목원대학교 전자정보통신공학부 전자기학 5-1 Chapter 5. Conductors, Dielectrics, and Capacitance 1.Current and Current Density Current(A) : a rate of movement of charge

목원대학교 전자정보통신공학부 전자기학 5-1

Chapter 5. Conductors, Dielectrics, and Capacitance

1. Current and Current Density• Current(A) : a rate of movement of charge passing a

given reference point (or crossing a reference plane).

SJ

J

SJI

dt

dQI

N

)(A/m :density Current 2

SJ dI

xvx

xvv

v

vJ

Svt

xS

t

QI

xSQ

vJ v


2. Continuity of Current• The principle of conservation of charge: charges

can be neither created nor destroyed.

• The current, or charge per second, diverging from a small closed surface per unit volume is equal to the time rate of decrease of charges per unit volume at every point.

• A numerical example: p. 123

vt

v

dvt

dvdt

ddvd

dt

dQdI

v

v

S v

S

i

)(

surfaceconstant :)(

surface closed theinside charge positive of decrease :

volvolvol

J

JSJ

SJ

tv

)( J


3. Metallic Conductors

)0(electron ofmobility the:

)city drift velo ( velocity averageconstant

structure lattice ecrystallin excited thermally with thecollisions : material eCrystallin

velocity.its increasely continuous and accelerate ouldelectron w the:space Free

:electrons ,conductors In the

ee

Ev

v

EF

ed

d

eeQ

EJ e e vJ v density chargeelectron -free the: 0e


EJ tyconductivi :The point form of Ohm’s law

Isotropic: same properties in every direction

Anisotropic: not isotropic

Resistivity: reciprocal of the conductivity

Superconductivity: the resistivity drops abruptly to zero at a few kelvin

Higher temperature→greater crystalline lattice vibration→lower drift velocity →lower mobility →lower conductivity →higher resistivity

e e

ELV

ddV

JSdI

a

b

a

bab

S

abba

uniform : ,

LELELELE

SJ

EJ

IS

LV

L

VE

S

IJ

S

LRRIV

where :law sOhm'

S

a

bab

dσ

d

I

VR

SE

LE


4. Conductor Properties and Boundary Conditions

• Suppose that there suddenly appear electrons in the interior of a conductor→Electric fields by these electrons →The electrons begin to accelerate away from each other →The electrons reach the surface of the conductor

• Good conductor: zero charge density within a conductor and a surface charge density resides on the exterior surface

No charge, no electric field within a conducting material

Relate external fields to the charge on the surface of the conductor• The external electric field intensity is decomposed into tangential comp

onent and normal component to the conductor surface.• Static condition: tangential one may be zero. If not, there will result in

a movement of electrons.


Guass’s law: The electric flux leaving a small increment of surface must be equal to the charge residing on that incremental surface.

• The flux must leave the surface normally!

• The flux density per square meter leaving the surface normally is equal to the surface charge density per square meter

SND

0 :finitebut small keeping ,0

02

1

2

1

conductor e within th0 gRememberin

0 0

aat ,bat ,

wEwh

hEhEwE

d

t

NNt

a

d

d

c

c

b

b

a

E

LE

0tE

SNSN

S

DSQSD

QQd

sidesbottomtop

SD


Boundary conditions for the conductor-free space boundary in electrostatics

0 tt ED SoN ED

Summary: p. 132

5. The Method of Images

• The dipole field: the infinite plane at zero potential that exists midway between the two charges.

Remove conducting plane and locating a negative charge (image)


6. Semiconductors• Current carriers: electrons (conduction band), holes (valence band)

• Temperature↑: mobility↓, charge density ↑(more rapidly) Conductivity ↑

• Doping • Donors: additional electrons, n-type• Acceptors: extra holes, p-type

(holes) , ),(electrons , he ee hhe e-


7. The Nature of Dielectric Materials

• Bound charges: bound in place by atomic and molecular forces. Only shift positions slightly in response to external fields.

