Alternative routes to the retarded potentials

Abstract

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.

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

“…In the cited papers it has been argued that charge conservation and causality, respectively represented by the local continuity equation and the retarded time are the cornerstones on which Maxwell's equations are based and therefore they can be considered to be the two fundamental postulates for these equations. It is worth mentioning that although Maxwell's equations are universally accepted, the question of what their fundamental physical postulates are remains a topic of discussion and debate [13,14,15,16,17,18,19]. The derivation of Maxwell's equations presented here in its three-dimensional and fourdimensional versions, which considers potentials as primary quantities and fields as derived quantities, may be useful to grasp the background of Maxwell's theory and may be presented in undergraduate courses of electromagnetism.…”

“…A question arises: could there be a set of first order field equations + Equation (7) is proved in reference [6]. As pointed out in reference [18] this identity is true for functions F such that the quantities [F ]/R have not the form…”

“…We have obtained two equivalent versions of electromagnetic field equations. The first one is represented by equations (6) and (8) which are expressed in terms of the retarded scalar and vector potentials defined in (5) and the second one is represented by equations (13)- (16) which are expressed in terms of the retarded fields defined by equations (17) and (18). This second version of the equations is identified with the familiar Maxwell's equations.…”

“…where (E) i and (B) k are the Cartesian components of the fields E and B, then (29), (31) and (32) reproduce (13)- (18). We have obtained two equivalent covariant versions of the electromagnetic field equations in the Minkowski spacetime.…”

Relativistic and electromagnetic molecular dynamics simulations for a carbon–gold nanotube accelerator

“…The electromagnetic interactions are calculated by a set of Maxwell equations in rectangular coordinates. Usually, the retarded potential is used for the special relativity [39,40]. Although this potential is difficult to solve for a system with a large number of particles, such as N > 10 5 particles, the large Coulomb potential field and the transverse electric field are separated.…”