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Dong-A University
• Electrostatic field : stuck charge distribution
• E, D field to H, B field
• Moving charge (velocity = const)
• Bio sarvart’s law and Ampere’s circuital law
Magnetostatic Fields
DDisplay isplay DDevice evice LLabab
Dong-A University
• Bio-Savart’s law
I
dl
H field RandIbtwangle
R
lengthntdisplacemedl
currentI
:
distance:
:
:
32 44 R
RdlI
R
adlIdH r
Experimental eq.
Independent on material property
R
DDisplay isplay DDevice evice LLabab
Dong-A University
• The direction of dH is determined by right-hand rule• Independent on material property• Current is defined by Idl (line current)
Kds (surface current)
Jdv (volume current)Current element
IK
DDisplay isplay DDevice evice LLabab
Dong-A University
• Ampere’s circuital law
I
H
dl
encIdlH
I enc : enclosed by path
By applying the Stoke’s theorem
equationthirdsMaxwellJH
dsJIdsHdlH enc
':
)(
DDisplay isplay DDevice evice LLabab
Dong-A University
• Magnetic flux density
typermeabili
fieldticmagnetostaHB
fieldticelectrostaED
:
)(
)(
0
0
0
From this
)/(
:
)(
:
2mwb
densityfluxmagneticB
wb
fluxmagneticdsB
Magnetic flux line always has same start and end point
DDisplay isplay DDevice evice LLabab
Dong-A University
• Electric flux line always start isolated (+) pole to isolated (-) pole :
• Magnetic flux line always has same start and end point : no isolated poles
QdsD
equationfourthsMaxwellB
fieldticmagnetostaforlawsGaussiandsB
':0
':0
DDisplay isplay DDevice evice LLabab
Dong-A University
• Maxwell’s eq. For static EM field
JH
E
B
D v
0
0
t
DH
t
BE
B
D
0
0
Time varient system
DDisplay isplay DDevice evice LLabab
Dong-A University
• Magnetic scalar and vector potentials
VEfrom
0)(
0
A
V 0)( mVJH
Vm : magnetic scalar potentialIt is defined in the region that J=0
BA 0)(
A : magnetic vector potential
DDisplay isplay DDevice evice LLabab
Dong-A University
• Magnetic force and materials
• Magnetic force
Q EEQFe
Bu
Q
BuQFm
Fm : dependent on charge velocity does not work (Fm dl = 0) only rotation does not make kinetic energy of charges change
DDisplay isplay DDevice evice LLabab
Dong-A University
• Lorentz force
• Magnetic torque and moment
Current loop in the magnetic field H
D.C motor, generator
Loop//H max rotating power
)( BuEQBuQEQFFF me
DDisplay isplay DDevice evice LLabab
Dong-A University
• Magnetic dipole
A bar magnet or small current loop
I
m
N
S
m
A bar magnet A small current loop
DDisplay isplay DDevice evice LLabab
Dong-A University
• Magnetization in materialSimilar to polarization in dielectric material
Atom model (electron+nucleus)
Ib
B
Micro viewpointIb : bound current in atomic model
DDisplay isplay DDevice evice LLabab
Dong-A University
• Material in B field
B
typermeabili
HH
H
HHB
r
m
m
:
)1(
)(
0
0
0
DDisplay isplay DDevice evice LLabab
Dong-A University
• Magnetic boundary materials
Two magnetic materials Magnetic and free space boundary
DDisplay isplay DDevice evice LLabab
Dong-A University
• Magnetic energy
dvHEWWW
dvHdvHBW
dvEdvEDW
me
m
e
