Page 286 - Electromagnetics

Here tan δ c is the loss tangent of the material, as first described in (4.107) for a material
without conductivity. For a good dielectric we have

c
k = β − jα = ˇω µ = ˇω µ [ + j ] = ˇω µ 1 − j tan δ c ,

c
hence

1

k ≈ ˇω µ 1 − j tan δ c (4.254)
2
bythe binomial approximation for the square root. Therefore

β ≈ ˇω µ (4.255)
and

β σ µ ˇ ω
α ≈ tan δ c = 1 − . (4.256)
2 2 σ
We conclude that α β. Using this and the binomial approximation we establish
ˇ ωµ ˇ ωµ 1 ˇ ωµ α
η = = ≈ 1 + j .
k β 1 − jα/β β β
Finally,
ˇ ω 1
v p = ≈ √
β µ
and

−1
dβ 1
v g = ≈ √ .
dω µ
To first order, the phase constant, phase velocity, and group velocity are the same as
those of a lossless medium.

Uniform plane waves in a good conductor. We classifya material as a “good
conductor” if
σ
tan δ c ≈ 1.
ˇ ω
ˇ
In a good conductor the conduction current σE is much greater than the displacement
ˇ
current j ˇω E, and is usuallyignored. Now we mayapproximate

k = β − jα = ˇω µ 1 − j tan δ c ≈ ˇω µ − j tan δ c .
√ √
Since − j = (1 − j)/ 2 we find that

β = α ≈ π f µσ. (4.257)
Hence

ˇ ω 2 ˇω 1 2
v p = ≈ = √ .
β µσ µ tan δ c
To find v g we must replace ˇω by ω and differentiate, obtaining

−1
−1
dβ 1 µσ 2 ˇω
v g = ≈ = 2 = 2v p .
dω 2 2 ˇω µσ
ω= ˇω
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