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Concept
Jefimenko's equations
Summary
In electromagnetism, Jefimenko's equations (named after Oleg D. Jefimenko) give the electric field and magnetic field due to a distribution of electric charges 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.
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:
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 and the Biot–Savart law to electrodynamics, which were originally true only for electrostatic and magnetostatic fields, and steady currents.
Jefimenko's equations can be found from the retarded potentials φ and A:
which are the solutions to Maxwell's equations in the potential formulation, then substituting in the definitions of the electromagnetic potentials themselves:
and using the relation
replaces the potentials φ and A by the fields E and B.
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, where it was used to introduce and describe the origin of electromagnetic radiation. The formula provides a natural generalization of the Coulomb's law 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.
Official source
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