3). Dielectric phenomena
• Microscopic picture – change in charge density when
field is applied
r(r) Electronic charge density
No E field
E
E field on
Dr(r) Change in electronic charge density
r
-
+
Note dipolar character
Dipole Moments of Atoms
• Total electronic charge per atom
Z = atomic number
 Ze 
• Total nuclear charge per atom
 Ze 
r
el
all space
r
(r )dr
nuc
all space
(r )dr
• Centre of mass of electric or nuclear charge distribution
r
• Dipole moment Zea

Ze a  Ze r
nuc
 r
el
   r
el/nuc

r
el/nuc
all space
(r ) r dr
r
el/nuc
all space
nuc
(r )  r el (r ) r dr
all space

r
el
all space
(r ) r dr if r
nuc
0
(r )dr
Electrostatic potential of point dipole
• +/- charges, equal magnitude, q, separation a
2
• axially symmetric potential (z axis)
a
 
r2  r 2     a r cos
2
z
q+
a/2
2


a a
2
 r 1    cos 
  2r 

r


r+
r
r-
x
1
 a a
 2
1
 r 1    cos 
  2r 

r
r


1 a
  2 cos
r 2r
qa cos p cos
 r  

2
4o r
4o r 2
2
p
a/2
q-
q  1 1
 (r ) 
  
4o  r r- 
Equipotential lines: dipole
• Contours on which electric potential is constant
p cos
 r  
4o r 2
• Equipotential lines perp. to field lines
Field lines: point dipole
• Generated from E  – grad 
r
k



    
 Cart.   , , 
 x y z 
j
i

 Sph.Pol.
 (r ) 
i, j, k 
 1 
1   ˆ ˆ ˆ
 (r, , )
  ,
,
 r r  r sin  
cos 
4o r 2
p
p  2cos  sin  
  (r, ) 
, 3 , 0

3
4o  r
r

← NB not a point dipole
Insulators vs metals
• Insulator
– Localised wave functions
• Metal
– Delocalised wave functions
No E field
E field on
Polarisation
• Polarisation P = dipole moment p per unit volume Cm/m3 =
Cm-2
p
E
• Mesoscopic averaging: P is a constant vector field for a
uniformly polarised medium
P
E
• Macroscopic charges sp in a uniformly polarised medium
sp
= ___?
dS
-
E
P
+
E
Depolarising electric field
• Depolarising electric field EDep in uniformly polarised ∞ slab
-
EDep = sP/2o +sP/2o
EDep = -P/o
EDep
+
• Macroscopic electric field EMac= E + EDep
EMac = E - P/o
E
-
EMac
+
E
Relative Permittivity and Susceptibility
• EMac = E – P/ o = (splates – P)/ o in magnitude
•
o E = o EMac + P
P = o cE EMac
•
o E = o EMac + o cE EMac = o (1 + cE)EMac = oEMac
– EMac = E /
– E =  EMac
• Dielectric constant (relative permittivity)  = 1 + cE
– Typical values: silicon 11.8, diamond 5.6, vacuum 1
• Dielectric susceptibilty cE
Polar dielectrics
•
•
•
molecules possess permanent dipole moment
in the absence of electric field, dipoles randomly oriented by
thermal motion
hence, no polarisation.
+
_
+
_
e.g. HCl and H2O
…..but not CS2
no net dipole moment
+
_
+
_
E
zero field, random
net P=0
preferential alignment
but P  Np
Effect of orientation on net field
+
Eappl
_
Edip
+
Eappl
_
•
Effect of alignment is to reduce the net field
•
Tendency to align is opposed by thermal effects
•
Balance is determined by Boltzmann statistics
•
Key factor is ratio of the potential energy of the dipole (U)
to the temperature (T), which enters as exp(-U/kT)
Potential energy of dipole in E field
•
Potential energy U (U c.f. W from before) when charge density
of molecule (r) is in slowly spatially varying external potential
(No factor of ½ c.f. W)
 ρ(r ) (r )dr
U
+
all space

