16.360 Lecture 28

Last lecture:

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

16.360 Lecture 28

The electromagnetic generator

),cos(

)cos(

00

0

CtAB

ABsdBS

),sin(

)cos(

00

00

CtBA

CtABdt

d

dt

dVemf

16.360 Lecture 28

Moving conductor in a time-varying magnetic field

,)( ldBusdt

BVVVldE

Cs

memf

tremfemfC

Example:

,101 AI ,ˆ5yu

10R

?2 I

16.360 Lecture 28

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 28

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

Today

• Boundary conditions for Electromagnetic• Charge-current continuity relation• Free charge dissipation in a conductor

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