16.360 Lecture 23

Static field

,vD

,0 E

Dynamic Field

,0 B

,JH

,

ˆ

4 2

l R

RldIH

,vD

,t

BE

,0 B

,t

DJH

,'

'ˆ4

1' 2 v

v

R

dvREdE

16.360 Lecture 23

Faraday’s Law

, s

sdB

,,t

sdBt

sdt

BldE

s

sC

16.360 Lecture 23

Electromotive force

,t

sdt

BldEV

sCemf

,memf

tremfemf VVV

Stationary Loop in a Time-varying Magnetic field

,

s

tremf sd

t

BV

,

s

i

tremf sd

t

B

RR

VI

Lenz’s law

16.360 Lecture 23

,)(

ssCemf sdEsd

t

BldEV

An example:

,sin)3ˆ2ˆ(0 tzyBB

Faraday’s law, differential form,t

BE

(a) The magnetic flux link of a single turn of the inductor.

(b) The transformer emf,.

(c) The polarity of the emf.

(d) The induced current.

16.360 Lecture 23

Example II

,3.0ˆ tzB

Determine the voltage drops across the two resistors

16.360 Lecture 24

The ideal Transformer properties:

• = • I = 0 in the core.• The magnetic flux is confined within the core

• I = ?, with applied voltage of V1and with RL • V2, and I2=?

Questions:

16.360 Lecture 24

,11 dt

dNV

,22 dt

dNV

,

1

2

1

2

N

N

V

V

Voltage transformer:

Power relations:

,21 PP Why?

,111 IVP ,2

1

1

2

N

N

I

I

Current transformer:

,222 IVP

Impedance transformer:

,/ 111 IVR ,/ 222 IVR ,)( 2

2

1

2

1

N

N

R

R

,)( 2

2

1Lin R

N

NR

16.360 Lecture 24

Moving conductor in a static magnetic field:

,t

sdt

BldEV

sCemf

,memf

tremfemf VVV

,)()(

ldBut

sdB

tV memf

,ldwsd

),()()( BACCABCBA

16.360 Lecture 24

Another way to look at it:

),( BuqFm

,q

FE mm

,)(2

1

2

1

ldBuldEVl

l

l

lmmemf

Next lecture:

• The electromagnetic generator • Moving conductor in a time varying magnetic field

16.360 Lecture 27

The electromagnetic generator

),cos(

)cos(

00

0

CtAB

ABsdBS

),sin(

)cos(

00

00

CtBA

CtABdt

d

dt

dVemf

16.360 Lecture 27

Moving conductor in a time-varying magnetic field

,)( ldBusdt

BVVVldE

Cs

memf

tremfemfC

Example:

,101 AI ,ˆ5yu

10R

?2 I

I

16.360 Lecture 27

Displacement current

,JH

• Ampere’s law in static electric field

,t

DJH

• Ampere’s law in time-varying electric field

• proof of Ampere’s law:

,vD

,)( sv

sdDdvDQ

,)()('

ssvS

sdDt

sdDt

dvDt

Qt

sdJI

t

DJ

' Displacement current density

16.360 Lecture 27

Displacement current

,t

DJH

• Ampere’s law in time-varying electric field

Example: ,cos0 tVVs

,sin01 tCVdt

dVCI C

c

,cosˆˆ 0 td

Vy

d

VyE c

,sin02 tCVsdt

DI

Sd

16.360 Lecture 28

• Boundary conditions for Electromagnetic

,t

BE

,t

DJH

,0 B

,vD

,21 vnn DD

,21 tt EE

,21 nn BB

,21 stt JHH

Maxwell equations boundary conditions

16.360 Lecture 28

• Charge-Current continuity Relation

charge current continuity equation

,

v vdvt

Qt

I

,

v vsdv

tsdJI

,

v vvsdv

tJsdJ

,vtJ

,0 sdJs

steady state integral form

,0i

iI Kirchhoff’s current law

16.360 Lecture 28

• Free-charge dissipation in a conductor

,vtJ

,EJ

,vtJD

,vv t

,/ rtvov e ,

r

16.360 Lecture 29

• Electromagnetic Potentials

Electrostatics: ,VE

,0 E

,0 B

,AB

Dynamic case:,t

BE

),( A

tE

,t

AVE

,AB

16.360 Lecture 29

• Retard Potentials

Electrostatics: ,''4

1)(

'

)(' dvR

RVv

Rv i

Dynamic case:

,''

),(

4

1),(

'

' dvR

tRtRV

v

iv

,''

)/',(

4

1),(

'

' dvR

uRtRtRV

v

piv

,''

)/',(

4),(

'dv

R

uRtRJtRA

v

pi

16.360 Lecture 29

• Time-Harmonic Potentials

,)(Re),( '

~

'

tj

iviv eRtR

,)(Re)(Re)/',( ''

~/'

'

~

'

jkRtj

iv

uRjtjivpiv eReRuRtR p

,''

)(

4

1Re'

'

)/',(

4

1Re),(

'

'~

'

'

'

dvR

eeRdv

R

uRtRtRV

v

jkRtjiv

v

piv

,''

)(

4

1)(

'

'~

'~

dvR

eRRV

v

jkRiv

,''

)(

4

1)(

'

'~

'~

dvR

eRJRA

v

jkRiv

16.360 Lecture 29

• Time-Harmonic Potentials

,~~

EjH

,~~

HjE

• example

,)()(~

22~

EEEE

,0~

2~

2 EE

,0 E if no free charge, trans-wave, why?

,)'( '0

~~jkReERE

,22 k

),10sin(10ˆ),( 10 kztxtzE

find k?

n1

n2

zikinin eExE 1ˆ

zik

rr eExE 1ˆ

ziktt eExE 2ˆ

x

z

0| ztrin EEE

BjE

ikz

ikz

Eeiky

Eezyx

xyx

E

)(ˆ

00

tr 1

21 )1( tkrk

tkrk 11 )1(

21

12

kk

kt

21

21

kk

kkr