[en] Textbooks frequently use the Helmholtz theorem to derive expressions for electrostatic and magnetostatic fields but they do not usually apply this theorem to derive expressions for time-dependent electric and magnetic fields, even when there is no formal objection to doing so because the proof of the theorem does not involve time derivatives but only spatial derivatives. Here we address the question as to whether the Helmholtz theorem is useful in deriving expressions for the fields of Maxwell’s equations. We show that when this theorem is applied to Maxwell’s equations we obtain instantaneous expressions of the electric and magnetic fields, which are formally correct but of little practical usefulness. We then discuss two generalizations of the theorem which are shown to be useful in deriving the retarded fields. (paper)

[en] Two procedures to introduce the familiar retarded potentials of Maxwell’s equations are reviewed. The first well-known procedure makes use of the Lorenz-gauge potentials of Maxwell’s equations. The second less-known procedure applies the retarded Helmholtz theorem to Maxwell’s equations. Both procedures are compared in the context of an undergraduate presentation of electrodynamics. The covariant form of both procedures is discussed for completeness. As a related discussion, two procedures to introduce the unfamiliar instantaneous potentials of Maxwell’s equations are also reviewed. The first procedure applies the standard Helmholtz theorem to Maxwell’s equations and the second one uses the Coulomb-gauge potentials of Maxwell’s equations. The retarded and instantaneous forms of the potentials of Maxwell’s equations are briefly commented upon. The retarded Helmholtz theorem is used to introduce the retarded potentials of Maxwell’s equations with magnetic monopoles. (paper)

[en] In this note we explicitly show how the Lorentz transformations can be derived by demanding form invariance of the d’Alembert operator in inertial reference frames. (paper)

[en] An extension of the Helmholtz theorem is proved, which states that two retarded vector fields ${\mathbf{F}}_1$ and ${\mathbf{F}}_2$ satisfying appropriate initial and boundary conditions are uniquely determined by specifying their divergences $\mathrm{\nabla <!--\; ???\; -->}\cdot <!--\; ???\; -->{\mathbf{F}}_1$ and $\mathrm{\nabla <!--\; ???\; -->}\cdot <!--\; ???\; -->{\mathbf{F}}_2$ and their coupled curls $-<!--\; ???\; -->\mathrm{\nabla <!--\; ???\; -->}\times <!--\; ??\; -->{\mathbf{F}}_1-<!--\; ???\; -->\mathrm{\partial <!--\; ???\; -->}{\mathbf{F}}_2/\mathrm{\partial <!--\; ???\; -->}t$ and $\mathrm{\nabla <!--\; ???\; -->}\times <!--\; ??\; -->{\mathbf{F}}_2-<!--\; ???\; -->(1/c^2)\mathrm{\partial <!--\; ???\; -->}{\mathbf{F}}_1/\mathrm{\partial <!--\; ???\; -->}t$, where c is the propagation speed of the fields. When a corollary of this theorem is applied to Maxwell’s equations, the retarded electric and magnetic fields are directly obtained. The proof of the theorem relies on a novel demonstration of the uniqueness of the solutions of the vector wave equation. (paper)

[en] 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 handle of this equation to obtain the retarded scalar and vector potentials, whose wave equations yield the homogeneous and inhomogeneous Maxwell’s equations. We also show how this axiomatic-heuristic procedure to obtain Maxwell’s equations can be formulated covariantly in the Minkowski spacetime.

‘He (Feynman) said that he would start with the vector and scalar potentials, then everything would be much simpler and more transparent.’

M A Gottlieb–M Sands Conversation.⁴ (paper)

[en] We reply to some comments made by Davis (2020 Eur. J. Phys. 40 018001) on our paper (2020 Eur. J. Phys. 41 035202), by arguing that Davis’s assertions are unsupported in some cases and are unsatisfactory in other cases. (reply)