Lesson 7Electrostatics in Materials

楊尚達 Shang-Da YangInstitute of Photonics TechnologiesDepartment of Electrical EngineeringNational Tsing Hua University, Taiwan

Introduction

Materials consist of atoms, which are made up of nuclei (positive charges) and electrons (negative charges).The “total” E-field must be modified in the presence of materials.Discuss the field behaviors inside some material, and on the interface of different materials.

Classification of materials − Classical view

Conductors: Electrons in the outermost shells of the atoms, very loosely held, can freely migrate due to thermal excitation ( ). These electrons are sharedby all atoms.Dielectrics: Electrons are confined within the inner shells, can hardly migrate.Semiconductors: Electrons are moderately confined, movable when applying an external electric field. Free electron density (conductivity) can be changed by doping or bias voltage.

23kTEk =

Classification of materials − Quantum view (1)

Classification of materials − Quantum view (2)

Electric properties of materials depend on the band structure, and how they are filled by the electrons (Pauli’s exclusion principle).

Outline

Static E-field in the presence of conductorsStatic E-field in the presence of dielectricsGeneral boundary conditions

Sec. 7-1Static E-field in the Presence of Conductors

1. Inside a conductor2. Air-conductor interfaces3. Example: Conducting shell

pushes free charges away from one another, modify by turns. The process continues until steady state is reached.

Inside a conductor

Ev

Ev

0=ρ0=E

v

sec10 19−≈t

0=t

Why no E-field inside a conductor? (1)

0≠Ev

1V

2V

,nonuniform is )(,21

rVVV

v>⇒

If

⇒ work has to be done to move free charges (contradiction!)

Why no E-field inside a conductor? (2)

Correspond to the lowest system energy, state of equilibrium

0=ρ0=E

v

Air-conductor interface-1

Charge is movable, not in equilibrium (contradiction!)

0≠tEIf

nnt EaEE vv==⇒ ,0

Air-conductor interface-2

tE

,0

=⋅∫CldEvv

00

0

=Δ⋅+Δ⋅=

⋅→Δ∫

wwE

ldE

t

habcda

vv

0=tE⇒

Air-conductor interface-3

⇒

, 0

εQsdE

S=⋅∫

vv

0

0

ερ SSE

sdE

sn

hS

Δ=Δ⋅=

⋅→Δ∫

vv

0ερ s

nE =

Example 7-1: Conducting shell (1)

Spherical symmetry:

)(REaE RRvv

=⇒

1. :iRR <

( )0

24)(ε

π QRRER =⋅=

=⋅∫3 S

sdE vv

204

)(R

QRER πε=⇒

:oi RRR <<

Example 7-1: Conducting shell (2)

2. ,0=Ev

0 2

=⋅∫SsdE vv

⇒ -Q induced on the inner surface⇒ +Q induced on the outer surface

Example 7-1: Conducting shell (3)

3. :oRR > =⋅∫1 S

sdE vv

0

2 )(4ε

π QQQRER R+−

==

204

)(R

QRER πε=⇒

Example 7-1: Conducting shell (4)

Conclusion

A “floating” conducting shell cannot isolate the effect of a charge.

Sec. 7-2Static E-field in the Presence of Dielectrics

1. Microscopic and macroscopic dipoles2. Polarization charge densities3. Electric flux density4. Example: Parallel-plate capacitor5. Example: Dielectric shell

Microscopic electric dipoles − Nonpolar molecules

H2 CH4

Non-polar covalent bond (electrons are equally shared by the atoms).

Symmetric arrangement of polar bonds.

Microscopic electric dipoles − Polar molecules

HF

Polar covalent bond

NH32 lone electrons

Asymmetric arrangement of polar bonds

H2O

Macroscopic electric dipoles-1

A dielectric bulk made up of a large number of randomly oriented molecules (polar or non-polar) typically has no macroscopic dipole moment in the absence of external E-field.

Macroscopic electric dipoles-2

In the presence of external E-field:

1. Non-polar molecule is polarized

2. Individual dipole moments are aligned

H2

Ev

dqpvv =

_

+Ev

E-torque Epvv //

Macroscopic electric dipoles-3

Each dipole (though electrically neutral) provides nonzero field, modifying the total E-field

(physics.udel.edu)

Strategy of analysis

It is too tedious to directly superpose the elementary fields:

( )θθπε θ sincos2

4)( 3

0

aaR

prE Rvvvv

+≈

Instead, we define polarization vector as:

vp

P k

v Δ≡ ∑

→Δ

vv

0lim

the kth dipole moment inside a differential volume Δv

⇒ Polarization charge ρp,

⇒ Electric flux density Dv

Polarization surface charge density-1

On the air-dielectric interface, is dis-continuous, net polarization exists.

Pv

Ev

Polarization surface charge density-2

Consider a differential parallelepiped Vbounded by Sb. The net polarization charge is:

( ) ( ) SaPSadnqQ nn Δ⋅=Δ⋅=Δ vvvv

Surface charge density:

⇒ΔΔ

= ,SQ

psρ

)mC( 2nps aP vv

⋅=ρ

Differential volume

Polarization volume charge density-1

In the interior of a dielectric, net polarization charge exists where is inhomogeneous:P

v

Polarization volume charge density-2

Consider two polarization vectors , in a differential volume V bounded by S.

