Mathematics of the Faraday cage

The amplitude of the gradient of a potential inside a wire cage is investigated, with particular attention to the 2D configuration of a ring of n disks of radius r held at equal potential. The Faraday shielding effect depends upon the wires having finite radius and is weaker than one might expect, scaling as |log r|/n in an appropriate regime of small r and large n. Both numerical results and a mathematical theorem are provided. By the method of multiple scales, a continuum approximation is then derived in the form of a homogenized boundary condition for the Laplace equation along a curve. The homogenized equation reveals that in a Faraday cage, charge moves so as to somewhat cancel an external field, but not enough for the cancellation to be fully effective. Physically, the effect is one of electrostatic induction in a surface of limited capacitance. An alternative discrete model of the effect is also derived based on a principle of energy minimization. Extensions to electromagnetic waves and 3D geometries are mentioned.