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(1831-1879)
April 2, 2013
(with Time-Varying Fields)
t
DJH
t
BE
B
D
0
Differential Form Integral Form
Gauss’s Law
Faraday’s Law
Gauss’s Law for
Magnetism
Ampere’s Law
vD
t
BE
S
QSdD
Sdt
BldE
SC
0 B
0S SdB
t
DJH
Sdt
DJldH
SC
Page 256
Ampere’s Law JH
Current Produces a magnetic field
I H
H
t
DJH
Sdt
DJldH
SC
PED
0
This term is present in material media and
in free space. It doesn't necessarily involve
any actual movement of charge, but it does
have an associated magnetic field, just as
does a current due to charge motion.
This term is associated with the
polarization of the individual
molecules of the dielectric
material. Polarization results
when the charges in molecules
move a little under the influence
of an applied electric field.
t
P
t
E
t
DJD
0Displacement
Current
Displacement Field
No conduction current enters cylinder surface R, while
current I leaves through surface L. Consistency of
Ampère's law requires a displacement current ID = I to
flow across surface R
Capacitor Imaginary
cylindrical surface
Example to illustrate the physical meaning of the displacement current Id
tVtVs cos)( 0
- Assume wires are perfect conductors and capacitor is filled with perfect dielectric
tCVtVdt
dC
dt
dVCI c
C sincos 001
dC II 21
tCVtVd
A
dsytd
Vy
tsd
t
DI
Sd
sinsin
ˆcosˆ
00
02
td
Vy
d
VyED c cosˆˆ 0
Page 300
Even though the displacement current
does not carry real charge, it nonetheless
behave like a real current.
Displacement current
?
)10sin(2
1
/102
9
7
d
c
r
I
mAtI
mS
Solution:
t
EAAJI dd
tAA
t
A
IE c
sin
10
102
sin102 10
7
3
ttId cos10885.0cos10 1210
Wire Ic
EAJAIc
Page 300
Field
Components
General Form Medium 1: Dielectric
Medium 2: Dielectric
Medium1: Dielectric
Medium 2: Conductor
Tangential E
Normal D
Tangential H
Normal B
0)(ˆ212 EEn
sDDn )(ˆ212
sJHHn
)(ˆ212
0)(ˆ212 BBn
tt EE 21
snn DD 21
tt HH 21
nn BB 21
021 tt EE
snD 102 nD
st JH 1 02 tH
021 nn BB
The boundary conditions derived previously for electrostatics
and magnetostatics remain valid for time-varying fields as well
Pages 301-303
s The surface charge density at the boundary;
Normal components of all fields are along , the outward unit vector of medium 2. 2n̂
Direction of Js is orthogonal to (H1-H2).
Surface current density at the boundary sJ
Implies that the tangential components are equal in magnitude and
Parallel in direction tt EE 21
Medium 1
Medium 2 2n̂
1E
tE1
nE1
2E
tE2
nE2
ED
Under static conditions, the charge density ρV and the current density J
at a given point in a material are totally independent of one another.
dvdt
ddv
dt
d
dt
dQI
v
v
vv
Net positive charge within v
Net current flowing across S out of v
vv
dvJsdJI
tJ v
Page 301
tJ v
0 J
0
t
Kirchhoff’s Current Law
0i
iI
Page 302
Charge-Current Continuity Relation
tJ v
tE v
0
v
v
dt
EJ
/vE
Relaxation time constant
rt
v
t
vv eet /)/(
00)(
Solution
t= 0 r r3
decay
t)(
ovof
%100
Copper
Mica
mS
mF
/108.5
/10854.8
7
12
0
mS /10
6
15
0
s
r
19105.1
s
r
4103.5
(Excess charges in a point within the material and how fast it decays)
Page 303
r
ovof
%37
ovof
%5
0)(&0)( AV
Dynamic Case Static Case 0/ t
ABB
VEE
0
0
0
B
t
BE
)( At
E
Vt
AE
)(
Relation between E & B and V & A ?
Page 303
t
A
Create an additional E field
'E
AB
t
AVE
0)(
t
AE
vdR
RRV
v
iv
)(
4
1)(
Static field
Retarded scalar potential
vdR
tRtRV
v
iv
),(
4
1),(
Dynamic field with no retardation
vdR
uRtRtRV
v
piv
)/,(
4
1),(
vdR
uRtRJtRA
v
pi
)/,(
4),(
Similarly:
Retarded vector potential
Page 304
vdR
uRtRtRV
v
piv
)/,(
4
1),(
vdR
uRtRJtRA
v
pi
)/,(
4),(
V and A are linearly dependent
on ρv and J, respectively.
AB
t
AVE
E and B are linearly dependent
on V and A.
tJ v
ρ and J have the same functional
dependence on time
V, A, E, D, B and H have the same functional dependence on time and the relationships
interconnecting all these quantities obey the rules of linear systems. We can use
sinusoidal function to determine the response of the system due to the source with any
type of time dependence Page 305
(Steady-State Sinusoidal Time Dependence)
Time harmonic responses of the retarded scalar and vector potentials
Suppose: tRtR iviv cos)(),(
tj
iviv eRetR )(~),(
In phasor notation
)/()(~)/,( puRtj
ivpiv eReuRtR
For retarded charge density:
tjRjk
ivpiv eeReuRtR )(~)/,(
Page 306
puk
Wavenumber
vdeR
eReeRVetRV tj
v
Rjk
ivtj
)(~
4
1)(
~),(
vdR
eRRV
v
Rjk
iv
)(~
4
1)(
~
vdR
eRJRA
v
Rjk
i
)(
~
4
1)(
~
Similarly:
AH~1~
Page 306
tjRjk
ivpiv eeReuRtR )(~)/,(
Ej
HorHjE
Hj
EorEjH
~1~~~
~1~~~
Page 307
t
DJH
t
BE
B
D
0
In a nonconducting medium (J=0)
tj
tj
eREetRE
eRHetRH
)(~
),(
)(~
),(
vdR
eRRV
v
Rjk
iv
)(~
4
1)(
~
vdR
eRJRA
v
Rjk
i
)(
~
4
1)(
~
AH~1~
Ej
HorHjE
Hj
EorEjH
~1~~~
~1~~~
Pages 305-307
VE~~
tjeRVetRV )(~
),(
tjeRAetRA )(~
),(
)10sin(10ˆ),( 1 0 k ztxtzE
?& kH jkzejxzE 10ˆ)(
~Phasor form
jkz
jkz
ek
jy
ej
zyx
zyx
j
Ej
zH
10ˆ
0010
///
ˆˆˆ1
~1)(
~
jexjexeextzE
eeeexkztxtzE
jkzjkzj
jkz
tjj
jkz
10ˆ)2
sin()2
cos(10ˆ10ˆ),(~
10ˆ)2
10cos(10ˆ),(
2
10210 10
)&16( 00
jkzek
jxHj
zE 2
210ˆ
~1~
jkzek
jyzH
10ˆ)(
~
jkzejxzE 10ˆ)(~
k=?
22 k
)/(133103
10444
8
10
00 mradc
k
ztyeek
jyeezHetzH tjjkztj 13310sin11.0ˆ10
ˆ)(~
),( 10
2
21010
k