Download Lecture Notes 09: Electrostatic Fields In Matter, Dielectric Materials and Their Properties

UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
LECTURE NOTES 9
ELECTROSTATIC FIELDS IN MATTER:
DIELECTRIC MATERIALS & THEIR PROPERTIES
Polarization
We have previously discussed the electrostatic properties of conductors of electricity. Now
we wish to discuss the opposite end of the spectrum – the electrostatic properties of insulators –
non-conductors (very poor conductors) of electricity.
Suppose we have a block of insulator material, or even a gas or liquid – non-conducting –
consisting of atoms, or atoms bound together as molecules (e.g. H2O, CO2, HCOOH, etc.) Solid
materials that are insulators are things like wood, plastic, glass (amorphous SiO2), rubber, etc.
All of these materials are very poor conductors of electricity – insulators. They are examples of
dielectric materials, generically known as dielectrics.
An “ideal” dielectric material is one which contains no free charges. Since microscopically, the
dielectric material consists of atoms, its macroscopic electrostatic dielectric properties arise from
the collective (sum total) contributions of its microscopic constituents – at the atomic scale.
Each atom consists of a central, positive-charged “pointlike” nucleus
Rnucleus  few fermi's 1 fm  1015 m  surrounded by “clouds” of electrons, bound to the


nucleus. The atomic electrons are not free → typical radius of orbiting electrons is  few
angstroms ( 1Å  1010 m ) ability to move / migrate!
Polarization of an Atom in an Externally-Applied Electric Field:
First, consider an atom (electrically neutral) in its ground state (lowest quantum energy level)
such as the hydrogen atom (simplest case). The single electron orbiting the nucleus (a single
proton) has a spherically–symmetric charge distribution (i.e. no  or  -dependence) in the

absence of any externally-applied electric field, i.e. Eext  0 :
The typical electric field strength “seen” by an electron orbiting the hydrogen nucleus (a single
proton) due to the nuclear electric charge is:
 e 
1 Qnucl
1
1.6  1019
ˆ

Enucl
r
rˆ
r  
4 o r 2
4 8.85  1012 1010 2
 e
which for r  1Å  1010 m gives a whopping Enucl
 r  1Å   1.44 1011 Volts / m (very large!!!)
n.b. The nucleus “sees” this same electric field strength, due to the electron’s electric charge.
This is a typical electric field strength internal to/in the vicinity of atoms (& molecules)!!!
Compare this internal electric field strength to the electric field strengths easily / routinely
 lab
 lab
 atom
 103  106 V/m. We realize that Eexternal
 Einternal
available in a laboratory setting of Eext
i.e.  103  106  1011 Volts/m !!!
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
1
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
Because of this, when an atom (e.g. hydrogen) is placed in an external electric field, because
Eext << Eint the atom “sees” the externally applied field as a perturbation to its internal electric
field.
Neutral atom, no externally-applied electric field:

Eext  0 :
Typical Atomic Size:
ratom ~ a few Ǻngstroms
(1 Ǻ = 10−10 m)
ratom
Spherically-symmetric electron “cloud”
charge distribution (negative)
Qnucl
e−
Point-like nucleus
has positive charge
Rnucl ~ few fm, i.e.
~ few x 10−15 m
A Neutral Atom in an Externally Applied Uniform / Constant Electric Field:
Suppose we place a neutral atom between the plates of parallel plate capacitor with

e.g. Eext  Eo xˆ  1000 Volts/meter in gap-region of capacitor plates:
hydrogen
Qnucl
 e


Atomic nucleus feels a net force of: Fnucl  Qnucl Eext   eEo xˆ


Electron “cloud” feels a net force of: Fe  Qe Ee  eEo xˆ


Thus we see that: Fe   Fnucl  eEo xˆ (forces are equal & opposite – i.e. Newton’s 1st Law!)

Fe  eEo xˆ
Qnucl

Fnucl   eEo xˆ
 xˆ
Electron “Cloud”, w/
Electric Charge −e
2
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede

The net effect of the externally-applied electric field Eext  Eo xˆ is that the nucleus is displaced a

tiny amount (a small fraction of size of atom) from its Eext  0 position in the  x̂ direction, and
the centroid (i.e. center) of the electron “cloud” is displaced a tiny amount (n.b. the size /
magnitude of the displacement of centroid of electron cloud is the same as that for nucleus) from

its Eext  0 position in the  x̂ direction:
+
+
+
+
+
+
+
+
+
+
+

Eext  Eo xˆ
Centroid
of e−
Cloud
-
+Qnucl
d
2
d
2
d
Displaced Electron Cloud
Charge Distribution
-

Eext  Eo xˆ

Original Eext  0
Atomic Configuration
−
−
−
−
−
−
−
−
−
−
−
x̂
Obviously, because the electrons of the atom are bound to the nucleus, the mutual attraction
of the atomic Coulomb force keeps the atom together – electron cloud is bound to nucleus.
Thus, because of the displacement of +Qnucl (= +e for hydrogen) in the  x̂ -direction by an
amount d/2 and the displacement of the centroid of the electron “cloud” charge distribution
(Qe = −e for hydrogen) to the  x̂ direction also by an amount d/2, caused by the application

of the externally-applied uniform / constant electric field Eext  Eo xˆ we see that an electric


dipole moment p  Qd  Qdxˆ is induced (i.e. created) in the atom by the application of the

external electric field Eext  Eo xˆ .



