({\vec {r}},t)=\int {\frac {{\vec {J}}({\acute {\vec {r}}},t_{r})}{Rc}}\mathrm {d} ^{3}{\acute {r}}$

• Please provide a proof that Jefimenko's equations are Lorentz invarient and so consistent with special relativity. It is clear that they should be, if they are going to be equivalent to Maxwell's equations. But it is not obvious to me that they are. Of course, they cannot be consistent with general relativity without modification because they implicitly assume a uniform metric for space-time.

• Please provide a proof that they are equivalent to Maxwell's equations as claimed.

Here's an abbreviated part of the proof (the easy part) in Gaussian units...

${\vec {B}}=\int {\vec {J}}\times {\frac {\vec {R}}{R^{3}c}}+{\frac {\partial {\vec {J}}}{\partial t}}\times {\frac {\vec {R}}{R^{2}c^{2}}}dr^{3}$ (Jefimenko's equation)

${\vec {B}}=\int {\vec {J}}\times {\frac {\vec {R}}{R^{3}c}}-{\frac {\partial {\vec {J}}}{\partial t}}\times {\frac {\nabla t_{r}}{Rc}}dr^{3}$ (gradient of the retarded time inserted)

${\vec {B}}=\int {\vec {J}}\times {\frac {\vec {R}}{R^{3}c}}+{\frac {1}{Rc}}\nabla \times {\vec {J}}dr^{3}$ (from the chain rule)

${\vec {B}}=\int \nabla \left({\frac {1}{Rc}}\right)\times {\vec {J}}+{\frac {1}{Rc}}\nabla \times {\vec {J}}dr^{3}$ (gradient of 1/R inserted)

${\vec {B}}=\int \nabla \times \left({\frac {\vec {J}}{Rc}}\right)dr^{3}$ (using the identify for the curl of a scalar times a vector)

$\nabla \cdot {\vec {B}}=0$ (divergence of a curl is zero)

${\vec {E}}=\int \rho {\frac {\vec {R}}{R^{3}}}+{\frac {\partial \rho }{\partial t}}{\frac {\vec {R}}{R^{2}c}}-{\frac {\partial {\vec {J}}}{\partial t}}{\frac {1}{Rc^{2}}}dr^{3}$ (Jefimenko's equation)

${\vec {E}}=\int -\rho \nabla \left({\frac {1}{R}}\right)+{\frac {\partial \rho }{\partial t}}{\frac {\vec {R}}{R^{2}c}}-{\frac {\partial {\vec {J}}}{\partial t}}{\frac {1}{Rc^{2}}}dr^{3}$ (gradient of 1/R inserted)

${\vec {E}}=\int -\rho \nabla \left({\frac {1}{R}}\right)-{\frac {\partial \rho }{\partial t}}{\frac {\nabla t_{r}}{R}}-{\frac {\partial {\vec {J}}}{\partial t}}{\frac {1}{Rc^{2}}}dr^{3}$ (gradient of the retarded time inserted)

${\vec {E}}=\int -\rho \nabla \left({\frac {1}{Rc}}\right)-{\frac {\nabla \rho }{R}}-{\frac {\partial {\vec {J}}}{\partial t}}{\frac {1}{Rc^{2}}}dr^{3}$ (from the chain rule)

${\vec {E}}=\int -\nabla \left({\frac {\rho }{R}}\right)-{\frac {\partial {\vec {J}}}{\partial t}}{\frac {1}{Rc^{2}}}dr^{3}$ (from the identity for the gradient of a product of scalars)

$\nabla \times {\vec {E}}=\int -\nabla \times {\frac {\partial {\vec {J}}}{\partial t}}{\frac {1}{Rc^{2}}}dr^{3}$ (the curl of a gradient is zero)

$\nabla \times {\vec {E}}=\int -{\frac {\partial }{\partial t}}\nabla \times \left({\frac {\vec {J}}{Rc^{2}}}\right)dr^{3}$ (curl and time differentiation commute)

$\nabla \times {\vec {E}}=-{\frac {1}{c}}{\frac {\partial {\vec {B}}}{\partial t}}$ (expression for the magnetic field in the above step inserted)

• I think you should mention that the equivalence of Jefimenko's equations plus the continuity equation with Maxwell's equations depends on some assumptions, including: (1) there are no cosmological electric or magnetic fields (i.e. none which extend infinitely in space), (2) there is no electromagnetic radiation coming in from the infinite past (and infinite distance), and (3) the integrals can be integrated, i.e. the distribution of charges and currents drops off rapidly enough to zero as the distance from the origin increases. JRSpriggs 05:53, 5 September 2006 (UTC)

• • Indeed. The "Jefimenko says" quote is the boundary condition under which J's eqs are derived, saying "they show it" is very unhelpful. I'm trying to fix that. -- Army1987 – Deeds, not words. 15:24, 6 January 2009 (UTC)

• These equations ignore induced current, because the conductivity parameter of Maxwell's equations is missing. Toolnut (talk) 01:55, 21 June 2011 (UTC)

