Scalar Potential Formulations for Magnetic Fields Produced by Arbitrary Electric Current Distributions in the Presence of Ferromagnetic Bodies

In this paper it is shown how to analyze stationary or quasistationary magnetic fields due to arbitrary distributions of electric current in terms of single-valued scalar potentials. The total scalar potential in a specified region outside the magnetic material bodies, not containing electric currents, is obtained by superposing two single-valued Laplacian scalar potentials. One is produced by the given current distribution in an unbounded space and is determined from the free-space values of the normal component of the magnetic field intensity over the region boundary. The other one is defined in the entire region outside the magnetic bodies and is employed in order to satisfy the boundary conditions. When the magnetic bodies can be considered to be linear and homogeneous or piecewise homogeneous, with no electric current within them, the field inside the bodies can also be expressed in terms of Laplacian scalar potentials. If the material of the bodies can be approximated to be ideal ferromagnetic, then the exterior field problem is a Dirichlet problem and is solved independently of the field inside the bodies. The Neumann boundary condition for the interior field problem is provided from the solution of the exterior problem. Differential equation and integral equation formulations, as well as an illustrative application example, are presented. Since the resultant field is computed in terms of only single-valued scalar potentials, the solution techniques based on the formulations presented in this paper are considerably more efficient than those in usual solution methods.