 (r )   (0)  r. (0)  ...
U
 ρ(r ) (0)  r. (0)  ... dr
all space
 ρ(r )dr  q
 r ρ(r )dr  E.p
all space
U  q (0)  p.E  ...
_
If q = 0, leading term is –p.E = -pEcos
all space
 (0).
E
Number Distribution function
•
•
•
Angular distribution function: N() (no E field), N’() (E field)
Number of dipoles oriented between  and +d : N()d
Total number of dipoles N
E||z
d


d  sin  d d
N( )d 2 sin  d

no E field
N
4
N( )d  N sin  d /2
N' ( )d  2 A sin  d e U/kT
e U/kT  1 U/kT
if U  kT
E field on
N' ( )d  2 A sin  1 U/kT  d
 2 A sin  1 pE cos/kT  d
Number Distribution function
• Total number of dipoles N
π
N   N' ( )d
0
2 A pE sin cos d
N' ( )d  2 A sin d 
kT
π
 N' ( )d  4 A
0
A  N / 4
Susceptibility of polar dielectric
• Molecules acquire induced dipole moment through:
- reorientation
- polarisation of molecular charge (polar or nonpolar molecules)
π
Pz reorientat ion   N'   p cos  d
0
N
 pEcos 

2 sin  1
 p cos d

4 0
  kT 
π
Pz polarisation
N p 2E
Np 2

 P
E
3kT
3kT
  o N E

p2 

P  c E o E  c E  N   
3 okT 

cE
Np 2
slope
p
3 ok
intercept  N
1/T
The Langevin Equation
When U/kT is not small, integration of N()d yields:
P

 pE  kT 
P  Np coth


 kT  pE 

Plotting P vs pE/kT
shows two distinct regimes:
Np

(1) High E, low T: all dipoles aligned:
pE/kT
P Np
(2) Low E, high T: small U/kT approximation:
Np2
P
E
3kT
Clausius-Mossotti equation
• Relationship between r and polarisability density Nincluding
local fields
• Neglected local field for polar dielectrics (dilute gases)
• Each molecule, atom, etc. located in spherical cavity
• C-M local field is external field + field due to polarisation
charges on cavity surface Eloc = E + Epol
• rpol = P.dS

dS
P


d
dS

cos- - cos 
R
+
+
+
-
+
P, E
P.dS= - P dS cos
ring area element = 2Rsin Rd
Clausius-Mossotti equation
• Charge on ring area element
- P dS cos-o cE E 2Rsin Rd cos
• Contribution to field at centre of cavity from rpol on ring
o cE E 2Rsin Rd cos/4o R2) = cE E sin cosd /2
• Field || P due to all charge on cavity surface Epol= cE E/3
• Local field Eloc = E + Epol= (1+ cE/3)E
• P = o N Eloc = o N (1+ cE/3)E (in cavity)
• P = o cE E
(in bulk)
• N (1+ cE/3) = cE
• N = cE / (1+ cE/3)
N/3 = (r – 1)/(r + 2) since r = 1 + cE
Non-uniform polarisation
• Uniform polarisation  surface charges only
P
E
P
-
+
• Non-uniform L polarisation  bulk charges also
+
-
+
-
Displacements of positive charges Accumulated charges
Non-uniform polarisation
• Box with origin of local axes at (x,y,z), volume DxDyDz
• Charge crossing area dS = P.dS
ẑ
• Charge entering LH yz face
ŷ
Px ΔyΔz
• Charge exiting RH yz face
Px 
Px
x̂
(x,y,z)
(x+Dx,y,z)
r
 .j
t
P
jpol 
t
Px
Δx
x
Px


P

Δx
 x
 ΔyΔz
x


• Net charge entering box
Px

ΔxΔyΔz
x
• Total charge including zx and xy
pairs of faces
 Px Py Pz 
 ΔxΔyΔz  .P volume 
 


y z 
 x
 ρpol bulk  .P
Electric displacement D
• What happens when a charge is added to a neutral dielectric ?
Added (free) charge
Polarisation (bound) charge
response of dielectric to added charge
• Two types of charge:
• Those due to polarisation
(bound charges) ρb  ρpol bulk  ρpol surf ace

• Those due to extra charges (free charges) ρf
(charge injection by electrode, etc)
ρt  ρf  ρb
• Total charge
Electric displacement D
• Gauss’s Law .E  ρ t  1 ρ f  ρb   1 ρf  .P 
o o
o
ρf   o.E  .P  . oE  P  .D
D   oE  P   oE  cE oE   o 1 cE E   o rE
• Displacement: a vector whose div equals free charge density
• Units: C·m-2 (same as P)
• D relates E and P
• D = oE + P is a constitutive relation