• Why is ${\vec {r}}_{s}(t_{r})$ used instead of plain ${\vec {r}}_{s}$? A displacement that is a function of time usually means motion -- is something like that implied? — Preceding unsigned comment added by 180.151.2.20 (talk) 05:02, 13 July 2011 (UTC)

Explanation needed

What does Jefimenko mean exactly in the last quotation? How does this come out of the equations? Also, what do these equations say differently about electric and magnetic fields than Maxwell's? Diagrams would be helpful for the less mathematically adept. JKeck (talk) 21:10, 20 January 2011 (UTC)

Causality

In the "Discussion" the notion that "electric and magnetic fields can cause each other" is mentioned as an often used explanation of electromagnetic waves. In an early version of the text this was followed by "Jefimenko's equations show otherwise". This can not be justified. Jefimenko's equations do nothing else than computing E and B.

Also, Jefimenko's own statement which is quoted: "...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields." is clearly not denying that the fields can cause each other. He only says that Maxwell's equations don't show that they cause each other. In the same way his own equations don't show that they do not cause each other. — Preceding unsigned comment added by Jos.Bergervoet (talk • contribs) 13:11, 18 December 2011 (UTC)

More Causality

There is much misunderstanding in this section. E and B are components of the electromagnetic field tensor. This object is sourced by the four-current density via Maxwell's equations. Done. The electromagnetic field does not carry electromagnetic charge, so the field cannot produce itself. This is well-understood stuff guys... 2607:F278:410E:5:9EB6:D0FF:FEF3:F2F1 (talk) 05:30, 17 January 2019 (UTC)

While Jos raises an important question with regard to whether or not E an B fields actually do in fact cause each other, clearly only experiment can answer such a question. The questions here only relate to Maxwell's equations and what they show. Jos says that Jefimenko only computes E and B. But clearly his solutions to Maxwell's equations show that a calculation for E in no way depends on B and a calculation for B in no way depends on E. To me saying they "cause each other" is like saying that the velocity of my car is caused by the price of eggs in China. There may even be correlation with the egg price, but correlation is not causality. — Preceding unsigned comment added by 99.159.251.126 (talk) 18:59, 15 February 2012 (UTC)

I think there's a problem with this article, because it says there is a long standing tradition that E and B "cause each other" in an electromagnetic wave, but the reference given for this statement (reference 2, Kinsler) does NOT say that. To the contrary, it says specifically that E causes B, and not the other way around. (Obviously if someone says E and B cause EACH OTHER, then they would be using the word "cause" in a different sense than is customary.) Kinsler's paper is making a sophisticated argument about criteria for causality. It says that E causes B based on a detailed and sophisticated criterion for the definition of causation, something which this Wikipedia article isn't even considering. So I think the article needs to have a reference for the claim about the long-standing tradition that E and B "cause each other". The existing reference definitely does NOT say that.

Also, the comments from user 99.159.251.126 above are not really valid, because one cannot infer causation from the functional dependence of expressions for physical quantities. For example, it's well known that different choices of gauge (Lorentz or Coulomb) give different equations for the electromagnetic forces that, viewed naively, would suggest either instantaneous (acausal) propagation or else retarded (causal) propagation. It is known (see the paper by Jackson, for example) that the propagation actually is causal, despite the fact that the equations can be written in a form that superficially appears non-causal. This Wikipedia article is perpetuating the error of thinking that the causality of phenomena can be inferred naively from the form of the equations. It's possible that Jefimenko himself believed this, and if so, it's fine to say he believed this, but it shouldn't be stated as fact. There is an abundance of reputable references explaining why that naive view is false. — Preceding unsigned comment added by Prion1 (talk • contribs) 15:31, 7 April 2012 (UTC)

It is easy to show that Jefimenko's equations imply nothing about causality: we can replace the retarded times by advanced times in the equations, and still obtain perfectly valid solutions to Maxwell's equations. Jefimenko's equations only provide *one* set of solutions. The difference in advanced and retarded solutions is of course a solution to the source-free ("vacuum") Maxwell equations. There are of course many many other solutions since we can add on any source-free solution to Jefimenko's equation, and still be valid. I would suggest correcting the article in all places where it refers to Jefimenko's as "The" solution to Maxwell. --Nanite (talk) 15:53, 14 July 2016 (UTC)

Errors

Several formulas use $\mathrm {d} ^{3}\mathbf {r} ',$ at the end of integral expressions, that is not correct. Even if changed to $(\mathrm {d} \mathbf {r} ')^{3},$ it is still incorrect because it should be a scalar volume and this is a vector, could author check and correct this, please. — Preceding unsigned comment added by 109.121.109.168 (talk) 20:12, 19 November 2021 (UTC)

Indeed a volume is a scalar and not a vector so I replaced $\mathrm {d} ^{3}\mathbf {r} '$ by the more common dV'. — Preceding unsigned comment added by Dodeluc (talk • contribs) 15:44, 20 November 2021 (UTC)