03/21/2014PHY 712 Spring 2014 -- Lecture 211 PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 21: Continue reading Chap. 9 A.Electromagnetic.

PHY 712 Spring 2014 -- Lecture 21 103/21/2014

PHY 712 Electrodynamics10-10:50 AM MWF Olin 107

Plan for Lecture 21:Continue reading Chap. 9

A. Electromagnetic waves due to specific sources

B. Dipole radiation examples



Maxwell’s equations

00

2

02

0

1

0 :monopoles magnetic No

0 :law sFaraday'

1 :law sMaxwell'-Ampere

/ :law sCoulomb':0) 0;( form or vacuum cMicroscopi

c

t

tc

B

BE

JEB

EMP


Formulation of Maxwell’s equations in terms of vector and scalar potentials:

JAA

A

02

2

22

02

2

22

2

1

/1

01 :require -- form gauge Lorentz

tc

tc

tc

LL

LL

LL

t

AEAB

0,, :condition continuity source

with theconsistent is gauge Lorentz that theNote

tt

t rJr


Electromagnetic waves from time harmonic sources

ti

ti

ti

ti

et

et

et

et

,~, :potentialVector

,~, :potentialScalar

,~, :densityCurrent

,~, :density Charge

rArA

rr

rJrJ

rr

,'~'

'4

1,~,~

:For

'3

00

rrr

rrrr

ikerd

ck

,'~'

'4

,~,~ '30

0 rJ

rrrArA

rr

ikerd


Electromagnetic waves from time harmonic sources – continued:

'ˆˆ'4

:expansion Useful

*'

rrrr

rr

lmlmllm

l

ik

YYkrhkrjike

'ˆ,'~',~

ˆ,~,~,~

*30

0

rrJa

rarArA

lmlllm

lmlm

lm

Ykrhkrjrdikr

Yr

'ˆ,'~',~

ˆ,~,~,~

*3

0

0

rr

rrr

lmlllm

lmlm

lm

Ykrhkrjrdikr

Yr


Forms of spherical Bessel and Hankel functions:

x

exx

iixhx

xxxx

xj

xe

xixh

xx

xxxj

ixexh

xxxj

ix

ix

ix

22232

121

00

331 cos3sin13

1 cossin

sin

x

eixhx

lxxjx

ixl

l

l

l

1 1

!!12 1

:behavior cAssymptoti



r

ekriωi

reωi

kr'

rdi

rdω

ikr

ikr

1ˆ4

,~4

,~

:source e within th1 and source ofextent outside fieldsFor

,~ 1,~

:frequency at moment dipole Define :caseradiation Dipole

0

0

33

rpr

prA

rJrrp



2

0

02

42

*

0

22

22

0

22

0

ˆˆ32

,~,~ˆ2

ˆ

:1for radiatedPower

11ˆ4

1

,~,~

1ˆˆ3ˆˆ

41

,~,~,~

rpr

rBrErSr

pr

rArB

pprrrpr

rArrE

ωkc

rrddP

krikr

ωkr

ec

ikrr

ωωωkr

e

i

avg

ikr

ikr


Properties of dipole radiation field for kr >>1:

2

0

02

422

22

0

2

0

ˆˆ32

ˆ

:1for radiatedPower

ˆ4

1,~

ˆˆ4

1,~

rprSr

prrB

rprrE

ωkcrddP

kr

ωkr

ec

ωkr

e

avg

ikr

ikr

Bx

y

Example: zr q p

E


Example of dipole radiation source

RrRr eRi

JeJ /0/0 cos,~ ˆ,~

q

rzrJ

''cos ,~

'' ˆ,~

10

1/'2

0

0

00

0/'2

00

krjkrhedrrRkJ

krjkrhedrrikJ

Rr

Rr

q

r

zrA

RkR

kri

rekJ

RkR

reJ

Rr

ikr

ikr

222

3

0

0

222

3

00

121 cos ,~

12ˆ,~

:for Evaluation

q

r

zrA


Example of dipole radiation source -- continued

RkR

kri

rekJ

RkR

reJ

Rr

ikr

ikr

222

3

0

0

222

3

00

121 cos ,~

12ˆ,~

:for Evaluation

q

r

zrA

Relationship to pure dipole approximation (exact when kR0)

r

ekriωi

reωi

iJRrd

irdω

ikr

ikr

1ˆ4

,~4

,~ :fields dipole ingCorrespond

ˆ8,~ 1,~

0

0

03

33

rpr

prA

zrJrrp


Alternative approach

'

