electromagnetic field and waves

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Electromagnetic Field and Waves

Gi-Dong Lee

Outline:Electrostatic FieldMagnetostatic FieldMaxwell Equation

Electromagnetic Wave Propagation


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Vector Calculus

• Basic mathematical tool for electromagnetic field solution and understanding.


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• 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


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• 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


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• 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


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• Divergence, Gaussian’s law

v

dsA

vAAdiv

0

lim

It is a scalar field

sink:0

source:0

A

A

theorysGaussiandvAdsA

equationthefrom

':


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• 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


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• Laplacian of a scalar

Practical solution method

harmonicVif

z

V

y

V

x

VV

divergenceandgradientofcompose

02

2

2

2

2

2

22

2


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• 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


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• Electric Field E

raR

Q

Q

F

Q

F

QE

04

0

lim

E : Field intensity to the normalized charge (1)

rr’

1QR


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• 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


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From this

dvdsD

dsDdvQ

Q

v

v

From the Gaussian’s law

equationfirstMaxwell'sD

dvDdsD

v :


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• 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


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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


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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

):(

,


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• Conduction current (current by electron : metal)

dlE

dlE

I

VR

lawsohmofformGeneral

sistivity

lawsOhms

l

s

lR

I

V

tyconductiviEJdt

dQI

c

c

'

Re:

':

:,


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• 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


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• 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


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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


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• Continuity equation

tJ v

Qinternal

time

0

t

Ct

v

v


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• Boundary condition

Dielectric to dielectric boundary Conductor to dielectric boundary Conductor to free space boundary


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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


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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


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• 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


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• 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


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• Ampere’s circuital law

I

H

dl

encIdlH

I enc : enclosed by path

By applying the Stoke’s theorem

equationthirdsMaxwellJH

dsJIdsHdlH enc

':

)(


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• 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


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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


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• 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


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• 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


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• 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


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• Slant loop

naISm

NmBmFrT

)(`

an

B

F0

F0


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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


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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


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• Material in B field

B

typermeabili

HH

H

HHB

r

m

m

:

)1(

)(

0

0

0


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• Magnetic boundary materials

Two magnetic materials Magnetic and free space boundary


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• Magnetic energy

dvHEWWW

dvHdvHBW

dvEdvEDW

me

m

e

)(2

1

2

1

2

1

2

1

2

1

22

2

2


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• 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


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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


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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


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• Stationary loop, time-varying B field

fieldyingtimeforequationMaxwellt

BE

dst

BdsEdlE

dst

BdsB

dt

ddlEVemf

var:

)(


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• 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


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• Time-varying loop and time-varyinjg B field

fieldyingtimetheinloopmotionalforequationMaxwell

Bvt

BE

dsBvdst

BdlEVemf

var:

)(

)(


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• 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


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• Therefore,

t

DJH

t

DJ

t

DJ

Dtt

JJ

JJH

dd

vd

d

)(

0)()(

Displacement current density


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• 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


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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


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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

)()(

)(

)(

)(

)(

)(


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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


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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


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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


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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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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


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• 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


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• 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

)(


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• 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


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• 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


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• 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


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• 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


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• 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


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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

electromagnetic field and waves

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