Inhomogeneous Wave Equation, Liénard-Wiechert Potentials, and Hertzian Dipole in Weber Electrodynamics

Aiming to bypass the Lorentz force, this study analyzes Maxwell's equations from the perspective of a receiver at rest. This approach is necessary because experimental results suggest that the general validity of the Lorentz force might be questionable in non-stationary cases. Calculations in the receiver's rest frame are complicated and, thus, are rarely performed. In particular, the most important case is missing: namely, the solution of a Hertzian dipole moving in the rest frame of the receiver. The present article addresses this knowledge gap. First, this work demonstrates how the inhomogeneous wave equation can be derived and generically solved in the rest frame of the receiver. Subsequently, the solution for two uniformly moving point charges is derived, and the close connection between Maxwell's equations and Weber electrodynamics is highlighted. The gained insights are then applied to compute the far-field solution of a moving Hertzian dipole in the receiver's rest frame. The resulting solution is analyzed, and an explanation is presented regarding why an invariant and symmetric wave equation is possible for Weber electrodynamics and why the invariance could be the consequence of a quantum effect.