Field, Force, Energy and Momentum in Classical Electrodynamics (Revised Edition)

The concepts of scalar and vector fields, which are central to the theory of electrodynamics, are introduced. These fields are generally defined in 3-dimensional Euclidean space as complex-valued functions of the space-time coordinates (x, y, z, t). Integration and differentiation in time and space, leading to such operations as gradient, divergence, and curl, and subsequently to theorems of Gauss and Stokes, are developed. The intuitive approach taken here avoids mathematical formalism in favor of physical understanding. Throughout the chapter, examples based on complex-valued scalar and vector plane-waves help to illustrate the various mathematical operations. The end-of-chapter problems should help refresh the reader’s memory of elementary mathematical tools needed in this as well as in subsequent chapters.

The sources of electromagnetic fields are electric charge, electric current, polarization and magnetization. The relationships among the fields and their sources, all of which represented by functions of space and time, are described by Maxwell’s macroscopic equations. The fields carry energy, whose rate-of-flow at each point in space at any instant of time is given by the Poynting vector. At any location where one or more fields and one or more sources reside simultaneously, there could occur an exchange of energy between the fields and the sources. The time-rates of such exchanges are uniquely specified by the Poynting theorem, which is a direct consequence of Maxwell’s macroscopic equations in conjunction with the definition of the Poynting vector. Electromagnetic fields also carry momentum and angular momentum, whose densities at all points in space-time are simple functions of the local Poynting vector. A generalized version of the Lorentz law of force dictates the time-rate of exchange of momentum between the fields and the sources in regions of space-time where they overlap. There also exists a simple expression for the torque exerted by the fields on the sources, which defines the time-rate of exchange of angular momentum between them. This chapter is devoted to a precise and detailed description of the relations among the fields and their sources, as well as their interactions involving electromagnetic force, torque, energy, momentum, and angular momentum.

In preparation for a Fourier analysis of Maxwell’s equations in the following chapter, we describe here the mathematics of Fourier transformation, exploring certain properties of the forward and reverse Fourier operators. Several special functions are also discussed—notable among them, Dirac’s delta-function and various Bessel functions—which appear frequently in Fourier analysis and elsewhere. Simple charge- and current-density distributions serve as exemplary electromagnetic systems that can be readily transformed into the Fourier domain.

We solve Maxwell’s macroscopic equations under the assumption that the sources of the electromagnetic fields are fully specified throughout space and time. Charge, current, polarization, and magnetization are thus assumed to have predetermined distributions as functions of the space-time coordinates (r, t). In this type of analysis, any action by the fields on the sources will be irrelevant, in the same way that the action on the sources by any other force-be it mechanical, chemical, nuclear, or gravitational-need not be taken into consideration. It is true, of course, that one or more of the above forces could be responsible for the presumed behavior of the sources. However, insofar as the fields are concerned, since the spatio-temporal profiles of the sources are already specified, knowledge of the forces would not provide any additional information. In this chapter, we use Fourier transformation to express each source as a superposition of plane-waves. Maxwell’s equations then associate each planewave with other plane-waves representing the electromagnetic fields. Inverse Fourier transformation then enables us to express the electric and magnetic fields as functions of the space-time coordinates.

The problem addressed in the present chapter is the same problem as discussed in the preceding one, namely, the determination of fields for given distributions of charge, current, polarization and magnetization. Here, however, we will derive expressions for the scalar and vector potentials as functions of space and time coordinates for the given source distributions, which are also specified in space-time. Once the potentials are obtained, the electromagnetic fields will be calculated by straightforward differentiation. The integrals will look very different from those encountered in Chapter 4, but the final results will be exactly the same.

The electric and magnetic dipoles, which produce the polarization and magnetization of material media, are frequently induced by the action of electromagnetic fields on the atomic and molecular constituents of these media. The electric field, for example, may draw the electron cloud surrounding the nucleus of an atom slightly to one side or the other, thus creating a small separation between the centers of positive and negative atomic charges. Or the magnetic field might speed up or slow down the motion of electrons circling a nucleus, thus producing a change in the magnetic dipole moment of individual atoms. Numerous other possibilities exist as well, such as the reorientation of permanent dipoles in the presence of electromagnetic fields. As it turns out, a simple mass-and-spring model, first proposed by the Dutch physicist H. A. Lorentz, captures the essence of induced polarization and magnetization in many situations of practical interest. In this chapter, we describe the classical Lorentz oscillator model and explore its fascinating properties.

Material media typically react to electromagnetic fields by becoming polarized or magnetized, or by developing charge- and current-density distributions within their volumes or on their surfaces. The response of a material medium to the fields could be complicated, as would be the case, for instance, when the relation between induced polarization and the electric field is non-local, non-linear, or history-dependent, or when the induced magnetization is an anisotropic function of the local magnetic and/or electric fields. In many cases of practical interest, however, the media are homogeneous, isotropic, and linear, with the electric dipoles responding only to the local E-field (and magnetic dipoles responding only to the local H-field) in accordance with the Lorentz oscillator model of the preceding chapter. Irrespective of the manner in which the charge-carriers or the dipoles of the medium respond to the fields, there is always an additional complication that the fields are not merely those imposed on the medium from the outside. The motion of the charges and/or the oscillation of the dipoles in response to the fields give rise to new electromagnetic fields, which must then be added to the external fields before the induced charge, current, polarization, or magnetization can be computed. In other words, the entire system of interacting fields and sources, whether originating outside or induced within the media, must be treated self-consistently. This chapter provides a detailed analysis of plane-wave propagation within the simplest kind of material media, namely, those that are homogeneous and isotropic, whose induced electric dipoles are linear functions of the local E-field, and whose induced magnetic dipoles are linear functions of the local H-field.

