Preamble

There are two different descriptions of the electromagnetic response of a medium. One, which is the older description, involves introducing the polarization and magnetization of the medium and was originally applied to the response to static fields. This method may be generalized to apply to fluctuating fields but it becomes cumbersome and ill-defined for sufficiently general media. The other description is based on the use of Fourier transforms and so applies only to fluctuating fields. The Fourier transform description is used widely in plasma physics, and although it is less familiar than the other description when applied to dielectrics and magnetizable media, it is simpler and no less general than the older description.

Static Responses

The response of a medium to a static uniform electromagnetic field is described in terms of induced dipole moments. On a microscopic level a static uniform electric field polarizes the atoms or molecules. The polarizationP is defined as the induced electric dipole moment per unit volume. A medium which becomes polarized in this way is called a dielectric.

The response of a magnetizable medium to a static uniform magnetic field is attributed to induced magnetic dipole moments and is described in terms of the magnetizationM, which is defined as the induced magnetic dipole moment per unit volume. Magnetizable media are classified as paramagnetic or diamagnetic depending on whether the magnetization is parallel or antiparallel, respectively, to the applied magnetic field.

In an undergraduate physics course it is common practice to introduce electromagnetic theory in two stages with a third stage at senior undergraduate level or higher. In a first course the integral forms of Maxwell's equations are introduced and used to treat a variety of problems relating electric and magnetic fields to their sources. The essential part of a second course is the introduction of the differential forms of Maxwell's equations, with a major ingredient being the development of the necessary mathematical tools of the differential vector calculus and the integral theorems of Gauss and Stokes. The physical content of this part of the second course differs little from that of the first course, and usually some additional chapters such as electromagnetic responses of media, propagation of electromagnetic waves in waveguides, Lorentz transformation of electromagnetic fields and so on, are included to add some new physical content. A third course in electromagnetic theory starts with the differential forms of Maxwell's equations, and is usually at senior undergraduate or first year graduate level. The present book is intended to be at the level of such a third course in electromagnetic theory.

There is an approach to the teaching of electromagnetic theory at this level that has become almost traditional due to the availability of some excellent textbooks that present a similar approach. Notable examples are Stratton (1941), Landau and Lifshitz, (1951) Jackson (1975) and Panofsky and Phillips (1962), as cited in the Bibliographic Notes.

Preamble

Plasmas can support a great variety of wave motions. For many purposes it suffices to have knowledge of three classes of waves, two of which are discussed here. These are waves in isotropic thermal plasmas, and waves in cold magnetized plasmas. The third class of waves are the MHD waves (MHD is short for magnetohydrodynamics), which are derived within the framework of a fluid model for the plasma. The MHD waves are not discussed in detail here.

Waves in Isotropic Thermal Plasmas

An isotropic plasma is defined to be a plasma (a) with no ambient magnetic field (it is unmagnetized), and (b) in which all species of particles have a Maxwellian distribution of velocities (or its relativistic generalization if relativistic effects are included). In any isotropic medium the waves are either longitudinal or transverse (§12.1). The longitudinal waves satisfy the longitudinal dispersion equations (12.6), viz., KL(ω, k) = 0, and the transverse waves satisfy the transverse dispersion equations (12.7), viz., n2 = KT(ω, k). The longitudinal and transverse parts of the dielectric tensor for an isotropic thermal plasma are given by (10.23) and (10.24), respectively.

Langmuir Waves

There are two solutions of the longitudinal dispersion equation that are important in practice. These are for Langmuir waves, which involve only the motion of the electrons, and ion sound waves (also called ion acoustic waves) that are associated with motion of the ions. As mentioned in §10.3, both these wave modes were identified by Tonks and Langmuir in 1929 in what is now recognized as the first major article in the development of modern plasma theory.

