DDisplay isplay DDevice evice LLabab
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Electromagnetic Field and Waves
Gi-Dong Lee
Outline:Electrostatic FieldMagnetostatic FieldMaxwell Equation
Electromagnetic Wave Propagation
DDisplay isplay DDevice evice LLabab
Dong-A University
Vector Calculus
• Basic mathematical tool for electromagnetic field solution and understanding.
DDisplay isplay DDevice evice LLabab
Dong-A University
• Line, Surface and Volume Integral
– Line Integral :
Circulation of A around L
( )
Perfect circulation :
– Surface Integral :
dl
A
Path L
dlA
dlAdlA //
dsAds
A
A dsA
Net outward flux of A
DDisplay isplay DDevice evice LLabab
Dong-A University
• Volume Integral :
• Del operator :
Gradient
Divergence
Curl
Laplacian of scalar
dvv
azz
ayy
axx
azz
Vay
y
Vax
x
VVV
:
)(:z
Az
y
Ay
x
AxAA
azy
Ax
x
Ayay
x
Az
y
Axax
z
Ay
y
AzAA )()()(:
)(:2
2
2
2
2
22
z
V
y
V
x
VVV
DDisplay isplay DDevice evice LLabab
Dong-A University
• Gradient of a scalar → azz
Vay
y
Vax
x
VV
V1
V2
dV = potential difference btw the scalar field V
)0(
cos
)(
max
Gdl
dV
GdldlG
dlazz
Vay
y
Vax
x
V
dzz
Vdy
y
Vdx
x
VdV
DDisplay isplay DDevice evice LLabab
Dong-A University
• Divergence, Gaussian’s law
v
dsA
vAAdiv
0
lim
It is a scalar field
sink:0
source:0
A
A
theorysGaussiandvAdsA
equationthefrom
':
DDisplay isplay DDevice evice LLabab
Dong-A University
• Curl, Stoke’s theorem
s
dlA
sAACurl
0
lim
0)(.3
0)(.2
:.1
V
A
sensenomakesV
theorysStokedsAdlA
equationthefrom
': ds
Closed path L
A
)()
//
currentIdlHex
componentcurlgeneratesdlAif
DDisplay isplay DDevice evice LLabab
Dong-A University
• Laplacian of a scalar
Practical solution method
harmonicVif
z
V
y
V
x
VV
divergenceandgradientofcompose
02
2
2
2
2
2
22
2
DDisplay isplay DDevice evice LLabab
Dong-A University
• Classification of the vector field
0,0
0,0
0,0
AA
AA
AA
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• Time-invariant electric field in free space
Electrostatic Fields
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• Coulomb’s law and field intensity
– Experimental law
– Coulomb’s law in a point charge
Q1 Q2
0
221
221
4
1
k
r
QQkF
r
QQF
– Vector Force F12 or F21
Q1 Q2
F21
r1 r2
F12
12
122
120
21122
0
2112
44 R
R
R
QQa
R
QQF
DDisplay isplay DDevice evice LLabab
Dong-A University
• Electric Field E
raR
Q
Q
F
Q
F
QE
04
0
lim
E : Field intensity to the normalized charge (1)
rr’
1QR
DDisplay isplay DDevice evice LLabab
Dong-A University
• Electric Flux density D
EDr
QD
dsDQ
024
Flux density D is independent on the material property (0)• Maxwell first equation from the Gaussian’s law
theoremdivergence
v
dsA
vA
:
0
lim
DDisplay isplay DDevice evice LLabab
Dong-A University
From this
dvdsD
dsDdvQ
Q
v
v
From the Gaussian’s law
equationfirstMaxwell'sD
dvDdsD
v :
DDisplay isplay DDevice evice LLabab
Dong-A University
• Electric potential Electric Field can be obtained by charge
distribution and electric potential
E
A
Q
B
B
A
dlEQW
BtoAfromQmovetoouterfromWork
lQElFWWork
In case of a normalized charge Q
AB
B
A
VdlEQ
W
+ : work from the outside
- : work by itself
DDisplay isplay DDevice evice LLabab
Dong-A University
Absolute potential
E
r
O : origin point
Q=1
r
QV
04
• Second Maxwell’s Equ. From E and V
)'(0
:0
theoremsStokedsEdlE
dlEdlEVV
nCirculatioVVVVA
B
B
A
BAAB
BAABBAAB
DDisplay isplay DDevice evice LLabab
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• Second Maxwell’s Equ
• Relationship btn. E and V
0 E
VE
z
VEz
y
VEy
x
VEx
dzz
Vdy
y
Vdx
x
V
EzdzEydyExdxdlEdV
,,
)(E
3 4 5
3,4,5 : EQUI-POTENTIAL LINE
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• Energy density We
dvED
dvVD
VdvD
VdvWe v
)(2
1
)(2
1
)(2
12
1
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• E field in material space ( not free space)
Material
Conductor
Non conductor
Insulator
Dielctric material
Material can be classified by conductivity << 1 : insulator >> 1 : conductor (metal : )Middle range of : dielectric
