Page 271 - Electromagnetics

and (4.222) as
ˆ
˜
˜
E =−ηk × H.
Here

ω ˜µ ˜ µ
η = = c
k ˜
is the complex intrinsic impedance of the medium.
Equations (4.223) and (4.221) show that the electric and magnetic fields and the wave
vector are mutually orthogonal. The wave is said to be transverse electromagnetic or
TEM to the direction of propagation.

The phase and attenuation constants of a uniform plane wave. For a uniform
plane wave we may write
ˆ ˆ ˆ ˆ
k = k k + jk k = kk = (β − jα)k

where k = β and k =−α. Here α is called the attenuation constant and β is the phase

constant. Since k is defined through (4.206), we have
2 2 2 2 2 c 2

c
c
k = (β − jα) = β − 2 jαβ − α = ω ˜µ˜ = ω ( ˜µ + j ˜µ )(˜ + j ˜ ).
Equating real and imaginary parts we have
2 2 2 2
c
c
c
c
β − α = ω [ ˜µ ˜ − ˜µ ˜ ], −2αβ = ω [ ˜µ ˜ + ˜µ ˜ ].

We assume the material is passive so that ˜µ ≤ 0, ˜ c ≤ 0. Letting
2 2 2 2

c

c
c
c
β − α = ω [ ˜µ ˜ − ˜µ ˜ ] = A, 2αβ = ω [| ˜µ |˜ + ˜µ |˜ |] = B,
we may solve simultaneously to find that
1 1
2 2 2 2 2 2
β = A + A + B , α = −A + A + B .
2 2
4
2
2
2
2
Since A + B = ω (˜ c 2 + ˜ c 2 )( ˜µ + ˜µ ),wehave
$
%

% 1 ˜ c 2 ˜ µ 2 ˜ µ ˜ c

˜
β = ω µ ˜ 1 + 1 + + 1 − , (4.224)
c &
2 ˜ c 2 ˜ µ 2 ˜ µ ˜ c

$
%

% 1 ˜ c 2 ˜ µ 2 ˜ µ ˜ c

˜
α = ω µ ˜ 1 + 1 + − 1 − , (4.225)
c &
2 ˜ c 2 ˜ µ 2 ˜ µ ˜ c

c
where ˜ and ˜µ are functions of ω.If ˜ (ω) = , ˜µ(ω) = µ, and ˜σ(ω) = σ are real and
frequency independent, then
$

%
√ % 1 σ 2
α = ω µ & 1 + − 1 , (4.226)
2 ω
$

%
√ % 1 σ 2
β = ω µ & 1 + + 1 . (4.227)
2 ω
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