Because Eext  1036  Eint  1011 Volts/meter, the externally-applied field Eext is seen as a

(very) small perturbation on the internal electric field Eint ; hence a linear relationship exists


between the induced electric dipole moment p and the externally-applied electric field Eext :


p   Eext ← n.b. vector relation
The constant of linear proportionality  is known as the atomic electric polarizability.
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
3
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede

p
Coulomb  meters
Coulombs 2

SI units of atomic polarizability:    
Newtons / Coulomb Newtons / m
Eext
The atomic polarizabilities  of atoms are often expressed in terms of  4 o which has SI
units of  4 o  3  meters 3  , since  o  8.85 1012 Farads / m ( Coulombs / Newton  m 2 )
The table below summarizes the (normalized) atomic polarizabilities  4 o of some of the
lighter atoms, in units of 10−30 m3:
Atomic
Element
Nuclear Charge
Znucleus
 4 o
# e− in
outer shell
electron
configuration
( ) denotes
closed
shell
“Alkali” (single e− in
Metal
outer shell)
H
1
0.667
“1”
Noble
Gas
He
2
0.205
(closed shells)
Li
3
24.3
1
Be
4
5.60
(1s2) (2s2)
C
6
1.76
(1s2) (2s2) 2p
Ne
10
0.396
Alkali (single e− in
Metal outer shell)
Na
11
24.1
1
Noble
Gas
Ar
18
1.64
(closed
shells)
Alkali (single e− in
Metal outer shell)
K
19
43.4
1
(1s2)(2s2)(2p6)(3s2)(3p6)4s
Alkali (single e− in
Metal outer shell)
Cs
55
59.6
1
(1s2)(2s2)(2p6)(3s2)(3p6)(4s2)
*(4p6)(4d10)(5s2)(5p6)6s
(closed
shells)
Alkali (single e− in
Metal outer shell)
Noble
Gas
4
(closed
shells)
(closed
shells)
1s
(1s2)
(1s2) 2s
(closed shells)
(1s2) (2s2) (2p6)
(1s2) (2s2) (2p6) 3s
(1s2)(2s2)(2p6)(3s2)(3p6)
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
It can be seen that atomic polarizability  has dependence on outer-most shell/valence electron!
(i.e. the least tightly bound electron to nucleus - due to screening effects of inner shell electrons).
It is (certainly) possible to obtain a theoretical relation - i.e. a theoretical “pre-diction”
(technically, a post-diction) of the atomic polarizability,  for a given atom (or molecule)


relating how p (the induced electric dipole moment) depends on Eext (the applied external
electric field). In order to do this, must “know” the atomic electron volume charge distribution /

electric charge density   r  .
Griffiths Example 4.1, pages 161-62:
A crude model for atom, due to J.J. Thompson (c.a. ~ 1910) is his “plum pudding” model of
the atom – i.e. a point nucleus of positive charge +Q surrounded by a uniformly charged
spherical electron cloud of total charge –Q of radius a. This means assuming a constant volume
charge density  eatomic for the atomic electrons orbiting the nucleus – i.e. one which is flat, out to a
radius r = a (and zero after that) as shown in the figure below:
e
atomic
3Q

4 a 3
e
 Q
 Coulombs 


m3


e

atomic
0
atomic
r=a
Vatom
3Q
4 a 3

Q
4 3
 a 
3

Coulombs m 
3
r
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
5
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
This is a (crude, but simple) theoretical model of an actual atom (i.e. far from reality), but it
works somewhat well - i.e. it is accurate to a factor of ~ 4.5 – c.f. with actual data – see below!
If the nucleus is displaced a relative distance d from the centroid of the atomic electron charge
density distribution, then the electric field intensity that the nucleus “sees” at that point is:
 nucl 
3Q
1
  rˆ 


   2  d  where   r     eatomic  r    
Einternal

r
 constant, for r  a.
r  


4 a 3
4 o v
r 

    
r  r  r   d  r   dxˆ  r 

(since here: d  dxˆ )
Then: r 2   d  x   y2  z 2
2

The source point r  moves around
the interior of the sphericallysymmetric atomic electron charge
density “cloud” in carrying out the
above volume integral.
One can see that explicitly doing this volume integral is a pain – it certainly can be done though!
However, by use of the divergence theorem, we find there is an easier way to obtain what we
 nucl 
want Einternal
 r  from Gauss’ Law:
  
  Qencl

E
r
d
E
r







dA 




v
S
o
So we take a fictitious, spherical Gaussian surface S of radius d < a centered on the centroid of
the spherically-symmetric atomic electron charge density distribution as shown below:
Then: Qencl = charge enclosed by S, radius d
 3Q  4 3 
Qencl  V  
d 
3 
 4 a  3

d 
Qencl  Q  
a
3
 nucl    nucl
Q
 d 2  encl
E
 S  internal  r dA  Einternal  d  4
Area of
Gaussian
Sphere, S'
3
o
d 
Q    o
 nucl
a
rˆ ( r̂ direction is due to intrinsic symmetry of problem)
Then: Einternal  r  d  
4 d 2
6
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
 nucl  
Qd
Einternal
r d 
xˆ
4 o a 3
Now the force on the nucleus due to the centroid-displaced spherically-symmetric atomic
electron charge distribution is:
 
@ Nucleus, i.e. r  d  dxˆ :


 nucl  
 nucl  
Q2d
Finternal
r  d  Qnucl Einternal
r d 
xˆ
4 o a 3

Qnucl = +Q:



Since this is an electrostatics problem, this force on the nucleus must be precisely balanced by
the force on it due to the externally applied field, i.e.
 nucl  

 
Finternal
r  d  Qnucl Eexternal r  d  Qnucl Eo xˆ


 nucl  
Finternal
r d




Eexternal  Eo xˆ

 nucl  
Fexternal
r d

Qnucl

x̂

 nucl
i.e. Finternal
 nucl
 Fexternal
 Q nucl

 nucl


  m a  0  a  0
 r  d   F  r  d   0



 r  d    F  r  d 








E
 r  d    Q E  r  d  or: E  r  d    E  r  d    4Qda
external
nucl
nucl
nucl
internal
nucl
external
nucl

But: p  p  Qd

 
 Eexternal r  d 


Turning this around:
  pp 
nucl
p
external
nucl
internal
external
nucl
internal
3
xˆ
o
p
4 o a 3
p   4 o a 3  Eexternal
 4 o a 3

