PHY 712 Spring 2015 -- Lecture 22 103/20/2015

PHY 712 Electrodynamics9-9:50 AM MWF Olin 103

Plan for Lecture 22:Start reading Chap. 9

A. Electromagnetic waves due to specific sources

B. Dipole radiation patterns

PHY 712 Spring 2015 -- Lecture 22 203/20/2015

PHY 712 Spring 2015 -- Lecture 22 303/20/2015

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

PHY 712 Spring 2015 -- Lecture 22 403/20/2015

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

t

t

tt

AE

AE

AEBE

ABB

or

0 0

0

PHY 712 Spring 2015 -- Lecture 22 503/20/2015

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

41

:formequation General

1

/1

01 :require -- form gauge Lorentz

2

2

22

02

2

22

02

2

22

2

ftc

tc

tc

tc

LL

LL

LL

JAA

A

PHY 712 Spring 2015 -- Lecture 22 603/20/2015

Solution of Maxwell’s equations in the Lorentz gauge -- continued

cttttG /'''

1',';, rrrr

rr

',''1''

1''

,, :, fieldfor Solution

3

0

tfc

ttdtrd

ttt

f

rrrrr

rrr

PHY 712 Spring 2015 -- Lecture 22 703/20/2015

Electromagnetic waves from time harmonic sources

0,~,~ 0,,:condition continuity that theNote

,~, :densityCurrent

,~, :density Charge

rJrrJr

rJrJ

rr

itt

t

et

etti

ti

,~4

,~ or

,~4

1,~ For

,~, :source General

0

0

rr

rr

rr

i

ti

Jf

f

eftf

PHY 712 Spring 2015 -- Lecture 22 803/20/2015

,'~

'' ,~

,'~'1''

1''

,~,~

',''1''

1''

,,

'3

0

'3

0

3

0

tii

tif

ti

tif

ti

f

eferde

efc

ttdtrd

ee

tfc

ttdtrd

tt

c

rrr

r

rrrrr

rr

rrrrr

rr

rr

Electromagnetic waves from time harmonic sources – continued:

PHY 712 Spring 2015 -- Lecture 22 903/20/2015

Electromagnetic waves from time harmonic sources – continued:

,'~'

'4

1,~,~

) gauge, (Lorentz potentialscalar For

'3

00

rrr

rrrr

ikerd

ck

,'~'

'4

,~,~

) gauge, (Lorentz potential For vector

'30

0

rJrr

rArArr

ikerd

ck

PHY 712 Spring 2015 -- Lecture 22 1003/20/2015

Electromagnetic waves from time harmonic sources – continued:

:function Hankel Spherical

:function Bessel Spherical

'ˆˆ'4

:expansion Useful

*'

krinkrjkrhkrj

YYkrhkrjike

lll

l

lmlmllm

l

ik

rrrr

rr

'ˆ,'~',~

ˆ,~,~,~

*3

0

0

rr

rrr

lmlllm

lmlm

lm

Ykrhkrjrdikr

Yr

PHY 712 Spring 2015 -- Lecture 22 1103/20/2015

Electromagnetic waves from time harmonic sources – continued:

:function Hankel Spherical

:function Bessel Spherical

'ˆˆ'4

:expansion Useful

*'

krinkrjkrhkrj

YYkrhkrjike

lll

l

lmlmllm

l

ik

rrrr

rr

'ˆ,'~',~

ˆ,~,~,~

*30

0

rrJa

rarArA

lmlllm

lmlm

lm

Ykrhkrjrdikr

Yr

PHY 712 Spring 2015 -- Lecture 22 1203/20/2015

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

PHY 712 Spring 2015 -- Lecture 22 1303/20/2015

Electromagnetic waves from time harmonic sources – continued:

'ˆ,'~',~

ˆ,~,~,~

*3

0

0

rr

rrr

lmlllm

lmlm

lm

Ykrhkrjrdikr

Yr

'ˆ,'~',~

ˆ,~,~,~

*30

0

rrJa

rarArA

lmlllm

lmlm

lm

Ykrhkrjrdikr

Yr

'ˆ','~',~

'ˆ','~',~source) of(extent For

*30

*3

0

rrJa

rr

lmlllm

lmlllm

Ykrjrdkrhikr

Ykrjrdkrhikr

r

PHY 712 Spring 2015 -- Lecture 22 1403/20/2015

Electromagnetic waves from time harmonic sources – continued:

'ˆ','~',~

'ˆ','~',~source) of(extent For

*30

*3

0

rrJa

rr

lmlllm

lmlllm

Ykrjrdkrhikr

Ykrjrdkrhikr

r

'ˆ'','~'

'ˆ','~',~0,~,~ :condition continuity

the viaconnected are ,'~ and ,'~ that Note

*3

0

*3

0

rrJ

rr

rJr

rJr

lmll

lmlllm

Ykrjrdkrhk

Ykrjrdkrhikr

i

PHY 712 Spring 2015 -- Lecture 22 1503/20/2015

Electromagnetic waves from time harmonic sources – continued:

'ˆ,'~',~

'ˆ3

','~',~

:expansions in onscontributi trivial)-(nonLowest !!12

'' 1'

1

:ionsapproximat Various

00*3

000

1*3

01

1

rrJa

rr

Yrdkreiikr

Ykrrdkreikr

llkrkrjkr

kreikrhkr

ikr

m

ikr

m

l

l

ikrl

l

PHY 712 Spring 2015 -- Lecture 22 1603/20/2015

Electromagnetic waves from time harmonic sources – continued:

r

ekriωi

reωi

rdi

rdω

ikr

ikr

1ˆ4

,~4

,~

,~ 1,~

:frequency at moment dipole Define :radiation dipole on;contributiorder Lowest

0

0

33

rpr

prA

rJrrp

Note: in this case we have assumed a restricted extent of the source such that kr’<<1.

PHY 712 Spring 2015 -- Lecture 22 1703/20/2015

Electromagnetic waves from time harmonic sources – continued:

2

0

02

42

*

0

22

22

0

22

0

ˆˆ32

,~,~ˆ2

ˆˆ

:1for radiatedPower

11ˆ4

1

,~,~

1ˆˆ3ˆˆ

41

,~,~,~

rpr

rBrErrSr

pr

rArB

pprrrpr

rArrE

ωkc

rrddP

krikr

ωkr

ec

ikrr

ωωωkr

e

i

avg

ikr

ikr

PHY 712 Spring 2015 -- Lecture 22 1803/20/2015

Example of dipole radiation source

RrRr eRi

JeJ /0/0 cos,~ ˆ,~

rzrJ

''cos ,~

'' ˆ,~

10

1/'2

0

0

00

0/'2

00

krjkrhedrrRkJ

krjkrhedrrikJ

Rr

Rr

r

zrA

RkR

kri

rekJ

RkR

reJ

Rr

ikr

ikr

222

3

0

0

222

3

00

121 cos ,~

12ˆ,~

:for Evaluation

r

zrA

PHY 712 Spring 2015 -- Lecture 22 1903/20/2015

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

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