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
