Shielding of a Perfectly Conducting Circular Disk: Exact and Static Analytical Solution

Abstract

The problem of the shielding evaluation of an infinitesimally thin perfectly conducting circular disk against a vertical magnetic dipole is here addressed. The problem is reduced to a set of dual integral equations and solved in an exact form through the application of the Galerkin method in the Hankel transform domain. It is shown that a second-kind Fredholm infinite matrix-operator equation can be obtained by selecting a complete set of orthogonal eigenfunctions of the static part of the integral operator as expansion basis. A static solution is finally extracted in a closed form which is shown to be accurate up to remarkably high frequencies.

Shielding of an Imperfect Metallic Thin Circular Disk: Exact and Low-Frequency Analytical Solution

“…The electromagnetic problem is axial-symmetric so that all the fields depend only on ρ and z. Time-harmonic sources and fields are assumed with an implicit e jωt dependence. As already shown in [18], the incident electric field produced by a vertical magnetic dipole of moment m z and placed at z = h is…”

“…The presence of a finite conductivity σ deeply changes the nature of the problem with respect to the perfectly conducting (PEC) case [18]: in fact, the static limit for R 0 = 0 produces a current which is identically zero. Moreover, the finite conductivity implies that the current is not singular anymore at the edges of the disk [25].…”

“…(19) is a second-kind Fredholm equation in the space 2 of the square-summable sequences; as such, it does not require any regularization scheme [10]. Therefore, compared with the PEC case [18], the finite conductivity has the role of a regularizing parameter. The fact that the integral equation is Fredholm second-kind type guarantees that a unique solution exists, which can be obtained through any discretization or truncation scheme.…”

“…In this work, we extend the analysis presented in [18] by still considering the interaction of a Vertical Magnetic Dipole (VMD) with a circular metallic disk, but having a finite conductivity and a finite thickness. It is worth noting that such a configuration has received much less attention than its ideal counterpart [13,[19][20][21].…”

“…Recently, the interaction of an infinitesimally thin PEC disk and a vertical magnetic point source (representing a small electric loop current) has been deeply investigated [18] obtaining the exact representation for the electromagnetic field, valid for any frequency range, through the solution of a set of dual integral equations by means of a numerical Galerkin Method-of-Moments approach and an analytical-regularization scheme. Moreover, surprisingly, a static solution has been obtained in a closed form which has been shown to be accurate up to very high frequencies.…”