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Chapter 4 Electrostatic Fields in Matter
4.1 Polarization
4.2 The Field of a Polarized Object
4.3 The Electric Displacement
4.4 Self-Consistance of Electric Field and Polarization; Linear Dielectrics
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4.1.1 Dielectrics
4.1.2 Induced Dipoles and Polarizability
4.1.3 Alignment of polar molecules
4.1.4 Polarization and Susceptubility
4.1 Polarization
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4.1.1 Dielectrics
According to their response to electrostatic field,most everyday objects belong to one of two large classes:
conductors and insulators (or dielectrics): all charges are attached to
specific atoms or molecules.
Stretching and rotating are the two principle mechanisms by which electric fields can distort the charge distribution of a dielectric atom or molecule.
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4.1.2 Induced Dipoles and Polarizability
A simple estimate:The field at a distance d from the center of a uniform charge sphere
At equilibrium,
(v is the volume of the atom)
A neutral atom
applied
Polarized atom
Induced dipole
p
qdp Ep
Atomic polarizability
3
2 2 3 30 0 0
1 1 14 4 4
inq q qdde
d d a aE
0 0
14 3 34
qd pe
a aE E
30 04 3a v
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Usually
the most general linear relation
or in (x,y,z)
ijdepend on the orientation of the axis you chose.
It’s possible to select axes such that ij = 0 for ji
4.1.2
P E E
E 40 22 10 m
Nc
E
40 24.5 10 mNc
P E �
( is polarizability tensor) �
x xx x xy y xz z
y yx x yy y yz z
z zx x zy y zz z
P E E E
P E E E
P E E E
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4.1.3 Alignment of Polar Molecules
Polar molecules have built-in, permanent dipole moments
applied electric field (uniform)
Torque
s
2 2
S S
N r F r F
qE qE qS E
N P E qS P
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If the applied electric field is nonuniform,the total force is not zero.
for short dipole,
4.1.3
F F F q E E q dE
x x
y y
z z
dE E S
dE E S
dE E S
dE S E
F P E
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4.1.4 Polarization and Susceptibility
The polarization of a polarized dielectric
dipole moment per unit volume
is electric susceptibility tensor
for linear dielectric
is electric susceptibility
P
0 eP X E �
eX�
0
0
0
( )
( )
( )
xx xy xz
yx yy yz
zx zy zz
x e x e y e z
y e x e y e z
z e x e y e z
P X E X E X E
P X E X E X E
P X E X E X E
0 eP X E
eX
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4.2 The Field of a Polarized Object
4.2.1 Bound charges
4.2.2 Physical Interpretation of Bound Charge
4.2.3 The Field Inside a Dielectric
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4.2.1 Bound charges
bound charges: surface charge
volume charge
0
0
0 0
1 14
1 1 14
1 1 14 4
( )
1
[ ]
RV
R RV V
Rsurf V
V P d
P d P d
P da P dR
2ˆ1( )R
R R
ˆb
b
P n
P
0 0
1 1 1 14 4b bR Rsurf V
V da d
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4.2.2 Physical Interpretation of Bound Charge
and represent perfectly genuine accumulations of charge.
for uniformly distributed dipoles
for a uniform polarization and perpendicular cut
s
b b
0b
b
qb A
q PA
P
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for a uniform polarization and oblique cut
for the polarization is nonuniform ,
4.2.2 (2)
ˆcosend
qb A
q PA
P P n
ˆb P n
?b
P Ppx
q b x
d Pd
b P
b P
bV surf
V
d P da
P d
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Ex.2
Ex.9 of chapter 3
potential of a dipole moment
4.2.2 (3)
?E
ˆ cosb P n P
( , )V r
0
3
2cos for
3P R r R
r cosr z
0for
3Pr R V z
0 0
1ˆ3 3PE V z P
0cos for
3P r r R
343
p R P 1
20
ˆ4
p rV
r
320
41 1 ˆfor 4 3
r R V R Prr
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4.2.3 The Field Inside a Dielectric
microscopic
microscopic field is defined as the averaged field over region large enough.
