Introduction

The electromagnetic fields produced by point charges in motion play some role in practically every sub-discipline of physics. The key issues are not new because retardation and radiation were the main subjects of Chapter 20. Indeed, all the topics studied in this chapter could have been treated immediately after Section 20.3.4 when we wrote down the retarded integrals for the electromagnetic potentials in the Lorenz gauge. The value added by delaying the discussion until now is that the methods and insights of special relativity simplify calculations and help build intuition.

The first section below derives the potentials and fields produced by a point charge that moves along a specified trajectory. Subsequent sections look into the details for simple trajectories with and without particle acceleration. We will be particulary interested in the changes that occur when the particle speed increases from non-relativistic to ultra-relativistic values. The experimentally important frequency spectrum of emitted power emerges when we Fourier analyze the time dependence of the emitted fields. The emission of radiation implies energy loss by a moving particle and thus some perturbation of its trajectory. We treat this problem using the concept of radiation reaction. The chapter concludes with a brief introduction to Cherenkov radiation.

The Liénard-Wiechert Problem

Figure 23.1 shows the trajectory r0(t) of a point charge q. The instantaneous velocity of the charge is v(t) = dr0(t)/dt. Our task is to compute the exact electromagnetic fields associated with this moving charge.

Introduction

This chapter provides an introduction to the use of Lagrangian and Hamiltonian methods in classical electrodynamics. Our goal is to demonstrate that the powerful variational methods developed to derive the equations of motion and conservation laws for conventional mechanical systems can be extended to describe electrodynamics. By its nature, the material in this chapter is rather formal and most of our attention focuses on deriving the Maxwell equations and the Coulomb-Lorentz force law from a single Lagrangian or Hamiltonian. The new physics we will encounter bears principally on the gauge invariance of the theory. At the Lagrangian level, we will show that gauge invariance implies conservation of charge and vice versa. At the Hamiltonian level, we will show that electrodynamics is an example of a constrained dynamical system and that the maintenance of the constraints exploits gauge invariance in an essential way.

Our main theoretical tool is Hamilton's principle of stationary action. Originally conceived in the context of geometrical optics—and then extended to include mechanical systems—Hamilton's principle determines the equations of motion for any system where generalized coordinates can be sensibly defined. In the most familiar examples, a small number of degrees of freedom are sufficient to characterize the system of interest.

Introduction

The colored bands of a rainbow are well separated in space (dispersed) because water droplets in the atmosphere refract light with different wavelengths through different angles. Snell's law predicts this behavior because the index of refraction of water is a function of frequency. The simple conducting matter studied in Section 17.6.1 had a frequency-dependent index of refraction also. In this chapter, we argue that all real matter has this property of frequency dispersion and we discuss both its origins and consequences. Among the latter, we show that a deep connection exists between frequency dispersion and the dissipation of energy in matter. We also show that no electromagnetic information can be communicated faster than the speed of light. Otherwise, we follow tradition and use simple classical models to develop archetypes of frequency dispersion. This is perfectly adequate for a classical thermal plasma, but it is manifestly inadequate for quantum mechanical condensed matter systems. Nevertheless, with suitable caution there is much to learn from these models, even when applied to solids, liquids, and gases.

Frequency Dispersion

The frequency dispersion of the index of refraction (and other constitutive parameters) occurs because matter cannot respond instantaneously to an external perturbation. This is not a new idea. We encountered it in Section 14.13, when the inevitable time delay between voltage stimulus and current response in AC circuit theory led us to define a complex, frequency-dependent impedance, Ẑ(ω), as the generalization of DC resistance.

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.