View source for Jefimenko's equations

{{Short description|Equations of electromagnetism}}
{{Electromagnetism|cTopic=Electrodynamics}}
In [[electromagnetism]], '''Jefimenko's equations''' (named after [[Oleg D. Jefimenko]]) give the [[electric field]] and [[magnetic field]] due to a distribution of [[electric charge]]s and [[electric current]] in space, that takes into account the propagation delay ([[retarded time]]) of the fields due to the finite [[speed of light]] and relativistic effects.  Therefore, they can be used for ''moving'' charges and currents.  They are the particular solutions to [[Maxwell's equations]] for any arbitrary distribution of charges and currents.<ref>[[Oleg D. Jefimenko]], ''Electricity and Magnetism: An Introduction to the Theory of Electric and Magnetic Fields'', Appleton-Century-Crofts (New-York - 1966). 2nd ed.: Electret Scientific (Star City - 1989), {{ISBN|978-0-917406-08-9}}. See also: [[David J. Griffiths]], Mark A. Heald, ''Time-dependent generalizations of the Biot–Savart and Coulomb laws'', American Journal of Physics '''59 (2)''' (1991), 111-117.</ref>

==Equations==

===Electric and magnetic fields===

[[File:Universal charge distribution.svg|250px|right|thumb|Position vectors '''r''' and '''r'''′ used in the calculation]]

Jefimenko's equations give the [[electric field]] '''E''' and [[magnetic field]] '''B''' produced by an arbitrary charge or current distribution, of [[charge density]] ''ρ'' and [[current density]] '''J''':<ref name=":0">Introduction to Electrodynamics (3rd Edition), D. J. Griffiths, Pearson Education, Dorling Kindersley, 2007, {{ISBN|81-7758-293-3}}.</ref>

<math display="block">\mathbf{E}(\mathbf{r}, t) = \frac{1}{4 \pi \varepsilon_0} \int \left[\frac{\mathbf{r}-\mathbf{r}'}{|\mathbf{r}-\mathbf{r}'|^3}\rho(\mathbf{r}', t_r) + \frac{\mathbf{r}-\mathbf{r}'}{|\mathbf{r}-\mathbf{r}'|^2}\frac{1}{c}\frac{\partial \rho(\mathbf{r}', t_r)}{\partial t} - \frac{1}{|\mathbf{r}-\mathbf{r}'|}\frac{1}{c^2}\frac{\partial \mathbf{J}(\mathbf{r}', t_r)}{\partial t} \right] dV',</math>
<math display="block">\mathbf{B}(\mathbf{r}, t) = -\frac{\mu_0}{4 \pi} \int \left[\frac{\mathbf{r}-\mathbf{r}'}{|\mathbf{r}-\mathbf{r}'|^3} \times \mathbf{J}(\mathbf{r}', t_r) + \frac{\mathbf{r}-\mathbf{r}'}{|\mathbf{r}-\mathbf{r}'|^2} \times \frac{1}{c} \frac{\partial \mathbf{J}(\mathbf{r}', t_r)}{\partial t} \right] dV',</math>

where '''r'''′ is a point in the [[charge distribution]], '''r''' is a point in space, and
<math display="block">t_r = t - \frac{|\mathbf{r}-\mathbf{r}'|}{c}</math>
is the [[retarded time]]. There are similar expressions for '''D''' and '''H'''.<ref>Oleg D. Jefimenko, ''Solutions of Maxwell's equations for electric and magnetic fields in arbitrary media'', American Journal of Physics '''60 (10)''' (1992), 899–902.</ref>

These equations are the time-dependent generalization of [[Coulomb's law]] and the [[Biot–Savart law]] to [[electrodynamics]], which were originally true only for [[electrostatic]] and [[magnetostatic]] fields, and steady currents.