microscopic field at P
or
averaged
V
out
dielectric
out inE E E
out inV V V
20
ˆ1
4
P RV doutsideout R
p Pd
p
VP
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srr
Ex.3 as Ex.2
inE E E
30
1 1 [ ]4
qr qrR
30
1 [ ]4
qr r
R
30
14
qs
R
( )s r r
0
13inE P
34( )3
qs p R P
for points outside, 20
ˆ14
p rV
r
4.2.2 (4)
r
2( ) 4E r da E r
3
30
43
43
rq
R
304
rqE
R
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4.2.3
30
1
4
pEin R
33
0
1 1 4( )34
R PR
0
1
3P
20
ˆ1
4
P RV dinin R
20
ˆ1
4out inP R
V V V dR
20
ˆ1
4
p rV
r
or
Ex.3(averaged field)
( )P
uniform
P
uniform
V pP
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4.3 The Electric Displacement
4.3.1 Gauss’s Law in The Presence of Dielectrics
4.3.2 A Deceptive Parallel
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4.3.1 Gauss’s Law in The Presence of Dielectrics
bound free
Gauss’s Law
electric displacement
Gauss’s Law
b f
0 b f fE P
0( ) fE P
fD
0D E P
.encfsurfaceD da Q
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4.3.1
Ex.4
inside
outside
need to know
Rrfor >
Rr
.encfsurfaceD da Q
?D
(2 )D rL L
ˆ2
D rr
?E
P
0 0
1ˆ
2E D r
r
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2
ˆ1
4f fR
D D dR
4.3.2 A Deceptive Parallel
one equation need 2 more equations [ i.e., ] to get unique solution
one unknown V, one equation.
?
One equation: ?
or
0E
02
ˆ1
4
RE d
R
3 , ,x y zE E Eunknown 0E
0E
E 0E
2
0:V V
3 , ,x y zD D Dunknown
0 ( )D E P P
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4.4 Self-Consistence of Electric Field and Polarization; Linear Dielectrics
4.4.1 Permittivity, Dielectric Constant
4.4.2 Special Problems Involving Linear Dielectrics
4.4.3 Energy in Dielectric System
4.4.4 Forces on Dielectrics
4.4.5 Polarizability and Susceptibility
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4.4.1 Permittivity, Dielectric Constant
self- consistence
for linear dielectrics
permittivity permittivity of free space
dielectric constant
x y
y x
( ) 2y f x x
( ) 1x f y y
2y
1x
0 eP x E
0 0 0 0(1 )e eD E P E x E x E
D E
0 (1 )ex
0
1 eK x
0( ) ( ) ( ) ( )
1e e
b fe
x xr P D r
x
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4.4.1 (2)
Ex.5 ?
for r < a
for r > a
for b > r > a
for r > b
dielectric
centerV
0E P D
2ˆ
4
QD r
r
02 24 4
a b acenter a b
Q QV V E d dr dr
r r
0
1 1 1( )
4
Q
b a b
0
2
2
ˆ4
ˆ4
Qr
rE
Qr
r
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4.4.1 (3)
for b > r > a
at r = b
at r = a
00 2
ˆ4
ee
x QP x E r
r
0b P 2
21
( 0)rP r Prr
0
0
2
2
4ˆ
4
e
be
x Q
bP nx Q
a
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4.4.1 (4)
? even in linear dielectric
may or may not
?
If the surface or the loop is entirely in a uniform linear dielectric,
0D
0( ) 0s
D da D d E P d P d
0 D
ExP e
0 0
dP
vacvac EK
ED
E
10
But for
0
dP 0D
D E
0 vacD E
0P
0P
vacuum
Dielectric
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4.4.1 (5)
QvacE
EK
vacVV
K
vacvac
QC
V vac
vac
Q QC K KC
V V
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4.4.2 Special problems Involving Linear Dielectrics
Ex.7
E
?
Method 1.
0 0 0eP x E
0 00
11
( )3 3
exE P E
1 0
0
22
1( )
3 3ex
E P E
1 0 1 0 0( )( )3e
e ex
P x E x E
0( )3n
nexE E
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4.4.2 (2)
dielectric constant
Method 2.
