Nick Trefethen, University of Oxford

Everybody has heard of the Faraday cage effect, in which a wire mesh does a good job of blocking electric fields and electromagnetic waves.  Surely the mathematics of such a famous and useful phenomenon has been long ago worked out and written up in the textbooks?

It seems to be not so.  One reason may be that that the effect is not as simple as one might expect: it depends on the wires having finite radius.  Nor is it as strong as one might imagine: the shielding improves only linearly as the wire spacing decreases.

This talk will present results by Jon Chapman, Dave Hewett and myself on the electrostatic version of the Faraday cage: (a) numerical simulations, (b) a theorem confirming the $O((\log r)/n)$ dependence on wire radius $r$ and number of wires $n$ (proved by estimating harmonic measure via conformal maps), and (c) a homogenized boundary condition that a Faraday cage effectively imposes (derived by multiple scales analysis). We also explain an interesting connection, whose details are unexpected, with the exponential convergence of the periodic trapezoidal quadrature rule for analytic integrands.