===Origin from retarded potentials===

Jefimenko's equations can be found<ref name=":0" /> from the [[retarded potential]]s ''φ'' and '''A''':
<math display="block">\begin{align}
& \varphi(\mathbf{r},t) = \dfrac{1}{4\pi \varepsilon_0} \int \dfrac{\rho(\mathbf{r}',t_r)}{|\mathbf{r}-\mathbf{r}'|} dV',\\
& \mathbf{A}(\mathbf{r},t) = \dfrac{\mu_0}{4\pi} \int \dfrac{\mathbf{J}(\mathbf{r}',t_r)}{|\mathbf{r}-\mathbf{r}'|} dV',
\end{align}</math>
which are the solutions to [[maths of EM field#Potential field approach|Maxwell's equations in the potential formulation]], then substituting in the definitions of the [[electromagnetic potential]]s themselves:
<math display="block"> \mathbf{E} = -\nabla\varphi - \dfrac{\partial \mathbf{A}}{\partial t}\,, \quad \mathbf{B} = \nabla \times \mathbf{A} </math>
and using the relation
<math display="block">c^2 = \frac{1}{\varepsilon_0\mu_0} </math>
replaces the potentials ''φ'' and '''A''' by the fields '''E''' and '''B'''.

==Heaviside–Feynman formula==
[[Image:Explanation of variables in Heaviside-Feynman formula.svg|thumb|400px|right|Explanation of the variables relevant for the Heaviside–Feynman formula.]]

The '''Heaviside–Feynman formula''', also known as the Jefimenko–Feynman formula, can be seen as the [[point particle|point-like]] electric charge version of Jefimenko's equations. Actually, it can be (non trivially) deduced from them using [[Dirac function]]s, or using the [[Liénard-Wiechert potentials]].<ref>[https://feynmanlectures.caltech.edu/II_21.html#Ch21-S5 The Feynman Lectures on Physics - 21.5 The potentials of a moving charge; the general solution of Liénard and Wiechert]</ref> It is mostly known from ''[[The Feynman Lectures on Physics]],'' where it was used to introduce and describe the origin of [[electromagnetic radiation]].<ref name=FeynmanLectures>[https://feynmanlectures.caltech.edu/I_28.html#Ch28-S1-p10 The Feynman Lectures on Physics Vol. I Ch. 28: Electromagnetic Radiation]</ref> The formula provides a natural generalization of the [[Coulomb's law]] for cases where the source charge is moving:
<math> \mathbf{E} =  \frac{-q}{4\pi \varepsilon_0} \left[ \frac{\mathbf{e}_{r'}}{r'^2} + \frac{r'}{c} \frac{d}{dt} \left(\frac{\mathbf{e}_{r'}}{r'^2}\right)  +\frac{1}{c^2} \frac{d^2}{dt^2} \mathbf{e}_{r'}  \right] </math>
<math display="block"> \mathbf{B} = - \mathbf{e}_{r'} \times \frac{\mathbf{E}}{c} </math>
Here, <math> \mathbf{E} </math> and <math>\mathbf{B}</math> are the electric and magnetic fields respectively, <math> q</math> is the electric charge, <math>\varepsilon_0</math> is the [[vacuum permittivity]] (electric field constant) and <math>c</math> is the [[speed of light]]. The vector <math>\mathbf{e}_{r'}</math> is a unit vector pointing from the observer to the charge and <math>r'</math> is the distance between observer and charge. Since the [[electromagnetic field]] propagates at the speed of light, both these quantities are evaluated at the retarded time <math> t - r'/c </math>.

[[Image:Diagram explaining charge position in the retarded time.svg|thumb|400px|right|Illustration of the retarded charge position for a particle moving in one spatial dimension: the observer sees the particle where it was, not where it is.]]
The first term in the formula for <math>\mathbf{E}</math> represents the Coulomb's law for the static electric field. The second term is the time derivative of the first Coulombic term multiplied by <math>\frac{r'}{c}</math> which is the propagation time of the electric field. Heuristically, this can be regarded as nature "attempting" to forecast what the present field would be by linear extrapolation to the present time.<ref name=FeynmanLectures /> The last term, proportional to the second derivative of the [[Unit vector|unit direction vector]] <math>e_{r'}</math>, is sensitive to charge motion perpendicular to the line of sight. It can be shown that the electric field generated by this term is proportional to <math> a_{t}/r' </math>, where <math>a_t</math> is the transverse acceleration in the retarded time. As it decreases only as <math>1/r'</math> with distance compared to the standard <math>1/r'^2</math> Coulombic behavior, this term is responsible for the long-range electromagnetic radiation caused by the accelerating charge.

