A rectangular core has fixed permeability r >> 1, a square cross section of dimensions a

A rectangular core has fixed permeability μ_r >> 1, a square cross section of dimensions a × a, and has centerline dimensions around its perimeter of b and d. Coils 1 and 2, having turn numbers N₁ and N₂, are wound on the core. Consider a selected core cross-sectional plane as lying within the xy plane, such that the surface is defined by 0 < x < a, 0 < y < a.

(a) With current I1 in coil 1, use Ampere’s circuital law to find the magnetic flux density as a function of position over the core cross-section.

(b) Integrate your result of part (a) to determine the total magnetic flux within the core.

(c) Find the self-inductance of coil 1.

(d) Find the mutual inductance between coils 1 and 2.