Calculation of laplace and helmholtz potentials in two-phase problems

In this thesis, two phase models in a magnetostatics context using the Maxwell-Maxwell (MM) model and the Maxwell-London (ML) model are investigated. The vector equations are transformed in terms of scalar potentials leading to mixed boundary value problems for Laplace-Laplace and Laplace Helmholtz equations in the respective cases. Exact analytic solutions for the exterior and interior potentials Fe(r;q;f) and Fi(r;q;f), where r;q;f are the spherical coordinates, are obtained as infinite series and in closed forms for the MM model. The general solutions are found as a theorem. Several illustrative examples for specific externally imposed magnetic fields including a magnetic monopole and dipole are discussed based on our analytic solutions. It is shown that the magnetic permeability parameter k = me me+mi , where me and mi are magnetic permeabilities in the exterior and interior phases, has a significant impact on the magnetic induction fields and the forces acting on the sphere. A new relation for the multipole coefficients of the external phase is derived as well. Exact solutions for the ML model involving a superconducting sphere are derived in terms of the magnetic flux density functions Ye(r;q) and Yi(r;q) in the respective phases. The general solutions are established as a theorem for this model as well. The non-dimensional penetration depth parameter l is found to dictate the induction fields in ML model. Our results are of interest in various topics in mathematical physics where two phase models are used.