Maxwell's Equations

Abstract

Microwaves and RF is a branch of electrical engineering which deals ultimately with special cases of the physics of electrically charged particles and their interactions via electromagnetic waves. The fundamental branch of science describing the physics of electrically charged particles is electromagnetism. Electromagnetism deals with the electromagnetic force and is based on the concept of electric and magnetic vector fields, (r, t) and H(r, t), respectively. The fields (r, t) and H(r, t) were first introduced to resolve the issues of the “action at a distance” experienced between charges. Maxwell's equations are four coupled partial differential equations describing the electromagnetic field in terms of its sources, the charges and their associated currents (charges in motion). Electromagnetic waves are one special solution of Maxwell's equations that microwave engineering is built upon. In engineering, of course, depending on the technology of interest, we deal with a full range of special circumstances of electromagnetism. At one end of the spectrum are applications such as solid-state devices where electromagnetics is applied to just a few charges, albeit in a phenomenological sense and in conjunction with quantum mechanics. In this realm the forces on individual charges are important. At the other end, we have applications where the wavelength of the electromagnetic waves is much smaller than the dimensions of the problem and electromagnetics is reduced to optics where only simple, plane wave phenomena are at play. In the middle of the spectrum, we deal with structures whose size is comparable to the wavelength and electromagnetics is treated as a rigorous mathematical boundary value problem. The majority of microwave applications is somewhat in the middle of this spectrum with some having connections to either end. When studying time-harmonic events in microwaves, the frequency domain version of Maxwell's equations is very convenient. In the frequency domain, we have developed a number of high level descriptions of electromagnetic phenomena and several specialized disciplines such as circuits, filtering, antennas, and others have been created to efficiently address the engineering problems at hand. This chapter of the handbook will describe Maxwell's equations and their solution in order to establish the connection between the various microwave and rf topics and their basic physics, electromagnetism.