Theoretical “post-diction” for  using J.J. Thompson’s
“plum-pudding” model of an atom (ca ~ 1910)
A better theoretical model of the atom: use the Schroedinger Wave Equation (i.e. use quantum
mechanics) which describes the wave-nature of electrons bound to the heavy / massive “pointlike” nucleus.
Electron wave function (“probability amplitude”)
H  E
Electron wave function:
Appropriate
Energy Eigenvalue(s)
  r , ,    R  r  Y  ,  
Hamiltonian
Operator
(bound state energy /
energy spectrum of electrons
bound to nucleus)
Spherical
harmonic
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
7
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
Electron Probability Density:
P  r , ,     *  r , ,    r , ,      r , ,  
Electron Volume Charge Density:
e
atomic
 r , ,    eP  r , ,  
2
  e   r , ,  
2
For an atomic electron in the lowest energy / ground state of e.g. the hydrogen atom:
Lz orbital angular quantum #
e1,0,0  r , ,    e  1,0,0  r , ,  
2
 n ,l , m
atomic
Ground state: n, l, m = 1, 0, 0
Principal quantum # Total orbital angular
(=Radial quantum #) momentum L quantum #
1
For hydrogen atom in its ground state:  1,0,0  r , ,   
a
3
o
e  r / ao (n.b. spherically symmetric!)
o
c
 4   c 
0.53A


 0.53 1010 m
Where: ao  Bohr radius   2 o 
2
2
e
m
c
m
c


 e
em e
2
And where: h = Planck’s constant = 6.626 x 10-34 Joule-sec
h
 1.054 1034 Joule  sec
2
e2  1 
1
 em  fine structure "constant" 
 
4 o  c  137.036
(  em = EM interactions’ dimensionless coupling strength)
And: c  197.327 MeV - fm
( 1 MeV  106 electron volts, 1 fm  1015 m )
Rest mass energy of electron:
me c 2  0.511 MeV
And: c = speed of light = 3 x 108 meters / sec
 For an atomic electron in the lowest energy / ground state of e.g. the hydrogen atom, the
electron volume charge density is:
e1,0,0  r ,  ,    e  1,0,0  r , ,   
2
atomic
e 2 r / ao
e
 ao3
  Qencl
e 2 r / ao
E
e
and
Gauss’
Law
 S  dA   o for a Gaussian sphere of
 ao3
radius r = d centered on the centroid of the atomic electron charge density distribution, the result
from this more sophisticated / quantum-mechanically motivated model, for the atomic
polarizability  QM is:
If we use e1,0,0
 r, ,  
atomic
 QM 
p
 3 o ao3 
Eext
 QM 3 3
 ao  0.112 1030 m3
4 0 4
See also Griffiths
problem 4.2
Compare this to J.J. Thompson’s (relatively crude/simple) “plum-pudding” model of the atom,
3e
with electron volume charge density eatomic  
 constant, and with a  ao , which gave:
4 ao3
8
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
 pp 
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
 pp
 ao3  0.149 1030 m3
4 0
p
 4 o ao3 
Eext
They differ by ~ 30% from each other (for the hydrogen atom / simplest atom) - not bad for J.J.!!
However, compare these two “post-dictions” e.g. to the actual experimental measurement for
(supposedly atomic, i.e. not molecular) Hydrogen (from the above table):
H
 expt
 0.667 1030 m3 which is 4.5 → 6 larger than that predicted by either theory!!!
4 0
2
 Eeek !!! 


Note that when Eext becomes comparable to, or larger than, Einternal then there no longer exists


a linear relationship between p and Eext . It becomes increasingly non-linear:

  2   3

i.e. p   Eext   Eext
  Eext  ....H .O.T .' s
  
linear
quadratic
cubic
(See S.Errede’s UIUC Physics 406POM lecture notes on distortion for more details if interested.)
Electrostatic Molecular Polarizability
Molecules, consisting of groups of (two or more) atoms bound together (by the
electromagnetic force / interaction), because of their intrinsic individual atomic structure, are
often highly non-spherical in their geometrical shapes / configurations.
- An example of one such highly non-spherical molecule is the (linear) carbon dioxide (CO2)
molecule:
O
C
O
ẑ
Axis of CO2
Molecule
Because of its shape (linear, and axially-symmetric) it should come as no surprise that:
 Molecular polarizability 
 Molecular polarizability 

  

  to CO 2 molecule axis 
  to CO 2 molecule axis 
 CO
i.e.
2
 CO  4.5 1040 C
2
2
N /m
>
 CO
>
 CO  2.0 1040 C
2
2
2
N /m
 CO 4.5

 2.25
 CO 2.0
2
or:
2
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
9
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede

Thus for non-spherical molecules, the net induced electric dipole moment pmol may not be



parallel to Eext , i.e. pmol   mol Eext .
Rather, a more correct mathematical description (i.e. for the more general case), one that does


properly relate pmol to Eext is given by:
 

pmol   mol Eext

 mol
 Molecular Electrostatic 


  Polarizability Tensor
:
 Rank-2 Tensor/3  3 Matrix 


 pxmol    xxmol  xymol  xzmol   Exext 
 mol   mol
mol
mol   ext 
 p y     yx  yy  yz   E y 
 pzmol    zxmol  zymol  zzmol   Ezext 
  



 pmol
 mol
 Eext
Now ẑ is the symmetry axis of molecule. It is always possible to choose the principal axes

xˆ , yˆ , zˆ of the molecule such that off-diagonal elements of  mol vanish (i.e.  xymol ,  yzmol ,  xzmol , etc.)
For the linear, axially-symmetric CO2 molecule,  zz    and  mol   xxmol   yymol .
pxmol   xxmol Exext   xymol E yext   xzmol Ezext   xxmol Exext   mol Exext
Writing things out completely:
p ymol   yxmol Exext   yymol E yext   yzmol Ezext   yymol E yext   mol E yext
pzmol   zxmol Exext   zymol E yext   zzmol Ezext   zzmol Ezext   mol Ezext
 

pmol   mol Eext
Alignment (Polarization) of Polar Molecules in an Externally-Applied Electrostatic Field
A neutral atom has (and many types of neutral molecules have) no intrinsic, permanent



electric dipole moment – i.e. if Eext = 0 then patom (or pmol ) = 0. Such molecules are known as


non-polar molecules. However, when an external electric field is applied Eext  0 , then patom