In order to relate the electromagnetic field to its sources one needs to solve Maxwell's equations. One method of solving Maxwell's equations is to introduce potentials (the scalar and vector potentials) which effectively reduces the number of independent equations. For static sources and fields one may reduce Maxwell's equations to Poisson's equation for the electromagnetic potentials, and for time-dependent sources and fields one may reduce Maxwell's equations to d'Alembert's equation for the electromagnetic potentials. An alternative approach is to Fourier transform. This approach applies only to fluctuating fields, and hence it is necessary to distinguish between fluctuating fields and any static field, which is regarded as an ambient field when describing the response to fluctuating fields using Fourier transforms. The use of Fourier transforms allows one to reduce Maxwell's equations for time-dependent sources and fields to a single algebraic equation, called the wave equation. A specific advantage of this approach is that it allows one to include the effect of an ambient medium in a simple but general way.

Maxwell's equations are written down and the electromagnetic potentials are introduced in Chapter 1. There are two mathematical tools that are required in the treatment of electromagnetic theory adopted here. One of these is tensor algebra, which is introduced in Chapter 2. Some electromagnetic applications of tensor algebra are described in Chapter 3 with particular emphasis on multipole moments. The other mathematical tool is the Fourier transformation, which is introduced in Chapter 4.

Preamble

Once the amplitude of a wave is defined, the electric and magnetic energies in the waves are calculated in terms of this amplitude. However, the total energy in the waves cannot be identified in any simple way in general. The damping of waves is used to identify the total energy by relating the damping to the dissipative part of the response tensor in two ways. One way involves calculating the work done by the dissipative process and equating this to the energy lost by the waves. The other way involves including damping in terms of an imaginary part of the frequency (§11.4). The equivalence of the two ways of treating damping provides an explicit expression for the total energy in the waves. A semiclassical description of a distribution of waves is useful for both formal and practical purposes; the semiclassical description is based on regarding the waves as a collection of wave quanta.

The Electric and Magnetic Energies in Waves

In any physical theory, energy is defined in terms of its mechanical equivalent. In §15.2 this is achieved by calculating the work done by an arbitrary dissipative process and equating it to the energy lost by the waves. An important preliminary step is to define the amplitude of the waves and to calculate the electric energy in waves in Fourier space by using the fact that the electric energy density in coordinate space is given by ½ε0∣E∣2. The magnetic energy in waves is calculated in an analogous way.

Preamble

The response tensor for a dielectric depends on the polarizability of the individual atoms and molecules. In a “dense” isotropic medium, where “dense” is not a well-defined concept, the relation between the dielectric constant and the polarizability is given by the Lorenz–Lorentz relation. The polarizability needs to be calculated quantum mechanically, but many of the features of the response of a dielectric may be inferred from a classical model of forced oscillators.

The Polarizability of Atoms and Molecules

A simple model for the response of dielectric materials is based on assuming that the response results from induced electric dipole moments in the medium. One distinguishes between three classes of polarization on a microscopic level in different types of media. One class consists of media in which the polarization is attributed to deformation of atoms or molecules so that the mean centers of the positive and negative charges become slightly separated, implying that the atoms or molecules develop induced dipole moments. A second class of media consists of those in which the individual particles have intrinsic dipole moments. These moments are randomly oriented in the absence of an external field and become partially aligned in the presence of an external field. The responses for these two classes of media exist for static fields as well as for oscillating fields. The third class of media consists of charges that are free to move and such media become polarized in the sense that there is a net displacement between the positive and negative charges.

Preamble

“Bremsstrahlung” is used both as the generic name to describe emission due to an accelerated charged particle and as the specific name for the emission when this acceleration is due to the Coulomb field of another particle. Here we are concerned with bremsstrahlung in the latter more restrictive sense. Bremsstrahlung can result in the emission of waves in any wave mode in a plasma, but most interest is in the emission of transverse waves. Emission of Langmuir waves is of less interest because, unlike transverse waves in a plasma, Langmuir waves are generated efficiently by the Cerenkov process. The absorption process corresponding to bremsstrahlung is called collisional damping or, in some astrophysical literature, free–free absorption.

