Low – frequency electromagnetic scattering by a perfectly conducting torus. The Rayleigh approximations

Abstract

The paper investigates the propagation of a plane electromagnetic wave in the exterior of a perfectly conducting torus. Using the fact that in toroidal coordinates the vector Helmholtz equation does not admit separation of variables we apply the low-frequency method and the electromagnetic scattering problem reduces to a sequence of potential problems. The incomplete R-separation leads to infinite system of three-diagonal form. More precisely, the boundary conditions for the magnetic field produce a third-order recurrence relations, due to the Riemannian radical and the non-orthogonality of the toroidal harmonics in the angular variable. The three-diagonal infinite systems of linear algebraic equations are solved analytically via the appropriate use of finite continuous fractions.