Electromagnetic Field inside a Current-Carrying Region

Abstract

The improper integrals, appearing in the course of evaluating the vector potential A and the electric field E inside a current-carrying region, are carefully examined. It is found that the integrals exist and have a well-defined meaning only when the current density function satisfies a Holder condition. A definite and precise way of evaluating them is derived and A, E are shown to satisfy the usual inhomogeneous equations.

Scalar Potential Depolarizing Dyad Artifact for a Uniaxial Medium

“…jωµ zẑ are the well-known electric and magnetic field depolarizing dyads resulting from the longitudinal current densities and are mathematically and physically consistent with prior well-documented findings [25][26][27][28][29][30][31][32][33][34][35]. In these findings, it is discussed that the source region of volume V is split into two subregions V − V δ and V δ , where V δ is a small cavity excavated around the source point (in the limit as δ → 0).…”

Electromagnetic field in the source region of continuously varying current density

“…As in [3,4] the approach is based on a direct integration of the field equation for A containing Helmholtz's kernel $(i?) = /R and provides an extension of previous results by the author, for constant and certain special radial current distributions, to, practically, any continuous J(r) in spherical regions.…”

“…For constant Ju this integral has been evaluated explicitly in [3,4], for some special radial distributions Ju(r) in [4] and for even more general ones, as pointed out below, in [6]. In this paper Au(v) and its first and second derivatives and, therefore, the fields H(r) = x A and E(r) = ~^(V x H -J), will be explicitly evaluated for, practically, any continuous Ju(r) in v providing the full generalization of the subjects investigated in [1,2,3,4].…”

“…In this paper Au(v) and its first and second derivatives and, therefore, the fields H(r) = x A and E(r) = ~^(V x H -J), will be explicitly evaluated for, practically, any continuous Ju(r) in v providing the full generalization of the subjects investigated in [1,2,3,4]. When r is interior to v, the integral in (1) becomes a convergent improper one.…”

“…(20) and (21) in [3], which indicate the correct way of evaluating the electric field E(r) at points r inside v. In particular, the more general relation (21) of [3], for any point r in reads as follows:…”