We all know the Faraday cage effect: light waves get through the screen on the front of your microwave oven, because their wavelength is much less than the size of the holes, whereas the microwaves don't get out, because their wavelength is much greater. Yet despite many hours of looking around and dozens of discussions with individuals, I've been unable to find an analysis of the mathematics of this effect.

Presumably there's a simple argument that shows some kind of exponential attenuation depending on the ratio of wavelength of hole size. Can anyone point me to the literature on this subject?

• I'd look at the physics of diffraction gratings. Commented Jul 6, 2013 at 9:42
• I always take a factor of 10 as a starting point. Microwaves in the oven have a wavelength of about 122mm. A tenth of this is 12.2mm (much bigger than the hole size) - it will not pass through. Visible light is about 0.5 micrometre so 10 times that is 5 micrometres. Its much smaller than the hole so no significant diffraction effects - it will pass straight through. Commented Jul 6, 2013 at 11:03

The following forum post might be interesting for you: Mathematical derivation of the Faraday cage from the Maxwell Equations.

Especially post #4 from Astronuc (please have look in the link above for the complete citation):

Well trivially, it's Gauss's law. Inside the hollow conductor there is not charge, so the enclosed charge is zero, so the electric field is zero everywhere.

Now more directly, consider the most trivial case of the center of a hollow sphere, with 'uniform' charge on the surface. For each charge, there is an equal charge diametrically opposed, and therefore at the center the electrical fields (vectors) are equal and opposite, so they cancel.

Now, consider any point, off-center. One cannot apply the opposite point charge, but rather one must consider opposing surfaces, $$\dA\$$, which would have charges $$\σ_1dA_1\$$ and $$\σ_2dA_2\$$. Now think if two cones with vertices touching (and having same solid angles) and colinear (parallel) axes, with heights $$\r_1\$$ and $$\r_2\$$. The $$\E\$$ from one is just $$\\dfrac{σ_1dA_1}{r_1^2}\$$ and the other is $$\\dfrac{σ_2dA_2}{r_2^2}\$$, but realize that $$\dA_i\$$ is proportional to $$\r_i^2dΩ_i\$$, where $$\dΩ\$$ is the solid angle enveloped by cones and subtended by $$\dA_i\$$.

So $$\E_i\$$ is proportional to $$\\dfrac{1}{r_i^2}\$$, and $$\dA_i\$$ is proportional to $$\r_i^2\$$, and the term cancel which then leaves equal charges ($$\σdΩ\$$) opposing each other, and therefore the electric fields cancel, i.e. $$\\vec{E}=0\$$.

Thought this might help: https://people.maths.ox.ac.uk/trefethen/chapman_hewett_trefethen.pdf

S. JONATHAN CHAPMAN, DAVID P. HEWETT, AND LLOYD N. TREFETHEN

Abstract.

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.

Key words. Faraday cage, shielding, screening, homogenization, harmonic function

AMS subject classifications. 31A35, 78A30

A recent publication on various aspects of the effect. Interesting is that it invalidates Feynman's discussions.

• The original asker is one of paper's authors. Commented Aug 3, 2016 at 20:24