Dr. Hugh Blanton ENTC 3331. Electrostatics in Media.

Dr. Hugh Blanton

ENTC 3331

Electrostatics in Media

Dr. Blanton - ENTC 3331 - Energy & Potential 3

• A medium (air, water, copper, sapphire, etc.) is characterized by its relative permittivity, (r).

Medium rvacuum 1

air 1.0006

conductors 1

glass 4.5 - 10


• Media can be grouped in two classes:conductors dielectrics

(insulators, semiconductors,

etc.)

free charges no free charges

charges will move until the conductor is field free

charges in the material are polarized by external fields

everywhere (this assumes we are dealing with an electrostatic problem with

0E

PED

oelectric flux density

field strength

polarization field

0 t

xE


+

+

+ +

+

+

+

+

xE

xE

xE

E

oP

D

0S

0S

Orientation of dipoles inside medium


• and are defined to be parallel.• A dielectric with field has positive and

negative surface charges on opposite sites.

E

P

++++

dielectric


• The polarization field is antiparallel to the polarization .

• The field inside the medium is smaller than the external field.

xE

P

oP

PE

mediumx EE


Microscopic Reasons for Induced PolarizationMicroscopic Reasons for Induced Polarization

• Deformation polarization in non-polar materials such as glass:

+

atom

+

polarized

atom

xE


• Orientation polarization in polar materials.

O

HH

O

H

H

xE

before dipoles line up

+ +

xE

after dipoles line up


• Note:• Isotropic implies that the , , and

fields are in the same direction. • Anisotropic implies that the , , and

fields may have different directions.• We limit the media to those that are linear,

isotropic, and homogeneous.• For such media, the polarization field is:

DE

P

DE

P

EP

eo

Electric susceptibility


• Since

• It follows that

• Materials with large permittivity also have a large susceptibility!

PED

o

EEED

eooeoo

ED

oe 1

r


Boundaries Between DielectricsBoundaries Between Dielectrics

• Maxwell’s equations are of general validity• In particular

dielectric1

dielectric2

Different amounts of surface charge at the

boundary.

What fields are at the boundary?

0ˆ0 C

drot lEE


• Construct a suitable path, C, about the boundary.

• and split the field into normal (n) and tangential (t) components.

ld

ld

a b

cd

2h

2h

medium1

medium2

0ˆˆˆˆˆ c

b

d

c

a

d

b

aC

ddddd lElElElElE

t

2E

n

2E

2E

t

1E

n

1E

1E

1

2


• Now make h smaller and smaller• This implies

• and

• Which implies

0ˆ c

bdlE

0ˆ

a

ddlE

0ˆˆ12

d

c

b

add lElE

Below boundary Above boundary


• Now, make l smaller and smaller, but not zero

tttt

12120ˆˆ EElElE


• Boundary conditions for the tangential components of the fields.• Across the boundary between any

media, the tangential component of is unchanged

• in all cases

tt EE12

E

tt DE111

tt DE 222


• However

• because

tt DD 22

11

ED


• Now use

• Construct suitable volume, V

QdVDdivsdDDdivS V

ˆGauss’s Law

3h

h32

medium1

medium2

n1D

upn

downn

S

S2

1

n2D

dVdS V

s sD ˆ

The only charge inside V is the surface charge on the boundary area S


S

Sdown

S

up dsdsdsd nDnDsD ˆˆˆ 21

3h

h32

medium1

medium2

n1D

upn

downn

S

S2

1

n2D

Let h go to zero,

Now, make the Gaussian surface smaller and smaller, but not zero

sincedsdsds Snn 21 DD

downup nn ˆˆ


• This implies

Snn 21 DD


• Boundary conditions for the normal component of the fields across the boundary between any two media.

