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)

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.


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


• 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


• 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


-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.


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)

+qa

-q -q

E

d

+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

prs

r r


τd rs

r r

p

τ dPp

Total potential :

τ

πε τ

dr

rrPrV

s

s2

0

ˆ

4

1


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


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

Potential due to a surface charge density b

& a 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


BOUND (POLARIZATION) CHARGE DENSITIES

►Accumulation of b and b

Consequence of an external applied field

τ

τρσ 0dda bS b


P

E

nqdnpP ( n : number of atoms per unit volume )


P

E

A A A

2

d2

d

nAqdQ Net transfer of charge across A :


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.


P

E

N M

Q Q

When P is uniform :

… net charge entering the volume is ZERO


PA

Volume bound charge

Net transfer of charge across A :APPA

θcos

Pb

ρ


P

E

N

M G

n̂ n̂

n̂

n̂2

d2

d

Net accumulated charge between M & N :

APQ nPbˆ

σ

Surface bound charge


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 Esphere

rEP,rV sphere

ρθ

1


rrEsphere

03ε

ρ

At a point inside the sphere ( r < R )

rPrV

03

1,

εθ

z

PE

03ε

PE

03

1

ε


Field lines inside the sphere :

►► ► ► ►

P

PE

03

1

ε

( Inside the sphere the field is uniform )


r̂r

RrEsphere 2

3

03ε

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


► ►

Field lines outside the sphere :

P


►► ► ► ►► ►

Field lines of a uniformly polarized sphere :


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

fD ρ

enclfQadD

Defining Electric Displacement ( D ) :

PED

0ε

( Differential form )

( Integral form )


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 :


D & E :

0 PD

0 E

Field = - Gradient of a Scalar Potential

No Potential for Displacement


Boundary Conditions:

fbelowabove DD σ

||||||||belowabovebelowabove PPDD

On normal components:

On tangential components:


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:


(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:


(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:


The Equations of Electrostatics Inside Dielectrics

00

EandE

ε

ρ

0 EandD f

ρ

or

with

PEDandVE

0ε


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


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


EP e

χε0 ED

εand / or

Electric Constitutive Relations

Represent the behavior of materials


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.


But in a homogeneous linear dielectric :

00 DP

00 DP

Generally, even in linear(& isotropic) dielectrics :


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 :


Capacitor filled with insulating material of dielectric constant r :

vacr

EE

ε

1

vacr

VVε

1

vacrCC ε


So far…………source charge distribution at

restELECTROSTATICS

ρε0

1 E

0 E

1st/4 Maxwell’s Equations


Coming Up…..

MAGNETOSTATICS

ELECTROMAGNETISM

…source charge distribution at motion

A New Instructor

Dr. Champak B. Das (BITS, Pilani) Electric Fields in Matter Polarization Electric displacement Field of a polarized object Linear dielectrics.

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