electrostatic fields in material mediaeee.guc.edu.eg/courses/communications/comm402... ·...

ELECTROMAGNETIC PROF. A.M.ALLAM

3/4/2017 LECTURES 1 3/4/2017 LECTURES 1

ELECTROSTATIC FIELDS IN

MATERIAL MEDIA

EMF


3/4/2017 LECTURES 2

• The conductivity usually depends on temperature and

frequency

• A material with high conductivity (σ >>1) is a metal

• A material with low conductivity (σ <<1) is insulator

• A material with conductivities lies in between is a

semi-conductor

• If a material exhibits infinite conductivity at T=0o k it

is a superconductor

•A dielectric material (in which D = ε E applies) is

– Homogeneous if ε does not change from point to point

– Isotropic of ε does not change with direction

– Linear if ε does not change with the applied E field

Materials media may be classified in terms of their conductivity σ (S/m) as:

Non conductors (insulators “dielectrics”)

Conductors


Unlike the free charges in vacuum which produce an electrostatic field, the dielectric

medium does not have any free charges

1-Polarizationand electric flux density

This polarization gives rise to an electric fields in opposite direction to the applied field

In the absence of an external electric field The electrons form a symmetrical cloud around

the nucleus with the center coincide with the

nucleus( the electrons still bounded in motion)

Since the electron can not move freely, the

center of electron cloud will move away

from the nucleus (polarization)

When an external electric field is applied

The polarized atom or molecule is represented

by an electric dipole

The displacements are limited by strong restoring force set up by the charge

configuration in the molecule -

+

- - - - - - - -

-

-


3/4/2017 4

The state of polarization is described by the vector of

polarization [Coul/m2]. It is defined as the electric

dipole moment per unit volume as;

P

][Coul/m lim)( 2

0

V

p

rP v

i

V

where,

.... v

ip

Is the vector summation of the dipole moments counted over the

volume V

Note: for uniformly polarized dielectric, is constant , while in the general

media, it is function of position

)a ( oPP

P

)a )(( rPP

V Dielectric

V’

ip

0

r

extE


5

i.e., the material is

equivalent to a free space

(ε= εo ) with surface and

volume polarized charges

(or bounded charges)

≠

Uniform polarization

•Nonuniform polarization

- -

+ + - -

Eext

Eext

Pbv

. C/m3

where

ρvb

ρvb≠0


Let us have a medium in space containing both free and bounded charges

The electric field intensity due to both types can be calculated as:

o

pE

.

( and b are the free and bounded charges respectively [Coul/m3] )

Pb

.Since )P .(

1 .

o

E or )PE( .o

The mixed quantity is a definition for the electric flux density D

PEo

D

.][C oul/m P 2

ED o & General differential form of Gauss’s law

dV . VV

dVD

enQdV . VS

SdD

General integral form of Gauss’s law

Notes: The source of is the true free charge due to while the

polarization vector is stirring up D D .

D

The source of is the total charges (true free charge + polarization charge b )

due to . opE /)( .

E

results from the impressed field on the dielectric but it is a function of the

total field , the impressed one and the field results from polarized charges totalE

P


e is a dimensionless quantity called electric susceptibility

It measures how susceptible or sensitive a given dielectric

is to electric field

PED o

EE eoo

Eeo

)1( Ero

E

ED

= o (1 + e ) is called permittivity of the dielectric

re

o

1

EP o

)(

is the relative permittivity

EEEEDP ooo

)(

Now we can define the electric flux density D and its relation to the electric field intensity E

3/..... mCpvp


8

Basic equations of electrostatic field in material media:

Differential form Integral form

D .

0E

0.dEc

vs

dVSdD .

ED

EP o

)(

EP eo

Summary:

3/..... mCpvp

D .


2-Boundary conditions

s

s

s

dSSdD

.

ShfSnDSnDLim sh

)]( ˆ. ˆ.[ 120

0 (as h→0)

s12 )D-D( . ˆ

n or

Surfaces that dividing the space into regions of different permittivity and/or

carrying a surface distribution of free charges cause the lines of or to be

discontinuous and hence we call them boundaries

E

D

1 1

2 2

1D

S2

n

1n

h

2D

n

s

s1 n2 n D-D

Gauss’ law:

0. c

dE

0)]h(f ˆ.E ˆ.E[Lim12

0h

h

1 1

2 2

1E

2E

c

The equation:

0 (as h→0)

0)E-E( ˆ12

n or 12 EE

Applying the equation:

S2

n

1n

h

1 1

2 2

1P

2P

n

sb

v

b

s

dVSdP

.

S)]h(fS n.PS n.P[Limsb12

0h

0 (as h→0)

bP

. or

)]h[Lim(sbb

0h

sb121 2 )P-P( . ˆ

n or s b1 n2 n

P-P


For s = 0

1

2

1E

2E

1

2

s = 0

D2n = D1n 2 E2n = 1 E1n

2 E2 cos2 = 1 E1 cos1 ………….(1)

Also, we have E2 = E1

E2 sin2 = E1 sin1 ………….(2 )

Eqn. (2) Eqn. (1) gives, 2

1

2

1

tan

tan

Special case (s = 0):


εr=2.5

εr=2.5

εr=4

εr=2.5


12

3-Capacitance

It is an idealized circuit element representing the electrostatic

energy stored in the system and is characterized by coulomb-

voltage law V ≈ Q

Any two conducting bodies, regardless

of their shapes and sizes, when

separated by an insulating (dielectric )

medium, form a capacitor

V =(1/C)Q

C=Q/V

B

s

d . E

.

A

SdD

V

QC

[ Farad ]


13

plate capacitor-parallel Example:

From the definition of voltage

Taking into account that the capacity is given by


14

capacitance of coaxial line Example:

The electric field is described for a < r < b by:

The voltage V between inner and outer conductors is

The capacity is than given by


3/4/2017 LECTURES 15

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