Electromagnetism and special relativity

In 1905, when Einstein published his theory of special relativity, Maxwell’s work was already about forty years old. It is therefore both remarkable and ironic (recalling the old arguments about the aether being the ‘preferred’ reference frame for describing wave propagation) that classical electrodynamics turned out to be a relativistically correct theory. In this chapter, a range of questions in electromagnetism are considered as they relate to special relativity. In Questions 12.1–12.4 the behaviour of various physical quantities under Lorentz transformation is considered. This leads to the important concept of an invariant. Several of these are encountered, and used frequently throughout this chapter. Other topics considered include the transformationof E- and B-fields between inertial reference frames, the validity of Gauss’s law for an arbitrarily moving point charge (demonstrated numerically), the electromagnetic field tensor, Maxwell’s equations in covariant form and Larmor’s formula for a relativistic charge.

Electromagnetic radiation

This chapter begins by expressing the multipole expansion of the dynamic vector potential A ( r, t) in terms of electric and magnetic multipole moments. Differentiation of A ( r, t) leads directly to the fields E ( r, t) and B ( r, t), which have a component transporting energy away from the sources to infinity. This component is called electromagnetic radiation and it arises only when electric charges experience an acceleration. A range of questions deal with the various types of radiation, including electric dipole and magnetic dipole–electric quadrupole. Larmor’s formula is applied in both its non-relativistic and relativistic forms. Also considered are some applications involving antennas, antenna arrays and the scattering of radiation by a free electron.

Static magnetic fields in vacuum

Wherever possible, an attempt has been made to structure this chapter along similar lines to Chapter 2 (its electrostatic counterpart). Maxwell’s magnetostatic equations are derived from Ampere’s experimental law of force. These results, along with the Biot–Savart law, are then used to determine the magnetic field B arising from various stationary current distributions. The magnetic vector potential A emerges naturally during our discussion, and it features prominently in questions throughout the remainder of this book. Also mentioned is the magnetic scalar potential. Although of lesser theoretical significance than the vector potential, the magnetic scalar potential can sometimes be an effective problem-solving device. Some examples of this are provided. This chapter concludes by making a multipole expansion of A and introducing the magnetic multipole moments of a bounded distribution of stationary currents. Several applications involving magnetic dipoles and magnetic quadrupoles are given.

The electrostatics of conductors

This chapter begins by proving some important properties of (i) conductors in electrostatic equilibrium, and (ii) harmonic functions. These results underpin most of the remaining questions of Chapter 3. The coefficients of capacitance for an arbitrary arrangement of conductors are introduced at an early stage, and numerical calculations then follow in a number of subsequent questions. Some important techniques (both analytical and numerical) for finding solutions to Laplace’s equation are considered. These include: the Fourier method, the relaxation method, themethod of images and the method of conformal transformation. All of these are discussed in some detail, and with appropriate examples.

Some essential mathematics

This chapter introduces most of the important mathematics that will be used repeatedly throughout this book. Whilst scalars like the electric potential Ф and the vector fields E and B are familiar quantities in electricity and magnetism, it is not always known that they are examples of a mathematical form termed a tensor. More sophisticated tensors are required in some descriptions and theories in electrodynamics. As tensor notation is compact and usually allows for a simple derivation of a result, the chapter begins with a series of questions involving Cartesian tensors. The results obtained here will facilitate the solution of many questions in subsequent chapters. Readers who are not familiar with tensors are advised to consult Appendix A before proceeding. Towards the end of this Appendix a ‘checklist for detecting errors when using tensor notation’ is given, which will hopefully provide some help for the uninitiated.

Quasi-static electric and magnetic fields in vacuum

In this chapter, the transition from time-independent to time-dependent source densities and fields is made. It is here that Faraday’s famous nineteenth-century experiments on electromagnetic induction are first encountered. This important phenomenon—whereby a changing magnetic field produces an induced electric field (whose curl is now no longer zero)—forms the basis of most of the questions and solutions which follow. Some new and interesting examples—not usually found in other textbooks—are introduced. These are treated both from an analytical and numerical point of view. Also considered here is the standard yet important topic (at least from a practical standpoint) of mutual and self-inductance. Several questions deal with this concept.

Some applications of Maxwell’s equations in matter

This chapter comprises questions of a miscellaneous nature. They mostly have little in common except that all processes are time-dependent and occur within matter. The first few questions introduce some important preliminaries. For example, modifying Maxwell’s equations to include the effect of matter. The behaviour of the electromagnetic field at the boundary between two media having different properties is an important topic. The matching conditions (as they are known) are derived from both the integral and differential forms of Maxwell’s equations. Certain specific examples then follow, including some simple applications involving conductors, dielectrics and tenuous electronic plasmas. Along the way, the connection between Maxwell’s electrodynamics and the laws of geometrical optics is demonstrated explicitly.

Static electric and magnetic fields in matter

The electric and magnetic fields at an arbitrary point inside matter fluctuate rapidly both in space and time. It is therefore necessary, in the chapter’s introduction, tomake the important distinction between microscopic and macroscopic fields. It is assumed throughout that the electric dipole approximation is valid, and several important properties which apply to matter in bulk are described early on. Along the way, it is convenient to define two new fields D and H, and these appear in a number of questions. Another feature which emerges during this treatment includes the concept of bound charge and current densities, and how these quantities are incorporated intoMaxwell’s inhomogeneous equations. Although Chapter 9 deals exclusively with static fields, it forms an important backdrop to Chapter 10 where some general timedependent aspects of macroscopic electromagnetism are considered.

The electromagnetic potentials

This chapter expresses the fields E ( r, t) and B ( r, t) in terms of the electromagnetic potentials Ф( r, t) and A ( r, t), and shows that these potentials are defined only up to a gauge transformation. This leads the reader naturally to the Coulomb and Lorenz gauges which are usually encountered in textbooks. The inhomogeneous wave equations whose solutions are the retarded electromagnetic potentials are also considered, as well as the Lienard–Wiechert potentials for an arbitrarily moving point charge. A few questions are included in which the Lagrangian and Hamiltonian of a point charge are expressed in terms of Ф and A. The chapter concludes by deriving a multipole expansion for the dynamic vector potential which will provide the starting point in our treatment of electromagnetic radiation later on in Chapter 11

Ohm’s law and electric circuits

This chapter considers various simple dc and ac circuits which contain at least one active element (always a voltage source) and passive elements (resistors, capacitors and inductors) arranged in different combinations to form a bilateral network. The notions of complex voltage, complex current and complex impedance are introduced and then used in the ensuing analysis. Some standard ‘network theorems’ including Kirchhoff’s rules, the delta-star transformation, Thevenin’s theorem and the superposition theorem are employed. Included in the questions are circuits involving bridges, filters, audio amplifiers and transformers. Important topics such as series and parallel resonance in LRC circuits are also considered along the way. Much of the laborious algebra involved in manipulating the complex quantities above is avoided by relegating this task to Mathematica.