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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.
The special theory of relativity owes its origin to Maxwell's equations of the electromagnetic field.
Albert Einstein (1949)
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):
It is well known that Maxwell's electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion.
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.
The solution of this problem presents mathematical difficulties which arise from the necessity of taking into account the geometrical shape of the obstacles on which the wave is falling.
Vladimir Fock (1948)
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.
It appears that the square of the index of refraction is equal to the product of the specific dielectric capacity and the specific magnetic capacity.
James Clerk Maxwell (1865)
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).
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.
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.