To this point in our discussions, we have dealt solely with static fields. We started with static electric fields, in which all charges are stationary. The electric fields produced by these charges are stationary as well. With electric fields, we developed the notion of the electric potential, the energy stored by electric fields, and the capacitance of a configuration of conductors. We then moved on to introduce static magnetic fields, which are produced by stationary currents. For magnetic fields, we have also introduced potential functions, one a vector function, the other a scalar, but we have not yet discussed the energy stored by a magnetic field, or the inductance of a configuration of current-carrying wires. We will, of course, treat these important topics, but before we do so, we find it useful to take a first look at some time-varying effects. In particular, we will develop a law known as Faraday’s Law, which is the basis for circuit elements such as inductors and transformers, as well as electrical generators and many other useful devices. After we have mastered Faraday’s Law, we will be in a much better position to discuss the energy stored in magnetic fields and inductances, and so we will return to these topics at that time.

Having examined many useful and interesting properties of first static electric fields, then static magnetic fields, and most recently the combination of electric and magnetic fields through the introduction of time-varying effects, we have reached a turning point in our studies. Specifically, we will introduce what is perhaps the most revolutionary concept in electromagnetism: propagation of electromagnetic waves. Electromagnetic waves can carry information and energy, and their properties are described in full using Maxwell’s Equations. We will explore these properties in detail in this and the following chapters.

In the previous chapter we introduced transmission lines, structures consisting of a pair of conductors that guide a high-frequency electromagnetic wave in the space between the conductors from a source to a load. We considered a number of different geometries (parallel plate, twin-lead, co-axial), each with a uniform cross-section, that guide the wave in the $\pm z$-direction, and we discussed the properties of signals carried by the transmission lines, largely in terms of the potential difference $\tilde{V}\left(z\right)$ between the conductors and the current $\tilde{I}\left(z\right)$ flowing in one conductor and returning in the other. For many high-power, high-frequency applications, waveguides are often used to guide electromagnetic waves from source to load. Unlike a transmission line, a waveguide typically consists only of a single conductor, which is hollow, with the wave existing in the interior of the waveguide. Waveguides are often used for frequencies ranging from 10 to 100 GHz, and suffer lower losses in this range than do transmission lines. The typical transverse dimension of a waveguide is a few centimeters, on the order of the wavelength of the wave inside. A few examples of waveguides are illustrated in Fig. 10.1. The parallel plate waveguide shown in Fig. 10.1(a) is not in common usage, but serves as a simple geometry for developing the properties of general waveguides. (Parallel plates differ from most other waveguides, in that they consist of two conductors and can function as transmission lines or waveguides.) Also shown in this figure are (b) rectangular and (c) circular waveguides. While the electromagnetic principles for the circular waveguide are similar to those of the rectangular waveguide, the mathematical functions needed for this geometry are not familiar to most undergraduate students, so we’ll need to introduce these functions. Also, while we devote our attention here to conducting waveguides, be aware that dielectric waveguides, such as optical fibers and photonic crystals can be treated in a similar manner. With the exception of TechNotes 10.1 and 10.2, we will not consider dielectric waveguides in this text. Optical fibers are common conduits for optical signals, and find application in medical instruments and communications systems.

In the previous chapter we discussed freely propagating waves: waves that propagate through a medium without any guidance from supporting conducting or dielectric surfaces. We can, however, effectively guide waves from a source to a load using a pair of parallel, uniform conductors known as a transmission line. Transmission lines consist of two or more conductors extending a long distance along one axis, and maintaining a uniform cross-sectional geometry in the plane transverse to this axis. In this chapter we will consider several different types of transmission line systems. Common examples that each of you has probably encountered in everyday experience are co-axial conductors, often used for delivering a signal from the cable provider to your TV; microstrip, often seen in printed circuit board applications; and twin-lead transmission lines, often used with portable “rabbit-ears” antennas for TV receivers. In this chapter we will explore the properties of waves on transmission line systems, including such important aspects as traveling waves, standing waves, characteristic impedances, impedance matching, losses, and much more.

Electrostatic fields form the basis for such everyday applications as air cleaners, photocopiers, spark plugs, and corona discharges, to name a few. In this chapter we will explore several techniques that can be used to determine the electric field produced by a single electric charge, or a distribution of electric charges, including two fundamental relations known as Coulomb’s Law and Gauss’ Law. Gauss’ Law is one of two differential properties obeyed by electric fields that we will explore. We will show that the electric field can be determined from an electric potential, and explore methods of determining the electric potential. Using the electric potential can often be a simpler approach for finding the electric field. We will then examine the influence of materials, including conductors and dielectrics (or insulators), on electric fields. This will require us to re-examine Gauss’ Law to include material effects. We will also develop a set of relations, known as boundary conditions, that relate the fields on the two sides of an interface between two materials. Our studies begin with empirical observations of the force between two charged bodies.

In the previous two chapters we introduced electric charges and electric fields, using primarily the force between charged bodies as the starting point. Charges in motion, of course, constitute electrical currents, with which you have a working knowledge from your previous study in circuit analysis. In this chapter we will take a closer look at electrical currents. Specifically, we’ll introduce current density, which describes the spatial distribution of a current. Then the relation between the ideas of electrostatics discussed in Chapters 1 and 2 and the laws that govern simple electric circuits, such as Kirchhoff’s Current Law, Kirchhoff’s Voltage Law, and Ohm’s Law, will be discussed. You may have already recognized the connection between these circuits concepts and some of the ideas from electrostatics that we introduced earlier. The circuit laws are but special cases of the more general treatment that we will now discuss in detail.