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Dr. Champak B. Das (BITS, Pilani)
Electric Fields in Matter
Polarization
Electric displacement
Field of a polarized object
Linear dielectrics
Dr. Champak B. Das (BITS, Pilani)
Matter
Insulators/Dielectrics
Conductors
All charges are attached to specific atoms/molecules and can only have a
restricted motion WITHIN the atom/molecule.
Dr. Champak B. Das (BITS, Pilani)
electron cloud
nucleus
• The positively charged nucleus is surrounded by a
spherical electron cloud with equal and opposite
charge.
A simplified model of a neutral atom
Dr. Champak B. Das (BITS, Pilani)
• The electron cloud gets displaced in a direction
(w.r.t. the nucleus) opposite to that of the applied electric field.
When the atom is placed in an external electric field (E)
E
Dr. Champak B. Das (BITS, Pilani)
• For less extreme fields
► an equilibrium is established
=> the atom gets POLARIZED
• If E is large enough
► the atom gets pulled apart completely
=> the atom gets IONIZED
Dr. Champak B. Das (BITS, Pilani)
-e +e
► The net effect is that each atom becomes a small charge dipole which
affects the total electric field both inside and outside the material.
Dr. Champak B. Das (BITS, Pilani)
Induced Dipole Moment:
Ep
α
Atomic Polarizability
(pointing along E)
Ep
Dr. Champak B. Das (BITS, Pilani)
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)
+qa
-q -q
E
d
+q
Dr. Champak B. Das (BITS, Pilani)
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)
Dr. Champak B. Das (BITS, Pilani)
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
Dr. Champak B. Das (BITS, Pilani)
Alignment of Polar Molecules:
when put in a uniform external field:
0netF
Ep
τ
Polar molecules: molecules having permanent dipole moment
Dr. Champak B. Das (BITS, Pilani)
Alignment of Polar Molecules: when put in a non-uniform external field:
FFFnet
d
F+
F- -q
+q
Dr. Champak B. Das (BITS, Pilani)
F-
d
F+
-q
+qE+
E-
EEqFnet
EpFnet
Dr. Champak B. Das (BITS, Pilani)
For perfect dipole of infinitesimal length,
Ep
τ
the torque about the centre :
the torque about any other point:
FrEp
τ
Dr. Champak B. Das (BITS, Pilani)
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
Dr. Champak B. Das (BITS, Pilani)
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)
Dr. Champak B. Das (BITS, Pilani)
A measure of this effect is POLARIZATION
defined as:
P dipole moment
per unit volume
Material becomes POLARIZED
Dr. Champak B. Das (BITS, Pilani)
The Field of a Polarized Object= sum of the fields produced by infinitesimal dipoles
2
0
ˆ
4
1
s
s
r
rprV
πε
prs
r r
Dr. Champak B. Das (BITS, Pilani)
τd rs
r r
p
τ dPp
Total potential :
τ
πε τ
dr
rrPrV
s
s2
0
ˆ
4
1
Dr. Champak B. Das (BITS, Pilani)
21 sss rr̂r
Prove it !
τπε τ
drPV s14
1
0
τπε
τπε
τ
τ
dPr
drPV
s
s
14
1
4
1
0
0
Dr. Champak B. Das (BITS, Pilani)
Using Divergence theorem;
τπε
πε
τ
dPr
adPr
V
s
S s
1
4
1
1
4
1
0
0
Dr. Champak B. Das (BITS, Pilani)
Defining:
nPbˆ
σ
Volume Bound Charge
Pb
ρ
Surface Bound Charge
Dr. Champak B. Das (BITS, Pilani)
τρ
πε
σ
πε
τ
dr
adr
V
s
b
S s
b
0
0
4
1
4
1
Potential due to a surface charge density b
& a volume charge density b
Dr. Champak B. Das (BITS, Pilani)
Field/Potential of a polarized object
Field/Potential produced by a
surface bound charge b
Field/Potential produced by a
volume bound charge b
+
=
Dr. Champak B. Das (BITS, Pilani)
Physical Interpretation of Bound Charges
…… are not only mathematical entities devised for calculation;
perfectly genuine accumulations of charge !
but represent
Dr. Champak B. Das (BITS, Pilani)
BOUND (POLARIZATION) CHARGE DENSITIES
►Accumulation of b and b
Consequence of an external applied field
τ
τρσ 0dda bS b
Dr. Champak B. Das (BITS, Pilani)
P
E
nqdnpP ( n : number of atoms per unit volume )
Dr. Champak B. Das (BITS, Pilani)
P
E
A A A
2
d2
d
nAqdQ Net transfer of charge across A :
Dr. Champak B. Das (BITS, Pilani)
PAQ Net charge transfer per unit area :
P is measure of the charge crossing unit area held normal to P when the dielectric gets
polarized.
