This preview shows pages 1–3. Sign up to view the full content.
Lecture 2 Notes, 95.658, Spring 2012, Electromagnetic Theory II
Dr. Christopher S. Baird, UMass Lowell
1. Dispersion Introduction
 An electromagnetic wave with an arbitrary waveshape can be thought of as a superposition of
singlefrequency plane waves.
 If the dielectric material responds in the exact same way to plane waves of any frequency, than
each component of the waveshape will travel at the same speed, and the overall waveshape will be
preserved.
 If the dielectric material responds differently to plane waves of different frequencies (i.e. its
dielectric constant is frequency dependent), then the superposed components of the waveshape will
travel at different speeds. The waveshape will change in time as it propagates through the material
and in general will spread out.
 This wave spreading is known as dispersion.
 We must explicitly describe the behavior of the frequencydependent permittivity
ε
(
ω
), or in other
words, the dielectric constant
ε
r
(
ω
) =
ε
(
ω
)/
ε
0
, before we can describe the dispersion in a detailed
way.
 An accurate prediction of a material's permittivity requires quantum mechanics, but we can find a
good approximation by building a classical model.
2. Classical Harmonic Model
 Remember that the dielectric constant describes a material's response to an applied electric field.
 The applied field induces dipoles in the material which then create their own electric fields that
add to the original field.
 So if we find the dipole moment induced in a single atom by an applied field, we can sum over all
atoms and find the dielectric constant.
 For simplicity, we will deal only with nonmagnetic dielectric materials, so that the magnetic
permeability equals the permeability of free space. We also ignore magnetic forces.
 Consider a single electron bound to an atomic nucleus pushed by a wave that passes by. Assume
the atomic nucleus is so much heavier than the electron that it stays fixed.
 Newton's Law states:
F
=
m
a
where
F
is the total force,
m
is the electron's mass
and
a
is the electron's acceleration.
F
wave
F
binding
=
m
d
2
x
dt
2
−
e
E
−
k
x
=
m
d
2
x
dt
2
where
k
is the spring constant of the harmonic binding force. For a harmonic system with a single
mass the spring constant is
k
=
m
ω
0
2
where
ω
0
is the resonant frequency of the spring (which is
here caused by the binding force in the atom).
+

e
F
wave
F
binding
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document−
e
E
−
m
0
2
x
=
m
d
2
x
dt
2
 In its current form, this equations describes an electron that would oscillate forever. In reality, the
oscillating electron interacts with other electrons, looses energy in the process, and slows down.
 Instead of going into the details of the damping, we can sum up the process in a
phenomenological damping parameter
γ
:
−
e
E
−
m
0
2
x
=
m
d
2
x
dt
2
m
d
x
dt
 Let us investigate the material response to a plane wave at a single frequency
ω
, then we can
always superimpose plane waves for more complex situations:
−
e
E
x
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '11
 Staff
 Mass

Click to edit the document details