## 1 Introduction  
The interaction of electromagnetic fields with a circular disk (and its dual counterpart, i.e., a circular aperture in a plate) is a canonical problem in electromagnetic theory; research on this subject started in the 1950s and continues to attract the attention of the electromagnetic research community, although most studies have considered plane-wave excitation and/or perfectly-conducting (PEC) obstacles, ideally with zero thickness (see, e.g., [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11]). In connection with this, we point out that in [12; 13; 14] the method of analytical preconditioning together with a Galerkin spectral Method of Moments with entire domain basis functions has been used to study the scattering from an infinitesimally thin PEC disk. Such a method has been adopted in those few studies which have considered thin disks, including those with dielectric properties [15; 16; 17]. On the other hand, the screening of a finite source (e.g., an electric or a magnetic dipole) by means of an infinite planar conductive shield is well-documented in both the frequency [18; 19; 20] and time [21; 22] domains. Recently, the shielding of a thin conductive disk in the presence of an ideal vertical magnetic dipole has been studied in details [23]; however, the main assumption in [23] is that the disk thickness has to be smaller than the skin depth. Such an assumption introduces an upper limit in terms of frequency, and makes the formulation essentially into a low-frequency analysis. It is worth mentioning that the dual problem (i.e., a circular aperture in a thin metallic plate) has only recently been addressed [24]. In general, for thick disks (and as such for higher frequencies) analysis requires the adoption of generalized boundary conditions (GBCs). In turn, such GBCs require the introduction of both equivalent magnetic and electric current densities along the discontinuity interface [25; 26; 27]. This implies that, for thick disks an integral formulation based on GBCs leads to the derivation of two sets of dual integral equations (DEs) to be solved. In the present work, we solve such sets of DEs (derived in the Hankel transform domain) by expanding the spectral unknowns in two Neumann series of Bessel functions. Moreover, for thick metallic non-magnetic disks, a new generalized boundary condition is derived which is a generalization of that used for thin metallic screens. This means that it is sufficient to introduce only an equivalent surface electric current density as unknown. In this way, only one set of DIEs needs to be solved, thereby reducing the problem in a similar way to that of thin screens (preliminary results of this work have been presented in [28]). Thus, we generalize the analysis presented in [23] considering a finite source (a coaxial loop with finite radius and constant current) and a conductive disk with finite thickness, now without any limitation in frequency and thickness. Finally, we want to mention a possible limitation of the present study. For simplicity, the electric current in the loop is assumed to be constant. In principle, the extension of the proposed analysis to non-constant currents could be feasible; however, introducing an azimuthal dependence of the exciting current would destroy the azimuthal invariance of the problem, thereby making the incident and scattered field hybrid. We leave this analysis for a future work. It is clear that the focus of the paper is more on modeling than on the performance of different materials. We considered copper as a typical conductive material for the purpose of presenting numerical results that illustrate the proposed method; however, the validity of the method is general and has been tested on different values of the screen parameters. The rest of this paper is organized as follows.