4.5.1 Potentials and Actions at a Distance
Does a Body Act Where It Is Not? Weberian Electrodynamics
Potentials: We may be running short on hands, but not theories, because we may alternatively consider potentials as primary. The previous chapter looked at scalar electric potentials as an equivalent way to understand how electrostatics works. It’s trickier to understand magnetic fields as the result of potentials, because the magnetic potential is a vector field. Furthermore, nearly all the physical effects we perceive from electricity and magnetism are due to the forces: the slope of the potential. That slope is the same if you add or subtract an arbitrary constant from the potential, making it difficult to claim any particular value is the “true” potential. This so-called “gauge invariance” is a fundamental concept in modern field theory.
Action at a Distance: It is entirely possible to describe electromagnetism in terms of forces acting on charges at a distance in analogy to Newtonian central forces. This was the original method of Coulomb and Ampère. However, to capture the behavior of magnetism, forces then become functions not only of distance, but also of velocity and acceleration. Further, the forces are no longer “central,” pointing to and from the charges in question, but instead may point in other directions. This approach has a long history dating back at least to 1857 when Gustav Robert Kirchhoff (1824–1887) expounded “On the Motion of Electricity in Wires” [[i]], and when Wilhelm Eduard Weber (1804–1891), who we met previously as a collaborator with famed geophysicist Johann Carl Friedrich Gauss (1777–1855), proposed his electrodynamics in 1864 [[ii]]. These works and explanatory notes are compiled in an excellent volume edited by André Koch Torres Assis (1962– ) [[iii]]. Weber’s model deserves further discussion.
In Weber’s model, force (F) between two charges (Q and q) depends upon free space permittivity (εo) and is a function of the charges’ relative distances (r), relative velocity (v), and relative acceleration (a), normalized by the speed of light (c). Emerging evidence that the transmission of electromagnetic effects is causal, progressive, and propagates at the speed of light eroded confidence in action-at-a-distance models, like Weber’s. Even today, work is underway to revive and apply Weberian thinking.
One can derive a telegrapher’s equation [[vii]] and wave equations from Weber’s electrodynamics [[viii], [ix]]. Steffen Kühn and others seek a synthesis between Weber and Maxwell [[x]].
There is no reason I know of why expressing electromagnetic interactions as direct forces between particles could not be mathematically equivalent to the Maxwellian field approach, provided the actions are retarded to account for the speed of light. The action-at-a-distance model, by focusing on interactions on charged particles enforces Newtonian action and reaction and may help expose weaknesses or flaws in the Maxwellian paradigm.
My personal objection to the action-at-a-distance model is that while it may be mathematically equivalent and useful in certain contexts, it prevents an understanding of the causal mechanism by which one charge influences another. Like Newton’s Universal Theory of Gravitation, Weber’s electrodynamics and similar theories describe what happens but frame no hypothesis for why. I find this story unsatisfying. The Maxwellian field model, as extended by Poynting and Heaviside to describe the flow of energy, holds the promise of understanding, at least in part, electromagnetism by meticulously following the progressive motion of energy from a starting cause to an ending effect as fields guide the flow of energy.
Next time: 4.5.2 Jefimenko & Lorentz: A Middle Ground Between Maxwell & Weber?
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]] Kirchhoff, Gustav, “On the motion of electricity in wires,” Philosophical Magazine, 13, 1857, pp. 393–412. See: https://archive.org/stream/londonedinburghp13maga#page/392/mode/2up.
[[ii]] Weber, Wilhelm, “Elektrodynamische Maassbestimmungen insbesondere über elektrische Schwingungen. Abhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, mathematisch-physischen Classe,” vol. 6, 1864, pp. 571–716. Reprinted in Wilhelm Weber’s Werke, Vol. 4, H. Weber (ed.), Berlin: Springer, 1894, pp. 105-241.
