Calculation of laplace and helmholtz potentials in two-phase problems


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A thesis Submitted in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE in MATHEMATICS from Texas A&M University-Corpus Christi from Corpus Christi, Texas.
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.
Mathematics and Statistics
College of Science and Engineering