Historical research guide · 0455 · 2023
Deriving an Electric Wave Equation from Weber’s Electrodynamics
Qingsong Li · Simon Maher
In one minute
Qingsong Li and Simon Maher ask whether Weber’s charge-to-charge force law can produce an electric wave in free space. Their calculation begins with a specific model: overlapping positive and negative vacuum charges separate slightly when an electric field acts on them. They sum Weber forces from a thin spherical shell onto a charge parcel, set the net force to zero, and derive an equation resembling Maxwell’s electric-field wave equation. To reach that form they assume a uniform, linear vacuum, low charge speeds, and a displacement field with no curl contribution. The paper also argues that its more general displacement equation can describe longitudinal electric waves. This is a mathematical derivation from the proposed vacuum model, not a measurement of those vacuum charges.
Connection
This is chapter 2's proposition — that empty space is a real medium with structure — used as the working assumption of a derivation rather than as a philosophical aside, and chapter 10's question of what the field really is answered from the direct-action side. It also leaves a loose thread the site should follow: the curl term the authors set aside may describe vortex structures in the electric field.
From the source
Starting from Weber's force and the single postulate that the vacuum is polarizable, Li and Maher derive an electric wave equation for free space in which the Laplacian of the electric field equals its second time derivative divided by the squared speed of light — an expression that bears remarkable resemblance to the electric field wave equation derived from Maxwell's equations, and the first such derivation from Weber's force.Abstract; Section 4, Derivation of Electric Wave Equation, Eq. (28)
In Li and Maher’s proposed vacuum model, positive and negative charges overlap rather than forming fixed dipoles. With no external electric field their displacement is zero and the modeled vacuum is neutral; an applied field separates them, and charge density varies with the divergence of displacement, as volumetric strain does in continuum mechanics.Section 3, Vacuum Polarization, Eq. (2) and figure 1
The derived equation does not rely on Gauss's law but on the non-zero divergence of the electric displacement — if that divergence is zero the field is static and no wave propagates — so the theory is compatible with the reported observations of longitudinal electric waves in free space, during the eclipse of the sun by the moon and in spherical antenna experiments, and with the Monstein-Wesley scalar potential wave equation.Section 5, Longitudinal Electric Wave, Eqs. (25) and (29)
The historical record runs the other way from the textbook order: it was Weber's electrodynamics that first established the relationship between the speed of light and electromagnetic waves, Weber who first hypothesised in his force law the quantity Maxwell introduced as the displacement current and who first measured it with Kohlrausch, and Weber with Kirchhoff who first deduced that signals within an electrical circuit propagate at light velocity.Section 1, Introduction; Section 6, Discussion
The familiar form of the result depends on an assumption the authors flag: it follows when the displacement field is irrotational, and they judge that the curl term is most likely not always zero, which would give differences from the Maxwell wave equation — noting that the equation then has some similarity to fluid dynamics, that the curl term may indicate vortex structures in the electric field, and that ball lightning is a potential candidate for studying them.Section 6, Discussion, following Eqs. (26) and (28)
The stated range of validity is a homogeneous vacuum at low relative charge-pair speed, with higher-order terms, the acceleration force, the polarization force and velocity terms neglected and no sources included; the named next steps are to incorporate sources, to treat varying source and medium velocities, to extend to cases where charged particles such as electrons are present and where charge pairs move fast, and to build a polarization expression sophisticated enough to cover all six components of the Weber force.Section 6, Discussion; Section 7, Conclusions
The way in
https://doi.org/10.3390/foundations3020024How to cite it
Qingsong Li, Simon Maher (2023) Deriving an Electric Wave Equation from Weber’s Electrodynamics doi:10.3390/foundations3020024