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6/3/2020
1
Advanced Electromagnetics:
21st Century Electromagnetics
Lorentz Oscillator Model
Lecture Outline
•High level picture of dielectric response•Qualitative description of resonance•Derivation of Lorentz oscillator model
2
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High Level Picture of Dielectric
Response
Slide 3
Moving Charges Radiate Waves (1 of 2)
4
outward travelling wave
This is called the single‐charge radiation model (Heaviside, 1894).
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Moving Charges Radiate Waves (2 of 2)
5
Dielectric Slab
6
It is desired to understand why a dielectric exhibits an electromagnetic response.
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Atoms at Rest
7
Without an applied electric field 𝐸, the electron “clouds” around the nuclei are symmetric and at rest.
Applied Wave
8
The electric field 𝐸 of a electromagnetic wave pushes the electrons away from the nuclei producing “clouds” that are offset.
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Secondary Waves
9
The motion of the charges emits secondary waves that interfere with the applied wave to produce an overall slowing effect on the wave.
Qualitative Description of
Resonance
Slide 10
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Visualizing Resonance – Low Frequency
11
• Driving force is able to modulate amplitude
• Displacement is in phase with driving force
• There exists a DC offset
Visualizing Resonance – on Resonance
12
• Driving force can cause large displacements
• Displacement is 90° out of phase with the driving force (i.e. peaks of push correspond to nulls of displacement)
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Visualizing Resonance – High Frequency
13
• Displacement has vanishing amplitude
• Displacement is 180° out of phase with driving force in order to perfectly oppose it.
Response of A Harmonic Oscillator
14
Amplitude P
hase Lag
180°
0° 0
> 0
>> 0
res
90°
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Impulse Response of a Harmonic Oscillator
15
Amplitude
Time, t
Excitation Ball Displacement
Damping loss
Derivation of Lorentz Oscillator
Model
Slide 16
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Lorentz Oscillator Model
17
Mass on a Spring
Atomic Model
nucleus
damper spring
mass
electron cloud
Electric
field 𝐸
Equation of Motion
18
2202
r rm m m r qEt t
electric force
restoring force
0
K
m
naturalfrequency
acceleration force
em m mass of an electron
frictional force
damping rate (loss/sec)
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Fourier Transform the Equation of Motion
19
2202
r rm m m r qEt t
2 20m j r m j r m r qE
2 20m j m m r qE
Fourier transform
Simplify
Charge Displacement 𝑟 𝜔
20
2 2
e 0
Eqr
m j
r
2 20m j m m r qE
Solve for r
The displacement 𝑟 𝜔 describes how far charge is displaced from its equilibrium position.
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Electric Dipole Moment �⃗� 𝜔
21
qr
r
Definition of Electric Dipole Moment:
** Sorry for the confusing notation, but μ here is NOT permeability.
2
2 2e 0
Eq
m j
charge
distance from center
The electric dipole moment �⃗� 𝜔 is a measure of the strength and separation of positive and negative charges.
Lorentz Polarizability 𝛼 𝜔
22
E
Definition of Lorentz Polarizability:
** Sorry for the confusing notation, but here is NOT absorption.
2
2 2e 0
1q
m j
𝛼 𝜔 is a tensor quantity for anisotropic materials. For simplicity, the scalar form will be adopted here.This is the Lorentz polarizability for a single atom.
The Lorentz polarizability 𝛼 𝜔 is a measure of how easily electrical charges are displaced. Charge may be more easily displaced in some directions that others.
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Polarization Per Unit Volume 𝑃 𝜔
23
1i
V
PV
Definition:Average dipole moment over all atoms in a material.All billions and trillions of them!!!
2
2 2e 0
ENqP N
m j
There is some randomness to the polarized atoms so a statistical approach is taken to compute the average.
Number of atoms per unit volume Statistical volume averageN
Unpolarized Polarized with some randomness Equivalent uniform polarization
AppliedE‐Field
AppliedE‐Field
Electric Susceptibility 𝜒 𝜔 (1 of 2)
24
0 eP E
A material becomes polarized 𝑃 in the presence of an electric field 𝐸 according to
This leads to an expression for the electric susceptibility:
2
e 2 20 0 e 0
1N Nq
m j
e() is called the electric susceptibility and is a measure of how easily an electric field 𝐸 can polarize a material.
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Electric Susceptibility 𝜒 𝜔 (2 of 2)
25
The electric susceptibility of a dielectric material is:
2p
e 2 20 j
22p
0 e
Nq
m
•Note this is the susceptibility of a dielectric which has only one resonance.•The location of atoms is important because they can influence each other. This was ignored.•Real materials have many sources of resonance and all of these must be added together.•Electric susceptibility is the transfer function of the oscillator system.
plasma frequency
19
120
31e
1.60217646 10 C
8.8541878176 10 F m
9.10938188 10 kg
q
m
Plot of Electric Susceptibility 𝜒 𝜔
Slide 26
0
2p
e 2 20 j
2
pe
0
0
e 0 0 e 0
e 180
e 0 90
2p
e 00
Γ is FWHM for 𝜒 𝜔 .