• which implies

Snn 21 DD

Snn 2211 EE


Application of Boundary ConditionsApplication of Boundary Conditions

• Given that the x-y plane is a charge-free boundary separating two dielectric media with permittivities 1 and 2.• If the electric field in medium 1 is

• Find• The electric field in medium 2, and• The angles 1 and 2.

zyx 1111 ˆˆˆ EzEyExE

2E


• What are the angles between 1 and 2 between and , as well as between and .• For any two media:

• With no charges (charge free) on the boundary plane

1E

z2E

z

1E

t1E

n1E

t2E

2E

n2E

x-y plane

z

tt21 EE

nnS

nn22112211 0 EEEE


• It follows that:

• since the z-component of the field is the normal component of the field.

zyxzyx EzEyExEzEyExE 12

1112222 ˆˆˆˆˆˆ


• The tangential components for and are:

• Then

1E

2E

21

211 yx

t EE E

22

222 yx

t EE E

and

z

yx

E

EE

1

21

21

1tan

z

yx

E

EE

2

21

21

2tan

and


z

yx

E

EE

1

21

21

1tan

z

yx

z

yx

E

EE

E

EE

12

1

21

21

12

1

22

22

2tan

and


• The relation looks very similar to Snell’s law of Refraction

1

2

12

1

21

21

1

22

22

1

2

tan

tan

z

yx

z

yx

E

EE

E

EE

2

1

22

11

2

1

sin

sin

n

n


Dielectric with Conductor BoundaryDielectric with Conductor Boundary

• Very important practically:• Capacitor• Coaxial shielded cable

• External field cannot penetrate inside the shield.

shield


• Boundary conditions:

• Since a conduct is free field

tt21 EE

Snn 21 DD

conductor

dielectric

02 tE

02 nD

02 tE


• Field lines at a conductor surface have no tangential component.• They are always perpendicular to the

conductor surface!

• In addition

• The surface charge on the conductor defines the field in the surrounding dielectric

Snn 21 DD


• Conducting slab

• Bottom surface:• Normal and field are in opposite directions.

• Top surface:• Normal and field are in same directions.

+ + + + + + + + + + + + E

11 n

n

conductor

0bs

0ts


• Since the conductor is field-free

• And since , the magnitude of the surface charge densities is given by the product of permittivity and field strength.

bs

ts

Snn 21 DD


• Dielectric slabcapacitor

• Most general capacitor

Parallel plate

capacitor

++ + +

++

++ +

V

Conductor 1

Conductor 2

E


• Because the conductors must have inside,

• To achieve this, the charges distribute on the two surfaces.• There are equilibrium currents until

everything is stationary.

• Very fast—speed of light.

0E

0

t

E


• The surface charges on conductor 1 and conductor 2 give rise to the field with

• This implies that the total charge on either conductor is:

E

nS E

dsdsdsQSS

n

S

S EE

Definition of surface charge density

Boundary conditions (no tangential

component


• The potential difference V along any one of the field lines is given by:

C

dV lE ˆ


• Capacitance is the charge per potential difference.

FFaradsd

A

Ed

EA

d

d

V

QC

C

S

lE

sE

ˆ

ˆ


• The capacitance of a parallel-plate capacitor is• proportional to area A.• inversely proportional to separation, d.• proportional to the permittivity of the

dielectric filling.• independent of E


Summary of ElectrostaticsSummary of Electrostatics

• The sources of the electrostatic field are time-independent charge distributions.• That is, the charge distributions are static

(derivative is zero).• Electrostatics follows from the empirical facts

of• Coulomb’s law• The principle of linear, vectorial superposition of

forces and fields.• Energy conservation.


Summary of ElectrostaticsSummary of Electrostatics

• Electrostatics can be based on two fundamental Maxwell equations;• •

• The electric field is free from circulation ( ) and can always be expressed as the gradient of a potential ( ).

0 EE

rotD

div

0E

rot

VE


• Potential and Fields can be calculated for a given charge distribution, • from the field definition• using Gauss’s Law• using image charges

• Conducting and dielectric media can be distinguished.

• At boundaries between media, the following conditions hold:• •

tt21 EE

S

nn 21 DD

Dr. Hugh Blanton ENTC 3331. Electrostatics in Media.

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