In the treatment of EQS and MQS systems, we started in Chaps. 4
and 8, respectively, by analyzing the fields produced by specified
(known) sources. Then we recognized that in the presence of
materials, at least some of these sources were induced by the fields
themselves. Induced surface charge and surface current densities
were determined by making the fields satisfy boundary conditions.
In the volume of a given region, fields were composed of
particular solutions to the governing quasistatic equations (the
scalar and vector Poisson equations for EQS and MQS systems,
respectively) and those solutions to the homogeneous equations (the
scalar and vector Laplace equation, respectively) that made the
total fields satisfy appropriate boundary conditions.
We now embark on a similar approach in the analysis of
electrodynamic fields. Chapter 12 presented a study of the fields
produced by specified sources (dipoles, line sources, and surface
sources) and obeying the inhomogeneous wave equation. Just as in the
case of EQS and MQS systems in Chap. 5 and the last half of Chap. 8,
we shall now concentrate on solutions to the homogeneous
source-free equations. These solutions then serve to obtain the fields
produced by sources lying outside (maybe on the boundary) of the
region within which the fields are to be found. In the region of
interest, the fields generally satisfy the inhomogeneous
wave equation. However in this chapter, where there are no sources in
the volume of interest, they satisfy the homogeneous wave equation.
It should come as no surprise that, following this systematic
approach, we shall reencounter some of the previously obtained
In this chapter, fields will be determined in some limited region
such as the volume V of Fig. 13.0.1. The boundaries
might be in part perfectly conducting in the sense that on their
surfaces, E is perpendicular and the time-varying H is
tangential. The surface current and charge densities implied by these
conditions are not known until after the fields have been found. If
there is material within the region of interest, it is perfectly
insulating and of piece-wise uniform permittivity and
1 If the region is one of free space,
o and o.
Sources J and are specified throughout the volume
and appear as driving terms in the inhomogeneous wave equations,
(12.6.8) and (12.6.32). Thus, the H and E fields obey the
Figure 13.0.1 Fields in a limited region are in
part due to sources induced on boundaries by the fields themselves.
As in earlier chapters, we might think of the solution to these
equations as the sum of a part satisfying the inhomogeneous
equations throughout V (particular solution), and a part satisfying
the homogeneous wave equation throughout that region. In principle,
the particular solution could be obtained using the superposition
integral approach taken in Chap. 12. For example, if an electric
dipole were introduced into a region containing a uniform medium,
the particular solution would be that given in Sec. 12.2 for an
electric dipole. The boundary conditions are generally not met by
these fields. They are then satisfied by adding an appropriate
solution of the homogeneous wave equation.
2 As pointed
out in Sec. 12.7, this is essentially what is being done in
satisfying boundary conditions by the method of images.
In this chapter, the source terms on the right in (1) and (2) will be
set equal to zero, and so we shall be concentrating on solutions to
the homogeneous wave equation. By combining the solutions of the
homogeneous wave equation that satisfy boundary conditions with the
source-driven fields of the preceding chapter, one can describe
situations with given sources and given boundaries.
In this chapter, we shall consider the propagation of waves in
some axial direction along a structure that is uniform in that
direction. Such waves are used to transport energy along pairs of
conductors (transmission lines), and through waveguides (metal tubes
at microwave frequencies and dielectric fibers at optical
frequencies). We confine ourselves to the sinusoidal steady state.
Sections 13.1-13.3 study two-dimensional modes between plane parallel
conductors. This example introduces the mode expansion of
electrodynamic fields that is analogous to the expansion of the EQS
field of the capacitive attenuator (in Sec. 5.5) in terms of the
solutions to Laplace's equation. The principal and higher order
modes form a complete set for the representation of arbitrary
boundary conditions. The example is a model for a strip transmission
line and hence serves as an introduction to the subject of Chap. 14.
The higher-order modes manifest properties much like those found
in Sec. 13.4 for hollow pipe guides.
The dielectric waveguides considered in Sec. 13.5 explain the guiding
properties of optical fibers that are of great practical
interest. Waves are guided by a dielectric core having permittivity
larger than that of the surrounding medium but possess fields
extending outside this
core. Such electromagnetic waves are guided because the dielectric
core slows the effective velocity of the wave in the guide to the
point where it can match the velocity of a wave in the surrounding
region that propagates along the guide but decays in a direction
perpendicular to the guide.
The fields considered in Secs. 13.1-13.3 offer the opportunity to
reinforce the notions of quasistatics. Connections between the EQS
and MQS fields studied in Chaps. 5 and 8, respectively, and their
corresponding electrodynamic fields are made throughout Secs.