[[iii]] Assis, André Koch Torres, ed., Wilhelm Weber’s Main Works on Electrodynamics Translated into English Volume III: Measurement of Weber’s Constant c, Diamagnetism, the Telegraph Equation and the Propagation of Electric Waves at Light Velocity, Montreal: Apeiron, 2021. See: https://www.ifi.unicamp.br/~assis/Weber-in-English-Vol-3.pdf and https://amzn.to/3yyVBOL for very reasonably priced hardcopies.
[[iv]] Portrait of Gustav Kirchhoff, Smithsonian libraries (SIL-SIL14-k002-03). See: https://infogalactic.com/info/File:Gustav_Robert_Kirchhoff.jpg
[[v]] See: https://upload.wikimedia.org/wikipedia/commons/5/55/Wilhelm_Eduard_Weber_II.jpg
[[vi]] Assis, A.K.T.; Tajmar, M. Rotation of a superconductor due to electromagnetic induction using Weber’s electrodynamics. Ann. Fond. Louis Broglie 2019, 44, 111–123. From Li & Maher, Op. Cit.
[[vii]] Assis, A.K.T.; Hernandes, J., “Telegraphy equation from Weber’s electrodynamics,” IEEE Trans. Circuits Syst. II Express Briefs 2005, 52, pp. 289–292.
[[viii]] Kühn, Steffen, “Inhomogeneous wave equation, Liénard-Wiechert potentials, and Hertzian dipoles in Weber electrodynamics,” Electromagnetics, vol. 42, no. 8, 2022, pp. 571-593. DOI: 10.1080/02726343.2022.2161709
[[ix]] Li, Qingsong, and Simon Maher, “Deriving an Electric Wave Equation from Weber’s Electrodynamics,” Foundations 2023, 3, pp. 323-334. DOI: 10.3390/foundations3020024.
[[x]] Kühn, Steffen, “Weber–Maxwell electrodynamics: classical electromagnetism in its most compact and pure form,” Electromagnetics, 29 July, 2024. DOI: 10.1080/02726343.2024.2375328
I realize this topic isn’t really the point of this book. But there is a lot said in this post in a very short space. It assumes an understanding of many things that are not readily accessible to a layman.
If I may, I would like to suggest more elaboration on this issue. I think this it is a very important discussion. I’m not sure that the criticisms of Weber’s theory are unavoidable or perhaps to some extent misunderstandings of the physical meaning of the math, etc. Important discoveries could be made by hashing it out point by point.
The relatively stable behavior of the solar system indicates there is something operating with gravity that is not just action at a distance or modern gravitational theory. The mathematics of Mechanics shows that you can't have stable inverse-square orbits in the presence of a non-zero torque on the system. You can demonstrate that the fundamental force equation from Newton's work, namely
F = -GMm(r^-2)
produces stable orbits, and that if you introduce non-zero torques that orbit becomes unstable.
Noting that the fundamental law is not time dependent, and recognizing that modern gravitational interactions are tensor not vector based, what happens when we introduce the condition of the speed of light being a constant and nowhere close to being "instantaneous" in traversing the distances from the Sun to Pluto? [Yes, IAU. Pluto. Lowell remembers; Lowell shall repay.]
In the case of the Earth, if the force of gravitation is mediated by gravitons, then the speed of that information exchange is light-speed limited. This implies that the total information exchange cycle between Sun and Earth is 17 minutes. How does the Sun "know" where to throw its graviton after it has received one from the Earth, considering that the Earth has moved several diameters away from the position it threw its first graviton at the Sun? To get those gravitons to their intended targets would imply some type of non-radial force acting on the system (torque-like), which should be detrimental to stable planetary motion. This effect should be even greater for planets more distant from the Sun, as the delay in exchange is longer.
But, the solar system has not yet flown apart, which indicates that there is something we don't yet understand about gravitation within modern theories. Is it because of an as-yet undiscovered medium than can communicate at superluminal speeds, action at a distance, or something else? It bears additional investigation and better explanation.
People in the "shut up and calculate" camp should get on with their own shutting up and calculating while the interested investigators continue to investigate.
Regardless of the reason, it's certainly not wrong to question these long-held, supposedly settled notions of Science, and especially those of Physics. This is why I follow our host's postings.