The Study Of Analytical Method About The Boundary Value Problem Of Electromagnetism Field

This thesis introduced theoretic approaches dealt with the boundary value problem of electromagnetism field, mainly about electrostatic field, with the methods of Conformal Mapping and Green Function. In addition, this thesis gained several results adopting Conformal Mapping method in special boundary conditions, have been drew equipotent line and electric field line of two dimensions or three dimensions. The computational methods of electromagnetism field boundary value problem have a few, thereinto, the conformal mapping method and Green function methods are both in common use. Adopting analytic function transform W = f ( Z), the field area with complicated boundary shape in Z plane can transform into field area with simple boundary shape in another complex plane .Thus the boundary value problem defined in new complex plane get results easily. The boundary that the new field area is imposed the boundary condition of formerly field area and get the new potential function afterward. Then, using analytic function transform, the independent variable belonged to new potential function ? ( u ,v) can transform back to former independent variable. This potential function ? ?? u ( x , y ) , v ( x ,y)?? is the solution of the former potential field; Green function method is the approach that arbitrarily response by the source excitation is expressed as superposition of excitation source at all space point. For the sake of simplifying the solver, we may through unit-excitation source response to solving the problem of arbitrarily excitation source response. The main work in the thesis is making some basal researches and study to conformal mapping method and Green function method.This thesis had studied a few basic theories problems about the conformal mapping method, which include its applied process and Schwarz-Christoffel transformation. Apply these theories in solved three typical models: The static electricity field problem of the neighborhood slit of the infinite conductor. According to the principle of all plane static electricity fields can describe with a certain analytic function, we found out the analytic function ( )1 ( 21)f z = 2i z ? z+ means that field of this problem. Thus so far as real part and imaginary part are separated, we can get the expression of potential and electric field; the boundary value problem of angle-shape conductor area after inside a linear electric charge. Using the analytic function transform W = Z2, the inner area of angle-shape conductor can transform to upper half-plane on another complex plane. Then we can solve the problem in this complex plane with image method, at last back to former complex plane we get the result we need; The solution of unit length capacitor problem of confocal ellipses cylinder. Using inverse trigonometrically function transform W ( Z ) = u + jv = arcsin DZ, elliptic region can transform to the line that parallels real axis on W plane. Thus capacity problem of confocal ellipses cylinder changed into parallel-plate capacitor, this is an easy problem.Discussed a few different forms of the Green Function in brief such as a limited form, series expansion form and integral form etc.and derivate among a few forms is equivalent at the foundation of compare with these different forms.At the end of this thesis, the problem of charge confinement put forward as a special boundary value problem and analog quark confinement. We use separated variables method determine potential of charge confinement in the condition of linear dielectric medium of sphericity and discussed two kinds of possibilities that realize the potential. We can prove that the confinement potential of point charge in electrostatic field requests anti-dielectric or mirror point charge out of confined region.