Does the Helmholtz theorem of vector decomposition apply to the wave fields of electromagnetic radiation?

Abstract

The derivation of the Helmholtz theorem of vector decomposition of a 3-vector field requires that the field satisfy certain convergence properties at spatial infinity. This paper investigates if time-dependent electromagnetic radiation wave fields of point sources, which are of long range, satisfy these requirements. It is found that the requirements are satisfied because the fields give rise to integrals over the radial distance r of integrands of the form sin(kr)/r and cos(kr)/r. These Dirichlet integrals converge at infinity as required.

Helmholtz decomposition theorem and Blumenthal's extension by regularization

“…One may be surprised that all calculations for the radiation field could be performed without regularization as expected according to the order of the decay of the vector field. Anyway, the integrals converge in an explicit calculation [20]. This might happen in other cases too (G 0 instead of G 1 etc.).…”

“…However, the discussion went on considering the expressions of different options to choose the potentials for electromagnetic fields and it was shown by Jackson (see [19] and references therein) that quasistatic potentials can also be used. Nevertheless, the question was taken up again quite recently in a paper by Stewart [20] with the title "Does the Helmholtz theorem of vector decomposition apply to the wave fields of electromagnetic radiation?". Since also in this paper Blumenthal's proof is not mentioned, the validity of Helmholtz decomposition is performed explicitly.…”

“…Radiation fields, which decay asymptotically as 1/r , are rarely connected with the decomposition theorem. If one starts with the assumption that the asymptotic behavior of the field should be stronger than 1/r , additional properties of the field are needed in order to prove the decomposition of the radiation fields [20]. Let us now show decomposition as an example of an oscillating point dipole.…”

Longitudinal and transverse components of a vector field

“…( 32) and (37). They have been shown to be valid for the long-range radiation fields of electromagnetism [28].…”

Role of the nonlocality of the vector potential in the Aharonov–Bohm effect

“…The forms of the radiation fields of, for example, a charge oscillating in the z direction (θ = 0), are given by [30,31]. The dominant radiation terms at infinity, in spherical coordinates {r', θ, φ} go as…”