The telegraph model of Weber-Gauss, relying on Weber's electrodynamics, modeled instantaneous action at a distance of the electric scalar potential (Coulomb potential) manifesting as propagation of signals in a resistanceless wire, through the collective forces of charge carriers in the wire.

Despite seeing many allusions to the notion that there was work on modeling wireless transmission of signals with Weber's electrodynamics I have been unable to locate such a model even for a simple Hertzian dipole acting on a remote test charge. Does such a wireless signaling model exist?

Some relevant background: Maxwell cited Weber's model in a positive light and objected to it because of a faulty critique by Helmholtz to the effect that it necessarily violated conservation of energy. Weber debunked this critique and later Maxwell retracted his critique but, presumably due to the absence of a way to describe electromagnetic energy propagation through space, he left it at that. This is most ironic not only because the speed of light was first described in Weber's work, but also because the vector potential originated with Weber's colleague, Kirchhoff. And that, presumably, contributed to Maxwell's development of his equations.

Maxwell did indeed criticize Weber’s law for electric force in 1864 based on a confusion: “mechanical difficulties, however, which are involved in the assumption of particles acting at a distance with forces which depend on their velocities are such as to prevent me from considering this theory as an ultimate one”. The “mechanical difficulties” refer to the fact that velocity only dependent forces must violate the conservation of energy. Weber’s force however also depends on acceleration, and so is consistent with conservation. After realizing his mistake Maxwell shifted his reasoning to what we now call the principle of locality, which he inherited from Faraday and favored already in 1864. The issue was that in Weber’s electrodynamics the electromagnetic energy and momentum emitted by one body and absorbed by another had no place to be in between, there was no medium of transmission:”I have therefore preferred to seek an explanation of the facts… without assuming the existence of forces capable of acting directly at sensible distances”. It is worth mentioning that the speed of light plays a different role in Weber’s electrodynamics since there are no waves in it, and the forces act instantaneously at "sensible" distances.

According to Wesley’s Weber Electrodynamics (which is very sympathetic to Weber) it was exactly its inability to explain Hertz’s electromagnetic signaling that led to Maxwell’s field theory superseding it. So presumably there weren’t satisfactory models at the time. Wesley himself develops what he calls “Weber field theory” by introducing potentials, and develops Weber style model of Pappas-Vaughan experiment with Z-antenna.

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    Do what about the answer to the question "Does (did) such a wireless signaling model exist? – Alexandre Eremenko Dec 4 '15 at 1:23

No matter how much I admire Weber's potential and force laws, these laws don't predict the physics of optics (for instance reflection and refraction, interference, etc ...). The Maxwell field theory predicts laws of optics as well.

However, Weber's force law in case of magnetostatics is exactly Ampere's force law which satisfies Newton's third principle of motion, where the Lorentz force violates this principle in case of magnetostatics. The violation by Grassmann's force law of momentum conservation is NOT compensated by the momentum of fields. Magnetostatica experiments are such that the momentum of fields do not compensate for the inbalance in the forces on massive current carrying objects (metalic conductors).

Perhaps it is wise to reconsider the Lorentz force law, and replace it for a better electromagnetic field based force law, which agrees with Newtonean classical mechanics. Whittaker was working on this problem.

The retardation of fields and the finite velocity of near fields and far fields can be measured, however, no experiment clearly confirms the Coulomb near fields also spread with TEM wave far field velocity. On the contrary, several Coulomb field speed measurements show super luminal velocity, for instance, Wheatstone's famous rotating mirror experiment shows super luminal velocity of 'electricity' (partly wired partly wireless path).

Since Weber's laws relies only on the relative distance between two charges, it is reference frame independent, also independent from relative rotation of frames. I wonder if all electrodynamics in a rotating lab frame is the same as electrodynamics in a non-rotating lab frame.

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