Introduction

This chapter begins our exploration of the single most important fact of electromagnetic life. The Maxwell equations have wave-like solutions which propagate from point to point through space carrying energy, linear momentum, and angular momentum. Electromagnetic fields of this kind transport life-giving heat from the Sun, reveal the internal structure of the human body, and facilitate communication by radio, television, satellite, and cell phone. The propagating solutions we will study are often called “free fields” because they are not “attached” to distributions of charge or current. Their electric field lines do not terminate on charge and their magnetic field lines do not encircle current. For that reason, electromagnetic waves bear very little relationship (both physically and mathematically) to the electrostatic, magnetostatic, and quasistatic fields we have studied to this point. This chapter focuses on the basic structure and surprising variety of propagating waves in vacuum. Chapter 20 takes up the question of how one produces them.

Electromagnetic waves are solutions of the Maxwell equations in the absence of sources. Such waves are also solutions of a vector wave equation which appears repeatedly through the course of the chapter. We analyze plane wave solutions first because they are simple, important, and provide a convenient setting for discussing polarization. We then superpose plane waves to form wave packets and demonstrate their fundamental properties of complementarity and free-space diffraction.

Introduction

This chapter explores the propagation of monochromatic plane waves in simple matter where the electric permittivity, magnetic permeability ∈, and ohmic conductivity µ are all constants. When a σ = 0 this model for matter is non-dispersive in the sense that plane waves with different frequencies all have the same phase velocity. This contrasts with real matter, which is frequency-dispersive because ∈ = ∈(ω) is a function of frequency and plane waves with different frequencies propagate with different phase velocities. Nevertheless, by focusing on one frequency at a time—and by not superposing waves with different frequencies—many important effects of wave propagation in real matter can be captured using a non-dispersive model. We will be particularly interested in the reflection, refraction, and interference that occur when waves interact with planar boundaries which separate regions of dissimilar simple matter.

The mathematics of wave propagation in linear and isotropic non-dispersive matter is nearly identical to the mathematics of wave propagation in vacuum. This has the virtue of generating results very quickly (by analogy) and the vice of masking some important physics associated with the matter. In this chapter, we can do little more than name this “hidden” physics; a proper discussion must wait until the reader has acquired an appreciation of retardation and radiation (Chapter 20).

Introduction

Special relativity is the theory of how different observers, moving at constant velocity with respect to one another, report their experience of the same physical event. This description is completely accurate, but it conceals the fact that special relativity radically altered physicists' conceptions of space and time. It also obscures the deep connection between special relativity and electromagnetism, a connection Albert Einstein chose to emphasize in the opening paragraph of his ground-breaking paper on the subject (1905):

The issue that concerned Einstein was the perceived difference between “transformer” EMF and “motional” EMF when a conductor and a magnet move relative to one another (see Section 14.4.1). From the point of view of the conductor, the moving magnet produces an electric field at every point in space, including within the body of the conductor, where it induces a current.

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.

In this chapter . . .

In this chapter we will introduce you to a system of thinking about, and then describing, English words and sentences. We will see that most words can have different forms. Combined according to rules of grammar (morpho-syntax), words can form larger units such as phrases, clauses and sentences. Many words can also be segmented into smaller meaningful units, which are called morphemes. We will discover that the most important shared characteristic of these units is that they all have internal structure. As we will see, in order to work systematically with English words and sentences, we need to analyse both: the structure that lies behind words (morphology), and the organising principles according to which native speakers assemble words into sentences (syntax). And we will see that, without this internal structure, we would not be able to communicate with language, and this is, after all, its purpose. We will not give you stylistic or prescriptive rules, but together we will discover the conventions underlying the use of standard English for communication.

Introduction

This chapter describes how linguists realized they needed to study ongoing changes. In the early part of the twentieth century, language change was assumed to happen too slowly to be observable. In their grammars, linguists had unwittingly omitted the variations which indicated that changes were under way. The American linguist William Labov was the key figure in this twentieth-century linguistic breakthrough. He made a preliminary ingenious attempt to study current changes in a now-famous department store survey in New York. Later, he proceeded to other New York changes. He found he could reliably predict the percentage of the features he was investigating in the speech of different ethnic groups, sexes, ages and socio-economic classes, in different speech styles. In his later work, he has refined his early techniques and produced a solid methodological framework for future researchers. His later work in Philadelphia again proved groundbreaking. Meanwhile, in Northern Ireland, Jim and Leslie Milroy explored speech varieties in different social networks. These seminal studies, on each side of the Atlantic, kick-started work on ongoing changes, which are now a standard part of language change investigations.

In this chapter . . .

In this chapter we will introduce the main word classes: nouns, determiners, pronouns, adjectives, verbs, adverbs, prepositions and conjunctions. We said that in order to work systematically with English words and sentences, we need to analyse both the structure that lies behind words (morphology) and the organising principles according to which sentences are assembled (syntax). In accordance with this principle, we will introduce you to morphological, syntactic and semantic criteria for word-class identification. We will see that word meaning is of course important for words to fit into a sentence; but their form characteristics are equally important. Each of the sections begins with a discussion of the shared properties of the word class under consideration. Once we have identified the typical characteristics of a word class, we will introduce you to different sub-types within it, if there are any. Activities on the fuzziness of different word classes, and on how distributional criteria allow us to shed some light on this fuzziness, will be relegated to the end of the chapter. In this chapter you will learn to work out which word class any word in any given sentence belongs to.