Page 400 - Engineering Electromagnetics, 8th Edition

382 ENGINEERING ELECTROMAGNETICS

J ′E

′
J ′ E

J E

Figure 11.2 The time-phase relationship
between J ds , J σs , J s , and E s . The tangent of θ
is equal to σ/ω , and 90 − θ is the common
◦

power-factor angle, or the angle by which J s
leads E s .
Considering a conductive material, for which = σ/ω, (43) becomes

σ

jk = jω µ 1 − j (59)
ω
We may expand the second radical using the binomial theorem
n(n − 1) n(n − 1)(n − 2)
n 2 3
(1 + x) = 1 + nx + x + x +· · ·
2! 3!
where |x| 1. We identify x as − jσ/ω and n as 1/2, and thus

1 2
σ σ

jk = jω µ 1 − j + +· · · = α + jβ
2ω 8 ω
Now, for a good dielectric,

. σ σ µ
α = Re( jk) = jω µ − j = (60a)
2ω 2
and

1
. σ 2
β = Im( jk) = ω µ 1 + (60b)
8 ω
Equations (60a) and (60b) can be compared directly with the transmission line α and
β under low-loss conditions, as expressed in Eqs. (54a) and (55b)in Chapter 10.
In this comparison, we associate σ with G, µ with L, and with C. Note that in
plane wave propagation in media with no boundaries, there can be no quantity that is
analogous to the transmission line conductor resistance parameter, R.In many cases,