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{{Electromagnetism|cTopic=Electrodynamics}}
In [[electromagnetism]], '''Jefimenko's equations''' (named after [[Oleg D. Jefimenko]]) describe the behavior of the [[electric field|electric]] and [[magnetic field]]s in terms of the [[electric charge|charge]] and [[electric current|current]] distributions at [[retarded time]]s.

Jefimenko's equations<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> are the solution of [[Maxwell's equations]] for an assigned distribution of electric charges and currents, under the assumption that there is no electromagnetic field other than the one produced by those charges and currents, that is no electromagnetic field coming from the infinite past.

== Equations ==

===Electric and magnetic fields===

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

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

:<math>\mathbf{E}(\mathbf{r}, t) = \frac{1}{4 \pi \epsilon_0} \int \left[ \left(\frac{\rho(\mathbf{r}', t_r)}{|\mathbf{r}-\mathbf{r}'|^3} + \frac{1}{|\mathbf{r}-\mathbf{r}'|^2 c}\frac{\partial \rho(\mathbf{r}', t_r)}{\partial t}\right)(\mathbf{r}-\mathbf{r}') - \frac{1}{|\mathbf{r}-\mathbf{r}'| c^2}\frac{\partial \mathbf{J}(\mathbf{r}', t_r)}{\partial t} \right] \mathrm{d}^3 \mathbf{r}'</math>

:<math>\mathbf{B}(\mathbf{r}, t) = \frac{\mu_0}{4 \pi} \int \left[\frac{\mathbf{J}(\mathbf{r}', t_r)}{|\mathbf{r}-\mathbf{r}'|^3} + \frac{1}{|\mathbf{r}-\mathbf{r}'|^2 c}\frac{\partial \mathbf{J}(\mathbf{r}', t_r)}{\partial t} \right] \times (\mathbf{r}-\mathbf{r}') \mathrm{d}^3 \mathbf{r}'</math>

where '''r'''' is a point in the [[charge distribution]], '''r''' is a point in space, and

:<math>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>Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3</ref> from the [[retarded potential]]s φ and '''A''':

:<math>\begin{align}
& \varphi(\mathbf{r},t) = \dfrac{1}{4\pi \epsilon_0} \int \dfrac{\rho(\mathbf{r}',t_r)}{|\mathbf{r}-\mathbf{r}'|} \mathrm{d}^3 \mathbf{r}'\\
& \mathbf{A}(\mathbf{r},t) = \dfrac{\mu_0}{4\pi} \int \dfrac{\mathbf{J}(\mathbf{r}',t_r)}{|\mathbf{r}-\mathbf{r}'|} \mathrm{d}^3 \mathbf{r}'\\
\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>- \mathbf{E} = \nabla\varphi + \dfrac{\partial \mathbf{A}}{\partial t}\,, \quad \mathbf{B} = \nabla \times \mathbf{A} </math>

and using the relation

:<math>c^2 = \frac{1}{\epsilon_0\mu_0} </math>

replaces the potentials φ and '''A''' by the fields '''E''' and '''B'''.

==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
| page=1687
| doi=10.1088/0143-0807/32/6/022
| arxiv=1106.1792 |bibcode = 2011EJPh...32.1687K }}
</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 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 Panofsky and 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> 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> The second feature is that the expression for '''E''' does not depend upon '''B''' and vice versa. Hence, it is impossible for '''E''' and '''B''' fields to be "creating" each other. Charge density and current density are creating them both.

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

== Notes ==

<references/>

{{DEFAULTSORT:Jefimenko's Equations}}
[[Category:Electrodynamics]]
[[Category:Electromagnetism]]