EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
EE243 Advanced Electromagnetic Theory
Lec # 7: Dielectric Materials
• Geometry for Homework Problem 3.1
• Multipoles
• Interaction of Multipoles
• Boundary Value Problems for Dielectrics
• Relationship of dielectric constant to molecules
Reading: Jackson Ch 4
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Copyright 2006 Regents of University of California
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Geometry HW 3.1
y
(0,0.5)
(0,0)
+λs1
−λ1
Observation
(1,0)
−λ2
(2,0)
x
• Positive line source induces negative line sources on
cylinders that can be approximated as being at their centers
• Find the potential at the observation point
• See if reciprocity holds when source and observation are
interchanged
2
Copyright 2006 Regents of University of California
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Overview
• Conductivity is produced by free carriers
(holes and electrons) moving to surfaces.
• But charges (electrons) bound to nuclei
(protons) can be displaced slightly by E.
• This creates dipole and possibly higher
spherical expansion eigenfunction moments.
• They in turn create
– Surface charge; reduction in internal field;
internal lattice fields; stored energy; force on
dielectrics
3
Copyright 2006 Regents of University of California
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Dielectrics
+
charge 1 = q1
- - Dielectric
Object
Surface charges due
to polarization P
P
+
+
+
• Charge produces E field
• E field produces Polarization and surface charge
• E field inside dielectric is lowered
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Copyright 2006 Regents of University of California
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Eigenfunctions in Cylindrical Coordinates
Jackson pp. 112
x and y become ρ and φ
Azimuthal = sin (mφ) and cos (mφ)
Radial = Bessel functions J (oscillatory like)
Hankel functions and Modified Bessel
functions I and K (exp. like)
• Boundary Conditions
•
•
•
•
– Wedge: Eigenfunction values of m (could be
fractional)
– Cylinders: Zeros of Bessel functions
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Copyright 2006 Regents of University of California
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Eigenfunctions in Spherical Coordinates
Jackson pp. 108
• x, y, z become ρ, φ and θ
• Azimuthal uniform = sinusoidal φ
• Or azumuthal and θ combine on sphere surface as
spherical harmonics
2l + 1 (l − m )! m
Ylm (θ , φ ) =
Pl (cos θ )e imφ
4π (l + m )!
∞
i
g (θ , φ ) = ∑ ∑ AlmYlm (θ , φ )
l =0 m = − l
Alm = ∫ Ylm (θ , φ )g (θ , φ )dΩ
*
15
Y21 (θ , φ ) = −
sin θ cos θe iφ
8π
Example
6
Copyright 2006 Regents of University of California
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Spherical Harmonic Expansion
• Representation outside the charge distribution
• Take eigenfunction moments over distribution
Ylm (θ , φ )
4π
qlm
Φ( x) =
∑
∑
r l +1
4πε 0 l =0 m = − l 2l + 1
1
Φ( x) =
1
4πε 0
∞
i
ρ ( x' )
∫ | x − x |ρ ( x )
'
( )
qlm = ∫ Ylm (θ , φ )r 'l ρ x ' dΩ
*
xi x j
⎤
1 ⎡q p ⋅ x 1
Φ ( x) =
⎢ + 3 + ∑ Qij 5 + ...⎥
r
r
4πε 0 ⎣ r
2 i, j
⎦
(
)
Qi , j = ∫ 3 xi' x 'j − r '2δ ij ρ ( x ' )d 3 x '
Copyright 2006 Regents of University of California
7
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Results Spherical Harmonic Expansion
• Negative gradient gives Eρ, Eθ and Eφ
• Energy
• Dipole case
∂Ei
1
(0) + ...
W = qΦ (0) − p ⋅ E (0) − ∑∑ Qij
∂x j
6 i j
W12 =
E (x ) =
3n( p ⋅ n) − p
4πε 0 | x − x0 |
p1 ⋅ p2 − 3(n ⋅ p1 )(n ⋅ p2 )
4πε 0 | x1 − x2 |3
• For dipole the average over a spherical volume is
the value at the center of the sphere
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Copyright 2006 Regents of University of California
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Ponderable Media
• Apply Maxwell’s equations locally and then
average to get macroscopic ME.
• Curl E = 0 added up over volume remains
the same => - Gradient of Potenetial
• Media free charges and dipole moments
contribute to potential P = sum pMOLECULE
• Add up charge and dipole contributions
• Integrate dipole term by parts to move
differential operator from 1/r to P.
• Obtain
Ponderable = having serious weight
Copyright 2006 Regents of University of California
9
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Ponderable Media: Results
• Div P is like charge
• Displacement
∇⋅E =
1
ε0
[ ρ − ∇ ⋅ P]
D = ε0E + P
∇⋅D = ρ
• Boundary Conditions
– D normal discontinuous
– E tangential continuous
ρ
∇⋅E =
ε
( D2 − D1 ) ⋅ n21 = σ
( E2 − E1 ) × n21 = 0
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Copyright 2006 Regents of University of California
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Image Charge Geometry
a
q
q’’
q’’ is necessary
to create the
fields for z < 0.