0 1 2 00
( )3
ne
n
xE E E E E
0 01 3
( ) ( )21
3e
E E Ex K
1 eK x
0 eP x E
0
1
3pE P
0 pE E E
3e
px
E E
0 ( )3ex
E E E
01
( )1
3e
E Ex
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Ex.8
4.4.2 (3)
? ( )qF force on q
0ˆ ( 0)b z e zP n P x E z
0
0
, 2 2
2 2 3/2
1cos
4 ( )
1
4 ( )
z due to qq
Er d
qd
r d
00 0
2 2 3/21
[ ]4 2( )
bb e
qdx
r d
2 2 3/21
2 2 ( )e
be
x qd
x r d
, , bz z due to q z due toE E E
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Total bound charge
Method 1.
4.4.2 (4)
2 2 3/20
2 2 1/20
2 ( )
1[ ]
2 ( )
2
eb b
e
e
e
e
e
x rdrq r drd qb
x r d
xq d
x r d
xq
x
?qF
02
ˆ1( )
4q bR
E daR
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Method 2. method of image
q
0z
b
0z
4.4.2 (5)
(0,0, )q at d
(0,0, )bq at d
z
b
(0,0, )q at d(0,0, )bq at d
b
2 2 20
2 2 2
1( 0)
4 ( )
( )
[
]b
qV z
x y z d
q
x y z d
2 2 20
1( 0) [ ]
4 ( )
bq qV z
x y z d
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• Satisfy Poission’s eq
•
•
•
The potential shown at above is the correct solution.
4.4.2 (6)
( ) 0V z 0
( 0) ( 0)z
V z V z
2 2 2 3/20 0
2 2 2 3/2 2 2 2 3/2
2( ) 2 ( ) 2( )1 1( ) ( )
4 2 ( )
21 1
4 2 2( ) ( )
b b
z z
b eb
e
q d q d q q dV V
z z x y d
q d x qd
xx y d x y d
2
2 20 0
1 1ˆ ˆ( )
4 4 2(2 ) 4b e
qe
qq x qF z z
xd d
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4.4.3 Energy in Dielectric System
Suggestion:
to charge a capacitor up
with linear dielectric (constant K)
energy stored in any electrostatic system (vacuum)
with linear dielectric
212
W CV
vacC KC
20
2W E d
20 1
2 2W KE d D Ed
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0 , for linear dielectric,
derivation:
4.4.3 (2)
( )
( )
( )
f
s
W Vd
D Vd
DV d D Vd
DV da D Ed
D E
( ) ( ) ( )
fD
DV D V D V
21 12 2
( ) ( )D E E E E D E
12
12
( )
[ ]
D E d
D Ed
12
W D Ed
s
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4.4.4 Forces on dielectrics
with Q constant
w
x
l
d Dielectric
us usF F dW F dx
,2
21 1
2 2
QW CV
C
0 ( )r e
Q AC
V dw
l X xd
22
21 1
2 2
dW Q dC dCF V
ds dx dxC
0 eX wdC
dx d
20
2eX w
F Vd
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with V constant, such as with a battery that does work,
4.4.4 (2)
212
0 e
W CV
X wdC
dx d
2 2
2 20
1
21 1
2 2 e
dW dQ dC dCF V V V
dx dx dx dxdC w
V X Vdx d
usdW F dx VdQ
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1. polarization for linear dielectric
induced dipole moment of each atom or molecular
susceptibility
atom polarizability
• What is the constant between α and Xe ?
non-self-consistent field
5 equations, 5 variables
4.4.5 Polarizability and Susceptibility
0 eP X E
p E
P Np N E
343
1N
R
self elseE E E
selfp E
30
14
pself
RE
elseP N E
2.
3.4.
5.
0e
NX
[ , , , , ]self elseP p E E E
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4.4.5 (2)
3 30 0
300
1 1
4 4
(1 ) (1 )34
else else else
else else
pE E E E
R R
NE E
R
0
0 03 3
, (1 ) 1
N
else eN NN
P E X
30
3433
3 34 43 3
4
(4 ) 3
11 ( )e
if a
aN a fX f
fN a R
0
00 03
31
N
e e eNN N
X X X
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Clausius-Mossott formulaLorentz-Lorenz equation in its application to optics
4.4.5 (3)
0
0 0
3
0
(1 )3
3
31
3 11
2
e
ee
e eX
e
r e
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