Chapter 3 - Electromagnetic Field Equations

Abstract

In this chapter, I present a parallel development of electromagnetic theory based on the traditional Gibbsian vector formalism (using grad, div, and curl), and using the methods of differential forms. In the framework of this new approach, Maxwell's equations appear naturally from the basic equations of the field theory for the differential forms. I provide the physical interpretation of Maxwell's equations as well. The boundary conditions for the vector fields are introduced based on Maxwell's equations. This chapter also develops the basic aspects of electromagnetic theory which are common to the many techniques of electrical exploration that will be discussed later in the book. We introduce the basic theorems characterizing the distribution of the energy of electromagnetic field (Poynting's theorems) and the concept of electromagnetic Green's tensors. The chapter concludes with a description of the reciprocity relations and Lorentz lemma.