Preamble

The electromagnetic field may always be described in terms of two basic fields: the electric field strength E and the magnetic induction B. The sources for these fields are the charge density ρ and the current density J. The sources and the fields are related by Maxwell's equations. Maxwell's equations are four simultaneous first order differential equations and one cannot solve them directly to find the fields given the source terms. One way of solving Maxwell's equations is to introduce an alternative description of an electromagnetic field in terms of the scalar potential φ and vector potential A. These potentials are not uniquely defined and a specific choice for them satisfies a gauge condition.

Maxwell's Equations

The electric fieldE is defined as the force per unit charge acting on a test charge. Thus, if q is an arbitrarily small test charge (so that the electric field that it generates is negligible) located at a point a distance x from the origin at time t, then the electric force on it is qE(t,x). (For simplicity in notation, and where no confusion is likely to arise, the dependences on t and x are not shown explicitly.) The units of electric field strength are thus those of a force (kg m s−2) per unit charge; the unit of charge is the Coulomb (C) so that the magnitude of E has units kg m s−2C−1.

The emission of waves is treated by solving the inhomogeneous wave equation and deriving an emission formula. The emission formula derived here may be used to describe the emission of waves in an arbitrary medium by an arbitrary source, described by an extraneous current. The current corresponding to electric and magnetic dipoles and electric quadrupoles is discussed in detail, and the electric dipole case is used to derive the Larmor formula, which describes emission by an accelerated nonrelativistic charge in vacuo. The more conventional treatment of emission based on the Lienard–Wierchert potentials is then developed. The back reaction of the radiating system to the emission of radiation may be taken into account in two different ways, depending on the context: by the use of quasi-linear theory, and through a radiation reaction force. These two procedures are discussed critically here.

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.