(or pmol ) ≠ 0. Such electric dipole moments are known as induced electric dipole moments.
Some molecules, such as the all-important water molecule (H2O) actually do have permanent


electric dipole moments (i.e. even when Eext = 0, pmol ≠ 0)! The non-vanishing permanent
electric dipole moment for some molecules arises because of how outer-shell atomic electrons
are (non-democratically) shared amongst the individual atoms making up the molecule.
For example, in the water molecule (H2O) the outer-shell (most loosely-bound) atomic
electrons tend to cluster preferentially around the oxygen atom, and since the H2O molecule is
not axially symmetric (the two hydrogen atoms form an opening angle between them of 105o
with oxygen atom at the vertex) leaving net negative charge at oxygen vertex and net positive
charge at hydrogen atom end.
10
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede

pH 2O
H+
H+

pH 2O  6.1 1030 Coulomb  meters
 = 105o
(very large!!!)
O−
one reason why H2O is very good solvent!
Molecules with permanent electric dipole moments are known as polar molecules.
(e.g. water, benzene, toluene, . . . )
If polar molecules, with permanent electric dipole moments are placed in a uniform applied

external field Eext , two things can/will happen:
1.) Permanent electric dipole moments tend to align / line-up with the external field:
 permanent 
 Eext
pmol

2.) The external applied electric field Eext can / does induce a (non-permanent) electric dipole
 induced  
moment: pmol
  mol Eext
 total  permanent  induced  permanent  
Thus: pmol
 pmol
 pmol  pmol
  mol Eext



 induced
 pmol

 induced
 permanent
 pmol
Usually, for typical values of Eext  1000 Volts/meter, thus pmol
for polar
 atomic

molecules with permanent electric dipole moments. Why?? Because Einternal  Eext .
So to “zeroth order” when a molecule with a permanent electric dipole moment is placed in an

external electric field Eext :
 atomic

 total  permanent
 induced
 permanent
pmol
 pmol
because pmol
if Einternal
 Eext .
 pmol
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
11
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
What happens when polar molecules (and/or pure physical electric dipoles) are placed in a

uniform external electric field Eext ?

Eext  Eo xˆ


F  QEext  QEo xˆ
+Q
 
 
d , p  Qd , r
d/2
d
d/2


F  QEext  QEo xˆ

Force on +Q(@ r ):

Force on –Q(@ r ):


r
−Q

Eext  Eo xˆ
 
 
F  r   QEext  r   QEo xˆ
 
 
F  r   QEext  r   QEo xˆ

 


Net force on polar molecular dipole: Fnet  F  F  QEext  QEext  0

→  No net force on polar molecular dipole!!! (for uniform / constant Eext  Eo xˆ )
  
However,  a net torque on the polar molecular dipole: N  r  F
The net torque acting on the polar molecular dipole is:



 
 
N net  N   N   r  F  r  F
Take torques about the midpoint  of dipole


d    d  
   F      F 
2
  2



 
 
   d
  1
d
1

   QEext      QEext   Q d  Eext  Q d  Eext
p
2
2
  2
 2

 

 Q d  Eext but p  Qd



p  Qd

Eext  Eo xˆ
 
Thus: N net  p  Eext



Eext  Eo xˆ
And: N net  pEext sin   pEo sin 

n.b.: N net points in direction dˆ  xˆ

 









(Use right-hand rule for -product!!!)
12
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I

Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede



The torque N net acting on the electric dipole (with pure electric dipole moment p  Qd )



due to uniform / constant electric field Eext  Eo xˆ acts such that p lines up parallel to Eext .
 
The polar molecule is free to rotate, and will do so until p  Eext  Eo xˆ , as shown below:

E-field-aligned polar molecule / electric dipole moment p :
+
+
+

Eext


p  Qd

Eext
+
+
+



Eext
−
−
−
−
 Eo xˆ −
−
−
+

Eext


n.b. when polar molecule / electric dipole moment p is aligned with Eext , note that the
 
 


net torque on the dipole, N net vanishes: N net  Qd  Eext but when d  Eext the cross product

 
vanishes! N net  0  pEo sin  because   0 when p  Eext   o xˆ.
We have shown (P435 Lect. Notes 8) that the potential energy U p of a pure electric dipole


p in an external electric field Eext is:
 
W  P.E.   p  Eext   pEext cos 
Work/Potential Energy vs. Angle Θ of Electric Dipole In External Electric Field:
+pEext
W(Θ)=P.E. (Θ)
(Joules)
0
−pEext
0

p

N ext

Eext
 
 0, p  Eext
90o
(π/2)

p
Θ
180o
(π)
 
p  Eext

Eext

p

Eext


p anti-  Eext
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
13
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
What happens to polar molecules (and / or pure physical electric dipoles)

when they are placed in a non-uniform externally-applied electric field Eext ?

1 Qs
rˆ  polar molecule in E-field associated with a point charge, −Qs :
e.g. Eext  
4 o r
+Q

1 Qs
Eext  
rˆ
4 o r

r


p  Qd
−Qs

r


Eext  @ dipole 
θ
(local to dipole)
–Q
 
 
 
 
Force on –Q: F  r   QEext  r 
Force on +Q: F  r   QEext  r 
 
 
But here:
Eext  r   Eext  r  !!!

