Page 407 - Engineering Electromagnetics, 8th Edition

CHAPTER 11 The Uniform Plane Wave 389

We may tie this field in the conductor to an external field at the conductor surface.
We let the region z > 0be the good conductor and the region z < 0bea perfect
dielectric. At the boundary surface z = 0, (80) becomes
E x = E x0 cos ωt (z = 0)
This we shall consider as the source field that establishes the fields within the con-
ductor. Since displacement current is negligible,
J = σE

Thus, the conduction current density at any point within the conductor is directly
related to E:
√
J x = σ E x = σ E x0 e −z π fµσ cos ωt − z π f µσ (81)
Equations (80) and (81) contain a wealth of information. Considering first the
negative exponential term, we find an exponential decrease in the conduction current
density and electric field intensity with penetration into the conductor (away from the
source). The exponential factor is unity at z = 0 and decreases to e −1 = 0.368 when
1
z = √
π fµσ
This distance is denoted by δ and is termed the depth of penetration,or the skin depth,

1 1 1
δ = √ = = (82)
π f µσ α β
It is an important parameter in describing conductor behavior in electromagnetic
fields. To get some idea of the magnitude of the skin depth, let us consider copper,
7
σ = 5.8 × 10 S/m, at several different frequencies. We have
0.066
δ Cu = √
f
At a power frequency of 60 Hz, δ Cu = 8.53 mm. Remembering that the power density
carries an exponential term e −2αz ,we see that the power density is multiplied by a
factor of 0.368 = 0.135 for every 8.53 mm of distance into the copper.
2
At a microwave frequency of 10,000 MHz, δ is 6.61 × 10 −4 mm. Stated more
generally, all fields in a good conductor such as copper are essentially zero at distances
greater than a few skin depths from the surface. Any current density or electric field
intensity established at the surface of a good conductor decays rapidly as we progress
into the conductor. Electromagnetic energy is not transmitted in the interior of a
conductor; it travels in the region surrounding the conductor, while the conductor
merely guides the waves. We will consider guided propagation in more detail in
Chapter 13.
Suppose we have a copper bus bar in the substation of an electric utility company
which we wish to have carry large currents, and we therefore select dimensions of 2
by 4 inches. Then much of the copper is wasted, for the fields are greatly reduced in