chapter 4 electrostatic fields in matter 4.1 polarization 4.2 the field of a polarized object 4.3...
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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
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
4.3 The Electric Displacement
4.3.1 Gauss’s Law in The Presence of Dielectrics
4.3.2 A Deceptive Parallel
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
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
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
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
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
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
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
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
4.4.1 (5)
QvacE
EK
vacVV
K
vacvac
QC
V vac
vac
Q QC K KC
V V
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
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
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
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
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
• 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
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
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
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
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
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
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
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
XNX
X X
N N X
KX
N K
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