Plane-waves are the building blocks of arbitrary electromagnetic waves residing in homogeneous, linear media. A basic understanding of plane-wave properties and their behavior upon arrival at a flat interface between two such media is sufficient for the analysis of a number of interesting electrodynamic problems. This chapter presents several exemplary problems that are easy to set up and to describe, yet the understanding and appreciation of their full impact requires subtle arguments involving the properties of plane-waves and those of the Fresnel reflection and transmission coefficients. These examples reveal certain interesting as well as useful features of optical and electromagnetic systems that are frequently encountered in practical applications.

We describe the solution of Maxwell’s equations in systems of homogeneous, isotropic, and linear media that exhibit cylindrical symmetry around a given axis. The solutions are generally expressed in terms of Bessel functions of various kinds and integer orders. The properties of these solutions are discussed in some detail throughout the chapter, where several practical applications are also pointed out.

Material systems in which homogeneous, linear, isotropic media exhibit spherical symmetry around a given point in space support electromagnetic waves that can be expressed as a superposition of certain eigen-modes of Maxwell’s equations. These eigen-modes, known as vector spherical harmonics, are expressed in terms of Bessel functions of various types and orders, associated Legendre functions, and ordinary sinusoidal functions. In contrast to the integer-order Bessel functions which describe the radial dependence of the eigen-modes in systems of cylindrical symmetry, the Bessel functions representing the radial dependence of vector spherical harmonics are of half-integer order. In this chapter, we derive the exact solutions of Maxwell’s equations for transverse electric (TE) as well as transverse magnetic (TM) modes of the electromagnetic field in systems of spherical symmetry. Several applications such as the excitation of whispering gallery modes within dielectric spheres and the scattering of plane-waves from spherical particles will then be discussed.

Electromagnetic fields are capable of exerting force and torque on material media. This is the mechanism by which linear and angular momenta are exchanged between the fields and the media. The fundamental principles that govern such exchanges were described in some detail in Chapter 2. The present chapter provides several examples that demonstrate the application of these principles in certain situations of practical interest.

The reaction of a material body to electromagnetic fields depends not only on the nature of its individual atoms and molecules – which are the seats of electric and magnetic dipoles – but also on the classical as well as quantum-mechanical interactions among its various atomic and molecular constituents. The interatomic interactions may be short-ranged, such as those associated with exchange-coupling among the near neighbors, or they could be mediumrange or even long-range, such as those mediated by conduction electrons. As useful and powerful as the Lorentz oscillator model of Chapter 6 has been in many practical applications, its main shortcoming is that it represents dipoles that respond to nothing but local electromagnetic fields. When the states of adjacent dipoles, or those of dipoles that are further removed, happen to have a direct impact on the behavior of a given electric or magnetic dipole, the Lorentz oscillator model must be modified to account for such interactions. In this chapter we explore the consequences of nearest-neighbor couplings among the electric dipoles of a homogeneous, linear, isotropic medium. It is these interactions that lead to spatial dispersion.

We present a general proof of the optical theorem (also known as the optical crosssection theorem) that reveals the intimate connection between the forward scattering amplitude and the absorption-plus-scattering of the incident wave within the scatterer. The oscillating electric charges and currents as well as the electric and magnetic dipoles of the scatterer, driven by an incident plane-wave, extract energy from the incident beam at a certain rate. The same oscillators radiate electromagnetic energy into the far field, thus giving rise to well-defined scattering amplitudes along various directions. The essence of the proof presented here is that the extinction cross-section of an object can be related to its forward scattering amplitude using the induced oscillations within the object but without an actual knowledge of the mathematical form assumed by these oscillations.

This chapter provides a simple physical proof of the reciprocity theorem of classical electrodynamics in the general case of material media that contain linearly polarizable as well as linearly magnetizable substances. The excitation source is taken to be a point-dipole, either electric or magnetic, and the measured field at the observation point can be electric or magnetic, regardless of the nature of the source dipole. The electric and magnetic susceptibility tensors of the material system may vary from point to point in space, but they cannot be functions of time. In the case of spatially non-dispersive media, the only other constraint on the local susceptibility tensors is that they be symmetric at each and every point. The proof is readily extended to media that exhibit spatial dispersion: For reciprocity to hold, the electric susceptibility tensor χE_ mn that relates the complex-valued magnitude of the electric dipole at location rm to the strength of the electric field at rn must be the transpose of χE_nm. Similarly, the necessary and sufficient condition for the magnetic susceptibility tensor is χ M_mn=χ T M_nm.

Einstein’s special theory of relativity provides a unique insight into the fundamental properties of the electromagnetic field. It presents, from a new perspective, the intimate connections among the E, D, H, B fields and their sources-connections that we have heretofore examined only through Maxwell’s equations. To appreciate the simplicity and beauty of electrodynamics in the context of special relativity, one must first learn the language of tensor analysis, which is where this chapter begins. From there we move to a description of the flat, four-dimensional spacetime of H. Minkowski, and the transformation rule, named after H. A. Lorentz, that governs coordinate transformations among different inertial reference frames. We proceed to describe the electromagnetic fields and their sources as seen by different observers in uniform unaccelerated motion relative to each other, thereby uncovering some of the most intriguing features of classical electrodynamics.