Page 90 - Maxwell House

70 Chapter 2

dielectric constant is a real number. But that is not completely true in frequency domain

because the existing or induced electric dipoles
in dielectrics possess, for example, some tiny but
finite masses and therefore cannot align in
parallel instantaneously. Suppose for
simplification that the electric field is
monochromatic and has the in-phase component
only. Then according to (1.80) in Chapter 1 the
vector of some frequency can be presented
in the form

(, ) = |(, )|cos ( + ) (2.64)
Figure 2.4.1 Electric field vector
variations over time The plot of this vector at some point r over the
time is shown in Figure 2.4.1. Accordingly, the
polarization vector P must follow all these variations in an isotropic medium. In other words,
the dipoles in dielectric have to turn upside down each half the time period. During such flip-
flops, the dipoles interact through their fields with surrounding material lattice that obstructs
and delays the dipoles movement. As a result, some portion of dipoles kinetic energy is
dissipated being converted mainly into heat energy. To account this effect, we can include in
(2.11) the exponential factor −() reflecting the reduction in E-field magnitude and expected
−() in dipoles rotation delay
phase shift
(, ) = −()−() (, ) (2.65)
0
It is expected that the energy loss and delay rises as frequency increases because the dipoles
rotation cycle defines by the electric field frequency. Substituting (2.65) into (2.15) and (2.63)
we can introduce the complex dielectric constant () as

′
′′
() = 1 + −− = (1 + − cos ) − − sin = () − () (2.66)

′
Typically, the parameters () and () as well () and ′′ () are measured

experimentally or some cases may be estimated theoretically as we will demonstrate it later in
′′
this chapter. Evidently, the imaginary part () in (2.66) is attributed mostly by the energy
dissipation through the polarization effect only. Meanwhile, the polarization effect in dielectrics
is not the only source of energy dissipation. Let consider the sum + (2 Maxwell’s
nd

equation in Table 1.9 from Chapter 1). Since according to (2.28) = and = ()
0
according to (2.66) we have
′
′
+ = [ () − ′′ ()] + = [ () − ( ′′ () + )] (2.67)
0

0

0
The value in the square parentheses is called the relative complex dielectric constant
′
′
′′
′′
() = () − � () + � = () − () (2.68)

0
Then for monochromatic fields, 2 Maxwell’s equation and the constitutive relation in Table
nd
1.9 can be rewritten in the form
(, ) = ()(, ) � (2.69)
0
x (, ) = (, ) + (, )