In his Lectures on Physics, Richard Feynman asserts that “ten thousand years from now, there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics”. Whether this prediction is borne out or not, it is impossible to deny the significance of Maxwell's achievement to the history, practice, and future of physics. That is why electrodynamics has a permanent place in the physics curriculum, along with classical mechanics, quantum mechanics, and statistical mechanics. Of these four, students often find electrodynamics the most challenging. One reason is surely the mathematical demands of vector calculus and partial differential equations. Another stumbling block is the non-algorithmic nature of electromagnetic problem-solving. There are many entry points to a typical electromagnetism problem, but it is rarely obvious which lead to a quick solution and which lead to frustrating complications. Finally, Freeman Dyson points to the “two-level” structure of the theory.1 A first layer of linear equations relates the electric and magnetic fields to their sources and to each other. A second layer of equations for force, energy, and stress are quadratic in the fields. Our senses and measurements probe the second-layer quantities, which are determined only indirectly by the fundamental first-layer quantities.

Introduction

Many contemporary technologies exploit the fact that electromagnetic waves can be guided along specified paths through space and transiently stored in low-loss enclosures. The special configurations of conductors and dielectrics used to do this are called waveguides and resonant cavities. In this chapter, we show that electromagnetic fields can be guided and stored because they adjust themselves to satisfy the required boundary (or matching) conditions at the surfaces (or internal interfaces) of a guide or cavity. The nature and characteristics of the waves are fixed by the geometry and topology of the guiding and storage structures. Besides the familiar transverse electromagnetic (TEM) waves, where E and B are both transverse to the direction of propagation, we will find transverse electric (TE) waves where only E is transverse and transverse magnetic (TM) waves where only B is transverse. By and large, our discussion focuses on the applications of waveguides and cavities to specific problems of physics. Textbooks of engineering electro magnetics discuss applications to communication and power transmission.

Guided waves were discovered in 1842 by the Swiss physicist Colladon, who reported that total internal reflection could be exploited to trap light inside the parabolic streams of water produced by drilling holes in a water-filled vessel. Fifty-five years later, Hertz sought and observed meter-scale waves guided by a conducting wire.

Introduction

An incident electromagnetic wave is said to scatter or diffract from a sample of matter when the field produced by the sample cannot be described using Fresnel's theory of reflection and refraction from a flat interface (Section 17.3). In this chapter, we focus on the class of problems where this occurs because the wavelength of the incident monochromatic field is not small compared to the curvature of a material boundary. From a Fresnel point of view, the total field in these cases results from the interference of many different “reflected” and “refracted” waves propagating in different directions. We will encounter other points of view as we proceed. Figure 21.1 shows some typical geometries of interest. There is no universal naming practice, but many authors say that “scattering” occurs from objects with smooth boundaries and “diffraction” occurs from objects with sharp edges.

The physics that produces scattering and diffraction is identical to the physics that produces the Fresnel equations. An incident electromagnetic wave sets the charged particles of a medium into motion. Each accelerated charge produces a retarded field which is felt by, and thus affects the motion of, every other charge in the medium. The motion of every charge and the field it produces must be consistent with the total field each charge experiences.

ELECTRIC CHARGE

Electricity appeared to its early investigators as an extraordinary phenomenon. To draw from bodies the “subtle fire,” as it was sometimes called, to bring an object into a highly electrified state, to produce a steady flow of current, called for skillful contrivance. Except for the spectacle of lightning, the ordinary manifestations of nature, from the freezing of water to the growth of a tree, seemed to have no relation to the curious behavior of electrified objects. We know now that electrical forces largely determine the physical and chemical properties of matter over the whole range from atom to living cell. For this understanding we have to thank the scientists of the nineteenth century, Ampère, Faraday, Maxwell, and many others, who discovered the nature of electromagnetism, as well as the physicists and chemists of the twentieth century who unraveled the atomic structure of matter.

Classical electromagnetism deals with electric charges and currents and their interactions as if all the quantities involved could be measured independently, with unlimited precision. Here classical means simply “nonquantum.” The quantum law with its constant h is ignored in the classical theory of electromagnetism, just as it is in ordinary mechanics. Indeed, the classical theory was brought very nearly to its present state of completion before Planck'sdiscovery. It has survived remarkably well. Neither the revolution of quantum physics nor the development of special relativity dimmed the luster of the electro-magnetic field equations Maxwell wrote down 100 years ago.