Homogenized boundary conditions and resonance effects in Faraday cages

We present a mathematical study of two-dimensional electrostatic and electromagnetic shielding by a cage of conducting wires (the so-called `Faraday cage e ect'). Taking the limit as the number of wires in the cage tends to in nity we use the asymptotic method of multiple scales to derive continuum models for the shielding, involving homogenized boundary conditions on an e ective 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 polarisation of the incident eld. In the electromagnetic case there are resonance e ects, whereby at frequencies close to the natural frequencies of the equivalent solid shell, the presence of the cage actually ampli es the incident eld, rather than shielding it. By appropriately modifying the continuum model we calculate the modi ed resonant frequencies, and their associated peak amplitudes. We discuss applications to radiation containment in microwave ovens and acoustic scattering by perforated shells.