Magnetostatic field of a conductor of semicircular cross section in a highly permeable half-space: exact analytic solution for infinite permeability with perturbative correction

Poisson's equation for the magnetic vector potential is solved using complex Fourier (Laplace) transforms in bipolar coordinates, the natural system for the subject two-dimensional geometry. The source is a dc current uniformly distributed over the semicircular cross section of a long conductor that is buried in, and flush with, the otherwise planar boundary of an infinitely permeable material. Exact closed-form potentials are obtained in the conformal mapping of the Neumann boundary value problem that characterizes the case of an infinitely permeable magnetic medium. One term of a perturbative correction that accounts for finite permeability is constructed for both the uniform source distribution and for the associated Green's function.