Dynamics and the Emergence of Maxwell's Equations

This paper completes the constructive reformulation of classical electromagnetism initiated in Paper I and extended in Paper II. It addresses the question of time dependence, showing that it does not arise from acceleration—which is not a primitive in this theory—but from the cancellation condition \mathbf{E}_m = -\mathbf{E}_s in the atomic potential scenario. Choosing a fixed surface in the nucleus rest frame, Gauss's law together with the continuity equation yields the dynamical relation \partial_t \mathbf{E}_m = \mathbf{J} / \epsilon. Combining this with the steady-state curl equations derived in Paper II gives the full time-dependent Maxwell equations: \nabla \times \mathbf{E}_m = -\partial_t \mathbf{B} and \nabla \times \mathbf{B} = c^{-2} \partial_t \mathbf{E}_m. From these, the homogeneous wave equation \nabla^2 \mathbf{E}_m = c^{-2} \partial_t^2 \mathbf{E}_m follows directly. Key insights include: the displacement current emerges naturally rather than as an ad-hoc addition; the wave equation is homogeneous, separating generation from propagation; light is a wave of the field under the cancellation condition, not detached radiation; and the atomic potential provides a natural setting for discrete light emission (spectra). This paper sets the stage for applications to atomic structure and quantum theory.