• Can solve for D field and implicitly include E and P fields
Validity of expressions
•
•
Gauss’ Law for E, P and D
relation D = oE + P
Limited validity: Expressions involving r and cE
Always valid:
•
Have assumed that cE is a simple number: P = o cE E
only true in LIH media:
•
Linear: cE independent of magnitude of E
interesting media “non-linear”: P = cE oE + c2E oEE + ….
•
Isotropic: cE independent of direction of E
interesting media “anisotropic”: cE is a tensor (generates
vector)
•
Homogeneous: uniform medium (spatially varying r)
Boundary conditions on D and E
• Simplest example – charged capacitor with dielectric
+
-
E
Emac=E/r
-
D =  oE
E
+
D =  oE + P
D =  oE
= orEmac
• D is continuous ┴ boundaries (no free charges there)
• E is discontinuous ┴ boundaries
Boundary conditions on D
•
We know that .D  r f
 .D dv   D.dS  r
v
•
•
S
f
dv
v
Absence of free charges at boundary  D.dS 0
S
D1 cos1 S – D2 cos2 S = 0
1
D1 cos1 = D2 cos2
2
D1┴ = D2 ┴
Perpendicular component of D is continuous
1
(E1,D1)
2
Presence of free charges at boundary  D.d S   r f dv
S
v
D1 cos1 S – D2 cos2 S = S sf
D1┴ = D2 ┴ + sf
Discontinuity in perpendicular component of D is free
charge areal density
(E2,D2)
S
Boundary Conditions on E
• We know that for an electrostatic E field  E.d  0
C
• E and D are constant along the horizontal sides of C
in regions 1 or 2
1
B
dℓ1
(E1,D1)
E
.d


E
sin


 1 1 1 1 1
1B
A
A
A
 E2 .d 2  E2 sin  2  2
B
2
C
2
(E2,D2)
dℓ2
• Sides of C thin enough to make no contribution
 E.d  E sin 
C
1
1 1
 E2sin 2 2  0
E1sin1  E2sin 2
E1||  E2||
• Parallel component of E is continuous across boundary
Interface between 2 LIH media
LIH  D = roE
E and D bend at interface
1
σ f  0  D1  D2  D1cos1  D2cos 2
1
  r1 oE1cos1   r2  oE2cos 2
2
E||1  E||2  E1sin1  E 2sin 2
E1sin1
E 2sin 2
tan 1 tan  2




 r1 oE1cos1  r2  oE 2cos 2
 r1
 r2
tan 1  r1


tan  2  r2
2
Energy of free charges in dielectric
o
1
 (r )r (r )dv 
• In vacuum W 

2 all space
2
   dv 
2
all space
o
2
2
E
 dv
all space
• Assembling free charges in a dielectric
1
1
W    (r )r f (r )dv    (r ).D(r )dv
V  vol containing all r f
2V
2V
.D  D.  .D
1
W   .D  D. dv
2V
.D  .D  D.
1
1
  D.dS -  D. dv
2S
2V
 o r
1
W
D.E dv 

2 all space
2
2
E
 dv
all space
Method of Images
Derives from Uniqueness Theorem: “only one potential
Satisfies Poisson’s Equation and given boundary conditions”
Image charges reproduce BC
 or  specified
r
.E 
E  
o
r
2
 .     Poisson' s Equation
o
  2  0 ( r  0 within boundaries ) Laplace' s Equation
Can replace parts of system with simpler “image” charge
arrangements, as long as same boundary conditions satisfied
Method exploits:
(1) Symmetry
(2) Gauss’s Law
Basic Image Charge Example
Consider a point charge near an infinite, grounded,
conducting plate:
induced -ve charge on plate; potential zero at plate surface
Complex field pattern, combining radial (point charge +Q)
and planar (conducting plate) symmetries, can also be
viewed as half of pattern of 2 point charges (+Q and -Q)
of equal magnitude and opposite sign!
+Q
+Q
-Q
Basic Image Charge continued
Arrangement is equivalent because it keeps the same
boundary condition (potential zero on plate and zero
potential on the median line).
Point charge -Q is located same distance behind,
like an image in a plane mirror.
The resulting field is easy to calculate (vector sum of
fields of 2 point charges of equal and opposite sign)
Field lines must be normal to surface of conductor
Also easy to calculate the induced -ve charge on plate!
Distribution of induced charge
s
Induced charge is related to
E  ind
the outward E field at the surface:
o
E+
Find E using image charge


