The Helmholtz theorem and retarded fields

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.

The role of electron orbital angular momentum in the Aharonov–Bohm effect revisited

“…Here we are thinking of more general time-dependent gauges like the Lorentz gauge. To answer this question, it is useful to remember the generalization of the Helmholtz theorem in the 4dimensional Minkowski space-time [19], [20]. It is known that the 4-dimensional generalization of the Helmholtz is not necessarily unique, but practically most useful one is the following.…”

“…It is known that the 4-dimensional generalization of the Helmholtz is not necessarily unique, but practically most useful one is the following. It is called the causal Helmholtz theorem for time dependent vector potential A(x, t) and given in the form [20] :…”

“…The causal Helmholtz theorem above means that, the time-dependent vector potential is decomposed into three pieces, each of which is specified by its divergence, rotation, and time derivative, respectively. Just like that the standard Helmholtz theorem is simplified by taking the Coulomb gauge, the causal Helmholtz theorem is simplified if we take the Lorentz gauge specified by the condition [20] :…”

Commentary: How to teach me physics: Tradition is not always a virtue

“…3 This task was much more exciting than the usual assignment of calculating the length contraction of a rod. I understood Maxwell's equations better when I reviewed the Helmholtz theorem 4 and this task was far more thrilling than calculating the electric field of a charged sphere.…”

“…3 The second insight was Braunbek and Laukien's 1952 calculation 4 exhibiting Newton's eel-like undulations by plotting the trajectories of the Poynting…”