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An engaging writing style and a strong focus on the physics make this comprehensive, graduate-level textbook unique among existing classical electromagnetism textbooks. Charged particles in vacuum and the electrodynamics of continuous media are given equal attention in discussions of electrostatics, magnetostatics, quasistatics, conservation laws, wave propagation, radiation, scattering, special relativity and field theory. Extensive use of qualitative arguments similar to those used by working physicists makes Modern Electrodynamics a must-have for every student of this subject. In 24 chapters, the textbook covers many more topics than can be presented in a typical two-semester course, making it easy for instructors to tailor courses to their specific needs. Close to 120 worked examples and 80 applications boxes help the reader build physical intuition and develop technical skill. Nearly 600 end-of-chapter homework problems encourage students to engage actively with the material. A solutions manual is available for instructors at www.cambridge.org/Zangwill.
I have been looking for a tolerably simple way of expressing the radiation at a distance from an electron.
Oliver Heaviside (1904)
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.
Larmor had an intense, almost mystical devotion to the principle of least action … To [him] it was the ultimate natural principle—the mainspring of the Universe.
Arthur Eddington (1942)
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.
If we accept the electromagnetic theory of light, there is nothing left but to look for the cause of dispersion in the molecules of the medium itself.
Hendrik Lorentz (1878)
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.
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.
The theoretical discovery of an electromagnetic wave spreading with the speed of light is one of the greatest achievements in the history of science.
Albert Einstein and Leopold Infeld (1938)
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.