E E 
Q
Q
E

4or 2
2QD
E2
cos 
2
4or
4or 3
 s ind  o E
+Q

r
E-
D
2QD
 s ind   oE 
(-ve and varying with r)
3
4r
s ind negative since field lines point inwards
-Q
ds
Total induced charge
r
Introduce parameter s
s  r 2  D2
s
D
sind has azimuthal symmetry:
consider elemental annulus, radius s, thickness ds
Qind 

s
0

ind
2s ds   
0
2QD
4 s  D
2
2

3
2
2s ds

QD

2


0


d s2  D2 
1

  Q

QD
3
1
2
2 2
2
2 2 

s

D
s

D


 0

s
Total induced charge: implications
2 conclusions from the result:
Qind  Q
(1) Induced charge equals the negative of original point charge trivially true in this case only!
(2) Induced charge equals the image charge - generally true!
Consider Gauss’s Law, concept of enclosed charge
 E.dS 
S
Qenc
o
Must not try to determine E in the region of image charge!
In this case (behind infinite conductor) it is zero, which is
not the answer the image charge would yield
Point charge near grounded
conducting sphere
+Q
-
-
a
-
D
By comparison with previous example:
(1) Distance D to centre of symmetry, radius a
(2) Image (charge) location
(3) -ve induced charge predominantly on side facing +Q
(4) Boundary condition, zero potential on sphere surface
Expect image charge will be a point charge on centre line,
left of centre of sphere, magnitude not equal to Q, call it Q
Point charge near grounded
conducting sphere
Q
P2
Q
P1
b
D
Q distance b from centre, =0 at symmetry points P1 and P2
1  Q
Q 



  0 at P1 
4o  D  a a  b 
1  Q
Q 



  0 at P2 
4o  D  a a  b 
2
a
solving b 
D
and Q'   a
D
Qind  Q' by Gauss
Point charge near floating conducting
sphere
Q
Q Q
V
On its own, floating the sphere at V relative to ground
results in uniform +ve charge density over the surface.
In the presence of Q, induced -ve charge predominantly on
left; this complex system easily solved by 2 image charges:
2
a
a
Q  
Q at b 
and Q  4oaV at centre
D
D
Qind  Q  Q by Gauss
Point charge near isolated conducting
sphere
Q Q
Q
With no connection to ground, the sphere is at an unknown
non-zero potential ; easily solved by same 2 image charges:
the potential is still determined by Q but in this case, the
sphere is overall neutral: Q+Q=0

Q

Q

 a Q
D 

4oa
4oa
4oa
Qind  Q  Q  0 by Gauss
(same potential as if sphere was absent!)
Q
4oD
Point charge near LIH dielectric block
•
•
•
Q polarises dielectric and produces bound surface charge sb,ind
sb,ind = P.n = o(r-1)Eni Eni normal component of E inside
Q
sb,ind negative if Q positive
=1
D r
Eno
s
sb,ind
Eni
= r

s b,ind
  r  1  Q
D

  1

2
2 3/2


2 o


1
4

r
o s  D 


• Eno   4 2
2
o s D
s b,ind
  r  1  Q
D

  1

2
2 3/2


2 o


1
4

r
o s  D 


• Eni  
Q

D
4o s2  D2
Q

D
3/2

3/2
Point charge near LIH dielectric block
• Image charge for Eno
E+
Er

+Q
• Image charge for Eni
  r  1  Q
D
Eno   1

2
2




1
r

 4o s  D


3/2

E  rr 11 Q
2 r
Q
D

 r  1 4o s2  D2
E+
Er
 2rr1 Q
r

r

3/2
Point charge near LIH dielectric block
(see Lorrain,Corson & Lorrain pp 212-217)
Outside: remove dielectric block and locate image charge Q a
distance D behind
 1
Q  r Q
r  1
Inside: remove dielectric block and replace
original point charge

Q by Q
2
Q 
Q
r 1
Point charge near LIH dielectric block
contrasting E field patterns
conductor
dielectric
note dielectric distorted outside but radial inside