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Introduction
Practical, intellectual, and cultural reasons motivate the study of electricity and magnetism. The operation of electrical systems designed to perform certain engineering tasks depends, at least in part, on electrical, electromechanical, or electrochemical phenomena. The electrical aspects of these applications are described by Maxwell's equations. As a description of the temporal evolution of electromagnetic fields in three-dimensional space, these same equations form a concise summary of a wider range of phenomena than can be found in any other discipline. Maxwell's equations are an intellectual achievement that should be familiar to every student of physical phenomena. As part of the theory of fields that includes continuum mechanics, quantum mechanics, heat and mass transfer, and many other disciplines, our subject develops the mathematical language and methods that are the basis for these other areas.
For those who have an interest in electromechanical energy conversion, transmission systems at power or radio frequencies, waveguides at microwave or optical frequencies, antennas, or plasmas, there is little need to argue the necessity for becoming expert in dealing with electromagnetic fields. There are others who may require encouragement. For example, circuit designers may be satisfied with circuit theory, the laws of which are stated in terms of voltages and currents and in terms of the relations imposed upon the voltages and currents by the circuit elements. However, these laws break down at high frequencies, and this cannot be understood without electromagnetic field theory. The limitations of circuit models come into play as the frequency is raised so high that the propagation time of electromagnetic fields becomes comparable to a period, with the result that "inductors" behave as "capacitors" and vice versa. Other limitations are associated with loss phenomena. As the frequency is raised, resistors and transistors are limited by "capacitive" effects, and transducers and transformers by "eddy" currents.
Anyone concerned with developing circuit models for physical systems requires a field theory background to justify approximations and to derive the values of the circuit parameters. Thus, the bioengineer concerned with electrocardiography or neurophysiology must resort to field theory in establishing a meaningful connection between the physical reality and models, when these are stated in terms of circuit elements. Similarly, even if a control theorist makes use of a lumped parameter model, its justification hinges on a continuum theory, whether electromagnetic, mechanical, or thermal in nature.
Computer hardware may seem to be another application not dependent on electromagnetic field theory. The software interface through which the computer is often seen makes it seem unrelated to our subject. Although the hardware is generally represented in terms of circuits, the practical realization of a computer designed to carry out logic operations is limited by electromagnetic laws. For example, the signal originating at one point in a computer cannot reach another point within a time less than that required for a signal, propagating at the speed of light, to traverse the interconnecting wires. That circuit models have remained useful as computation speeds have increased is a tribute to the solid state technology that has made it possible to decrease the size of the fundamental circuit elements. Sooner or later, the fundamental limitations imposed by the electromagnetic fields define the computation speed frontier of computer technology, whether it be caused by electromagnetic wave delays or electrical power dissipation.
Overview of Subject
As illustrated diagrammatically in Fig. 1.0.1, we start with Maxwell's equations written in integral form. This chapter begins with a definition of the fields in terms of forces and sources followed by a review of each of the integral laws. Interwoven with the development are examples intended to develop the methods for surface and volume integrals used in stating the laws. The examples are also intended to attach at least one physical situation to each of the laws. Our objective in the chapters that follow is to make these laws useful, not only in modeling engineering systems but in dealing with practical systems in a qualitative fashion (as an inventor often does). The integral laws are directly useful for (a) dealing with fields in this qualitative way, (b) finding fields in simple configurations having a great deal of symmetry, and (c) relating fields to their sources.
Chapter 2 develops a differential description from the integral laws. By following the examples and some of the homework associated with each of the sections, a minimum background in the mathematical theorems and operators is developed. The differential operators and associated integral theorems are brought in as needed. Thus, the divergence and curl operators, along with the theorems of Gauss and Stokes, are developed in Chap. 2, while the gradient operator and integral theorem are naturally derived in Chap. 4.
Figure 1.0.1 Outline of Subject. The three columns, respectively for electroquasistatics, magnetoquasistatics and electrodynamics, show parallels in development. Static fields are often the first topic in developing an understanding of phenomena predicted by Maxwell's equations. Fields are not measurable, let alone of practical interest, unless they are dynamic. As developed here, fields are never truly static. The subject of quasistatics, begun in Chap. 3, is central to the approach we will use to understand the implications of Maxwell's equations. A mature understanding of these equations is achieved when one has learned how to neglect complications that are inconsequential. The electroquasistatic (EQS) and magnetoquasistatic (MQS) approximations are justified if time rates of change are slow enough (frequencies are low enough) so that time delays due to the propagation of electromagnetic waves are unimportant. The examples considered in Chap. 3 give some notion as to which of the two approximations is appropriate in a given situation. A full appreciation for the quasistatic approximations will come into view as the EQS and MQS developments are drawn together in Chaps. 11 through 15.
