Exact solutions for certain axisymmetric problems of induction in thin nonuniform sheets

A general relation between the axisymmetric induction problems for the infinite plane and the spherical sheets is obtained. By using this relation and starting with the uniform plane and spherical sheets and an axial inducing dipole, exact solutions are obtained for induction in these two sheets when they have special axisymmetric distributions of conductivity. These distributions are given by σ=σ0a²/ρ², σ=2σ0a²/(a² + ρ²) in the case of the plane sheet, where ρ is the distance from a fixed point in the sheet, and σ=σ1/(1 + ϵ cosθ), 0 ≤ ϵ ≤ 1 in the case of the spherical sheet, where θ is the colatitude (σ0, σ1, a, and ϵ are constants). The conductivity in each case decreases gradually as we go away from a fixed point in the sheet. The sheets are thus suitable models for representing the ionosphere (neglecting the nonisotropic character of its conductivity) or a large ocean. The exact solutions obtained are particularly useful in testing the accuracy of the iteration techniques, which are applied in the numerical solutions of the equations to which induction problems are reduced. As an example, numerical results for induction in the spherical shell by a uniform inducing field are given in the two cases when the time variations of the field are either periodic or aperiodic.