Page 207 - Electromagnetics

or
c
c
c
c
˜ (r, −ω) = ˜ (r,ω), ˜ (r, −ω) =−˜ (r,ω).
ij ij ij ij
For an isotropic material it takes the particularly simple form
˜ σ ˜ σ
c
˜ = + ˜ = + 0 + 0 ˜χ e , (4.26)
jω jω
and we have
c
c
c
c
˜ (r, −ω) = ˜ (r,ω), ˜ (r, −ω) =−˜ (r,ω). (4.27)
4.4.2 High and low frequency behavior of constitutive parameters
At low frequencies the permittivity reduces to the electrostatic permittivity. Since ˜
is even in ω and ˜ is odd, we have for small ω

˜ ∼ 0 r , ˜ ∼ ω.
If the material has some dc conductivity σ 0 , then for low frequencies the complex per-
mittivity behaves as
c
˜ ∼ 0 r , ˜ c ∼ σ 0 /ω. (4.28)
If E or H changes very rapidly, there may be no polarization or magnetization effect at
all. This occurs at frequencies so high that the atomic structure of the material cannot
respond to the rapidly oscillating applied field. Above some frequency then, we can
assume ˜ ¯χ = 0 and ˜ ¯χ = 0 so that
e
m
˜
˜
P = 0, M = 0,
and
˜
˜
˜
˜
D = 0 E, B = µ 0 H.
In our simple models of dielectric materials (§ 4.6) we find that as ω becomes large
2
3

˜ − 0 ∼ 1/ω , ˜ ∼ 1/ω . (4.29)

Our assumption of a macroscopic model of matter provides a fairly strict upper frequency
limit to the range of validity of the constitutive parameters. We must assume that the
wavelength of the electromagnetic field is large compared to the size of the atomic struc-
ture. This limit suggests that permittivity and permeability might remain meaningful
even at optical frequencies, and for dielectrics this is indeed the case since the values of
˜
˜
P remain significant. However, M becomes insignificant at much lower frequencies, and
˜
˜
at optical frequencies we may use B = µ 0 H [107].
4.4.3 The Kronig–Kramers relations
The principle of causality is clearly implicit in (2.29)–(2.31). We shall demonstrate
that causality leads to explicit relationships between the real and imaginary parts of the
frequency-domain constitutive parameters. For simplicity we concentrate on the isotropic
case and merely note that the present analysis may be applied to all the dyadic com-
ponents of an anisotropic constitutive parameter. We also concentrate on the complex
permittivity and extend the results to permeability by induction.

© 2001 by CRC Press LLC