This appendix derives the momentum carried by electromagnetic fields and rewrites the Lorentz force law as a conservation law for total momentum in matter plus fields. Starting from the volume integral of ρE + J × B, it uses Maxwell’s equations to express the mechanical momentum change in terms of the fields alone, identifying a field momentum density ε0 E × B and introducing the Maxwell stress tensor as the flux of electromagnetic momentum through a surface. The resulting continuity equation states that changes in mechanical plus field momentum inside a volume are balanced by stress-tensor fluxes across its boundary. The appendix interprets this structure in terms of forces exerted on material bodies and the momentum carried by light, connecting the momentum density to the Poynting vector via g = S/c2.

This chapter develops the far-field, or radiation-zone, approximation of classical electromagnetism and uses it to analyze how accelerated charges radiate energy and momentum. Starting from the retarded integral solutions of Maxwell’s equations, it isolates the 1/𝑟 radiation fields, derives compact expressions for 𝐸rad, 𝐵rad, and the radiative Poynting vector, and shows how total radiated power follows from angular integration. Specializing to localized sources leads to the Larmor formula for nonrelativistic point charges and to explicit formulas for radiation from a one-dimensional harmonic oscillator. The chapter then confronts the back-reaction of radiation on the source, introducing the radiation–reaction force, the radiation relaxation time, and a local self-interaction picture, and it discusses the “classical radiation catastrophe” for bound electrons. Finally, it contrasts radiating, time-dependent currents with steady currents, introduces the energy density carried by radiation from extended current distributions, and describes the geometric–optics picture of radiation in terms of phase fronts and plane waves, before closing with problems.

This chapter develops harmonic plane waves as central solutions of the source-free Maxwell equations and connects them to radiation from localized sources. Starting from the homogeneous equations in vacuum, it introduces complex exponential plane-wave solutions, derives their dispersion relation and transversality, and relates the wave vector, electric field, and magnetic field. The far-field radiation of an oscillating electric dipole is then recast in plane-wave form, illustrating how radiation looks locally like a plane wave. The chapter proceeds to construct an idealized current sheet that produces an exact plane wave and evaluates the fields from the Jefimenko integrals. It goes on to analyze polarization by superposing plane waves, identifying linear, circular, and elliptical polarization states. Finally, it defines the intensity of an electromagnetic wave via the time-averaged Poynting vector and works out interference of crossing plane waves, showing how spatial intensity fringes arise from the superposition principle.

Chapter 1 presents a summary of the brief history of the geometric optics method for light-scattering computation. Then, basic physical variables and constants of quantifying light scattering by a particle are introduced. In particular, Maxwell’s equations are briefly discussed to introduce the Poynting vector, which is used in later discussions of electromagnetic energy conservation concerning extinction, scattering, and absorption. We also introduce the concepts of the Stokes parameters, amplitude scattering matrix, optical theorem, phase matrix, and scattering/absorption/extinction cross sections. The simplification of the scattering matrix is discussed with respect to particle orientations in both the particle and laboratory coordinate systems. We also discuss chirality and mirror symmetry associated with light scattering by nonspherical particles. In addition, we discuss the extinction matrix associated with a specifically oriented particle. The remaining portion of this chapter is devoted to the discussion of Snell’s law, Fresnel equations, and Fraunhofer diffraction. The scalar and vector Kirchhoff diffraction formulas are discussed in detail. Furthermore, we discuss the applicability of Babinet’s principle to the diffraction computation of a particle.

The real fun of the Maxwell equations comes when we understand the link between electricity and magnetism. A changing magnetic flux can induce currents to flow. This is Faraday’s law of induction. We start this chapter by understanding this link and end this chapter with one of the great unifying discoveries of physics: that the interplay between electric and magnetic fields is what gives rise to light.

Having examined many useful and interesting properties of first static electric fields, then static magnetic fields, and most recently the combination of electric and magnetic fields through the introduction of time-varying effects, we have reached a turning point in our studies. Specifically, we will introduce what is perhaps the most revolutionary concept in electromagnetism: propagation of electromagnetic waves. Electromagnetic waves can carry information and energy, and their properties are described in full using Maxwell’s Equations. We will explore these properties in detail in this and the following chapters.

We have introduced Gaussian optics and used a matrix formalism to describe light rays through optical systems in Chapter 1. Light rays are based on the particle nature of light. Since light has a dual nature, light is waves as well. In 1924, de Broglie formulated the de Broglie hypothesis, which relates wavelength and momentum. In this chapter, we explore the wave nature of light, which accounts for wave effects such as interference and diffraction.