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.