Page 220 - Electromagnetics

where W e + W m is the total time-average electromagnetic energy stored in the volume
region V . This is known as the energy theorem. We shall use it in § 4.11.3 to determine
the velocity of energy transport for a plane wave.

4.6 Some simple models for constitutive parameters

Thus far our discussion of electromagnetic fields has been restricted to macroscopic
phenomena. Although we recognize that matter is composed of microscopic constituents,
we have chosen to describe materials using constitutive relationships whose parameters,
such as permittivity, conductivity, and permeability, are viewed in the macroscopic sense.
By performing experiments on the laboratory scale we can measure the constitutive
parameters to the precision required for engineering applications.
At some point it becomes useful to establish models of the macroscopic behavior of
materials based on microscopic considerations, formulating expressions for the consti-
tutive parameters using atomic descriptors such as number density, atomic charge, and
molecular dipole moment. These models allow us to predict the behavior of broad classes
of materials, such as dielectrics and conductors, over wide ranges of frequency and field
strength.
Accurate models for the behavior of materials under the influence of electromagnetic
fields must account for many complicated effects, including those best described by quan-
tum mechanics. However, many simple models can be obtained using classical mechanics
and field theory. We shall investigate several of the most useful of these, and in the
process try to gain a feeling for the relationship between the field applied to a material
and the resulting polarization or magnetization of the underlying atomic structure.
For simplicity we shall consider only homogeneous materials. The fundamental atomic
descriptor of “number density,” N, is thus taken to be independent of position and time.
The result may be more generally applicable since we may think of an inhomogeneous
material in terms of the spatial variation of constitutive parameters originally deter-
mined assuming homogeneity. However, we shall not attempt to study the microscopic
conditions that give rise to inhomogeneities.

4.6.1 Complex permittivity of a non-magnetized plasma
A plasma is an ionized gas in which the charged particles are free to move under
the influence of an applied field and through particle-particle interactions. A plasma
differs from other materials in that there is no atomic lattice restricting the motion of
the particles. However, even in a gas the interactions between the particles and the fields
give rise to a polarization effect, causing the permittivity of the gas to differ from that
of free space. In addition, exposing the gas to an external field will cause a secondary
current to flow as a result of the Lorentz force on the particles. As the moving particles
collide with one another they relinquish their momentum, an effect describable in terms
of a conductivity. In this section we shall perform a simple analysis to determine the
complex permittivity of a non-magnetized plasma.
To make our analysis tractable, we shall make several assumptions.

1. We assume that the plasma is neutral: i.e., that the free electrons and positive ions
are of equal number and distributed in like manner. If the particles are sufficiently

© 2001 by CRC Press LLC