The Heaviside–Feynman formula can be derived from [[Maxwell's equations]] using the technique of the [[retarded potential]]. It allows, for example, the derivation of the [[Larmor formula]] for overall radiation power of the accelerating charge.

==Discussion==
There is a widespread interpretation of Maxwell's equations indicating that spatially varying electric and magnetic fields can cause each other to change in time, thus giving rise to a propagating electromagnetic wave<ref>{{cite journal
| author=Kinsler, P.
| year=2011
| title=How to be causal: time, spacetime, and spectra
| journal=Eur. J. Phys.
| volume=32
| issue=6
| page=1687
| doi=10.1088/0143-0807/32/6/022
| arxiv=1106.1792 |bibcode = 2011EJPh...32.1687K | s2cid=56034806
}}
</ref> ([[electromagnetism]]). However, Jefimenko's equations show an alternative point of view.<ref>Oleg D. Jefimenko, ''Causality Electromagnetic Induction and Gravitation'', 2nd ed.: Electret Scientific (Star City - 2000) Chapter 1, Sec. 1-4, page 16 {{ISBN|0-917406-23-0}}.</ref> Jefimenko says, "...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously created by their common sources: time-variable electric charges and currents."<ref>[[Oleg D. Jefimenko]], ''Causality Electromagnetic Induction and Gravitation'', 2nd ed.: Electret Scientific (Star City - 2000) Chapter 1, Sec. 1-5, page 16 {{ISBN|0-917406-23-0}}.</ref>

As pointed out by [[Kirk Thomas McDonald|McDonald]],<ref>Kirk T. McDonald, ''The relation between expressions for time-dependent electromagnetic fields given by Jefimenko and by Panofsky and Phillips'', American Journal of Physics '''65 (11)''' (1997), 1074-1076.</ref> Jefimenko's equations seem to appear first in 1962 in the second edition of [[Wolfgang K. H. Panofsky|Panofsky]] and [[Melba Phillips|Phillips]]'s classic textbook.<ref>Wolfgang K. H. Panofsky, Melba Phillips, ''Classical Electricity And Magnetism'', Addison-Wesley (2nd. ed - 1962), Section 14.3. The electric field is written in a slightly different - but completely equivalent - form. Reprint: Dover Publications (2005), {{ISBN|978-0-486-43924-2}}.</ref> [[David J. Griffiths|David Griffiths]], however, clarifies that "the earliest explicit statement of which I am aware was by Oleg Jefimenko, in 1966" and characterizes equations in Panofsky and Phillips's  textbook as only "closely related expressions".<ref name=":0" /> According to [[Andrew Zangwill]], the equations analogous to Jefimenko's but in the Fourier [[frequency domain]] were first derived by [[George Adolphus Schott]] in his treatise Electromagnetic Radiation (University Press, Cambridge, 1912).<ref>Andrew Zangwill, Modern Electrodynamics, Cambridge University Press, 1st edition (2013), pp. 726—727, 765</ref>

Essential features of these equations are easily observed which is that the right hand sides involve "retarded" time which reflects the "causality" of the expressions. In other words, the left side of each equation is actually "caused" by the right side, unlike the normal differential expressions for Maxwell's equations where both sides take place simultaneously. In the typical expressions for Maxwell's equations there is no doubt that both sides are equal to each other, but as Jefimenko notes, "... since each of these equations connects quantities simultaneous in time, none of these equations can represent a causal relation."<ref>[[Oleg D. Jefimenko]], ''Causality Electromagnetic Induction and Gravitation'', 2nd ed.: Electret Scientific (Star City - 2000) Chapter 1, Sec. 1-1, page 6 {{ISBN|0-917406-23-0}}.</ref>

==See also==
* [[Liénard–Wiechert potential]]

==Notes==
{{reflist}}

{{DEFAULTSORT:Jefimenko's Equations}}
[[Category:Electrodynamics]]
[[Category:Electromagnetism]]
[[Category:Eponymous equations of physics]]