The Helmholtz theorem and retarded fields

TL;DR

This paper investigates the application of the Helmholtz theorem to derive time-dependent electromagnetic fields, revealing limitations with instantaneous expressions and proposing generalizations for retarded fields.

Contribution

It demonstrates that the standard Helmholtz theorem yields only instantaneous field expressions and introduces generalized theorems that effectively derive retarded electromagnetic fields.

Findings

Standard Helmholtz theorem produces instantaneous field expressions.

Generalizations of the theorem enable derivation of retarded fields.

Instantaneous expressions are formally correct but practically limited.

Abstract

Textbooks frequently use the Helmholtz theorem to derive expressions for the electrostatic and magnetostatic fields but they do not usually apply this theorem to derive expressions for the 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 to derive 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 to derive the retarded fields.