)(2
1
2
1
2
1
2
1
2
1
22
2
2
DDisplay isplay DDevice evice LLabab
Dong-A University
• Maxwell equations– In the static field, E and H are independent on
each other, but interdependent in the dynamic field– Time-varying EM field : E(x,y,z,t), H(x,y,z,t)– Time-varying EM field or waves : due to accelated
charge or time varying current
Maxwell equations
currentsyingtimefieldneticElectromag
CDcurrentstaticfieldticMagnetosta
echstaticfieldticElectrosta
var
.).(
arg
DDisplay isplay DDevice evice LLabab
Dong-A University
• Faraday’s law– Time-varying magnetic field could produce
electric current
unitinflux
numberN
fieldyingtimeby
forceiveelectromotvoltageinducedV
dt
dN
dt
dV
emf
emf
:
:
var
)(:
Electric field can be shown by emf-produced field
DDisplay isplay DDevice evice LLabab
Dong-A University
• Motional EMFs
dsBdt
ddlEV
dt
dV
emf
emf
E and B are related
B(t):time-varying
IE
Bfieldyingtimeandloopyingtime
fieldBstaticandloopyingtime
Bfieldyingtimeandloopstationary
varvar.3
var.2
var.1
DDisplay isplay DDevice evice LLabab
Dong-A University
• Stationary loop, time-varying B field
fieldyingtimeforequationMaxwellt
BE
dst
BdsEdlE
dst
BdsB
dt
ddlEVemf
var:
)(
DDisplay isplay DDevice evice LLabab
Dong-A University
• Time-varying loop and static B field
loop varying-for timeequation sMaxwell':)(
)()(
theoremsstoke' applyingBy
field electric motional:
chargeaon:
BvE
dsBvdsEdlE
dlBvdlEV
EBvQ
FEm
BvQF
m
mm
memf
mm
m
DDisplay isplay DDevice evice LLabab
Dong-A University
• Time-varying loop and time-varyinjg B field
fieldyingtimetheinloopmotionalforequationMaxwell
Bvt
BE
dsBvdst
BdlEVemf
var:
)(
)(
DDisplay isplay DDevice evice LLabab
Dong-A University
• Displacement current→ Maxwell’s eq. based on Ampere’s
circuital law for time-varying field
In the static field
JHJH 0)(
In the time-varying field : density change is supposed to be changed
d
v
JJ
equationcontinuityJt
H
eq. smaxwell' esatisfy th order toIn
)(0
DDisplay isplay DDevice evice LLabab
Dong-A University
• Therefore,
t
DJH
t
DJ
t
DJ
Dtt
JJ
JJH
dd
vd
d
)(
0)()(
Displacement current density
DDisplay isplay DDevice evice LLabab
Dong-A University
• Maxwell’s Equations in final forms
t
DJH
t
BE
B
D v
0
s
s
v
v
dst
DJdlH
dsBt
dlE
dsB
dvdsD
)(
0
Gaussian’s law
Nonexistence ofIsolated M charge
Faraday’s law
Ampere’s law
Point form Integral form
DDisplay isplay DDevice evice LLabab
Dong-A University
• Time-varying potentials
VE
stationary E field
In the tme-varying field ?
t
AVEV
t
AE
potentialelectricScalarVV
t
AEA
tE
potentialmageneticVectorAB
Att
BE
)(:0)(
0)()(
)(
)(
DDisplay isplay DDevice evice LLabab
Dong-A University
Poisson’s eqation in time-varying field
vV 2
poisson’s eq. in stationary field
poisson’s eq. in time-varying field ?
v
v
At
V
At
Vt
AVE
)(
)()(
2
2
Coupled wave equation
DDisplay isplay DDevice evice LLabab
Dong-A University
• Relationship btn. A and V ?
t
VA
t
A
t
VJAA
AA
AB
t
A
t
VJ
t
AV
tJ
t
EJ
t
DJHB
2
22
2
2
2
)()(
)(
)(
)(
)(
)(
DDisplay isplay DDevice evice LLabab
Dong-A University
→ From coupled wave eq.
