Digital Signal Processing Reference
In-Depth Information
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(a)
(b)
Figure 6-1
(a) Atoms with no external electric field applied; (b) electric dipoles induced
as the externally applied electric field distorts the electron cloud.
the electron clouds of the atoms are distorted by the applied electric field, similar
to stretching a spring as depicted in Figure 6-1b. This gives the material the
ability to store electric energy (as potential energy).
To demonstrate the relationship between the applied electric field and the
distortion of the electron cloud, a very simplified model of an atom can be
employed by assuming a point positive charge of
q
e
+
representing the nucleus
encompassed by a sphere, with a charge of
q
e
−
representing the electron cloud.
When an electric field is applied, the electron cloud is displaced until the attractive
force between the positively charged nucleus and the negatively charged sphere
is equal to the force of the applied electric field. By equating these two forces,
the displacement of the electron cloud can be estimated. Note that this model
assumes that the electron cloud remains spherical when it is displaced. In reality,
when the electron cloud is moved by the external electric field, it is elongated
and stretched. However, this simple derivation will provide valuable insight into
the mechanisms that cause electronic polarization.
Consider the spherical cloud of uniform charge density as in Figure 6-2a. The
charge density per unit volume is
Q
V
=
Q
ρ
=
4
3
πr
e
where
Q
is the total charge distributed over the volume of the sphere
V
and
r
e
is the radius of the electron cloud. The electric field is solved using the integral
form of Gauss's law (2-59):
ε E
·
d
s
=
ρ dV
=
Q
enclosed
S
V
In this case, only the electric field inside the sphere is relevant because the force
required to move the electron cloud away from the positive nucleus must be
calculated. Assuming that the electron cloud encompasses free space,
ε
=
ε
0
,
Gauss's law reduces to
ε
0
E
r
4
πr
2
4
3
πr
3
ρ
=
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