In
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
, Jefimenko's equations (named after
Oleg D. Jefimenko) give the
electric field and
magnetic field due to a distribution of
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
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
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
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, and electric circuits.
...
for any arbitrary distribution of charges and currents.
Equations
Electric and magnetic fields
Jefimenko's equations give the
electric field E and
magnetic field B produced by an arbitrary charge or current distribution, of
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in ...
''ρ'' and
current density J:
[Introduction to Electrodynamics (3rd Edition), D. J. Griffiths, Pearson Education, Dorling Kindersley, 2007, .]
where r′ is a point in the
charge distribution, r is a point in space, and
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 law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
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
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
, which were originally true only for
electrostatic
Electrostatics is a branch of physics that studies electric charges at rest ( static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amb ...
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 equati ...
fields, and steady currents.
Origin from retarded potentials
Jefimenko's equations can be found
from the
retarded potential
In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light ''c'', so th ...
s ''φ'' and A:
which are the solutions to
Maxwell's equations in the potential formulation, then substituting in the definitions of the
electromagnetic potential
An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. ...
s themselves:
and using the relation
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
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up ...
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 some lectures by Richard Feynman, a Nobel laureate who has sometimes been called "The Great Explainer". The lectures were presented before undergraduate students at the Californ ...
,'' where it was used to introduce and describe the origin of
electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) li ...
.
[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 law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
for cases where the source charge is moving:
Here, and are the electric and magnetic fields respectively, is the electric charge, is the vacuum permittivity (electric field constant) and is the speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
. The vector is a unit vector pointing from the observer to the charge and 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 .
The first term in the formula for represents the Coulomb's law for the static electric field. The second term is the time derivative of the first Coulombic term multiplied by 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 , 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 , where is the transverse acceleration in the retarded time. As it decreases only as with distance compared to the standard 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, and electric circuits.
...
using the technique of the retarded potential
In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light ''c'', so th ...
. It allows, for example, the derivation of the Larmor formula
In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light.
When any charged ...
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. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
). 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
Erwin Panofsky (March 30, 1892 in Hannover – March 14, 1968 in Princeton, New Jersey) was a German-Jewish art historian, whose academic career was pursued mostly in the U.S. after the rise of the Nazi regime.
Panofsky's work represents a hig ...
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 physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
were first derived by George Adolphus Schott
George Adolphus Schott (also referenced as George Augustus Schott) FRS (25 January 1868 – 15 July 1937) was a British mathematician. He is best known for developing the full theory of radiation from electrons travelling at close to the sp ...
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