1Pv

2Pv

Polarization charges on top and bottom surfaces:

( )∫ ⋅=+S npsps dsaPQQ

21vv

( )∫∫ ⋅∇=⋅=VS

dvPsdP

vvv

psρ

divergence theorem

Polarization volume charge density-3

By definition:

⇒

To maintain the electric neutrality:

( ) ( ) ( )∫∫ ⋅∇−=⋅∇−=+−=VVpspsp dvPdvPQQQ 21

vv

∫= V pp dvQ ρ

( ) )mC( 3Pp

v⋅∇−=ρ

Comment-1

can be regarded as a special case of , where( )Pp

v⋅∇−=ρ

nps aP vv⋅=ρ

∞→⋅∇ Pv

∞→⋅∇ Pv

Comment-2

Equivalent charge densities:

can be substituted into the formulas:

to evaluate the influence of polarized dielectrics.

( )Pp

v⋅∇−=ρ,nps aP vv

⋅=ρ

∫ ′′

′′

=V

vR vd

rrRrarE

20 ),(

)(4

1)( vv

vvvv ρ

πε

vdrrRrrV

Vv ′

′′

= ∫ ′ 0 ),(

)(4

1)( vv

vv ρ

πε

Electric flux density-definition

Total E-field is created by free & polarization charges:

0ερ

=⋅∇ Ev

0ερρ pE

+=⋅∇

v( )P

v⋅∇−

( ) ( )PEvv

⋅∇−=⋅∇⇒ ρε 0

ρ=⋅∇ Dr

⇒PEDvvv

+= 0ε (C/m2)

…Gauss’s lawQsdDS

=⋅∫ vv

only free charge

Electric flux density - Application (1)

For linear, homogeneous, and isotropicdielectrics, the polarization vector is proportional to the external electric field:

EP e

vvχε 0=

0EeE vv=

qdep vv =

02EeE vv=

qdep 2vv =

0=Ev

0=pv

Susceptibility, independent of magnitude, position, direction of E

v

Electric flux density-application (2)

( )EPED e

vvvvχεε +=+= 100

Ee

vχε 0

EDvv

ε=⇒

( ) 01 εχε e+= …permittivity of the dielectric

In this way, a single constant replaces the tedious induced dipoles, polarization vector, equivalent polarization charges in determining the total electric field.

ε

Example 7-2: Physical meanings of D, E, P

: free charge: polarization charge

: total charge

Dv

Pv

Ev

0ε

Example 7-3: Dielectric shell (1)

Spherical symmetry:

)(RDaD RRvv

=⇒

For all :0>R

( ) QRRDR =⋅= 24)( π

=⋅∫SsdD

vv

24)(

RQRDR π

=⇒

freecharge

Dv

Example 7-3: Dielectric shell (2)

By ,EDvv

ε=

,4

)( 2RQRER πε

=⇒

⎩⎨⎧ <<

=otherwise ,

for ,

0

00

εεε

εRRRir

Ev

Example 7-3: Dielectric shell (3)

decay slower

Example 7-3: Dielectric shell (4)

By ,

EDvv

ε=

EDPvvv

0ε−=

( )⎪⎩

⎪⎨⎧ <<−

=

−

otherwise ,0

for ,4

1

)(

021 RRR

RQ

RP

ir

R

πε

Pv

Example 7-3: Dielectric shell (5)

By ,Pv

nps aP vv⋅=ρRn aa vv =

Pv

Rn aa vv −=

( )

( )⎪⎪⎩

⎪⎪⎨

⎧

=>−

=<−−

=

−

−

oo

r

ii

r

ps

RRR

Q

RRR

Q

for ),0(4

1

;for ),0(4

1

21

21

πε

πε

ρ

Example 7-3: Dielectric shell (6)

By ,SdsQ psS psps ρρ == ∫

( )( )⎪⎩

⎪⎨⎧

=>−

=<−−

=

−

−

or

ir

ps

RRQ

RRQ

Q

for ),0(1

;for ),0(11

1

ε

ε

_

_ _

_

+

+

+ +

+ +

+ +

_ _

_ _Ev

Polarization charges create a radially inward E-field to reduce the total field.

Sec. 7-3General Boundary Conditions

1. Tangential boundary condition2. Normal boundary condition

General interface-1

tE

,0

=⋅∫CldEvv

( ) ( ),0

21

0

=Δ⋅+Δ−⋅=

⋅→Δ∫

wEwE

ldE

tt

habcda

vv

⇒ tt EE 21 =

General interface-2

⇒

( )( )S

SaDaD

sdD

s

nn

hS

Δ⋅=Δ⋅+⋅=

⋅→Δ∫

ρ1221

0

vvvv

vv

QsdDS

=⋅∫ vv

( ) sn DDa ρ=−⋅ 212

vvv

snn DD ρ=− 21⇒

only free charge

Comment-1

0=tE

0ερ s

nE =

tt EE 21 =

( ) sn DDa ρ=−⋅ 212

vvv{ } 0, 22 =DE

vv

If M2 is a conductor:

Comment-2

Only free surface charge density counts in:

( ) sn DDa ρ=−⋅ 212

vvv

If the two interfacing media are dielectrics:

nns DD 21 ,0 =⇒=ρ

Comment-3

tt EE 21 =

( ) sn DDa ρ=−⋅ 212

vvv

remain valid even the E-fields are time-varying.