Page 588 - Engineering Electromagnetics, 8th Edition

570 ENGINEERING ELECTROMAGNETICS

Withthewavesinthisform,timedifferentiationproducesafactorof jω.Consequently
(E.11) can be simplified and rewritten in phasor form:
e
2 2
−ω d s + jωγ d d s + ω d s =− m E s (E.13)
0
where (E.4) has been used. We now solve (E.13) for d s , obtaining
−(e/m)E s
d s = 2 2 (E.14)
ω − ω + jωγ d
0
The dipole moment associated with displacement d s is
p s =−ed s (E.15)
The polarization of the medium is then found, assuming that all dipoles are identical.
Eq. (E.1) thus becomes

P s = Np s
which, when using (E.14) and (E.15), becomes
2
Ne /m
P s = 2 2 E s (E.16)
ω − ω + jωγ d
0
Now, using (E.3) we identify the susceptibility associated with the resonance as
Ne 2 1

χ res = = χ res − jχ res (E.17)
2
0 m ω − ω 2 + jωγ d
0
The real and imaginary parts of the permittivity are now found through the real
and imaginary parts of χ res : Knowing that

= 0 (1 + χ res ) = − j
we find
= 0 (1 + χ ) (E.18)

res
and

= 0 χ res (E.19)

The preceding expressions can now be used in Eqs. (44) and (45) in Chapter 11 to
evaluate the attenuation coefficient, α, and phase constant, β, for the plane wave as it
propagates through our resonant medium.
The real and imaginary parts of χ res as functions of frequency are shown in
.
Figure E.2 for the special case in which ω = ω 0 . Eq. (E.17) in this instance becomes
. Ne 2 j + δ n
χ res =− 2 (E.20)
0 mω 0 γ d 1 + δ n
where the normalized detuning parameter, δ n ,is
2
(ω − ω 0 ) (E.21)
γ d