Page 410 - Engineering Electromagnetics, 8th Edition

392 ENGINEERING ELECTROMAGNETICS

which may be written as
√
2 45 ◦ (1 + j)
η = = (85)
σδ σδ
Thus, if we write (80) in terms of the skin depth,
z

E x = E x0 e −z/δ cos ωt − (86)
δ
then
z π
σδE x0 −z/δ
H y = √ e cos ωt − − (87)
2 δ 4
and we see that the maximum amplitude of the magnetic field intensity occurs one-
eighth of a cycle later than the maximum amplitude of the electric field intensity at
every point.
From (86) and (87) we may obtain the time-average Poynting vector by applying
(77),
1 σδE 2
π
e
S z = √ x0 −2z/δ cos
2 2 4
or
1
2
S z = σδE e −2z/δ
x0
4
We again note that in a distance of one skin depth the power density is only e −2 =
0.135 of its value at the surface.
The total average power loss in a width 0 < y < b and length 0 < x < L in
the direction of the current, as shown in Figure 11.3, is obtained by finding the power

Figure 11.3 The current density J x =
J x0 e −z/δ − jz/δ decreases in magnitude as the wave
e
propagates into the conductor. The average power
loss in the region 0 < x < L, 0 < y < b, z > 0,
2
is δbL J /4σ watts.
x0