Dr. Champak B. Das (BITS, Pilani)
P
E
N M
Q Q
When P is uniform :
… net charge entering the volume is ZERO
Dr. Champak B. Das (BITS, Pilani)
PA
Volume bound charge
Net transfer of charge across A :APPA
θcos
Pb
ρ
Dr. Champak B. Das (BITS, Pilani)
P
E
N
M G
n̂ n̂
n̂
n̂2
d2
d
Net accumulated charge between M & N :
APQ nPbˆ
σ
Surface bound charge
Dr. Champak B. Das (BITS, Pilani)
Field of a uniformly polarized sphere
Choose: z-axis || P
P is uniform
0 Pb
ρ
θσ cosˆ PnPb
z
P R
n̂
Dr. Champak B. Das (BITS, Pilani)
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
Dr. Champak B. Das (BITS, Pilani)
τ
τρ
περ ss
rr
dPV ˆ
4
112
0
Electric field of a uniformly charged
sphere Esphere
rEP,rV sphere
ρθ
1
Dr. Champak B. Das (BITS, Pilani)
rrEsphere
03ε
ρ
At a point inside the sphere ( r < R )
rPrV
03
1,
εθ
z
PE
03ε
PE
03
1
ε
Dr. Champak B. Das (BITS, Pilani)
Field lines inside the sphere :
►► ► ► ►
P
PE
03
1
ε
( Inside the sphere the field is uniform )
Dr. Champak B. Das (BITS, Pilani)
r̂r
RrEsphere 2
3
03ε
ρ
rPr
RrV ˆ
3
1, 2
3
0
εθ
At a point outside the sphere ( r > R )
Dr. Champak B. Das (BITS, Pilani)
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
Dr. Champak B. Das (BITS, Pilani)
► ►
Field lines outside the sphere :
P
Dr. Champak B. Das (BITS, Pilani)
►► ► ► ►► ►
Field lines of a uniformly polarized sphere :
Dr. Champak B. Das (BITS, Pilani)
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
Dr. Champak B. Das (BITS, Pilani)
At the top a cap of LEFTOVER positive charge and at the bottom a cap of negative charge
Bound Surface
Charge b
+ ++ + + + + +
+ +
+
-d
+ +
- - - - - - - -
Dr. Champak B. Das (BITS, Pilani)
Recall: Pr. 2.18
Two spheres , each of radius R, overlap partially.
dE
03ε
ρ+
-
_
+d
_
+
r r
d
Dr. Champak B. Das (BITS, Pilani)
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
Dr. Champak B. Das (BITS, Pilani)
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
Dr. Champak B. Das (BITS, Pilani)
The Electric Displacement
Polarization
Accumulation of Bound charges
Total field = Field due to bound charges + field due to free charges
Dr. Champak B. Das (BITS, Pilani)
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
Dr. Champak B. Das (BITS, Pilani)
Gauss’ Law
fD ρ
enclfQadD
Defining Electric Displacement ( D ) :
PED
0ε
( Differential form )
( Integral form )
Dr. Champak B. Das (BITS, Pilani)
D & E :
τρ drr
rKrE
s
s
2
ˆ
τρ drr
rKrD f
s
s
2
ˆ
… “looks similar” apart from the factor of 0 ( ! )
…….but :
Dr. Champak B. Das (BITS, Pilani)
D & E :
0 PD
0 E
Field = - Gradient of a Scalar Potential
No Potential for Displacement
Dr. Champak B. Das (BITS, Pilani)
Boundary Conditions:
fbelowabove DD σ
||||||||belowabovebelowabove PPDD
On normal components:
On tangential components:
Dr. Champak B. Das (BITS, Pilani)
Prob. 4.15:
A thick spherical shell is made of dielectric material with a “frozen-in” polarization
a
b
rr
krP ˆ
where k is a constant and r is the distance from the center. There is no free charge.
Find E in three regions by two methods:
Dr. Champak B. Das (BITS, Pilani)
(a) Locate all the bound charges and use Gauss’ law.
a
b
Prob. 4.15: (contd.)
For r < a : 0E
For r > b:
For a < r < b: rr
kE ˆ
0
ε
0E
Answer:
Dr. Champak B. Das (BITS, Pilani)
(b) Find D and then get E from it.
a
b
Prob. 4.15: (contd.)
0encfreeQ 0 D
)&(0 brarforE
)(ˆ0
braforrr
kE
ε
Answer:
Dr. Champak B. Das (BITS, Pilani)
The Equations of Electrostatics Inside Dielectrics
00
EandE
ε
ρ
0 EandD f
ρ
or
with
PEDandVE
0ε
Dr. Champak B. Das (BITS, Pilani)
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
Dr. Champak B. Das (BITS, Pilani)
In such 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
Dr. Champak B. Das (BITS, Pilani)
EP e
χε0 ED
εand / or
Electric Constitutive Relations
Represent the behavior of materials
Dr. Champak B. Das (BITS, Pilani)
Location ► Homogeneous
Magnitude of E
► Linear
Direction of E ► Isotropic
In a dielectric material, if e is independent of :
Most liquids and gases are homogeneous, isotropic and linear dielectrics at least at low electric fields.
Dr. Champak B. Das (BITS, Pilani)
But in a homogeneous linear dielectric :
00 DP
00 DP
Generally, even in linear(& isotropic) dielectrics :
Dr. Champak B. Das (BITS, Pilani)
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 .
vacED 0ε
0 DandD f
ρ
Free charges D , as:
In LD :
Dr. Champak B. Das (BITS, Pilani)
Capacitor filled with insulating material of dielectric constant r :
vacr
EE
ε
1
vacr
VVε
1
vacrCC ε
Dr. Champak B. Das (BITS, Pilani)
So far…………source charge distribution at
restELECTROSTATICS
ρε0
1 E
0 E
1st/4 Maxwell’s Equations
Dr. Champak B. Das (BITS, Pilani)
Coming Up…..
MAGNETOSTATICS
ELECTROMAGNETISM
…source charge distribution at motion
A New Instructor