Dr. Ahmed Said Eltrass

Electrical Engineering Department

Alexandria University, Alexandria, Egypt

Fall 2015

Part-I: Electro-Static

Lecture 8

Office hours: Sunday (10:00 to 12:00 a.m )

4th floor, Electrical Engineering Building

Chapter 5

Conductors and Dielectrics

Current (I) :is the number of charges passing/crossing a given

area per unit time (in one second) (Ampere or C/t)

It can also be defined as the rate of change of charge passing

a given area.

• Current Density (J) :Current passing a plane per unit area

(A/m2)

dt

dQI

t

QI

If J is known, we can find the total current (I) as follows:

S

sdJI

Derive an expression for the current density ?

xvx

xvv

vv

vS

IJ

Svt

xS

t

QI

SxvQ

t

QI

But

vJ v

In general:

)(A/mVector Density Current :J

(m/s) charges ofcity Drift velo :

)(C/mdensity charge Volume :

2

3

v

v

Continuity Equation

Q

v

S

Flow in

(Iin)

Flow out

(Iout)

• Suppose that we have a closed surface (S), then:

Flow out – Flow in= rate of decrease of charge inside (S)

v

v

v

vS

v

v

S

S

inout

dvdt

ddvJ

dvJSdJ

dvdt

dSdJ

dt

dQSdJ

dt

dQII

: theoremdivergence theFrom

dt

dJ v

Continuity Equation

Ohm’s Law

EσJ

:law sOhm' From

(S/m) Unitsconductor. a ofty conductivi theis :

EJ

sdJ

ldE

S

A

B

But

I

VR AB

I

VR AB

S

A

B

sdE

ldE

This formula is used to

find the resistance of any

geometric shape

A B

S

- + VAB

I

potentialhigher A

potentiallower B

Examples

A B

S

- + VAB

x

L

v/m

given thatcylinder shown theof resistance theFind -1

0 xaEE

)v/m()/1(

given thatconductor shown theof resistance theFind -2

aE

x

y

z

B A

h

a b

• Suppose suddenly a number of electrons are injected in the

interior of a conductor:

- Forces of repulsion among electrons

- Electrons move towards conductor surface

Properties of Conductors

• Applying Gauss’s Law

- Electric field inside the conductor E= 0

- Electric field outside the conductor E≠ 0

• The potential at any point inside the

conductor is equal to the potential at the

surface

(Conductor is an equipotential surface)

( E is perpendicular to the conductor surface)

Example: Consider the sphere of radius R

Summary for the Properties of Conductors

1- There is no charge density inside the conductor ( ) and

all the charges are accumulated at the surface as

0v

s

2- E=0 inside the conductor, otherwise the current will flow

3- The conductor is an equipotential surface

Dielectric Materials • The dielectric material consists of atoms in which the centers of their

positive and negative charges do not coincide

• The slight shift between the centers forms an electric dipole whose

electric dipole moment (p) is given by:

xaQdp

x

• When the dielectric material is placed in an external electric field, the

electric dipoles are arranged in such a way that their dipole moments

are aligned with the external applied field.

p

+

+ +

+

+

- -

-

- -

+

+ +

+

+

- -

-

- -

+

+ +

+

+

- -

-

- -

field) electric applied (external E

pNP

Number of dipoles/m3 Dipole moment

Polarization

• Define the polarization : is the net dipole moments per unit volume

(C/m2) P

Relation between Polarization and Permittivity

EE

PED

EPED

r

r

0

0

0

00

1

: dielectricFor

erE

P

11

0

Electric Susceptibility Relative Permittivity

0

1

e

r

EEEDP

P

r

000

: required If

EP r

10

.density charge volume theand ,on polarizati the

,density flux electric the,density field electric thefind 2.1,with

material dielectric ain ,2050200V field, potential Given the -3

r

vρP

DE

yx

Example

Boundary Conditions

1- Conductor-Dielectric

Dielectric

1

2

Conductor

h

w

l

SE

nE

tE

0Ea b

c d

Et : Tangential component

EN : Normal component

Tangential Component

0022

00

00 :KVL Applyingabcd

lEwE

lE

ldE

ntn

a

d

d

c

c

b

b

a

0tE

nD

Dielectric

1

2

Conductor

h

w

l

SE

nE

tE

0Ea b

c d

Normal Component

SSD

Q

QSdD

sn

enc

enc

00

:cylinder a as surface closed with theLaw Gauss Applying

sidesbottomTop

S

nD

/ sn

sn

E

D

2- Dielectric-Dielectric

Dielectric

1

2

h

w

l

S

a b

c d

Dielectric

1r

2r

1E1nE

1tE

1

2

2E

2tE

2nE2nD

1nD

Tangential Component

02222

00 :KVL Applying

211122

abcd

lE

lEwE

lE

lEwE

ldE

nntnnt

a

d

d

c

c

b

b

a

21 tt EE

Tangential component of

electric field is continuous

Dielectric

1

2

h

w

l

S

a b

c d

Dielectric

1r

2r

1E1nE

1tE

1

2

2E

2tE

2nE2nD

1nD

Normal Component

SSDSD

QQSdD

snn

encenc

0

:cylinder a as surface closed with theLaw Gauss Applying

21

sidesbottomTopS

snn DD 21

EEE

DDDρ

nn

nns

usDiscontinoOr

Continous , given)not (or 0 If

21

21

21

boundary at the charge no is e when ther0sρ

1

2

1r

2r

1E

1

2

2E

Useful Relation from Refraction Law

)1(sinsin 2211

21

EE

EE tt

2tE

1tE

1nE

2nE

)2(coscos 222111

21 21

21

EE

EE

DD

nn

nn

222

22

111

11

cos

sin

cos

sin

(2)by (1) Divide

E

E

E

E

2

1

2

1

tan

tan

Examples

2121212

1

P and ,P , , , , , Find plane.y -at x isboundary The

)v/m(403020 Given that -4

DDE

aaaE zyx

1

2

21r

2.32r

1E

1

2

2E

Z

surface.conductor theofequation thealso and P,at and , , find

boundary, space-free-to-conductor aon lie tostipulated is that P(2,-1,3)point a and

100 potential, Given the -5

s

22

DV, E

)-y(xV