 
 
 
 
Net Force on dipole: Fnet  F  r   F  r   QEext r   QEext  r 

 
 
Fnet  Q Eext  r   Eext  r 

 
 


Fnet  QEext where Eext  Eext  r   Eext  r 
  
For polar molecules d  r  r  few Ǻngstroms – typical atomic distance scale!!!


 



  

Thus, since d is so small, here we may use: Eext  d  Eext



Exext  Exext d


E yext  E yext d


Ezext  Ezext d
See Griffiths

d
equation 1.35

d
  


  
 Fnet  QEext  Qd  Eext  p Eext


  
Fnet  p Eext



n.b.




Fnet  mdipole adipole (Newton’s 2nd Law of Motion)

 If external field Eext is non-uniform, a net force exists on an electric dipole / polar molecule
 
which is proportional to the spatial gradient of the externally applied electric field Eext which
causes the dipole / polar molecule to accelerate!
14
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
Suppose we have a polar liquid – e.g. water (H2O) with pmol  pH 2O  6.1 1030 coulomb-meters
permanent electric dipole moment for each / associated with each water molecule.
Why don’t polar water molecules spontaneously align with each other, thereby minimizing their
overall energy?? (e.g. like magnetic dipoles (atomic) inside a permanent magnet)
Consider a universe in which there are only two such polar molecules (and nothing else).
Then the overall/total potential energy (by the Principle of Linear Superposition) is:
  
  
W p1 p2  P.E. p1 p2  U p1  U p2   p1  E2  r1   p2  E1  r2 

E -field of dipole2

@ dipole1  r1 

E -field of dipole1


@ dipole2  r2 (  r1 ) 

 
E -field of dipole2 (@ dipole1): E2  r1  


r2   r1
 

Define: r  r1  r2
1


 3 p2 rˆ1  rˆ1  p2 
3 
4 o r

 
1 1


E -field of dipole1 (@ dipole2): E1  r2  
 3 p1 rˆ2  rˆ2  p1 
3 
4 o r

p1
1


E2  @ r1 
 
r1  r



r2  r1  r

p2


E1  @ r2 
  
  
 W p1 p2  P.E. p1 p2   p1  E2  r1   p2  E1  r2 

1 1
1 1


 


 
 3   3 p2 rˆ1  p1 rˆ1   p1  p2  
 3   3 p1 rˆ2  p2 rˆ2   p1  p2 
4 o  r 
4 o  r 
 

Now: r  r1   r2
and
 
 
p1  p2  p2  p1
1 1


 


 
 3   3 p2 rˆ  p1 rˆ   p1  p2    3 p1 rˆ  p2 rˆ   p1  p2 
4 o  r 
2 1


 

 3  3  p1 rˆ  p2 rˆ   p1  p2 
4 o  r 
W  P.E.  
Then:




Notice that: W min / P.E.min occurs when p1 is anti-parallel to p2 , i.e. when p2   p1
 


and when: p1 , p2  to rˆ (or anti -  to r̂ ) then → p1 rˆ  0 and p2 rˆ  0.
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
15
UIUC Physics 435 EM Fields & Sources I
p1 p2

Wminp1 p2 P.E.min
1 1
  p1 p2
2 o  r 3 
Fall Semester, 2008
Lecture Notes 9
when we have:
Prof. Steven Errede


p1  Qd1

r

r


p2  Qd 2
So let’s put in some actual numbers:
2
 o  8.85  1012 C
F
m
N  m2


A typical intermolecular separation distance for the water (H2O) molecule is
rH 2O  H 2O ~ 10Ǻ = 10 x 10−10 m = 10−9 m = 1 nm
p1  p2  pH 2O  6.11030 Coulomb  meters
p1 p2
Wminp1 p2 P.E.min

2
1
1
6.1 1030   6.69  1022 Joules

3
12
2  8.85  10 109 
 Very small energy!!!
Now liquid water (room temperature T  300K) also has thermal energy associated with it.
Boltzmann Kinetic Theory: – get a contribution to thermal energy of ½ kBT for
a.) each kinetic degree of freedom
that are operative at that
b.) each rotational degree of freedom
absolute temperature T.
c.) each vibrational degree of freedom
kB = Boltzmann’s Constant = 1.38 x 10-23 Joules/K
For water (H2O) @ T = 300K:
v x , v y , v z  3 k BT
a.) kinetic:
2
b.) rotational: I  , I   2 k BT
2
c.) vibrational: none  0 (i.e. no quantum vibrations of water molecule H2O
no H-O-H vibrational excitations @ T = 300K)
5
H 2O
 k BT .
Thus, for a single H2O molecule: Wthermal
2
Since we’re considering two H2O molecules (here), then (using superposition principle):
H 2O
H 2O
Wthermal
 5k BT  5  1.38  1023  300  2.07  1020 Joules
TOT   2Wthermal
H 2O
Wthermal
TOT   2.07 1020 Joules  Wminp1 p2  6.69 1022 Joules
H 2O
Ratio: Wthermal
TOT  Wminp1 p2  30.9
H 2O
p1 p2
Sum: WHTOT
 Wthermal
 Wmin
 2.07 1020 J  6.69 1022 J  2.00  1020 Joules
2O
16
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
Thus, we see that the internal energy of water (@ the microscopic level) is dominated by thermal
p1 p2
energy at T  300K (room temperature). In order for the dipole-dipole term Wmin
 to actually
p1 p2
H 2O
dominate, the water temperature would have to be TH 2O  9.7 K for Wmin
to dominate WTOT
0.

So, typically (but not always) for polar materials (gases / liquids / solids), macroscopic amounts
of individual molecules   N A  6 1023 molecules/mol  the thermal energies overwhelm the
dipole-dipole interaction energies at room temperature.
No net macroscopic alignment of permanent microscopic electric dipoles occurs at T  300K.
When one places either non-polar or polar materials (at T  300K) in externally applied electric

field Eext , nevertheless, there is a net, partial macroscopic alignment of either induced
microscopic electric dipole moments (for non-polar materials) or permanent electric dipole
moments (for polar materials).
A“snapshot” of a dielectric material: (with either induced atomic / molecular electric dipole
moments or permanent electric dipole moments at T  300K)

No external applied field  ext  0
Random orientation(s) of induced / permanent
electric dipole moments at microscopic scale atomic or molecular at T

300K.