Jt
AA
Jt
A
tA
t
VV
t
V
tV
At
V
v
v
v
2
22
2
2
22
2
2
)(
)(
)(
Uncoupled wave eq.
npropagatiowaveofvelocity
v
:
1
DDisplay isplay DDevice evice LLabab
Dong-A University
• Time-harmonic fields Fields are periodic or sinusoidal with time
→
Time-harmonic solution can be practical because most of waveform can be decomposed with sinusoidal ftn by fourier transform.
tjett ,cos,sin
Im
Re
Explanation of phasor ZZ=x+jy=r
DDisplay isplay DDevice evice LLabab
Dong-A University
• Phasor form
If A(x,y,z,t) is a time-harmonic fieldPhasor form of A is As(x,y,z)
)Re( tjseAA
For example, if yakztAA )cos(0
ytj
s aeAA 0
)1
Re()Re(
)Re()Re(
tjs
tjs
tjs
tjs
eAj
dteAAdt
eAjeAtt
A
DDisplay isplay DDevice evice LLabab
Dong-A University
• Maxwell’s eq. for time-harmonic EM field
DjJH
BjE
B
D v
0
s
s
v
v
dsDjJdlH
dsBjdlE
dsB
dvdsD
)(
0
Point form Integral form
DDisplay isplay DDevice evice LLabab
Dong-A University
EM wave propagation
• Most important application of Maxwell’s equation
→ Electromagnetic wave propagation• First experiment → Henrich Hertz• Solution of Maxwell’s equation, here is
),,,(.4
),,0(.3
),,,0(.2
),,0(.1
00
00
00
00
orconductorgood
dielectriclossy
ordielectriclossless
spacefree
r
rr
rr
General case
DDisplay isplay DDevice evice LLabab
Dong-A University
• Waves in general form
t
DH
t
BE
B
D
0
0
Sourceless
EEE
t
E
t
E
t
Htt
BE
2
2
2
)(
)(
)()(
0
02
22
2
2
z
Eu
t
Eu : Wave velocity
DDisplay isplay DDevice evice LLabab
Dong-A University
• Solution of general Maxwell’s equation
)(),(cos),(sin,:
)()(
utzjkeutzkutzkutzsolution
utzgutzfE
Special case : time-harmonic
u
Ez
Es
s
022
2
ss EEjt
E 222
2
)(
DDisplay isplay DDevice evice LLabab
Dong-A University
• Solution of general Maxwell’s equation
)(
)(
ztj
ztj
BeE
AeE
)()( ztjztj BeAeEEE
A, B : Amplitudet - z : phase of the wave : angular frequency : phase constant or wave number
DDisplay isplay DDevice evice LLabab
Dong-A University
• Plot of the wave
E
z
t
0
0
/2 3/2
T/2 T 3T/2
A
A
uf
fT
fuT
)21
(
2,
numberwaveu
:2
DDisplay isplay DDevice evice LLabab
Dong-A University
• EM wave in Lossy dielectric material),,0( 00 rr
Time-harmonic field
ss
ss
s
s
EjH
HjE
B
D
)(
0
0
s
sss
Ejj
EEE
)(
)( 2
0
tconsnpropagatio
jj
EE ss
tan:
)(
02
22
DDisplay isplay DDevice evice LLabab
Dong-A University
• Propagation constant and E field
)1)(1(2
)1)(1(2
2
2
jIf z-propagation and only x component of Es
formtcons
azteEtzE
or
formphasor
EeEzE
xz
zezxs
tan:
)cos(),(
:
')(
0
00
DDisplay isplay DDevice evice LLabab
Dong-A University
• Propagation constant and H field
formconstant:
)Re(),(
:
')(
)(0
00
yztjz
zezxs
aeeHtzH
or
formphasor
HeHzH
impedenceintrinsic:
00
jej
j
EH
yz
yztjz
azteE
aeeHtzH
)cos(
)Re(),(
0
)(0
DDisplay isplay DDevice evice LLabab
Dong-A University
• EM wave in free space),,0( 00
377120
2,
1
,0
0
00
00
00
cu
cy
x
aztE
tzH
aztEtzE
)cos(),(
)cos(),(
0
0
0
kHEHEk aaaaaa
HEk
t
BE
DDisplay isplay DDevice evice LLabab
Dong-A University
• E field plot in free space
y
x
z
ak
aEaH
TEM wave(Transverse EM)
Uniform plane wave
Polarization : the direction of E field