• Dielectric materials can store electric energy (a shift in the relative positions of the internal, bound positive and negative charges against the normal molecular and atomic forces)

• Polar molecule: random dipole → alignment• Nonpolar molecule: dipole arrangement after a field is applied

• Define: Polarization as the dipole moment per unit volume

4.7 Sec.in Eq.(37)

1total n

Qvn

ii

pp

dp

0 ofn orientatio Random total pp idipoles per unit volume

vn

ii

v v 10

1lim pP


The net increase in the bound charge within the closed surface

SdS nQnQdn b cosQ :mmolecules/ 3

surface thecross surface theof cos2

1 within Charges

charges negative and positive ofmovement an Apply

moleculesnonpolar Assume

S

E

d

SP bQ

The net total charge which crosses the elemental surface

Sb dQ SP (*Resemblance to Gauss’s law)


• Generalize the definition of electric flux density

• Isotropic material: linear relationship between E and P

S obT

bT

bTS oT

dQQQ

QQQ

QQQdQ

SPE

SE

)(

charge free : charge, bound : charge, enclosed total:

where

PED o

v v

v TT

v bb

dvQ

dvQ

dvQ

To

b

E

P

v D

EP e o

)constant dielectricor y permitivit reletive : 1(

)1(

e

ee EEEEED

R

Roooo typermittivi : Ro


8. Boundary Conditions for Perfect Dielectric Materials

0 LE d

02tan 1tan wEwE

2tan 1tan EE

2

1

2tan

1tan

2

2tan 2tan 1tan

1

1tan or

D

DDEE

D

QdS

SD SQSDSD SNN 21

0h

SNN DD 21

0 :dielectrcsPerfect S 21 NN DD

2211 NN EE


21212

1

2

1

2211122

1

22

11

2tan

1tan

222111

, if tan

tan

sinsinor sin

sin

coscos

DDD

D

D

D

DDDD NN

12

2

2

11

212

12

2

1

21

212

cossin

sincos

EE

DD


The boundary conditions at the interface between a conductor and a dielectric

law sGauss' 3.

and 0satisfy tozero beboth must components field and l tangentiaThe 2.

conductor theinside zeroboth are and 1.

EDLEDE

ED

d

/ , , SNSNSEDQd SD

0 tt ED SNN ED

How any charge introduced within a conductor arrives at the surface

(surface charge)

)exp( ,

, ,

ttt

ttt

ovv

vv

vvv

D

DEJEJ

timerelaxation :


9. Capacitance

0charge total: at - and at 1 2 MQMQ

Surface charge, normal electric field, equipotential surface

12 to fromFlux Electric MM

oV

QC ecapacitanc : Define

LE

SE

d

dC S

difference Potentialintensity field Electricdensityflux Electricdensity Charge

constant is ratio The

The capacitance is a function only of the physical dimensions of the system of conductors and of the permittivity of the homogeneous dielectric.


SQ

ddzdzdV

S

S

d

Szd z

So

zSzS

00lower

upper

and

aaLE

aDaE

d

S

V

QC

o

The total energy stored in the capacitor

2

222

0 0 2

2

vol

2

2

1

2

1

2

1

2

1

d

d

SSddzdSdvEW SS

S dS

E

C

QQVCVW ooE

22

2

1

2

1

2

1


10. Several Capacitance Examples

A coaxial capacitor (inner radius a, outer radius b)

a

bdV

LLDQd

La

bL

a

b

Lo

L

S

LL

ln2

ln22

2

2 2

aEaDSD

)/ln(

2

ab

LC

A spherical capacitor (inner radius a, outer radius b)

baV

QC

o11

4

ba

Q

r

Qdr

r

QV

r

QE

a

b

a

bor

11

4

1

44

4 22

Capacitance of an isolated spherical conductor

aC 4


Coating this sphere with a different dielectric material

111

1111112

12

12121

2

1111

4

1111

4

11

4444

)( 4

)( 4

4

1

1

rra

C

rra

Q

ra

Q

r

Qdr

r

Qdr

r

QVV

rrr

QErra

r

QE

r

QD

o

oo

a

r

r

oa

orrr

Multiple dielectrics2

21

1221121

SSSS EEEEDD

SQddV SSS

o

22

11

212

2

1

1

22

11

1111

CCS

d

S

d

dd

S

V

QC

SS

S

o


11. Capacitance of a Two-Wire Line

120

210

2

20

1

10

2

20

1

10

ln2

lnln2

ln2

ln2

RR

RR

R

R

R

RV

R

RV

R

RV

LL

LL

2010 and at reference zero RR

plane) 0( Choose 2010 xRR22

22

22

22

)(

)(ln

4)(

)(ln

2 yax

yax

yax

yaxV LL

Documents

목원대학교 전자정보통신공학부 전자기학 5-1 Chapter 5. Conductors, Dielectrics, and Capacitance 1.Current and Current Density Current(A) : a rate of movement of charge