electric fields in matter
DESCRIPTION
Electric Fields in Matter. Polarization. Field of a polarized object. Electric displacement. Linear dielectrics. Conductors. Matter. Insulators/Dielectrics. All charges are attached to specific atoms/molecules and can only have a restricted motion WITHIN the atom/molecule. - PowerPoint PPT PresentationTRANSCRIPT
Electric Fields in Matter
Polarization
Electric displacement
Field of a polarized object
Linear dielectrics
Matter
Insulators/Dielectrics
Conductors
All charges are attached to specific atoms/molecules and can only have a
restricted motion WITHIN the atom/molecule.
When a neutral atom is placed in an external electric field (E):
… positively charged core (nucleus) is pushed along E;
• If E is large enough
► the atom gets pulled apart completely
=> the atom gets IONIZED
… centre of the negatively charged cloud is pushed in the opposite direction of E;
• For less extreme fields
► an equilibrium is established
=> the atom gets POLARIZED
……. the attraction between the nucleus and the electrons
AND
……. the repulsion between them caused by E
Induced Dipole Moment:
Ep
α
Atomic Polarizability
(pointing along E)
Ep
To calculate : (in a simplified model)The model: an atom consists of a point
charge (+q) surrounded by a uniformly charged spherical cloud of charge (-q).
At equilibrium, eEE ( produced by the negative charge cloud)
-q
E
d
+q+q
a
-q
304
1
a
qdEe πε
At distance d from centre,
304
1
a
qdE
πε
Eaqdp 304πε
va 03
0 34 επεα
(where v is the volume of the atom)
Prob. 4.4:
A point charge q is situated a large distance r from a neutral atom of polarizability .
Find the force of attraction between them.
Force on q :
Attractiverr
qF
1
42
2
20
πεα
Alignment of Polar Molecules:
when put in a uniform external field:
0netF
Ep
τ
Polar molecules: molecules having permanent dipole moment
Alignment of Polar Molecules: when put in a non-uniform external field:
FFFnet
d
F+
F- -q
+q
F-
d
F+
-q
+qE+
E-
EEqFnet
EpFnet
For perfect dipole of infinitesimal length,
Ep
τ
the torque about the centre :
the torque about any other point:
FrEp
τ
Prob. 4.9:A dipole p is a distance r from a point
charge q, and oriented so that p makes an angle with the vector r from q to p.
(i) What is the force on p?
(ii) What is the force on q?
rrppr
qF pon
ˆˆ.34
13
0
πε
prrpr
qF qon
ˆˆ3
4
13
0πε
Polarization:When a dielectric material is
put in an external field:
A lot of tiny dipoles pointing along the direction of the field
Induced dipoles (for non-polar constituents)
Aligned dipoles (for polar constituents)
A measure of this effect is POLARIZATION
defined as:
P dipole moment
per unit volume
Material becomes POLARIZED
The Field of a Polarized Object
= sum of the fields produced by infinitesimal dipoles
2
0
ˆ
4
1
s
s
r
rprV
πε
p
rs
Dividing the whole object into small elements, the dipole moment in each volume element d’ :
τ dPp
Total potential :
τ
πε τ
dr
rrPrV
s
s2
0
ˆ
4
1
2
ˆ1
s
s
s r
r
r
Prove it !
τπε τ
dr
PVs
1
4
1
0
FffFFf
Use a product rule :
τπε
τπε
τ
τ
dPr
dr
PV
s
s
1
4
1
4
1
0
0
Prr
P
rP
sss
11
Using Divergence theorem;
τπε
πε
τ
dPr
adPr
V
s
S s
1
4
1
1
4
1
0
0
Defining:
nPbˆ
σ
Volume Bound Charge
Pb
ρ
Surface Bound Charge
τρ
πε
σ
πε
τ
dr
adr
V
s
b
S s
b
0
0
4
1
4
1
surface charge density b
volume charge density b
Field/Potential of a polarized object
Field/Potential produced by a
surface bound charge b
Field/Potential produced by a
volume bound charge b
+
=
Physical Interpretation of Bound Charges
…… are not only mathematical entities devised for calculation;
perfectly genuine accumulations of charge !
