Page 395 - Engineering Electromagnetics, 8th Edition

CHAPTER 11 The Uniform Plane Wave 377

Two important mechanisms that give rise to a complex permittivity (and thus result
in wave losses) are bound electron or ion oscillations and dipole relaxation, both of
which are discussed in Appendix E. An additional mechanism is the conduction of
free electrons or holes, which we will explore at length in this chapter.
Losses arising from the response of the medium to the magnetic field can occur

as well, and these are modeled through a complex permeability, µ = µ − jµ =
µ 0 (µ − jµ ). Examples of such media include ferrimagnetic materials, or ferrites.

r
r
The magnetic response is usually very weak compared to the dielectric response in
most materials of interest for wave propagation; in such materials µ ≈ µ 0 . Con-
sequently, our discussion of loss mechanisms will be confined to those described
through the complex permittivity, and we will assume that µ is entirely real in our
treatment.
We can substitute (42) into (37), which results in

k = ω µ( − j ) = ω µ 1 − j (43)

Note the presence of the second radical factor in (43), which becomes unity (and
real) as vanishes. With nonzero , k is complex, and so losses occur which are

quantified through the attenuation coefficient, α,in (39). The phase constant, β (and
consequently the wavelength and phase velocity), will also be affected by . α and

β are found by taking the real and imaginary parts of jk from (43). We obtain:
1/2



2
µ
α = Re{ jk}= ω  1 + − 1  (44)
2
1/2



2
µ
β = Im{ jk}= ω  1 + + 1  (45)
2
We see that a nonzero α (and hence loss) results if the imaginary part of the
permittivity, ,is present. We also observe in (44) and (45) the presence of the ratio

/ , which is called the loss tangent. The meaning of the term will be demonstrated

when we investigate the specific case of conductive media. The practical importance
of the ratio lies in its magnitude compared to unity, which enables simplifications to
be made in (44) and (45).
Whether or not losses occur, we see from (41) that the wave phase velocity is
given by
ω
ν p = (46)
β
The wavelength is the distance required to effect a phase change of 2π radians

βλ = 2π