30, ' ',4 '

iked r

r r

A r J rr r

'

3

0

Fields from time harmonic source:

1, ' ',4 '

iked r

r r

r rr r

ˆ3 '0

ˆFor ' : | ' | 1 ' ...

, ' ',4

ikrik

r r r

e d r er

r r

r r r r

A r J r


Alternative approach -- continued

Linear center-fed antennaz

r

qd/2

ˆ3 '0

0

,

' ',4

',

ˆsin | | ( ) ( )2

ikrik

kd k

e d r er

z xI y

r r

A r

J r

r zJ


Alternative approach – linear center-fed antenna continued

/1

cos( '0

/2

)0

02

0

, ' 4

cos cos cos2 2 =

2 sin

ˆ

sin | ' |2

ˆ

dikrik

d

ikr

zI e dz er

kd kdI e

r

kd k zq

q

q

z

z

A r

00 2

2

2

0

Time averaged power:

cos cos cos1 2 2 =

8 sin

kd kddP Id

q q


Alternative approach – linear center-fed antenna continued

00

2

22

0

Time averaged power:

cos cos cos1 2 2 =

8 sin

kd kddP Id

q q

2

2 00 2 2

0

00 2 2

0

4

2

cos cos1 2for =

8 sin

cos cos4 2for 2

:

= 8 sin

:

dPkd Id

dPkd Id

q

q

q

q


Dipole radiation in light scattering by small (dielectric) particles

EincHinc

Hsc

Esc

sc0

sc2

0sc

inc0

incˆ

00inc

ˆ1 ˆˆ4

1

:ionapproximat dipole electricIn

ˆ1 ˆ

ErHrprE

EkHεE 0rk0

crek

ceE

ikr

ik


Dipole radiation in light scattering by small (dielectric) particles

EincHinc

Hsc

Esc

2

200

4

20

22

2

0

ˆ4ˆ

ˆ

ˆˆ

ˆ,ˆ;ˆ,ˆ

:section cross Scattering

pεEε

Eε

Sk

Srεkεr

00

Ekr

r

dd

inc

sc

avginc

avgsc


Estimation of scattering dipole moment:Suppose the scattering particle is a dielectric sphere with permittivity and radius a:

20

2

0

064

22

00

4

20

22

0

0

00

3

ˆˆ2/1/

ˆ4ˆ

ˆˆ,ˆ;ˆ,ˆ

:section cross Scattering

2/1/4

εε

pεEε

Eεεkεr

Ep

0

ak

Ekr

dd

a

inc

sc

inc


Scattering by dielectric sphere with permittivity and radius a:

q

22

0

064

20

2

0

0640

cos2/1/

ˆˆ2/1/ˆ,ˆ;ˆ,ˆ

ak

akdd εεεkεr 0

ε̂0ε̂

r̂

0k̂q

For Einc polarized in scattering plane:

q



2

0

064

20

2

0

0640

2/1/

ˆˆ2/1/ˆ,ˆ;ˆ,ˆ

ak

akdd εεεkεr 0

ε̂0ε̂r̂

0k̂q

For Einc polarized perpendicular to scattering plane:

1cos2/1/

2 ˆ,ˆ;ˆ,ˆ

:bygiven issection crossaverage likely,equally are onspolarizatiboth Assuming

22

0

064

0

q ak

dd εkεr 0



ε̂0ε̂r̂

0k̂q

24 620

00

/ 1ˆˆ ˆ ˆ, ; , cos 12 / 2

d k ad

q

0r ε k ε

q

03/21/2014PHY 712 Spring 2014 -- Lecture 211 PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 21: Continue reading Chap. 9 A.Electromagnetic.

Documents