16.360 Lecture 19

Maxwell equations

,vD

,t

BE

,0 B

,t

DJH

,ED

,HB

: electrical permittivity; :magnetic permativity

v: electric charge density per unit volume; J: current density per unit area.

,vD

,0 E

,0 B

,JH

Electrostatics Magnetostatics

E: electric field intensity

D: electric flux intensity

H: magnetic field intensity

B: magnetic flux intensity

16.360 Lecture 19

,vD

,0 E

Electrostatics

Volume charge density

,lim0 dv

dq

v

qv

v

Surface charge density

,lim0 ds

dq

s

qs

s

Line charge density

,lim0 dl

dq

l

ql

l

,dvQv v

16.360 Lecture 19

,tsuq v

,sut

qI v

Current density J

,uJ v

,sdJIs

16.360 Lecture 19

,4 2R

qRE

,'EqF

Coulomb’s law

,4

)(3

1

111

RR

RRqE

,

4

)(3

2

222

RR

RRqE

],)()(

[4

13

2

223

1

1121

RR

RRq

RR

RRqEEE

,)(

4

1

13

1

N

ii

i

RR

RRqE

16.360 Lecture 19

Electric field due to a charge distribution

,'4

'ˆ2R

dqREd

,'

'ˆ4

1' 2

v

v

R

dvREdE

,'

''ˆ

4

12

s

s

R

dsREdE

,'

''ˆ

4

12

l

l

R

dlREdE

16.360 Lecture 19

Gauss’s law

,vD

,QdvdvDv vv

, sv

sdDdvD

,QsdDs

Gauss’s law

16.360 Lecture 19

Electrical scalar potential

,ldEdV

,ldEqldFdW e

,qdVdW

,2

1

2

11221

P

P

P

PldEdVVVV

,00 CCldEdV

,0)( CsldEsdE

16.360 Lecture 19

Electrical potential due to point charge

,''4

1)(

'

' dvRR

RVv

v

,4 2R

qRE

,4

ˆ4

ˆ2 R

qdRR

R

qRldEV

RR

R

ldEV

Electrical potential due to continuous distributions

,4

)(1RR

qRV

,''4

1)(

'

' dsRR

RVs

s

,''4

1)(

'

' dlRR

RVl

l

16.360 Lecture 19

Electric field as a function of Electrical potential

Poison’s equation

,ldEdV

,ldVdV

,VE

,vD

,vE

,vV ,2

vV Poison’s equation

,02 V Laplace’s equation

16.360 Lecture 19

Electrical properties of material

• conductor• dielectric • semiconductor

16.360 Lecture 20

Conductors

Electron drift velocity Eu ee

Hole drift velocity Eu hh

Conducting current

,)( EuuJJJ hvhevehvhevehe

,hvheve

,EJ

Point form of Ohm’s law

16.360 Lecture 20

Resistance

,1

221 lEldEVVV x

x

x

General form

,AEsdEsdJI xAA

,A

l

I

VR

,

1

2

1

2

A

x

x

A

x

x

sdE

ldE

sdJ

ldE

I

VR

16.360 Lecture 20

Joule’s law

,hhee lFlFW

General form

,

)(

vEJ

vEuEuuFuF

t

lF

t

lF

t

WP

hvhevehhee

hh

ee

v

dvEJP ,

16.360 Lecture 20

Dielectrics

Electrical field induced polarization

16.360 Lecture 20

Dielectrics

,0 PED

P: electric polarization field

For homogeneous material:

,0 EP e

,000 EEEPED e

),1(0 e

),1(0

er Relative permittivity:

Electric susceptibility

Dielectric breakdown

16.360 Lecture 20

Electric boundary condition

;0][ 120

lim

ldEldEldEd

c

b

ah

C

,111 nt EEE

,222 nt EEE

,021 lElE tt

,21 tt EE

the tangential component is continuousacross the boundary of two media.

16.360 Lecture 20

Electric boundary condition

;][lim0

ssdDsdDsdD sbottomtoph

C

,21 ssDsD snn

the normal component of D changes, theamount of change is equal to the surfaceCharge density.

,21 snn DD

16.360 Lecture 20

Dielectric-Conductor boundary

,1 snD

,021 tt EE

16.360 Lecture 20

Conductor-Conductor boundary

,221121 snnnn EEDD

,21 tt EE

,2

2

1

1

tt JJ

,

2

22

1

11 s

nn JJ

,)(2

2

1

11 snJ

16.360 Lecture 20

Capacitance

, s

sdEQ

,V

QC

l

ldEV

,

RC

,

l

s

ldE

sdEC

,

1

2

1

2

A

x

x

A

x

x

sdE

ldE

sdJ

ldE

I

VR

16.360 Lecture 20

Electrostatic Potential Energy

,ldWldFdW ee

,

2

1EDWe

,eWF

Image Method

Any given charge above an infinite, perfect conducting plane is electrically equivalent to the combination of the give charge and it’s image with conductingplane removed.