Page 232 - Electromagnetics
P. 232
nuclei provides a restoring force F r . In the absence of loss the restoring force causes
the electron cloud (and thus the induced dipole moment) to oscillate in phase with the
applied field. In addition, there will be loss due to radiation by the oscillating molecules
and collisions between charges that can be modeled using a “frictional force” F s in the
same manner as for a mechanical harmonic oscillator.
We can express the restoring and frictional forces by the use of a mechanical analogue.
The restoring force acting on each electron is taken to be proportional to the displacement
from equilibrium l:
2
F r (r, t) =−m e ω l(r, t),
r
where m e is the mass of an electron and ω r is a material constant that depends on the
molecular structure. The frictional force is similar to the collisional term in § 4.6.1 in
that it is assumed to be proportional to the electron momentum m e v:
F s (r, t) =−2 m e v(r, t)
where is a material constant. With these we can apply Newton’s second law to obtain
dv(r, t)
2
F(r, t) =−q e E (r, t) − m e ω l(r, t) − 2 m e v(r, t) = m e .
r
dt
Using v = dl/dt we find that the equation of motion for the electron is
2
d l(r, t) dl(r, t) 2 q e
+ 2 + ω l(r, t) =− E (r, t). (4.101)
r
dt 2 dt m e
We recognize this differential equation as the damped harmonic equation. When E = 0
we have the homogeneous solution
2
l(r, t) = l 0 (r)e − t cos t ω − 2 .
r
2
2
Thus the electron position is a damped oscillation. The resonant frequency ω − is
r
usually only slightly reduced from ω r since radiation damping is generally quite low.
Since the dipole moment for an electron displaced from equilibrium by l is p =−q e l,
and the polarization density is P = Np from (93), we can write
P(r, t) =−Nq e l(r, t).
Multiplying (4.101) by −Nq e and substituting the above expression, we have a differential
equation for the polarization:
2
d P dP 2 Nq e 2
+ 2 + ω P = E .
r
dt 2 dt m e
To obtain a constitutive equation we must relate the polarization to the applied field E.
We can accomplish this by relating the local field E to the polarization using the Mosotti
field (4.96). Substitution gives
2
d P dP 2 Nq 2 e
+ 2 + ω P = E (4.102)
0
dt 2 dt m e
where
Nq 2
2 e
ω 0 = ω −
r
3m e 0
© 2001 by CRC Press LLC