Jefimenko's equations

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Jefimenko's equations describe the behavior of the electric and magnetic fields in terms of the sources at retarded times. Combined with the continuity equation, Jefimenko's equations^[1] are equivalent to Maxwell's equations of electromagnetism.

1 Electromagnetic field in vacuum
2 Electromagnetic field in dielectric and magnetic media
3 Discussion
4 Notes

[edit] Electromagnetic field in vacuum

In the vacuum, the electric field $\vec{E}$ and the magnetic field $\vec{B}$ are given in terms of the charge density $\rho\,$ and the current density $\vec{J}$ as:

$\vec{E}(\vec{r},t) = \frac{1}{4\pi\epsilon_0}\int{\left(\frac{\rho(\vec{r'},t_r)\,\vec{R}}{R^3}+\frac{\vec{R}}{R^2c}\frac{\partial\rho(\vec{r'},t_r)}{\partial t} - \frac{1}{Rc^2}\frac{\partial \vec{J}(\vec{r'},t_r)}{\partial t}\right)\mathrm{d}^3\vec{r'}}$

$\vec{B}(\vec{r},t) = \frac{\mu_0}{4\pi}\int{\left(\frac{\vec{J}(\vec{r'},t_r)\times\vec{R}}{R^3}+\frac{1}{R^2c}\frac{\partial \vec{J}(\vec{r'},t_r)}{\partial t}\times\vec{R}\right)\mathrm{d}^3\vec{r'}}$

where $\vec{R} = \vec{r} - \vec{r'}$ , and $t_r = t - R/c \,$ (the retarded time).

[edit] Electromagnetic field in dielectric and magnetic media

The two former expressions for the electric and magnetic fields admit extensions to the case of the electromagnetic field in an arbitrary dielectric and magnetic medium^[2]. The macroscopic fields $\vec{E}$ , $\vec{D}$ , $\vec{B}$ and $\vec{H}$ are then expressed in terms of charge density $\rho\,$ , current density $\vec{J}$ , polarization $\vec{P}$ , and magnetization $\vec{M}$ .

[edit] Discussion

There is a widespread interpretation of Maxwell's equations to the effect that time variable electric and magnetic fields can cause each other. This is often used as part of an explanation of the formation of electromagnetic waves. However, Jefimenko's equations show otherwise. ^[3] 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^[4], Jefimenko's equations (in the vacuum case) seems to appear first in 1962 in the second edition of Panofsky and Phillips's classic textbook^[5].

[edit] Notes

^ 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 978-0917406089. See also: David J. Griffith, Mark A. Heald, Time-dependent generalizations of the Biot-Savart and Coulomb laws, American Journal of Physics 59 (2) (1991), 111-117.
^ 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.
^ Oleg D. Jefimenko, Causality Electromagnetic Induction and Gravitation, 2nd ed.: Electret Scientific (Star City - 2000) Chapter 1, page 16 ISBN 0-917406-23-0.
^ 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.
^ 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 slighlty different - but completely equivalent - form. Reprint: Dover Publications (2005), ISBN 978-0486439242.