Page 408 - Engineering Electromagnetics, 8th Edition

390 ENGINEERING ELECTROMAGNETICS

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one skin depth, about 8.5 mm. A hollow conductor with a wall thickness of about
12 mm would be a much better design. Although we are applying the results of an
analysis for an infinite planar conductor to one of finite dimensions, the fields are
attenuated in the finite-size conductor in a similar (but not identical) fashion.
The extremely short skin depth at microwave frequencies shows that only the sur-
face coating of the guiding conductor is important. A piece of glass with an evaporated
silver surface 3 µm thick is an excellent conductor at these frequencies.
Next, let us determine expressions for the velocity and wavelength within a good
conductor. From (82), we already have
1
α = β = = π f µσ
δ
Then, as

2π
β =
λ
we find the wavelength to be
λ = 2πδ (83)

Also, recalling that
ω
ν p =
β
we have

ν p = ωδ (84)

For copper at 60 Hz, λ = 5.36 cm and ν p = 3.22 m/s, or about 7.2 mi/h! A lot of us
can run faster than that. In free space, of course, a 60 Hz wave has a wavelength of
3100 mi and travels at the velocity of light.

EXAMPLE 11.6
Let us again consider wave propagation in water, but this time we will consider
seawater. The primary difference between seawater and fresh water is of course the
salt content. Sodium chloride dissociates in water to form Na and Cl ions, which,
+
−
being charged, will move when forced by an electric field. Seawater is thus conductive,
and so it will attenuate electromagnetic waves by this mechanism. At frequencies
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in the vicinity of 10 Hz and below, the bound charge effects in water discussed
earlier are negligible, and losses in seawater arise principally from the salt-associated
conductivity. We consider an incident wave of frequency 1 MHz. We wish to find the
skin depth, wavelength, and phase velocity. In seawater, σ = 4 S/m, and = 81.

r

6 This utility company operates at 60 Hz.