Page 228 - Electromagnetics

where ˜χ e is the dielectric susceptibility. In this section we shall model a homogeneous
dielectric consisting of a single, uniform material type.
We found in Chapter 3 that for a dielectric material immersed in a static electric field,
the polarization vector P can be viewed as a volume density of dipole moments. We
choose to retain this view as the fundamental link between microscopic dipole moments
and the macroscopic polarization vector. Within the framework of our model we thus
describe the polarization through the expression
1
P(r, t) = p i . (4.92)
V
r−r i (t)∈B
Here p i is the dipole moment of the ith elementary microscopic constituent, and we form
the macroscopic density function as in § 1.3.1.
We may also write (4.92) as

N B
N B 1
P(r, t) = p i = N(r, t)p(r, t) (4.93)
V N B
i=1
where N B is the number of constituent particles within V . We identify
1
N B
p(r, t) = p i (r, t)
N B
i=1
as the average dipole moment within V , and

N B
N(r, t) =
V
as the dipole moment number density. In this model a dielectric material does not require
higher-order multipole moments to describe its behavior. Since we are only interested
in homogeneous materials in this section we shall assume that the number density is
constant: N(r, t) = N.
To understand how dipole moments arise, we choose to adopt the simple idea that mat-
ter consists of atomic particles, each of which has a positively charged nucleus surrounded
by a negatively charged electron cloud. Isolated, these particles have no net charge and
no net electric dipole moment. However, there are several ways in which individual par-
ticles, or aggregates of particles, may take on a dipole moment. When exposed to an
external electric field the electron cloud of an individual atom may be displaced, resulting
in an induced dipole moment which gives rise to electronic polarization. When groups
of atoms form a molecule, the individual electron clouds may combine to form an asym-
metric structure having a permanent dipole moment. In some materials these molecules
are randomly distributed and no net dipole moment results. However, upon application
of an external field the torque acting on the molecules may tend to align them, creating
an induced dipole moment and orientation,or dipole, polarization. In other materials,
the asymmetric structure of the molecules may be weak until an external field causes
the displacement of atoms within each molecule, resulting in an induced dipole moment
causing atomic,or molecular, polarization. If a material maintains a permanent polar-
ization without the application of an external field, it is called an electret (and is thus
similar in behavior to a permanently magnetized magnet).
To describe the constitutive relations, we must establish a link between P (now describ-
able in microscopic terms) and E. We do this by postulating that the average constituent

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