# Anisotropy in electrostatics - Solutions for inclusions with canonical shapes

Research output: ThesisDoctoral ThesisCollection of Articles

## Abstract

Inclusions that take a canonical shape have a special place in electrostatics. Their analysis involves methods with great mathematical subtlety and rigor. Where an arbitrarily shaped inclusion can only be solved with a method that involves approximation, some of the canonical shapes permit an approach that leads to an exact solution in a closed form.What makes canonical shapes special, is that they have a particularly simple presentation in an important class of coordinate systems. There are 11 orthogonal coordinate systems where the coordinate curves are defined by equations of second degree. In these 11 systems, the Laplace equation is separable. The particular canonical shapes that the thesis studies are spheres, spheroids, and ellipses. In their corresponding coordinate systems, they correspond to surfaces with a fixed coordinate. Because these three systems allow the Laplace equation to separate, much work can be done analytically before involving approximation or computation. Some of the presented solution in the thesis are exact and others are semi-analytical. Although solutions for inclusions with the three canonical shapes exist in the literature, the thesis goes beyond the existing work by generalizing the materials of the inclusions. In all of the enclosed articles, the material of the inclusion is assumed to be anisotropic and inhomogeneous. More specifically, it is assumed that the axes of anisotropy correspond to the coordinate unit vectors of the relevant coordinate system. The term radial anisotropy has been used in the literature to refer to anisotropy in the spherical coordinates when the tangential components are equal but possibly differ from the normal component. The thesis employs a similar concept in spheroidal and elliptic systems. In the spherical coordinate system, the thesis relaxes the assumption that the two tangential components must be equal. The resulting inclusions is referred to as the systropic sphere.
Translated title of the contribution Anisotropia sähköstatiikassa English Doctor's degree Aalto University Sihvola, Ari, Supervising ProfessorWallén, Henrik, Thesis Advisor Aalto University 978-952-60-7184-8 978-952-60-7183-1 Published - 2016 G5 Doctoral dissertation (article)

## Keywords

• anisotropy
• Laplace equation
• potential

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