Lecture 12
Dielectrics: Dipole, Polarization
Sections: 4.7, 5.7
Homework: See homework file
LECTURE 12
slide
1
Electric Dipole and its Dipole Moment
• electric dipole: two point charges of equal charge but opposite
polarity in close proximity
• electric dipole is characterized by its moment p (a vector that
points from –Q to +Q!)
p  Qd, C  m
Q
E
p  Qd
Q
d
LECTURE 12
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2
Field of a Dipole: Far-field Assumption
distance from dipole to observation point is much larger than dipole
z
P(r ,  ,  )
R

Q
x
r
( d / 2) cos 
d
actual
R
y

z
R
Q

Q
rd
far-field approximation
x
r
( d / 2) cos 
d
P(r ,  ,  )
R
y

Q
LECTURE 12
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3
Field of a Dipole – 2
• the far-field assumption leads to
d
d
R  r  cos  , R  r  cos 
2
2
• using superposition
Q  1
1 
V ( P) 



4  R R 

1
1
Q 

V ( P) 

4 r  d cos  r  d cos 

2
2


 Q
d cos 
  4
2

d

 
2

 r   cos   

2
 

0
p  ar
Qd cos 
V ( P) 

2
4 r
4 r 2
LECTURE 12
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4
Field of Dipole – 3
• dipole potential decreases with distance as 1/r2
• dipole potential is zero in the azimuthal plane θ = 90°
• E field obtained from V (using SCS)
Qd
E  V 
2 cos  a r  sin  a 
3
4 r
• E field of dipole decreases
with distance as 1/r3
LECTURE 12
slide
5
The charges of a dipole are located at the points D1
(0,1,0) and D2 (0,−1,0).
(a) What is the direction of the dipole moment if the
charge at D2 is negative?
(b) Along which axis is the potential the strongest for a given r ≠ 0?
(c) How is the E vector oriented with respect to the dipole axis for
observation points in the xz plane?
LECTURE 12
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6
Torque on Dipole in External Field


F

QE
F
 F  F sin  

Te
Q


Q
p



 E


• torque fulcrum is at dipole’s 
• dipole tends to rotate so
that its moment aligns
with the external field
center (dipole is bound)
torque definition: T  R F


d
| Te | 2   Q
| E |   sin    Te  p  E



2
 F

the dipole moment p determines
the torque exerted on the dipole
by an external electric field E
dielectric polarization is the tendency of bound dipoles to orient
along external electric fields
LECTURE 12
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7
Polarization in Dielectrics
• dielectrics have negligible DC conductivity and their charges are
assumed immobile, i.e., they are bound to a fixed location
• polarization involves the orientation of microscopic bound dipoles
so that their moments align more or less with the external E field
• there is a variety of mechanisms of polarization
 electronic polarization
E

(atomic level)


 ionic polarization
(molecular level)
p0

p
 orientational polarization occurs in materials that have
permanently polarized microscopic sub-domains (electrets:
quartz, polypropylene; polar liquids: water)
LECTURE 12
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8
Polarization Vector
polarization vector is the dipole moment per unit volume
p

i i
,
P  lim
v
v  0
C/m 2
dp  Pdv
• assume n (m−3) bound dipoles (atoms, molecules, sub-domains)
per unit volume of averaged distance and orientation given by
dav and dipole charge qd
p
N ( qd dav )

i i
P  lim

v 0
v
v
 P  n q
d d av
pav
number density of bound dipoles
LECTURE 12
slide
9
Polarization Vector and Surface Bound Charge – 1
L
specimen is homogeneous, isotropic, and
     
surrounded by vacuum (air)
    





Eext
is orthogonal to surface planes
unit normals an point inwards
Eext



















an an
P
A

Eext
Qsb
Qsb
uncompensated bound charge exists on surface only → ρsb =Qsb/A
p
|Q | L
|  A | L
P  total  sb a E  sb
a E |  sb | a E    sba n
v
v
A L
total dipole moment of specimen
P is along Eext (isotropic material) and points from –ρsb to +ρsb
P  Pna n , where Pn    sb
LECTURE 12
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10
Polarization Vector and Surface Bound Charge – 2
assume Eext and P are not orthogonal to the surface planes
p total | Qsb | L
|  sb A | L
|  sb |
P

aE 
aE 
aE
cos 
v
v
A  L
• boundary condition for
the normal P component
ext
|  sb |
E
P  an 
a E  a n    sb
cos 
Pn    sb
A
cos  
A
L
       A
    
A       a n
     

       a
    
n


A        A
P
holds regardless of surface orientation with respect to E
outside the specimen P = 0
compare with D  a n   s in PEC
LECTURE 12
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11
Polarization Vector and Surface Bound Charge – 3
• so far we have considered interface between air (non-polarizable
medium) and dielectric (polarizable medium)
• now we consider the interface between two dielectrics
 sb   sb(1)   sb(2)  | P (1) |  | P (2) | cos 
general boundary
condition for P at
dielectric interface
  sb   P (1)  P (2)   a n
#1 dielectric #2
Eext dielectric
     



































  


 
 
  

n

a
P (1)  sb P (2)
strongly polarizable
LECTURE 12
 
 
 



weakly polarizable
slide
12
Polarization Vector and Volume Bound Charge
make use of relation to surface charge
inside volume
ins |
| Pn ||  sb
flux of P is negative if bound charge
inside surface s is positive
ins s
 P  s    sb
Eext
L
 
 
P   
 
 
 
an 
s
s
 an
P

Qbins
take integral over s
ins
P

d
s


Q
b


net bound charge inside s
s

 P  ds    vb dv
s[ v ]
Gauss Theorem
v
   P   vb
LECTURE 12
Compare with   D  v
slide
13
Susceptibility and Permittivity – 1
• two types of charges: free charge (in conductors) and bound
charge (in dielectrics)
• from microscopic point of view both types of charge exist in vacuum
• vacuum flux density relates to the total charge density
  ( 0 E)   vf   vb
• relate D (flux density) vector to free charge only   D  vf
• at the same time   P    vb
   ( 0 E)    D    P
 D   0E  P
LECTURE 12
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14
Susceptibility and Permittivity – 2
• polarization P and E-field in isotropic materials: susceptibility
P   e ( 0 E)
 D   0 (1   e ) E

r
• relative permittivity
 r  1  e
D   0 r E   E
• anisotropic dielectrics (permittivity tensor)
 Dx   xx
 D   
 y   yx
 Dz   zx
 xy  xz   Ex 

 
 yy  yz  E y  D  εE
 
 zy  zz   Ez 
• nonlinear dielectrics [χe = f(E), ferroelectric materials]
LECTURE 12
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15
Dielectric Strength – Homework
We have implicitly assumed that a dielectric material preserves its
properties regardless of the field strength. In practice, if the field
magnitude exceeds certain critical value (the dielectric strength of
the material), the polarization forces become too strong, the
electrons break free from the atoms, accelerate through the
material, thereby heating it, damaging its molecular structure, and
ultimately causing even more electrons to break free. This
avalanche process leads to arcing/sparking, which destroys the
material (dielectric breakdown). The dielectric strength Eds is the
strongest E-field that the material can sustain without breakdown.
glass
25-40 MV/m
mica
200 MV/m
oil
12 MV/m
air
3 MV/m
LECTURE 12
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16
You have learned:
what a dipole is and how to calculate its potential and E field
what torque an external E field exerts on a dipole of moment p
what polarization is and how it is described by the polarization
vector P
how polarization P relates to surface and volume bound charge
what susceptibility and permittivity mean
what dielectric strength is
LECTURE 12
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17