With external applied field  ext  0
Partial alignment of induced / permanent electric dipole
moments at microscopic scale – atomic or molecular at
T

300K – when remove

Eext ,
alignment rapidly disappears.
Both pictures change from one instant to the next:
- Thermal fluctuations (fluctuations in thermal energy density)
- Quanta of thermal energy (EM energy – virtual photons) “traded” between constituents
(atoms / molecules) at microscopic scale.
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
17
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
 
Electric Dipole Moment Per Unit Volume of Material (a.k.a. Electric Polarization)   r 
For a single atom (or molecule or physical dipole), we have seen that the electric dipole moment
 
 
of the charge distribution is given by: p  r    r   r   d 
v
For atoms / molecules, the integration volume v is associated with the space “occupied” by that
atom / molecule, but it essentially can be over all space.
For a macroscopic sample of matter (gas, liquid or solid consisting of many, many atoms,
molecules (i.e. a sample containing N A ~  1023  molecules) it makes sense that we can define
a quantity / parameter known as the electric dipole moment per unit volume (also known as the
 
electric polarization)   r  :
Electric Dipole Moment Per Unit Volume (a.k.a. Electric Polarization):


 pimol  ri 
N
 
 r  
i 1
Volume Element d


Q
d
 i imol  ri 
N

i 1
Volume Element d


Where the infinitesimal volume element d is centered on the vector r , and the ri vectors
associated with each atom/molecule are contained within the infinitesimal volume element d .
 1 N 
Thus: r   ri .
N i 1
 
Note that the physical interpretation of   r  is that it is the macroscopic average of the
microscopically-summed-over electric dipole moment per unit volume. Important!!!
 
What are the S.I. units of the electric polarization,   r  ? From above, we see that they are:
Qlength
 length 
3

Q
length 2

Coulombs / meter2
 
Thus,   r  has the same S.I. units as surface charge density,  !
Note that if macroscopic dielectric / insulating material has no net electric charge associated with
N


it, i.e. dQ  r    Qi  ri   0 in each of the infinitesimal volume elements, d centered on the
i 1

vector r , then the monopole moment  M 0  of the charge distribution associated with the entire


dielectric material, integrating over the whole volume: M 0  QTOT   dQ  r       r  d   0.
v
18
v
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede

Thus, any potential V  r  that arises due to (microscopic, and thus macroscopic) polarization of
the dielectric material will be due to the next most important term in the multipole expansion of
the potential, namely the (macroscopic) electric dipole moment associated with that material!





We already know the potential Vdipole  r  (and corresponding Edipole  r   Vdipole  r  ) associated

with a single, microscopic pure electric dipole moment p is:

Vdipole  r  

1  p  rˆ 


4 o  r 2 

The electric dipole moment p  r   is located

at the source point r 
For a macroscopic dielectric material, an
infinitesimal volume element d  has a

net electric dipole moment p  r   associated
with it of:
 
 
 
p  r
 
p  r      r   d  (i.e.   r   
)
d 
 
 

The infinitesimal contribution to the potential dVdipole  r  due to this p  r      r   d  is thus:
 
 
1  p  r   rˆ 
1    r   rˆ 

dVdipole  r  

 d 


4 o  r 2  4 o  r 2 
Then integrating this expression over the volume v of the dielectric material, we have:
 
   r   rˆ 
1


Macroscopic potential due to
Vdipole  r    dVdipole  r  

 d  
2


v
v
a polarized dielectric material
4 o  r

Now:
 1   1 
         

r
 r  r 
rˆ  rˆ  rˆ 
 2   2
r
r  r
 

r  r 

r
 3  3
r
r  r

n.b.  refers to differentiation with
respect to source coordinates only!!!
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
19
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
  1  rˆ  rˆ  rˆ 
1

Thus, using     2    2 we can write Vdipole  r  
4 o
r r
r  r

Vdipole  r  
1
4 o

Prof. Steven Errede
 
   r   rˆ 
v  r 2  d  as:




1
   r   r  d 
v
{n.b. In practice, it is often extremely difficult to explicitly carry out this integration,
except for a
 
few cases with very simple geometry and / or simple polarization(s), e.g.   r   = constant.}
 
Note that in general that   r  (= electric dipole moment per unit volume / a.k.a. electric
polarization) is never known a-priori (i.e. before the fact), but is instead inferred a-posteriori
(i.e. after the fact) from known, physically measurable quantities / parameters.
How is this accomplished???
 
 fA 
1
f  
r
Note that:
Let:
i ntegrand in
above integral

  


f   A  A f
1
and:
 
r  r
Then:

Vdipole  r  
   1
1

 r    d  


v
4 o
4 o
r
1
 
 
 
Then: Af   fA  f  A
 
A

    r  
1
v '  r  d   4 o



 1     
v  r     r  d 


use the divergence theoremconvert to surface integral

Vdipole  r  
1
 1    
    r nˆ dA 



S
4 o
4 o

 r  

1


 B  r 
Thus:

1
4 o

S

1
   B  r   dA
r
+
 1     
  r  d 
v '  r  


1
4 o


  B  r  

v'

1
   B  r   d 
r
= Potential due to (bound)
surface charge density
 

 B  r      r  nˆ 
= Potential due to (bound)
volume charge density
  

 B  r      r  
(SI units: Coulombs/m2)
(SI units: Coulombs/m3)
n̂ = Outward-pointing
unit normal vector on
bounding surface S 
 

The bound surface charge density  B  r      r  nˆ  and bound volume charge density
  

