4.5.2 Jefimenko & Lorentz
A Middle Ground Between Maxwell & Weber?
There is, however a potential middle ground between a purely Maxwellian field model and the Kirchhoff-Weber action-at-a-distance model.
Oleg Jefimenko (1922–2009) demonstrated that electric fields (E) are a function of the charge density (ρ), the time rate of change of the charge density (∂ρ/∂t), and the time rate of change of the current density (∂J/∂t). Magnetic fields (H) are a function of the current density (J) and the time rate of change of the current density (∂J/∂t) [[i], [ii]]. The square backets denote that the sources are evaluated at “retarded” time, allowing for the speed-of-light time of propagation delay between activity at the source and its influence upon the field vector at a distant location. Further details are available in Jefimenko’s excellent introductory textbook [[iii]], and in a streamlined presentation, elsewhere [[iv]].
Jefimenko argued that the seeming link between electric and magnetic fields in free space electromagnetic waves is an artifact of their common origin in the charges and currents of the source, not a cause-and-effect relationship [[v]]. His approach is a good starting point for anyone who wants to treat charges and currents as primary within a Maxwellian framework.
To Hendrik Antoon Lorentz (1853-1928) we owe the Lorentz force law which captures how electric (E) and magnetic (B = μoH) fields exert a force (F) on a charge (q) moving at a particular velocity (v). The magnetic part of the force was earlier noted by Maxwell [[ix]] and Heaviside [[x], [xi]], but Lorentz was the first to combine them to determine the total force on a charged particle. The B-field, or magnetic flux density, describes the total magnetic field, including material effects which are captured by the magnetic permeability (in general μ, for free space, μo) as discussed in Section 3.4.6.
The B-field gives rise to the force on charges. Electrical engineers often use the H-field which better captures the free-space magnetic field due to antennas and other engineering applications.
By combining Jefimenko’s theory and the Lorentz force law, in principle one can obtain an action-at-a-distance approach equivalent to Weber electrodynamics, however, the answers the two approaches provide are not exactly the same. Researchers in this area argue that the Jefimenko-Lorentz forces are not correct and that they violate conservation of momentum [[xii]]. In addition, there is evidence for an additional electromagnetic force that is proportional to the velocity of a charge and acts parallel to the direction of motion rather than perpendicular [[xiii]]. These intriguing results would take us beyond the scope of our current investigations, however.
Next time: 4.5.3 A Synopsis: The Tripartite Model of How Electromagnetism Works
Full Table of Contents [click here]
Chapter 4 Electromagnetism Comes of Age
4.5 An Introduction to Electromagnetic Models
4.5.1 Potentials and Actions at a Distance
4.5.2 Jefimenko & Lorentz
4.5.3 A Synthesis
4.6 Hertz & Radiation Fields
4.7 How Does Radiation Work?
4.8 Summary & Conclusions
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References
[[i]] Griffiths, David J. and Mark A. Heald, “Time-dependent generalizations of the Biot-Savart and Coulomb laws,” American Journal of Physics Vol. 59, 1991, pp. 111 117.
[[ii]] Panofsky, Wolfgang K.H., and Melba Phillips, Classical Electricity and Magnetism, 2nd ed., New York: Dover, 2005.
[[iii]] Jefimenko, Oleg D., Electricity and Magnetism, New York: Appleton-Century-Crofts, 1966, §15-7, pp. 515 518.
[[iv]] Li, Shengchao Alfred, “Jefimenko Made Easy: Electromagnetic Fields through Radiation,” 25 June, 2023. See: https://arxiv.org/abs/2306.14930
[[v]] Jefimenko, Oleg D., Causality Electromagnetic Induction and Gravitation, Star City: Electret Scientific Company, 1992.
[[vi]] By kind permission of Dave Walker and Electret Scientific. See: http://electretscientific.com/author/author.html
[[vii]] Painting of Hendrik Lorentz by Menso Kamerlingh Onnes, 1916. See: https://en.wikipedia.org/wiki/Hendrik_Lorentz#/media/File:Hendrik_Antoon_Lorentz,_in_1916_geschilderd_door_Menso_Kamelingh_Onnes.jpg
[[viii]] Schantz, Hans G., The Art and Science of Ultrawideband Antennas, 2nd ed., Boston: Artech House, 2015, pp. 221-222.
[[ix]] Maxwell, James Clerk, A Treatise on Electricity and Magnetism, London: Oxford University Press, vol. II, 3rd ed., §602, 1892, pp. 243-246.
[[x]] Heaviside, Oliver, Electrical Papers, vol. 2, London: The Electrician Publishing Company, 1892, p. 506.
[[xi]] Nahin, Paul, Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age, Baltimore: The Johns Hopkins University Press, 2002, p. 120.
[[xii]] Kühn, Steffen, Private communication, 29 August, 2024. “I am convinced that the Lorentz force formula is only valid for the simplified Maxwell equations without Maxwell's addition. In other words, the Lorentz force formula is a model for forces generated by direct currents and for cases in which the test charge is so close to the field-generating current that the propagation time of the field/force can be neglected.”
[[xiii]] Kühn, Steffen, “Experimental investigation of an unusual induction effect and its interpretation as a necessary consequence of Weber electrodynamics,” Journal of Electrical Engineering, Vol. 72, Issue 6, December 2021, pp. 366-373. DOI: 10.2478/jee-2021-0052.