Page 587 - Engineering Electromagnetics, 8th Edition

APPENDIX E Origins of the Complex Permittivity 569

where the electron charge, e,is treated as a positive quantity. The applied force is
F a (z, t) =−eE(z, t) (E.6)
We need to remember that E(z, t)ata given oscillator location is the net field, com-
posed of the original applied field plus the radiated fields from all other oscillators.
The relative phasing between oscillators is precisely determined by the spatial and
temporal behavior of E(z, t).
The restoring force on the electron, F r ,is that produced by the spring which is
assumed to obey Hooke’s law:
F r (z, t) =−k s d(z, t) (E.7)
where k s is the spring constant (not to be confused with the propagation constant). If
the field is turned off, the electron is released and will oscillate about the nucleus at
the resonant frequency, given by

ω 0 = k s /m (E.8)
where m is the mass of the electron. The oscillation, however, will be damped since
the electron will experience forces and collisions from neighboring oscillators. We
model these as a velocity-dependent damping force:
F d (z, t) =−mγ d v(z, t) (E.9)
where v(z, t)is the electron velocity. Associated with this damping is the dephasing
process among the electron oscillators in the system. Their relative phasing, once
fixed by the applied sinusoidal field, is destroyed through collisions and dies away
exponentially until a state of totally random phase exists between oscillators. The 1/e
point in this process occurs at the dephasing time of the system, which is inversely
proportional to the damping coefficient, γ d (in fact it is 2/γ d ). We are, of course,
driving this damped resonant system with an electric field at frequency ω.Wecan
therefore expect the response of the oscillators, measured through the magnitude of
d,tobe frequency-dependent in much the same way as an RLC circuit is when driven
by a sinusoidal voltage.
We can now use Newton’s second law and write down the forces acting on the
single oscillator of Figure E.1. To simplify the process a little we can use the complex
form of the electric field:
E c = E 0 e − jkz e jωt (E.10)
Defining a as the acceleration vector of the electron, we have

ma = F a + F r + F d
or
2
m ∂ d c + mγ d ∂d c + k s d c =−eE c (E.11)
∂t 2 ∂t
Note that since we are driving the system with the complex field, E c ,we anticipate a
displacement wave, d c ,of the form
e
d c = d 0 e − jkz − jωt (E.12)