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• Convection current ( In the case of insulator)– Current related to charge, not electron– Does not satisfy Ohm’s law
l
s
dsJI
Jvs
I
velosityvsvt
ls
t
QI
dt
dQI
v
vv
):(
,
DDisplay isplay DDevice evice LLabab
Dong-A University
• Conduction current (current by electron : metal)
dlE
dlE
I
VR
lawsohmofformGeneral
sistivity
lawsOhms
l
s
lR
I
V
tyconductiviEJdt
dQI
c
c
'
Re:
':
:,
DDisplay isplay DDevice evice LLabab
Dong-A University
• Polarization in dielectric
Therefore, we can expect strong electric field in the dielectric material, not current
+
-
-
---
-
-
--
-
-
---- -
---
+After field is induced
Displacement can be occurred
– Equi-model
+
-
-
---
-
-
-+
-Q +Qd
dQP
Dipole moment
DDisplay isplay DDevice evice LLabab
Dong-A University
• Multiple dipole moments
- +D
P
E
litysusceptabidielectric
EP
PED
e
e
:0
0
EEE
EEPED
re
e
00
000
)1(
0 : permittivity of free space : permittivity of dielectricr : dielectric constant
DDisplay isplay DDevice evice LLabab
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• Linear, Isotropic and Homogeneous dielectric
• D E : linear or not linear
• When (r) is independent on its distance r
: homogeneous
• When (r) is independent on its direction
: isotropic anisotropic (tensor form)
Ez
Ey
Ex
Dz
Dy
Dx
zzzyzx
yzyyyx
xzxyxx
DDisplay isplay DDevice evice LLabab
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• Continuity equation
tJ v
Qinternal
time
0
t
Ct
v
v
DDisplay isplay DDevice evice LLabab
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• Boundary condition
Dielectric to dielectric boundary Conductor to dielectric boundary Conductor to free space boundary
DDisplay isplay DDevice evice LLabab
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• Poisson eq. and Laplacian• Practical solution for electrostatic field
solutionharmonic
LaplacianVif
eqpoissonV
ifV
VEED
v
v
v
)(00
.)(
)shomogeneou()(
,
2
2
DDisplay isplay DDevice evice LLabab
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• 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
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• 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
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• 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
• Slant loop
naISm
NmBmFrT
)(`
an
B
F0
F0
DDisplay isplay DDevice evice LLabab
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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)()(
)(
)(
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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
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• 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
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→ 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
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• 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
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• 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
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• 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
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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
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• E field plot of example
x
z
t=t0
t=t0+t
DDisplay isplay DDevice evice LLabab
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• 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
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• E field plot in free space
y
x
z
ak
aEaH
TEM wave(Transverse EM)
Uniform plane wave
Polarization : the direction of E field
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Reference
• Matthew N. O. Sadiku, “Elements of electromagnetic” Oxford University Press,1993
• Magdy F. Iskander, “Electromagnetic Field & Waves”, prentice hall