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Solution for electric field and magnetic field due to a distribution of moving electric charges and electric current in space.
replaces the potentials ? and A by the fields E and B.
Explanation of the variables relevant for the Heaviside-Feynman formula.
The Heaviside-Feynman formula, also known as the Jefimenko-Feynman formula, is a special case of Jefimenko's equations obtained when the source is a single point-like electric charge. 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 and is the speed of light. 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.
Illustration of the retarded charge position for a particle moving in one spatial dimension: the observer sees the particle where it was, not where it is.
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 Coulumbic behavior, this term is responsible for the long-range electromagnetic radiation caused by the accelerating charge.
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). 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 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." 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.
^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), ISBN978-0-917406-08-9. 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.
^ abcIntroduction to Electrodynamics (3rd Edition), D. J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN81-7758-293-3.
^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.
^ abFeynman, R. P., R .B. Leighton, and M. Sands, 1965, The Feynman Lectures on Physics, Vol. I, Addison-Wesley, Reading, Massachusetts
^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 slightly different - but completely equivalent - form. Reprint: Dover Publications (2005), ISBN978-0-486-43924-2.
^Andrew Zangwill, Modern Electrodynamics, Cambridge University Press, 1st edition (2013), pp. 726--727, 765