Page 246 - Electrical Properties of Materials

228 Dielectric materials

Hence, the average dipole moment is given as
net moment of the assembly
μ =
total number of dipoles
π
μE cos θ
0 A exp kT (μ cos θ)2π sin θdθ
= . (10.11)
π μE cos θ
0 A exp kT 2π sin θdθ
Equation (10.11) turns out to be integrable, yielding

a =(μE /kT), and L(a) is called μ 1
= L(a) = coth a – . (10.12)
the Langevin function. μ a
If a is small, which is true under quite wide conditions, eqn (10.12) may be
approximated by
2
μ E
μ = . (10.13)
3kT
That is, the polarizability is inversely proportional to the absolute temperature.
∗ The complex dielectric constant used
by electrical engineers is invariably in
the form = 0 ( –j ). We found a dif- 10.5 The complex dielectric constant and the refractive index

ferent sign because we had adopted the
physicists’ time variation, exp(–iωt). In engineering practice the dielectric constant is often divided up into real and
The loss tangent is defined as imaginary parts. This can be derived from Maxwell’s equations by rewriting

tan δ ≡ / . the current term in the following manner:
(a) (b) J –iω E = σE –iω E
σ

=–iω +i E , (10.14)
ω
where the term in the bracket is called the complex dielectric constant. The
usual notation is ∗
σ ε

ε = ε ε 0 , = ε ε 0 , and tan δ = . (10.15)
ω ε
The refractive index is defined as the ratio of the velocity of light in a
vacuum to that in the material,
n n n n n n
1 2 1 2 1 2 c
n =
ν
Fig. 10.5 √ √

Quarter wavelength layers used to = r μ r = , (10.16)
make dielectric mirrors.
† This is actually not true for a new set of since μ r = 1 in all known natural materials that transmit light. †
artificial materials called metamaterials, Conventionally, we talk of ‘dielectric constant’ (or permittivity) for the
which can have effective permeabilities lower frequencies in the electromagnetic spectrum and of refractive index for
well above unity even in the infrared
optical region (see Chapter 15). light. Equation (10.16) shows that they are the same thing—a measure of the
polarizability of a material in an alternating electric field.
A fairly recent and important application of dielectrics to optics has been
that of multiply reflecting thin films. Consider the layered structure represen-
ted in Fig. 10.5 with alternate layers of transparent material having refractive