Analysis Of Electromagnetic Fields Using Helmholtz's Theorem

Helmholtz's theorem summarizes the properties of the vector fields and has important meaning in studying the space distribution and transformation of the vector fields, and the relationships between the field character and the origins. So it draws a great significance in the theory of the electromagnetic fields. The way of directly analyzing the electromagnetic fields using Helmholtz's theorem is pretty original, which is different from the usual methods—the inductive method and deductive method. This way avoids the defects of repetitions between the inductive method and the physics, and avoids the much high starting point by using of deductive method. It takes some very special advantages in direct discussing the quality of the electromagnetic fields and the inner contacts among the electromagnetic fields and their origins. It is the Helmholtz's theorem that is used to analyze the electromagnetic fields systematically in the thesis.On the base of Helmholtz's theorem we deduced the equations of the electrostatic field in vacuum and in the dielectric, and make further discussion about the polarization of dielectric. We get the formula of the intensity of polarization P(r), make great points in discussing the electric displacement vector D(r), and deduce the formula of electric displacement vector D(r) in the isotropic linearity medium. In the course of deal with stable magnetic fields, the equations of stable magnetic fields in vacuum are reduced, which show the contacts between the Helmholtz's theorem and the time varying electromagnetic field and indicate the whole information of the vector fields through the divergence and the curl. Because the Helmholtz's theorem is a summary about the relations between the vector fields and its sources, so we also discuss the usage of the theorem in the time varying electromagnetic fields. In this outlook of this dissertation, the primary idea is to extend the Helmholtz's theorem to get the complete projection theorem of electromagnetic fields in vector wave space—the extend of the Helmholtz's theorem which can be used to study the eigenvalue problem of the vector partial differential operator. Firstly, the ideology that the vector field disassemble in the space of vector function is also available to the problem of electromagnetic wave propagate. Secondly, it's promising and feasible to use this method to process the specific software of the electromagnetic eigenvalue problems in the future. Directly qualitative analyzing electric displacement vector and magnetic field intensity are the achievement and innovation of this thesis.