 B  r      r   are called bound charge densities because these charges cannot move
– they are certainly not free charges – because they are bound to atoms / molecules.
20
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
  
 


Then, using these two relations,  B  r      r  nˆ  and  B  r      r   , we see that:
1
 1     
 1     
  r dA 


     r  d 

4 o S   r 
4 o v '  r 


  B  r  
  B  r  
1
1


 dA +

 d 
4 o  S   r 
4 o v '  r 

Vdipole  r  
1




Physically what this formula / above expression says is that the electric potential Vdipole  r 


 


and hence E-field Edipole  r   Vdipole  r  due to the macroscopic electric polarization   r   of
dielectric material is formally / mathematically
equivalent to that of a potential associated with a
 

bound surface charge density  B  r      r  nˆ  plus a contribution to the potential associated
  

with a bound volume charge density  B  r      r  
So instead of having to integrate over contributions from infinitesimal-sized point-like atomic
/ molecular dipoles contained within the volume v of the dielectric material, we simply find /
 

determine the bound charge densities  B and  B , and then calculate V  r  and E  r  the same
exact way(s) we have already done e.g. for the free charge densities  f and  f !!!

Consider an infinitesimal, single volume element d   r   inside e.g. a uniformly polarized
 
block of dielectric material - i.e. a dielectric with uniform / constant polarization   r     o xˆ
e.g. a block of dielectric material (such as plastic) inserted between the plates of a parallel plate
capacitor:
n.b: The number density of molecules is nmol = # molecules/unit volume (i.e. # molecules/m3).
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
21
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
Suppose the mean charge displacement, i.e. <charge displacement> of molecules, going from an

unpolarized state “
” to a polarized state, e.g. “
” is an average distance of d  dxˆ .
d
Then the (average/mean) amount of charge displaced in a volume element d  is (see above
figure on previous page) is:
 

= Infinitesimal surface area element of one
dQ  nmol Qd   nmol Qd dS  where: dS   nˆ dS   side of the infinitesimal volume element d 

 nmol Qd nˆ dS 
and: nmol = # molecules/m3
dQ = the amount of charge displaced in the polarization process, crossing an area element dS  .
Now we re-arrange / rewrite this a bit:


Since: p  Qd = average/mean electric dipole moment per molecule in the volume element d .


Then: dQ  nmol Qd nˆ dS   nmol p nˆ dS 
 
 
 
nmol p  r      r   {Since the electric polarization,   r   is defined as the (macroscopic)
But:
electric dipole moment per unit volume!!!}
 
 
  r    nmol p  r   where nmol = number density of molecules (# molecules/unit volume)
i.e.:
 
and p  r   = electric dipole moment per molecule, in volume element d 
 
 
{Check units/dimensions of   r    nmol p  r   = (#/m3 * Coulomb-m = Coulombs/m2 – Yes!!!)}
Thus we see that:
 
 
 
dQ  nmol Qd  r  nˆ dS   nmol p  r  nˆ dS     r  nˆ dS   dQB
= bound charge QB on the surface area dS  of one side of the infinitesimal volume element d  .
 
If we now integrate dQB    r  nˆ dS  (= amount of charge displaced in the polarization
process) over all 6 surfaces of the infinitesimal volume element d  (for simplicity, assume d 
to be a cube of side dS  ), and for again for simplicity’s sake, let us assume that the electric
 
polarization,   r   has negligible variation (i.e. is a constant) over the infinitesimal volume
 
element, d  such that e.g.   r     o xˆ .
Then:
 
 
 

QBnet (of d  @ r )    r  nˆ1dS     r  nˆ2 dS     r  nˆ3 dS 
 
 
 
   r  nˆ4 dS     r  nˆ5 dS     r  nˆ6 dS 
where:
nˆ1   nˆ4  xˆ
22
and
nˆ2  nˆ5  yˆ
and
nˆ3   nˆ6  zˆ
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
 
Then if:   r     o xˆ , and
 
 
 

QBnet (of d  @ r )    r  nˆ1dS     r  nˆ2 dS     r  nˆ3 dS 
 
 
 
   r  nˆ4 dS     r  nˆ5 dS     r  nˆ6 dS 
With nˆ1   nˆ4  xˆ
nˆ2  nˆ5  yˆ
and
nˆ3   nˆ6  zˆ , then:
and
QBnet  Q1  Q2  Q3  Q4  Q5  Q6


  o dS  xˆ  xˆ  xˆ  yˆ  xˆ  zˆ  xˆ  xˆ  xˆ  yˆ  xˆ  zˆ  0
Thus we see {here} that the net charge displacement across the entire surface S  enclosing the
infinitesimal volume element, d  is QBnet  0 , because Q4  Q1 and Q2  Q3  Q5  Q6  0 .

(Note that this is true only for the case of uniform/constant polarization,  throughout d 

i.e. in general, QBnet  0 for arbitrary polarization  throughout d . )
  
 

If   r     o xˆ , then (obviously) the bound volume charge density  B  r      r    0
  
 


because:  B  r      r     o xˆ   o  xˆ  0.



 
n.b. This result also holds for macroscopic volumes: QB     r  dA
S

i.e. there will be no bound volume charge density,  B  r    0 if the electric polarization
 
 
(electric dipole moment per unit volume)   r   is uniform / constant, e.g.   r     o xˆ
throughout the macroscopic dielectric medium.
 
