Title: Mathematical study of the Faraday cage in a simple three-dimensional configuration.

Summary: In this presentation, we are interested in the mathematical justification of the Faraday cage phenomenon in a simple three-dimensional case. We assume that Faraday’s cage consists of a flat layer of small, evenly distributed obstacles in two directions. The size of the obstacles and the distance between two consecutive obstacles are of the same order of magnitude $\epsilon$, $\epsilon$ supposedly small in front of the wavelength of the incident wave illuminating the obstacles. We then study the asymptotic behaviour of the solution of Maxwell’s equations (in harmonic mode) when $\epsilon$ tends towards 0. We begin by presenting well known results in the two-dimensional case: if we impose homogeneous Dirichlet conditions on the obstacles, then, at the limit, the waves do not penetrate under the layer of obstacles: this is the Faraday cage phenomenon. On the other hand, if one imposes homogeneous Neumann conditions, the layer of obstacles disappears at the limit. The three-dimensional case is more complex because the limit behaviour of the solution depends on the shape of the obstacles constituting the periodic layer. Indeed, if the slick consists of disjointed related obstacles, then the slick of obstacles disappears at the limit. If the structure consists of a layer of thin parallel wires, then, at the limit, only the normal (or tangential) component of the field is filtered through the periodic wall. Finally, if the structure is made up of a mesh of threads (of two layers of parallel threads oriented in two different directions), then the Faraday cage phenomenon is well observed.