Wave Motion & EM Waves (IV) Chih-Chieh Kang Electrooptical Eng.Dept. STUT email:[email protected].

Wave Motion & EM Waves (IV)

Chih-Chieh KangElectrooptical Eng.Dept. STUT

email:[email protected]

Electromagnetic wave Radiation field far from the antenna

Maxwell’s Equations

Differential form Integral form

0

Bt

DJH

Dt

BE

0

dsB

dsDdt

ddsJdlH

dvdsD

dsBt

dlE

S

SSc

VS

SC

B

HED

Maxwell’s EquationsE： electric field intensity [V/m] ： permittivity [s2C2/ m3kg]

D： electric flux density [C/m2] ： permeability [mkg / C2]

B： magnetic flux density [Wb/m2] ： charge density [C/m(2)]

H： magnetic field intensity [A/m] J： current density [A/m2]

kji

kji

zyx

zyx

HHHH

EEEE

in Cartesian coordinates

kjizyx

del operator

3-D Wave equations for EM waves In free space, Maxwell’s equations wave

equations

270

32290

800

2

2

22

2

2

22

2

2

002

2

2

002

kg/Cm 104kg,/mCs 1036

1

m/s 103/1 velocity phase

01

01

0

0

c

t

H

cH

t

E

cE

t

HH

t

EE

Solutions of 3-D Wave Equations for EM Waves Every component of EM field obeys

the 3-D scalar differential wave equation

Solutions for time-harmonic plane waves propagating in the +k direction

zyxzyx HHHEEE ,,,,,

01

2

2

22

2

2

2

2

2

tczyx

zkykxkrk

zyxr

zyx

kji

rktAeAtrtzyx

rktAeAtrtzyx

eAeAtrtzyx

rkti

rkti

zkykxktirkti zyx

sinˆIm,ˆIm;,,

cosˆRe,ˆRe;,,

ˆˆ,ˆ;,,ˆ

Solutions of 3-D Wave Equations for EM Waves Take

rktiti

rki

rkirkizxyyxx

zrki

xyrki

yxrki

xzzyyxx

rkixz

rkiyy

rkixx

tiz

tirkz

rktizz

tiy

tirky

rktiyy

tix

tirkx

rktixx

rkti

zyxzyx

eEerEtrEtzyxE

eErE

eEeaEaEaE

aeEaeEaeEarEarEarErE

eErEeErEeErE

ezEeeEeEtrE

ezEeeEeEtrE

ezEeeEeEtrE

eA

trHtrHtrHtrEtrEtrE

0

0

0000

000

000

00

00

00

ˆReˆRe,;,,

ˆˆ

ˆˆˆˆ

ˆˆˆˆˆˆˆ Let

ˆˆ,ˆˆ,ˆˆ:tionrepresentaphasor

ˆReˆReˆRe,

ˆReˆReˆRe,

ˆReˆReˆRe,

ˆRe:form theas

expressed be can ,,,,,,,,,,, trkAeAtrtr trki cosˆRe,ˆRe,

TEM Waves

BEBE

BEk

k

vBiEki

t

BE

kBE

rBkHHB

EkEkiE

eBtrBeEtrE rktirkti

ˆˆ :other each lar toperpendicu are ˆ and ˆ

ˆˆ1ˆˆˆ

ˆ

)(TEM waves wavesTransverse

npropagatio of direction thelar toperpendicu are ˆ and ˆ

ˆ0ˆ0ˆˆ

ˆ0ˆ0ˆ

ˆ,ˆ,ˆ,ˆ00

Relation between E and H in a Uniform Plane Wave In general, a uniform

plane wave traveling in the +z direction may have both x- and y-components

yyxx

yyxx

azHazHzH

azEazEzE

ˆˆˆ

ˆˆˆ

0xE ： amplitude of E-field, 0yH ： amplitude of H-field

phase constant 00

intrinsic impedance 00000 // yx HE

Energy Transport by EM Waves

Poynting theorem

electric energy density magnetic energy density Poynting vector determines the

direction of energy flow

32/

2

1mJEwe

32/

2

1mJHwm

v mevv

dvwwt

dvEJdsS

2/mWHES

Energy Transport by EM Waves Poynting theorem : Electromagnetic power flow into a

closed surface at any instant equals the sum of the time rates of increase of the stored electric and magnetic energies plus the ohmic power dissipated (or electric power generated, if the surface enclosed a source) within the enclosed volume.

Energy Transport by EM Waves

rktA

rktk

kE

v

rktEk

kEv

HES

rktEk

k

vH

rktEk

k

vE

k

k

vB

rktEeEE rkti

2

22

0

200

0

0

00

cos

cos1

cos1

cos1

cos11

cosRe

Energy Transport by EM Waves Time-averaged Poynting vector

light wave theodetector t theof timeresponse the:

light wave theoffrequency :

122

1

2

2sin2sin42

2cos12

1

cos

cos11

2

0

2

0

0

0

2

0

2

0

T

Tk

kE

v

k

kE

v

A

rkrkTT

AA

tdrktT

A

tdrktT

A

dtrktAT

dtST

S

T

T

TT

Energy Transport by EM Waves Irradiance I : average energy per unit

area per unit time

The intensity of light wave is proportional to the square of the amplitude of the (electric) field.

2

0

2

0

2

0 22

1

EI

vwEv

Ev

SI e

Radiation Pressure & Momentum

c

Ip

d

v

A

dt

A

dw

A

Fp

t

dvFdw

t

wA

Fp

///

,,

References E. Hecht, Optics, Addison-Wesley. F. T. Ulaby, Fundamentals of Applied

Electromagnetics, Prentice Hall. J. D. Cutnell, and K. W. Johnson, Physics, Wiley.