Homogenized boundary conditions and resonance effects in Faraday cages

Abstract

We present a mathematical study of two-dimensional electrostatic and electromagnetic shielding by a cage of conducting wires (the so-called ‘Faraday cage effect’). Taking the limit as the number of wires in the cage tends to infinity, we use the asymptotic method of multiple scales to derive continuum models for the shielding, involving homogenized boundary conditions on an effective cage boundary. We show how the resulting models depend on key cage parameters such as the size and shape of the wires, and, in the electromagnetic case, on the frequency and polarization of the incident field. In the electromagnetic case, there are resonance effects, whereby at frequencies close to the natural frequencies of the equivalent solid shell, the presence of the cage actually amplifies the incident field, rather than shielding it. By appropriately modifying the continuum model, we calculate the modified resonant frequencies, and their associated peak amplitudes. We discuss applications to radiation containment in microwave ovens and acoustic scattering by perforated shells.

Bloch Waves in an Arbitrary Two-Dimensional Lattice of Subwavelength Dirichlet Scatterers

“…in the 3D variant of this problem and for Neumann scatterers in both 2D and 3D). We already mentioned in the introduction the Biharmonic pinned-plate problem, and other important examples include wire media [5,40,44], Faraday cages [6,16,32], lattices of high-contrast rods [41,43], and plasmonic nanoparticle waveguides and metasurfaces [29]. In addition, ongoing research into photonic and mechanical analogies of topological insulators and the integer quantum hall effect has stirred interest in media breaking time-reversal symmetry.…”

“…(1) 0 (ωr), a being proportional to ψ(0) through (16). An alternative formulation is obtained by noting from (12) that φ satisfies the forced Helmholtz equation…”

On the homogenization of the acoustic wave propagation in perforated ducts of finite length for an inviscid and a viscous model

“…corresponding to a solid wall (see also [3] for the electromagnetic scattering on a cylindrical Faraday cage). The first corrector satisfies transmission conditions with a source term depending on the limit solution…”

Electromagnetic shielding by thin periodic structures and the Faraday cage effect

“…In particular it was shown in [2] how modern techniques of homogenization and matched asymptotic expansions could be used to derive effective interface conditions that accurately capture the shielding effect. These results were generalised to the 2D electromagnetic case (TE-and TM polarizations) in [6], and related approximations for similar problems have also been studied recently by other authors, e.g. [8,9].…”