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In
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
, Jefimenko's equations (named after Oleg D. Jefimenko) give the
electric field An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
and
magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
due to a distribution of
electric charge Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
s and
electric current An electric current is a flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. The moving particles are called charge c ...
in space, that takes into account the propagation delay ( retarded time) of the fields due to the finite
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
and relativistic effects. Therefore, they can be used for ''moving'' charges and currents. They are the particular solutions to
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
for any arbitrary distribution of charges and currents.


Equations


Electric and magnetic fields

Jefimenko's equations give the
electric field An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
E and
magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
B produced by an arbitrary charge or current distribution, of charge density ''ρ'' and current density J:Introduction to Electrodynamics (3rd Edition), D. J. Griffiths, Pearson Education, Dorling Kindersley, 2007, . \mathbf(\mathbf, t) = \frac \int \left frac\rho(\mathbf', t_r) + \frac\frac\frac - \frac\frac\frac \rightdV', \mathbf(\mathbf, t) = -\frac \int \left frac \times \mathbf(\mathbf', t_r) + \frac \times \frac \frac \rightdV', where r′ is a point in the charge distribution, r is a point in space, and t_r = t - \frac is the retarded time. There are similar expressions for D and H. These equations are the time-dependent generalization of
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that calculates the amount of force (physics), force between two electric charge, electrically charged particles at rest. This electric for ...
and the
Biot–Savart law In physics, specifically electromagnetism, the Biot–Savart law ( or ) is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the ...
to electrodynamics, which were originally true only for
electrostatic Electrostatics is a branch of physics that studies slow-moving or stationary electric charges. Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word (), mean ...
and
magnetostatic Magnetostatics is the study of magnetic fields in systems where the currents are steady (not changing with time). It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equat ...
fields, and steady currents.


Origin from retarded potentials

Jefimenko's equations can be found from the retarded potentials ''φ'' and A: \begin & \varphi(\mathbf,t) = \dfrac \int \dfrac dV',\\ & \mathbf(\mathbf,t) = \dfrac \int \dfrac dV', \end which are the solutions to Maxwell's equations in the potential formulation, then substituting in the definitions of the electromagnetic potentials themselves: \mathbf = -\nabla\varphi - \dfrac\,, \quad \mathbf = \nabla \times \mathbf and using the relation c^2 = \frac replaces the potentials ''φ'' and A by the fields E and B.


Heaviside–Feynman formula

The Heaviside–Feynman formula, also known as the Jefimenko–Feynman formula, can be seen as the point-like electric charge version of Jefimenko's equations. Actually, it can be (non trivially) deduced from them using Dirac functions, or using the Liénard-Wiechert potentials. It is mostly known from ''
The Feynman Lectures on Physics ''The Feynman Lectures on Physics'' is a physics textbook based on a great number of lectures by Richard Feynman, a Nobel laureate who has sometimes been called "The Great Explainer". The lectures were presented before undergraduate students ...
,'' where it was used to introduce and describe the origin of
electromagnetic radiation In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength ...
.The Feynman Lectures on Physics Vol. I Ch. 28: Electromagnetic Radiation
/ref> The formula provides a natural generalization of the
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that calculates the amount of force (physics), force between two electric charge, electrically charged particles at rest. This electric for ...
for cases where the source charge is moving: \mathbf = \frac \left \frac + \frac \frac \left(\frac\right) +\frac \frac \mathbf_ \right \mathbf = - \mathbf_ \times \frac Here, \mathbf and \mathbf are the electric and magnetic fields respectively, q is the electric charge, \varepsilon_0 is the
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
(electric field constant) and c is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
. The vector \mathbf_ is a unit vector pointing from the observer to the charge and r' is the distance between observer and charge. Since the
electromagnetic field An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
propagates at the speed of light, both these quantities are evaluated at the retarded time t - r'/c . The first term in the formula for \mathbf represents the Coulomb's law for the static electric field. The second term is the time derivative of the first Coulombic term multiplied by \frac 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. The last term, proportional to the second derivative of the unit direction vector e_, 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 a_/r' , where a_t is the transverse acceleration in the retarded time. As it decreases only as 1/r' with distance compared to the standard 1/r'^2 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 Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
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 (
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
). However, Jefimenko's equations show an alternative point of view. 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." As pointed out by McDonald, Jefimenko's equations seem to appear first in 1962 in the second edition of Panofsky and Phillips's classic textbook. 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". According to Andrew Zangwill, the equations analogous to Jefimenko's but in the Fourier
frequency domain In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
were first derived by George Adolphus Schott in his treatise Electromagnetic Radiation (University Press, Cambridge, 1912). 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." Oleg D. Jefimenko, ''Causality Electromagnetic Induction and Gravitation'', 2nd ed.: Electret Scientific (Star City - 2000) Chapter 1, Sec. 1-1, page 6 .


See also

* Liénard–Wiechert potential


Notes

{{DEFAULTSORT:Jefimenko's Equations Electrodynamics Electromagnetism Eponymous equations of physics