Download 4. Electric Fields in Matter

4. Electric Fields in Matter
4.1 Polarization
Insulators: All charge is attached to the atoms or molecules.
p  E
Point charge in a homogenously charged sphere
p  E,   3 ov
(v - Volume)
Molecules
p  ||E||   perpE perp
Polar molecules have a permanent dipole.
F  (d  )E
N  pE
p  E
4.2 The Field of a Polarized
Object
Polarization P (dipole moment per unit volume)
tells how strongly the atoms/molecules
are polarized and/or aligned with the electric field.
Np
P
v
Take a small volume v that contains, say,
N=1000 atoms.
Potential generated by the
microscopic dipoles:
V (r ) 
1
40

V
rˆ  P(r )
r
2
d '
Bound charges
V (r) 
1

40 S
 b (r' )
r
da'
volume charge density
surface charge density
1
40

V
b (r' )
r
d '
b    P
 b  P  nˆ ,
where nˆ is the normal on the surface.
Example 4.2
Electric field of a uniformly polarized sphere.
P
Vin 
z
3 0
Vout
rR
1 p  rˆ

4o r 2
4 3
p
RP
3
constant field
rR
dipole at the center of the
sphere
The field inside a dielectric
Deriving the expressions for the bound charges we considered
pure dipoles.
The real dielectric contains physical dipoles. The electric
field is much more complicated near the molecular dipoles.
The macroscopic field is the average over a small volume
that contains many molecules.
The average field of the pure and molecular dipoles is the same.
4.3 The Electric Displacement
Total charge
   f  b
Free charge (at our disposal)
f
Bound charge (induced, comes along)
b
Electric displacement (auxiliary field)
D   0E  P
D  f
 D  da  Q f ,enclosed
Boundary
conditions
Dabove  Dbelow   f
|
|
But
D  0
in general
||
||
||
||
Dabove
 Dbelow
 Pabove
 Pbelow
Example 4.4
Long straight wire with uniform line charge is surrounded
by a rubber insulation. Find the electric displacement.
4.4 Linear Dielectrics
Most macroscopic fields are weak as compared to the atomic
and molecular fields. The polarization is weak.
linear dielectric
P   o  eE
electric susceptibility
e
permittivity
   o (1   e )
permittivity of free space
o
relative permittivity,
dielectric constant

r 
o
D  E
Dielectric constants
Table
On may calculate D in the same way as E in the vacuum if
the different boundary conditions for E and D do not play
role. In this case, one simply replaces  o  
This is the case if:
a) When the space is filled with a homogenous dielectric.
b) When the symmetry of the problem makes
D ||  0
Charge embedded in a homogenous dielectric material.
E
1
q
rˆ
2
4 r
Bound charges partially screen q.
Parallel plate capacitor filled with a dielectric.
C
A
d
Qf
D
  rCvacuum
dielectric
 Qf
D=0
Dielectrics are used to:
a) Increase the capacity
b) Keep the plates apart
c) Increase the dielectric strength (field strength without a spark)
Air:
dielectric constant
dielectric strength
 r  1.00059
Ec  3kV / mm
Ceramic capacitors
r  7
Ec  5.7kV / mm
A cut section of a
multiplayer capacitor
with a ceramic dielectric.
Foil wound capacitor.
Frequently used dielectrics:
 r  3.7 Ec  16kV / mm
Paper
 r  5.4 Ec  110  100kV / mm
Mica
Polysterene  r  2.6 Ec  24kV / mm
Example 4.5
Metal sphere of radius a carries a charge Q. It is surrounded
by a linear dielectric material. Find the potential at the center.
Displacement at a boundary without free charge.
Dabove  Dbelow
|
|
||
||
Dabove
/  above  Dbelow
/  below
 below   above
4.4 Boundary Problems with
Linear Dielectrics
Within a homogenous linear dielectric,
Laplace’s equation holds.
2V  0
Boundary conditions on the surface between two
dielectrics:
Vabove  Vbelow
Vabove
Vbelow
 above
  below
  f
n
n
Example 4.7
A sphere of homogeneous
dielectric material is placed
into an otherwise uniform
electric field. Find the field
inside the sphere.
Example 4.8
Find the electric field
inside and outside the
dielectric and the force
on the charge.
4.5 Energy in a dielectric system
Capacitor
For linear dielectrics
W
1
CV 2
2
W
Cdielectric   rCvacuum
1
D  Ed

2
Force on a dielectric