-a
q’
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Copyright 2006 Regents of University of California
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
BVP Dielectrics: Image in Half-Space
• Charge q at z = a in material with ε1 above half
space of material with ε2
• Postulate two image charges
– q’ at z = -a for fields z > 0
– q’’ at z = a for fields z < 0
• Match D normal and E tangential
⎛ ε 2 − ε1 ⎞
'
⎟⎟q
• Find
q = −⎜⎜
⎝ ε 2 + ε1 ⎠
⎛ 2ε 1 ⎞
''
⎟⎟q
q = ⎜⎜
⎝ ε 2 + ε1 ⎠
σ POL
q ε 0 ⎛ ε 2 − ε1 ⎞
a
⎟
⎜
= −( P2 − P1 ) ⋅ n21 =
2π ε 1 ⎜⎝ ε 2 + ε 1 ⎟⎠ a 2 + ρ 2
Copyright 2006 Regents of University of California
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)
3/ 2
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EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Integral Representations & Equations
Volume integration points
Surface Charge Patches
ε1
ρ(x’)
Observation Point
ε2
outside moves to
surface
P Dielectric Object
Observation Point
inside moves to
Boundary Integration point
surface
δΦ
δG ( x , x ′) ⎤
1
1 ⎡
3
ρ ( x ′)G ( x , x ′)d x′ +
Φ( x ) =
− Φ ( x ′)
G ( x , x ′)
da ′
∫
∫
⎢
⎥
δn′
δn′ ⎦
4πε1 V
4π S ⎣
1 ⎡
δΦ
δG ( x , x ′) ⎤
Φ( x ) =
− Φ ( x ′)
ρ ( x ′)G ( x , x ′)d x′ +
G ( x , x ′)
da′
∫
∫
⎥
⎢
4πε 2 V
4π S ⎣
δn′
δn′ ⎦
1
3
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Copyright 2006 Regents of University of California
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Integral Representations for Dielectrics
• Separate representations inside and outside
homogeneous dielectrics
– Use a free space Green’s function for the media
in which the observation point is immersed
– Will require knowledge of both the potential
and its normal derivative on the boundary
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Copyright 2006 Regents of University of California
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Integral Equations for Dielectrics
• Use Integral Representations
• Form integral equations by
– Requiring potential to be continuous and D
normal discontinuous by div P.
• Solve integral equations
– Every point has two unknowns (potential and
Div P) and two boundary conditions (potential
continuous and D normal continuous)
15
Copyright 2006 Regents of University of California
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
BVP Dielectrics: Sphere
• Spherical Harmonic Expansion inside and
outside sphere with m = 0.
• Match Tangential E and normal D
• Result
Jackson 158
– Constant and lower E field in sphere
– Only moment is dipole = p times volume
3
=
E
– Div P gives surface charge IN ε / ε + 2 E APPLIED
0
• Hole in Dielectric
– Similar result
⎛ ε / ε 0 −1 ⎞
⎟⎟ E APPLIED
P = 3ε 0 ⎜⎜
⎝ ε / ε0 + 2 ⎠
σ POL
⎛ ε / ε 0 −1 ⎞
⎟⎟ E APPLIED cos θ
= 3ε 0 ⎜⎜
⎝ ε / ε0 + 2 ⎠
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Copyright 2006 Regents of University of California
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Molecular Level Models
• Electric field strength inside
a material must be corrected
– Macroscopic P (lowers)
– Near molecules add a discrete
sum (usually zero)
• Electric Susceptibility
• Molecular Polarizability
1 ⎞
⎛
P = Nγ mol ⎜ ε 0 E + P ⎟
3 ⎠
⎝
P = ε 0 χe E
Nγ mol
χe =
1
1 − Nγ mol
3
3 ⎛ ε / ε 0 −1 ⎞
⎟⎟
γ mol = ⎜⎜
N ⎝ ε / ε0 + 2 ⎠
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Copyright 2006 Regents of University of California
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Models for Molecular Polarizability
• Harmonic oscillator model for atom
• Displacement times charge is dipole
• Observations
– Induced dipole moment is temperature
independent
– Partial orientation of otherwise random
permanent dipole moments has higher
molecular polarizability that reduces with
temperature increase
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Copyright 2006 Regents of University of California
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Energy and Force
•
•
•
•
Start charge based
Substitute for charge
Integrate by parts
Chain rule derivative
δW = ∫ δρ (x )Φ(x )d 3 x
δρ = ∇ ⋅ (δD )
δW = ∫ E ⋅ δDd x
3
1
EδD = δ ( E ⋅ D)
2
1
W = ∫ E ⋅ Dd 3
2
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Copyright 2006 Regents of University of California
EE 210 Applied EM Fall 2006, Neureuther
Lecture #07 Ver 09/17/06
Energy with Motion of Dielectric
• Basic formula
1
3
W = ∫ E ⋅ Dd x
2
• Result: Insert a
1
dielectric into a field
w = − P⋅E
2
with
fixed
sources
1
δW = ∫ (ρδ Φ + Φ δρ )d 3 x
• Extend for fixed
2
1
Voltages as two steps
δW1 = ∫ (ρδ Φ 1 )d 3 x
2
– Without batteries
1
δW2 = ∫ (ρδ Φ 2 + Φ δρ 2 )d 3 x = −2δW1 – Reconnect batteries
2
Dielectrics are attracted to high field regions and fields
are attracted to dielectric regions
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Copyright 2006 Regents of University of California