ABSTRACT
We may now consider the medium as a lossy dielectric by writing Eqs.(23.4) in the usual Helmholtz form
022 =+∇ EE c GG µωε
(23.5) 022 =+∇ HH c
GG µωε
where the real permittivity is replaced by a complex permittivity cε written as
( ) ⎟⎟⎠ ⎞
⎜⎜⎝ ⎛ ++=⎟⎟⎠
⎞ ⎜⎜⎝ ⎛ +=+=+= ωε
σχεωε σεεεεεω
σεε 0
00 1 iiii erirc (23.6)
The constitutive parameters of a conducting medium, if regarded to be a lossy dielectric, are the conductivity the permeability ,σ µ and the complex permittivity cε . In this case, the optical properties become formally identical with those of transparent media, if the quantity is replaced by the complex quantity 200 nr εεεε ==
( )2020 icc innn +== εεε (23.7) which can be regarded as a definition of the complex refractive index Substituting Eq.(23.7) and then equating the real and imaginary parts, Eq.(23.6) becomes
.cn
, rinn ε=− 22 iinn εωε σ == 0
2 (23.8)
On solving these equations we obtain expressions for both n, called the refractive index and called the extinction index, in the form ,in
⎥⎥⎦ ⎤
⎢⎢⎣ ⎡
+⎟⎟⎠ ⎞
⎜⎜⎝ ⎛ += 11
ωε σε rn , ⎥⎥⎦
⎤ ⎢⎢⎣ ⎡
−⎟⎟⎠ ⎞
⎜⎜⎝ ⎛ += 11
ωε σε r
in (23.9)
One also may leave the constitutive parameters εσ and unspecified, so that the plane wave solutions of Eqs.(23.4) must be written in all but special cases in terms of complex wave vectors ck
G as
( ) ( ) ( ), ci k r ti tE r t E r e Ee ωω ⋅ −−= = G GG G GG G (23.10)
( ) ( ) ( ), ci k r ti tH r t H r e He ωω ⋅ −−= = G GG G GG G
Since the operator relations (20.10) are valid under the assumptions of Eq.(23.10), that is when ck
G are complex quantities, Eqs.(23.4) reduce to the following
algebraic plane wave equation for conducting media
EiEEkc GGG µσωεµω += 22
(23.11) HiHHkc GGG µσωεµω += 22
Each of Eqs.(23.11) requires that
⎟⎠ ⎞⎜⎝
⎛ += εω σεµω ikc 122 (23.12)
and hence, the wave number may be written in complex representation as ck ic ikkk += (23.13) Specific expressions for k and ki follow by squaring Eq.(23.13) and then equating the real and imaginary parts to the real and imaginary parts of Eq.(23.12), which gives 2 2ik k
2εµω− = , µσω=ikk2 (23.14)
The resulting expressions are
11 2 ⎥⎥⎦
⎤ ⎢⎢⎣ ⎡
+⎟⎟⎠ ⎞
⎜⎜⎝ ⎛ += ωε
σεµωk , 2/12/1
11 2 ⎥⎥⎦
⎤ ⎢⎢⎣ ⎡
−⎟⎟⎠ ⎞
⎜⎜⎝ ⎛ += ωε
σεµωik (23.15)
and their similarity with Eqs.(23.9) allows us to define the complex wave number in terms of the complex refractive index (23.7) as
( ) criric ncinncikkk ωµωµ =+=+= (23.16)
Using the configuration illustrated before in Figure 20.1 and Eq.(20.12), the transverse fields (23.10) can be written as
( ) ( ) ( ), i i k tki nk ikH r t e E e e ξ ωξµω −−+= ×G GG G
where .ner GG ⋅=ξ Equations (23.17) show that the imaginary part of the complex wave
number, called the absorption coefficient, is a measure of attenuation of a plane wave during propagation in metals, whereas its real part k must preserve the same relationship
with the wave vector nekk GG = as in the absence of attenuation. The planes of constant
phase are propagated with a velocity defined in terms of k and derived from Eqs.(23.15) in the form
( ) ( ) 2/12/1222 1/12/ ⎭⎬⎫⎩⎨⎧ ⎥⎦⎤⎢⎣⎡ ++
== ωεσµε
ω rr
c k
v (23.18)
where use has been made of Eq.(15.53) for c. Hence, conducting media must be referred to not only as absorbing but also as dispersive media, since waves of different frequencies are transmitted at different velocities. It is apparent that phase velocities decrease with increasing conductivity σ and increase with increasing frequency ω as long as εσ and are independent of .ω If is written in the form ck ϕiic ekkk
−+= 22 (23.19) it follows from Eq.(23.19) that the fields inside a conducting medium are out of phase by an angle ,ϕ given by
n n
k k ii ==ϕtan (23.20)
The first term on the right hand side of the plane wave equation (23.11) arises from the displacement current density / /D t E t i Eε ωε∂ ∂ = ∂ ∂ = −G G G whereas the second term arises from the conduction current density .j Eσ= GG The ratio of their magnitudes
εωσ / appears to be a macroscopic criterion for the classification of conducting media. In the case of poor conductors the displacement current is of importance, as ,1/ <<εωσ so that the components (23.15) of the complex wave number can be approximated by
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛ +≅⎟⎟⎠
⎞ ⎜⎜⎝ ⎛ +≅ 22
8 1
2 2
2 ωε σεµωωε
σεµωk (23.21)
ε µσ
ωε σεµω
=⎟⎟⎠ ⎞
⎜⎜⎝ ⎛≅ik
For good conductors it is the conduction current that is important, ,1/ >>εωσ so that the real and imaginary parts of are equal, as given by ck
ωµσ== ikk (23.22)
EXAMPLE 23.1. The skin effect
It is usual to define the so-called skin depth δ of a good conductor to be the distance the electromagnetic wave penetrates the surface before its amplitude drops to 1/e times its surface value. Using Eqs.(23.17) and (23.22) this may be expressed in terms of as ik
ωµσδ 21 ==
ik (23.23)
H
E
Free Space Conducting Medium
Figure 23.1. Field vector attenuation in a conducting medium High values for either the conductivity σ or frequency ω will lead to a small skin depth. The fields inside a good conductor, as represented in Figure 23.1, are out of phase by an angle
4/πϕ = and result from Eqs.(23.17) as
The complex representation (23.16) of the wave number, and therefore, the complex refractive index, are associated with the fact that waves are attenuated in conducting media. However, only a small part of the incident wave intensity on the surface of a conducting medium undergoes a true absorption (i.e., conversion into Joule heat). A large fraction is propagated back into the external medium as a high intensity reflected wave. Consider the reflection of a plane wave which falls normally on the surface of a conducting medium, lying in the xz-plane. The configuration illustrated in Figure 23.2 is a special case of that given before in Figure 21.3, where 0,θ ϕ= = thus the distinction between the TM and TE polarizations is lost.
