Page 396 - Engineering Electromagnetics, 8th Edition

378 ENGINEERING ELECTROMAGNETICS

which leads to the fundamental definition of wavelength,

2π
λ = (47)
β
Because we have a uniform plane wave, the magnetic field is found through
E x0 −αz − jβz
H ys = e e
η
where the intrinsic impedance is now a complex quantity,

1
µ µ
η = = √ (48)
− j 1 − j( / )

The electric and magnetic fields are no longer in phase.
A special case is that of a lossless medium, or perfect dielectric, in which = 0,

and so = . From (44), this leads to α = 0, and from (45),

β = ω µ (lossless medium) (49)
With α = 0, the real field assumes the form
E x = E x0 cos(ωt − βz) (50)

We may interpret this as a wave traveling in the +z direction at a phase velocity ν p ,
where

ω 1 c
ν p = = √ =
β µ µ r r
The wavelength is
2π 2π 1 c λ 0
λ = = √ = √ = = (lossless medium) (51)
β ω µ f µ f µ r r µ r r
where λ 0 is the free space wavelength. Note that µ r > 1, and therefore the wave-

r
length is shorter and the velocity is lower in all real media than they are in free
space.
Associated with E x is the magnetic field intensity

E x0
H y = cos(ωt − βz)
η
where the intrinsic impedance is

µ
η = (52)