Can Maxwell's equations be obtained from the continuity equation?

Linking Maxwell, Helmholtz and Gauss through the Linking Integral

We take the Gauss' linking integral of two curves as a starting point to discuss the connection between the equation of continuity and the inhomogeneous Maxwell equations. Gauss' formula has been discussed before, as being derivable from the line integral of a magnetic field generated by a steady current flowing through a loop. We argue that a purely geometrical result - such as Gauss' formula - cannot be claimed to be derivable from a law of Nature, i.e., from one of Maxwell's equations, which is the departing point for the calculation of the magnetic field. We thus discuss anew the derivatio…

On Two Complementary Types of Total Time Derivative in Classical Field Theories and Maxwell's Equations

Close insight into mathematical and conceptual structure of classical field theories shows serious inconsistencies in their common basis. In other words, we claim in this work to have come across two severe mathematical blunders in the very foundations of theoretical hydrodynamics. One of the defects concerns the traditional treatment of time derivatives in Eulerian hydrodynamic description. The other one resides in the conventional demonstration of the so-called Convection Theorem. Both approaches are thought to be necessary for cross-verification of the standard differential form of continui…

A New Formulation of Electrodynamics

A new formulation of electromagnetism based on linear differential commutator brackets is developed. Maxwell equations are derived, using these commutator brackets, from the vector potential $\vec{A}$, the scalar potential $ϕ$ and the Lorentz gauge connecting them. With the same formalism, the continuity equation is written in terms of these new differential commutator brackets. Keywords: Mathematical formulation, Maxwell's equations

基尔霍夫对电磁学的被遗忘贡献：连续性方程与位移电流

Kirchhoff`s Forgotten Contributions to Electromagnetism: Continuity Equation versus Displacement Currents

基尔霍夫在1857年首次提出连续性方程，比麦克斯韦的位移电流概念更早，且在电磁学中更为基础。

An Approach to Maxwell Equations in Uniformly Accelerated Spherical Coordinates by Newman-Penrose Method

Variables are separated in Maxwell equations by the Newman-Penrose method of isotropic complex tetrade in the uniformly accelerated spherical coordinate system. Particular solutions are obtained in terms of spin 1 spherical harmonics. PACS: 03.50.De

On Feynman's handwritten notes on electromagnetism and the idea of introducing potentials before fields

In his recently discovered handwritten notes on "An alternate way to handle electrodynamics" dated on 1963, Richard P. Feynman speculated with the idea of getting the inhomogeneous Maxwell's equations for the electric and magnetic fields from the wave equation for the vector potential. With the aim of implementing this pedagogically interesting idea, we develop in this paper the approach of introducing the scalar and vector potentials before the electric and magnetic fields. We consider the charge conservation expressed through the continuity equation as a basic axiom and make a heuristic hand…

Symmetric energy-momentum tensor in Maxwell, Yang-Mills, and Proca theories obtained using only Noether's theorem

The symmetric and gauge-invariant energy-momentum tensors for source-free Maxwell and Yang-Mills theories are obtained by means of translations in spacetime via a systematic implementation of Noether's theorem. For the source-free neutral Proca field, the same procedure yields also the symmetric energy-momentum tensor. In all cases, the key point to get the right expressions for the energy-momentum tensors is the appropriate handling of their equations of motion and the Bianchi identities. It must be stressed that these results are obtained without using Belinfante's symmetrization techniques …

Does the electromagnetic field of an accelerated charge satisfy Maxwell equations?

We considered the electromagnetic field of a charge moving with a constant acceleration along an axis. We found that this field obtained from the Lienard-Wiechert potentials does not satisfy Maxwell equations.

Higher order derivatives extension of Maxwell-Chern-Simons electrodynamics in the presence of field sources and material boundaries

In this paper we consider some new classical effects obtained for a planar electrodynamics with the presence of a higher order derivatives term. The model can be interpreted as a kind of extension for the $3d$ Maxwell-Chern-Simons electrodynamics with higher order derivatives. We consider setups with stationary field sources describing point-like charges and Dirac points. We also investigate this model with the presence of a conducting line (in $3d$ dimensions, it is the equivalent of a conducting plate in $4d$ dimensions). In this case we calculate the propagator for the gauge field and the i…

Some Arguments for the Wave Equation in Quantum Theory

We clarify some arguments concerning Jefimenko's equations, as a way of constructing solutions to Maxwell's equations, for charge and current satisfying the continuity equation. We then isolate a condition on non-radiation in all inertial frames, which is intuitively reasonable for the stability of an atomic system, and prove that the condition is equivalent to the charge and current satisfying certain relations, including the wave equations. Finally, we prove that with these relations, the energy in the electromagnetic field is quantised and displays the properties of the Balmer series.