This is an undergraduate textbook on the physics of electricity, magnetism, and electromagnetic fields and waves. It is written mainly with the physics student in mind, although it will also be of use to students of electrical and electronic engineering. We have aimed to produce a concise text which emphasises the meaning and significance of the concepts that appear in the theory, and the overall coherence and beauty of the Maxwell equations.

The theory is set out in a self-contained way, but we assume that the reader will already have some knowledge of the basic phenomena of electricity and magnetism. (At the University of Bristol there is an established tradition of demonstration experiments in the introductory first year physics lectures.) We also assume some familiarity with the mathematics of scalar and vector fields, and the properties of the ∇ operator. The basic theorems are set out for reference in the Mathematical Prologue. The Dirac δ-function is introduced in a non-rigorous way on the first page of Chapter 1, and used freely: in our experience, physics students readily accept this as an obviously useful mathematical device. A few other mathematical tools are developed in the text, as and when they are needed. To avoid impeding the flow of the main argument, the technical details of mathematical manipulations are sometimes relegated to the problems.

The relationship between the microscopic structure of matter and the macroscopic fields which are the main concern of the text is stressed from the start, albeit from a classical standpoint.

Phenomena associated with magnets have been known for many centuries. The fact that a compass needle experiences a torque which aligns it in a particular direction was known to Chinese mariners before 1100, and in Europe about a century later; it was vital to the navigators of the great age of exploration. We now interpret this phenomenon as due to the interaction of the needle with the earth's magnetic field. The torque on the compass needle is similar to that on an electric dipole in an electric field; the needle behaves like a magnetic dipole and the fact that it experiences no net force tells us that it carries no net ‘magnetic charge’.

Until 1819 the connection between magnetism and electricity was unknown, but in that year the Danish physicist Oersted observed that an electric current flowing in a wire deflected a nearby compass needle. Conversely, by Newton's law of action and reaction, the compass needle could be expected to exert forces on the current carrying wire. Oersted's discovery created great excitement in scientific academies throughout Europe and in particular stimulated the more detailed investigations of Biot and Savart, and of Ampère, in Paris. It was Ampère who found that two current carrying wires interacted by magnetic forces.

To account for the experimental phenomena it is natural to introduce a magnetic field B(r), which is determined by the magnets and current flows in the system under consideration, and through which different parts of the system interact.

In 1911 the Danish physicist Kamerlingh Onnes found that the electrical resistivity of mercury appeared to vanish completely below 4.2 K: a steady current flowed in a ring without need of a sustaining electric field. The discovery followed from his success in 1908 of liquifying helium, thereby making temperatures down to about 1 K accessible to experiment. This phenomenon of superconductivity was subsequently found in many other metals and alloys, but until recently the known critical temperatures Te at which the transition to the superconducting state occurs had not exceeded 24 K; all superconducting technology depended on the availability of (expensive) liquid helium. In 1986–7 new classes of ‘high-Te’ superconductors were discovered having critical temperatures which exceed the boiling point at atmospheric pressure of (cheap) liquid nitrogen, 77.4 K. These new superconductors are complex compounds, such as YBa2Cu3O7-δ; their properties and possible technological applications are being intensively studied.

The Meissner effect

Materials which become superconducting have the remarkable property of being ‘perfectly diamagnetic’ in their superconducting state. In a static magnetic field, up to a certain critical magnitude, magnetic flux is completely expelled from the inner regions of a large sample when the sample is cooled below its transition temperature (Fig. 14.1). An electric current is generated whose field exactly cancels the applied field in the interior of the sample. From the Maxwell equation ∇ × B = µ0J, the current flow is confined to a region close to the surface of the sample, since ∇ × B = 0 in its interior.

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.