but represent
-q +q
d
A
Surface Bound Charge
A dielectric tube
Dipole moment of the small piece:
AdP
PAq
=
Surface charge density:
PAq
b σ
P
AP
n̂
A’
θσ cosPAq
b
In general:
nPbˆ
σ
If the cut is not to P :
+
+
+
+
+
+
+
+
____
_
_
___
Volume Bound Charge
A non-uniform polarization
accumulation of bound charge within the volume
diverging P
pile-up of negative charge
Net accumulated charge with a volume
Opposite to the amount of charge pushed out of the volume through the surface
=
S
b adPd
τρτ
ττ
dP
Pb
ρ
Field of a uniformly polarized sphere
Choose: z-axis || P
P is uniform
0 Pb
ρ
θσ cosˆ PnPb
z
P R
n̂
Potential of a uniformly polarized sphere: (Prob. 4.12)
Potential of a polarized sphere at a field point ( r ):
τ
πε τ
dr
rrPV
s
s2
0
ˆ
4
1
P is uniform
P is constant in each volume element
τ
τρ
περ ss
rr
dPV ˆ
4
112
0
Electric field of a uniformly charged
sphere
rEPrV
ρ
θ1
,
rrE
03ε
ρ
At a point inside the sphere ( r < R )
rPrV
03
1,
εθ
z
PE
03ε
kP
E ˆ3 0ε
PE
03
1
ε Inside the sphere
the field is uniform
rr
RrE ˆ
3 2
3
0ε
ρ
rPr
RrV ˆ
3
1, 2
3
0
εθ
At a point outside the sphere ( r > R )
20
ˆ
4
1
r
rpV
πε
(potential due to a dipole at the origin)
prrpr
rE
ˆˆ31
4
13
0πε
Total dipole moment of the sphere: PRp 3
3
4π
Uniformly polarized Uniformly polarized sphere – A physical sphere – A physical
analysisanalysis Without polarization:
Two spheres of opposite charge, superimposed and canceling each other
With polarization:The centers get separated, with the positive
sphere moving slightly upward and the negative sphere slightly downward
At the top a cap of LEFTOVER positive charge and at the bottom a cap of negative charge
Bound Surface
Charge b
+ ++ + + + + +
+ +
+
-d
+ +
- - - - - - - -
Recall: Pr. 2.18
Two spheres , each of radius R, overlap partially.
dE
03ε
ρ+
-
_
+d
_
+
r r
d
dE
03ε
ρ
Electric field in the region of overlap between the two spheres+ +
+ + + + + + + +
+
-d
+ +
- - - - - - - - PE
03
1
ε
For an outside point:
20
ˆ
4
1
r
rpV
πε
Prob. 4.10:A sphere of radius R carries a polarization
rkrP
where k is a constant and r is the vector from the center.
(i) Calculate the bound charges b and b.
(ii) Find the field inside and outside the sphere.
kRb σ kb 3ρ
rkE inside
0ε 0outsideE
The Electric Displacement
Polarization
Accumulation of Bound charges
Total field = Field due to bound charges + field due to free charges
Gauss’ Law in the presence of dielectricsWithin the dielectric the total charge density:
fb ρρρ
bound charge free charge
caused by polarization
NOT a result of polarization
Gauss’ Law :fbE ρρρε
0
fPE ρε
0
Electric Displacement ( D ) :
PED
0ε
Gauss’ Law
fD ρ
enclfQadD
D & E :
τρ drr
rKrE
s
s
2
ˆ
τρ drr
rKrD f
s
s
2
ˆ
PD
0 E
Boundary Conditions:
fbelowabove DD σ
||||||||belowabovebelowabove PPDD
On normal components:
On tangential components:
For some material (if E is not TOO strong)
EP e
χε0
Electric susceptibility of the medium
Linear DielectricsRecall: Cause of polarization is an Electric field
Total field due to (bound + free) charges
Location ► Homogeneous
Magnitude of E
► Linear
Direction of E ► Isotropic
In a dielectric material, if e is independent of :
In linear (& isotropic) dielectrics; EED e
χεε 00
)1(0 ewithED χεεε
Permittivity of the material
The dimensionless quantity:
0
1ε
εχε er
Relative permittivity or Dielectric constant of the material
EP e
χε0 ED
εand / or
Electric Constitutive Relations
Represent the behavior of materials
But in a homogeneous linear dielectric :
0 DandD f
ρ
00 DP
Generally, even in linear dielectrics :
vacED 0ε
DE
ε
1 vacEE
ε
ε0
vacr
EE
ε
1
When the medium is filled with a homogeneous linear dielectric, the field is
reduced by a factor of 1/r .
Capacitor filled with insulating material of dielectric constant r :
vacr
EE
ε
1
vacr
VVε
1
vacrCC ε
Energy in Dielectric Systems
Recall: The energy stored in any electrostatic system:
τε
dEWspaceall 20
2
The energy stored in a linear dielectric system:
τdEDWspaceall
2
1