Although capacitors and inductors are examples in the electroquasistatic and magnetoquasistatic categories, respectively, it is not true that quasistatic systems can be generally modeled by frequency-independent circuit elements. High-frequency models for transistors are correctly based on the EQS approximation. Electromagnetic wave delays in the transistors are not consequential. Nevertheless, dynamic effects are important and the EQS approximation can contain the finite time for charge migration. Models for eddy current shields or heaters are correctly based on the MQS approximation. Again, the delay time of an electromagnetic wave is unimportant while the all-important diffusion time of the magnetic field is represented by the MQS laws. Space charge waves on an electron beam or spin waves in a saturated magnetizable material are often described by EQS and MQS laws, respectively, even though frequencies of interest are in the GHz range.
The parallel developments of EQS (Chaps. 4-7) and MQS systems (Chaps. 8-10) is emphasized by the first page of Fig. 1.0.1. For each topic in the EQS column to the left there is an analogous one at the same level in the MQS column. Although the field concepts and mathematical techniques used in dealing with EQS and MQS systems are often similar, a comparative study reveals as many contrasts as direct analogies. There is a two-way interplay between the electric and magnetic studies. Not only are results from the EQS developments applied in the description of MQS systems, but the examination of MQS situations leads to a greater appreciation for the EQS laws.
At the tops of the EQS and the MQS columns, the first page of Fig. 1.0.1, general (contrasting) attributes of the electric and magnetic fields are identified. The developments then lead from situations where the field sources are prescribed to where they are to be determined. Thus, EQS electric fields are first found from prescribed distributions of charge, while MQS magnetic fields are determined given the currents. The development of the EQS field solution is a direct investment in the subsequent MQS derivation. It is then recognized that in many practical situations, these sources are induced in materials and must therefore be found as part of the field solution. In the first of these situations, induced sources are on the boundaries of conductors having a sufficiently high electrical conductivity to be modeled as "perfectly" conducting. For the EQS systems, these sources are surface charges, while for the MQS, they are surface currents. In either case, fields must satisfy boundary conditions, and the EQS study provides not only mathematical techniques but even partial differential equations directly applicable to MQS problems.
Polarization and magnetization account for field sources that can be prescribed (electrets and permanent magnets) or induced by the fields themselves. In the Chu formulation used here, there is a complete analogy between the way in which polarization and magnetization are represented. Thus, there is a direct transfer of ideas from Chap. 6 to Chap. 9.
The parallel quasistatic studies culminate in Chaps. 7 and 10 in an examination of loss phenomena. Here we learn that very different answers must be given to the question "When is a conductor perfect?" for EQS on one hand, and MQS on the other.
In Chap. 11, many of the concepts developed previously are put to work through the consideration of the flow of power, storage of energy, and production of electromagnetic forces. From this chapter on, Maxwell's equations are used without approximation. Thus, the EQS and MQS approximations are seen to represent systems in which either the electric or the magnetic energy storage dominates respectively.
In Chaps. 12 through 14, the focus is on electromagnetic waves. The development is a natural extension of the approach taken in the EQS and MQS columns. This is emphasized by the outline represented on the right page of Fig. 1.0.1. The topics of Chaps. 12 and 13 parallel those of the EQS and MQS columns on the previous page. Potentials used to represent electrodynamic fields are a natural generalization of those used for the EQS and MQS systems. As for the quasistatic fields, the fields of given sources are considered first. An immediate practical application is therefore the description of radiation fields of antennas.
The boundary value point of view, introduced for EQS systems in Chap. 5 and for MQS systems in Chap. 8, is the basic theme of Chap. 13. Practical examples include simple transmission lines and waveguides. An understanding of transmission line dynamics, the subject of Chap. 14, is necessary in dealing with the "conventional" ideal lines that model most high-frequency systems. They are also shown to provide useful models for representing quasistatic dynamical processes.
To make practical use of Maxwell's equations, it is necessary to master the art of making approximations. Based on the electromagnetic properties and dimensions of a system and on the time scales (frequencies) of importance, how can a physical system be broken into electromagnetic subsystems, each described by its dominant physical processes? It is with this goal in mind that the EQS and MQS approximations are introduced in Chap. 3, and to this end that Chap. 15 gives an overview of electromagnetic fields.