Qualitative Discussion of Bremsstrahlung

Bremsstrahlung due to Coulomb interactions between electrons and ions is an important emission process in a wide variety of plasmas. We mention only three general applications. (i) For laboratory plasmas and many space plasmas, bremsstrahlung is the basic thermal emission process at radio frequencies. Radio frequency emission results from distant electron–ion encounters in which the motion of the electron is perturbed only slightly by the Coulomb field of the ion. (ii) High-frequency photons, with an energy comparable with the initial energy of the electron, can result from a close encounter between an electron and an ion. So-called non-thermal bremsstrahlung due to energetic electrons is an important source of X-rays from a plasma.

Specific emission and absorption processes may be classified in several different ways. The presentation here is based on the nature of the extraneous current that acts as the source term for the emission. The simplest emission processes are “direct” emission in which the current is due solely to the motion of the emitting particle. The specific direct emission processes discussed here are (i) Cerenkov emission, which occurs for a particle in constant rectilinear motion at greater than the phase speed of the emitted wave, (ii) bremsstrahlung, which results from the accelerated motion of an electron due to the influence of the Coulomb field of an ion, and (iii) gyromagnetic emission, which is due to the accelerated motion of a particle in a magnetostatic field. To each of these processes there is a corresponding absorption process, and negative absorption is possible for both the Cerenkov and gyromagnetic processes. Another class of emission processes corresponds to scattering of waves by particles. The unscattered wave perturbs the orbit of the particle, and the emission due to this perturbed motion corresponds to the generation of the scattered radiation. Non-linear plasma currents need to be taken into account in treating scattering in a plasma. The non-linear currents allow another class of emission processes that are attributed to wavewave interactions, which include a variety of processes such as frequency doubling in non-linear optics and radiation from turbulent plasmas.

Preamble

Synchrotron emission is gyromagnetic emission from ultrarelativistic particles. It is important in the laboratory as a source of radiation in synchrotrons and as an energy loss mechanism for relativistic electrons confined by a magnetic field. Synchrotron radiation is of particular importance in astrophysics because it is the dominant emission mechanism for the vast majority of radioastronomical sources.

Forward Emission by Relativistic Particles

Before discussing synchrotron emission in particular, it is appropriate to discuss emission by relativistic particles in general. Important features of the emission by relativistic particles may be determined by the special theory of relativity and are only weakly dependent on the specific emission mechanism involved. Emission by a particle which is highly relativistic in the laboratory frame K may be inferred from its emission pattern in its rest frame K0. Let the instantaneous velocity v of the particle in K be along the z axis. The frame K0, as viewed from K is moving along the z axis at velocity v, as illustrated in Figure 24.1, and the frame K, as viewed from K0, is moving along the z0 axis at velocity –v.

In K0 the particle is momentarily at rest, and so its emission pattern is dipolar. For present purposes the only important point is that the emission pattern in K0 is not highly anisotropic. Let w0, k0, ψ0 describe a plane wave in K0, with ψ0 the angle between k0 and the z0 axis.

A medium with electromagnetic properties modifies an electromagnetic field imposed on it. The response of some media may be described satisfactorily in terms of induced dipole moments, and this is particularly the case for the response to static fields. The response to a fluctuating field may also be described in terms of the induced current density. This alternative description is used widely in plasma physics and is emphasized in the approach adopted here.

In Part Two the nature of electromagnetic responses is discussed in Chapter 6 and general properties of response tensors are summarized in Chapter 7. An understanding of the material in these two chapters is important in the discussion of waves in media in Part Three. The remaining three Chapters in Part Two are more of the nature of reference material. Although much of the material in these Chapters is referred to and used in the remainder of the book, a detailed understanding of this material is not essential before proceeding to Parts Three, Four and Five.