If   r   is not uniform in the macroscopic volume v ,

 
 
then: QB     r  dA     r  nˆ dA  0
S
S
 

Note that if dQ  r      r  nˆ dS   0 exists on the entire/total surface dS  associated with a
volume element d  then this is also = to the net charge that flows out of (or into) the
infinitesimal volume element d  across/through the total surface area element dS  when the
dielectric material is polarized.
 Then a net charge –dQB must remain in the infinitesimal volume d , so then:

 

  

  
v  B  r d   QB   S    r dA  v    r d    B  r     r 
Thus, we see that  B and  B are (bound / induced / polarization) surface and volume charge
densities, respectively.
Synonyms of each other here
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
23
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
More on the Physical Interpretation of the Bound Charges:
We have seen that  B and  B represent real, physical accumulations of electric charges on
the surfaces and/or in the volume of dielectric materials, respectively.
However, the primary distinction associated with these two physical quantities is that
QB    B dA and QB    B d  are associated with bound charges – i.e. charges that are bound
S
v
to atoms / molecules – they are not free surface / volume charge densities.
c.f. bound charges  B and  B with free charges  free and  free (which can move freely on
bounding surfaces ( S  ) and volumes ( v )) .
Consider
a fully-polarized, very long dielectric rod of radius R with uniform polarization

   o zˆ ║ to the ẑ -axis of the polarized dielectric rod:

   o zˆ
R
QB−
R
ẑ
+QB+
d
L

p  QB d zˆ

   o zˆ
Carefully cut out a very thin disk
e.g. of thickness d  10Ǻ = 10−9 m
nˆ2   zˆ
nˆ1   zˆ
2
The thin disk has cross-sectional area A = πR :
d

ẑ
QB−
QB+
QB−
QB+
d

We can replace the polarization,  by bound surface charges QB− and QB+!


Thus, the thin disk has (macroscopic) electric dipole moment p  QB d  QB d zˆ
However, the macroscopic electric dipole moment of the disk is also:


 
p    Volume of disk   *( Ad )    R 2 d    o  R 2 d  zˆ  QB d zˆ

(Since the macroscopic polarization  = macroscopic electric dipole moment per unit volume!!!)
Therefore:  o  R 2 d   QB d or : QB   o R 2   o A

Or:
 B   QB  A  QB    B  A  nˆ1 A   o zˆ  zˆ  A   o A where: nˆ1   zˆ

 B   QB  A  QB    B  A  nˆ2 A   o zˆ  zˆ  A   o A and: nˆ2   zˆ

but:  B  nˆ




Thus we see that: QB   QB    o A and:  B    B  
24
QB 
Q
  B  o
A
A
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
oblique
A
Aend
If we instead make an oblique cut in the long, polarized rod:
cos 
n̂
θ

zˆ,    o zˆ
cross-sectional area A = R2
oblique
A
Aend
cos 

oblique
 B   end   nˆ   o zˆ nˆ   o cos 

oblique
QB   oblique
   B  end
 Aendoblique   o cos   A
end
cos 
   A  same result !
o
If the polarization is non-uniform, then  accumulations of bound charge within the polarized
dielectric material as well as on the surface of the dielectric material.
 
Diverging Polarization   r  :
 
R
e.g.   r    o   rˆ
r
  

 B  r     r   0
 

 B   r  R     r nˆ
P
r R
 
   r rˆ
r R
QB    B  Asphere  4 R 2 B 
R
We know by charge conservation that:
QB   QB   4 R 2 B 
  

QB     B  r   d      r   d 
v
v
Griffiths Example 4.2:
Determine the electric field associated with uniformly polarized sphere of radius R

   o zˆ  constant
ẑ
nˆ  rˆ

   o zˆ

ŷ
x̂
Here:
  


Volume bound charge density:  B  r     r    o zˆ  0
 
  o zˆ rˆ   o cos 
Surface bound charge density:  B  r  R     r nˆ
r R
z r  cos 
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
25
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede
In Griffiths Example 3.9 we obtained for   k cos  on a sphere of radius R that for k   o :
 o
 3 r cos 
 o
V r   
3
  o  R  cos 
 3 o  r 2 
r  R
r  R
Now z  r cos  , and inside the sphere  r  R  , the electric field intensity is:





Einside  r  R   V  r  R   
  o zˆ for r  R
3 o
3 o
Note that outside the sphere  r  R  , the potential V  r  R  is identical to that
of a point electric dipole at the origin!
 4
4
 
p    Volume of sphere   R 3   R 3 o zˆ
3
3
Electric dipole moment per unit volume

o  R3 
1 4 R 3
1  p rˆ 
 o  zˆ rˆ  
V r  R 
  cos  


3 o  r 2 
4 o 3r 2
4 o  r 2 


Then use Eoutside  r  R   V  r  R  to obtain electric dipole field intensity outside the sphere
(see P435 Lecture Notes 8).


1
Lines of Eoutside  r  R  and Einside  
 o zˆ :
3 o
ẑ

Note the discontinuity in the normal component of E @ r  R !!
This is due to the existence of surface bound charge density  B on surface of the sphere!!
26
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2008
Lecture Notes 9
Prof. Steven Errede

The Polarization Current Density J B
Suppose we have an initially un-polarized dielectric, which we then place in an external electric


field Eext – but one which varies extremely slowly with time. Then the (induced) polarization 
in the dielectric will also vary extremely slowly with time, simply tracking the externally-applied
electric field.
Using electric charge conservation (i.e. the using the empirical fact that electric charge cannot be
created, nor can it be destroyed), we obtain the so-called continuity equation for bound charge in
a dielectric, which we simply give here (for now – we will discuss in more detail, in the future):

 


J B  r  dA     B  r   d 
S
t v
  

Using the divergence theorem and  B  r      r   we obtain:


  
   r, t 

   

v  JB r , t d    t v  B  r d    t v   r , t  d   v  t d 
 
The above volume integral equation must be valid for arbitrary volumes, v (e.g. for ≈ a few
molecules in the dielectric → entire dielectric). Therefore the integrands in the volume integrals
must be equal to each other at each/every point in space, and at each/every instant in time:
 
 
  r , t 
Continuity equation for
J B r , t 

bound electric charge
t
 
SI units of Polarization Current Density, J B   t = Coulombs/m2-sec = Amperes/m2
(1 Ampere of current  1 Coulomb / second of charge passing through a  imaginary plane)
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
27