PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 7: Start reading Chapter 3

PHY 712 Spring 2014 -- Lecture 7 101/31/2014

PHY 712 Electrodynamics10-10:50 AM MWF Olin 107

Plan for Lecture 7:Start reading Chapter 3

Solution of Poisson equation in for special geometries –

A. CylindricalB. Spherical



Solution of the Poisson/Laplace equation in various geometries -- cylindrical geometry with z-dependence

rf

zZQRz

z

frfr

frrr

rr

,,

0 1 1

0 :equation Laplace

2

2

2

2

2

2

z


Cylindrical geometry continued:

kρNkρJ RmkddR

dRd

eQQmd

Qd

ekzkzzZZkdz

Zd

mm

im

kz

, 01

)( 0

),cosh(),sinh()( 0

2

22

2

2

22

2

22

2

rrrr

ff

f

rf

z

:ypossibilit One

,, 0 :equation Laplace 2

zZQRz frfr


Cylindrical geometry continued:

kρKkρI RmkddR

dRd

eQQmd

Qd

ekzkzzZZkdz

Zd

mm

im

ikz

, 01

)( 0

),cos(),sin()( 0

2

22

2

2

22

2

22

2

rrrr

ff

f

rf

z

:ypossibilitAnother

,, 0 :equation Laplace 2

zZQRz frfr


Solutions of Laplace equation inside cylindrical shapeExample with non-trivial boundary value at z=L

0)( where

sin)sinh()(),,(

boundariesother allon 0),,(),(),,(

,

akJ

mzkkJAz

zVLz

mnm

mnmnmnmnmmn frfr

frfrfr

kr

)( rkJm

m=0

m=1m=2


Solutions of Laplace equation inside cylindrical shapeExample with non-trivial boundary value at z=L

rrrff

frrrrff

ffr

frfrfr

frfr

2

0 0

22

2

0 0

,

)(cos)sinh(

),()(cos

:2/ that so offunction symmetric is ),( If

sin)sinh()(),,(boundariesother allon 0),,(

),(),,(

a

mnmmn

a

mnm

mn

mn

mnmnmnmnmmn

kJdmdLk

VkJdmdA

V

mzkkJAzz

VLz


mnmnmmn m

Lzn

LnIAz

zzVza

,

sinsin),,(


frfr

frffr

Solutions of Laplace equation inside cylindrical shapeExample with non-trivial boundary value at r=a

kr

)( rkIm m=0

m=1 m=2


ff

fff

ff

frfr

frffr

2

0 0

22

2

0 0

,

sincos

),(sincos

:/2 that so offunction symmetric a is ),( If

sinsin),,(


L

m

L

mn

mn

mnmnmmn

Lzndzmd

LanI

zVL

zndzmdA

αzV

mL

znL

nIAz

zzVza

Solutions of Laplace equation inside cylindrical shapeExample with non-trivial boundary value at r=a


Green’s function for Dirchelet boundary value inside cylindar:

LzS Lz

V

n m mnmnmmn

mnmnmnmmnmim

mnm

Vz

zzGdd

zzzGdzddz

LkakJkzLkzkkJkJe

πa

zzGakJ

zzaVLz

'; '

0

12

1

'

2

',''

',,',,','''41

',','',,',,',''''4

1),,(

sinh

sinhsinh'8

)',,',,',(0)( :zerosfunction Bessel of in termsExpansion

0)0,,( , 0),,(),(),,(

frffrrrrf

frrffrrrrf

fr

rr

ffrr

frfrfrfr

ff


Comments on cylindrical Bessel functions

)(),()(

)()()(),(),()(

0)(112

2

2

2

uKuIuF

uiNuJuHuNuJuF

uFum

dud

udud

mmm

mmmmmm

m

m=0

J0

K0 I0/50

N0


m=1

J1N1

K1I1/50


Some useful identities involving cylindrical Bessel functions

'2

1

2

'0

2

2

2

2

2

0) ( :zeros of in terms functions Bessel of Properties

integer for 0)(11

nnmnmmnmmnm

a

mnmmn

m

)(xJaρ/a)(xρ/a)J(xJρdρ

xJx

muJum

dud

udud


Poisson and Laplace equation in spherical polar coordinates

http://www.uic.edu/classes/eecs/eecs520/textbook/node32.html

cossinsincossin

rzryrx

http://www.uic.edu/classes/eecs/eecs520/textbook/node32.html


fff

ff

f

f

,1,sin

1sinsin

1

:functions harmonic Spherical

,)(,,

0sin

1sinsin

111

:,, potential ticelectrostafor equation Laplace

22

2222

2

lmlm

lmlmlm

YllY

YrRr

rr

rr

r

Poisson and Laplace equation in spherical polar coordinates -- continued


Properties of spherical harmonic functions

ff

ff

cos4

12

:spolynomial Legendre toipRelationsh

''coscos'ˆˆ''

:ssCompletene

sin

)conventionShortley -Condon (standard ,1,

0

*

ll

lm

*lmlm

mm'll'*

l'm'lm*

l'm'lm

mlm

lm

Plθ,φY

,φθYθ,φY

δδθ,φYθ,φYθ dθ dφ θ,φYθ,φYdΩ

YY

rr


Useful identity:

lm

*lmlml

l

,φθYθ,φYrr

l''

124

'1

1

rr

lm

*lmlml

l

,φθYθ,φYrr

lrd

rd

r

''12

4''4

1

'''

41

:for vanishingpotential ticelectrosta with density charge isolatedfor Example

13

0

3

0

r

r

r

r

rrrr

r


Example -- continued

r

arQ

ea

Qrrdrr

,φθYd

ea

Q

,φθYθ,φYrr

lrd

ar

ml*

lm

ar

lm

*lmlml

l

/erf4

''44

4'' '

' :Suppose

''12

4''4

1

0

/'2/331

0

0

2

0

00

/'2/33

13

0

22

22

r

r

r

r

rr


Useful identity:

lm

*lmlml

l

,φθYθ,φYrr

l''

124

'1

1

rr

0

22

2/12

'ˆˆ1

21'ˆˆ

23'ˆˆ11

'ˆˆ21

1'

1 :proof"" of Elements

ll

l

Prr

r

rr

rr

r

rr

rrr

rr

rrrr

rrrr


1ˆ12

4

1'ˆˆ,'ˆˆfor that Note

'ˆˆ12

4 'ˆˆ

:harmonics sphericalfor rule Sum :continued -- proof"" of Elements

2

*

Yl

P

YYl

P

l

-lmlm

l

lm

l

-lmlml

r

rrrr

rrrr

Useful identity:

lm

*lmlml

l

,φθYθ,φYrr

l''

124

'1

1

rr


Some spherical harmonic functions:

21cos

23

45ˆ

cos sin815ˆ

sin3215ˆ

cos43ˆ

sin83ˆ

41ˆ

220

12

2222

10

11

00

f

f

f

r

r

r

r

r

r

Y

eY

eY

Y

eY

Y

i

i

i

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PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 7: Start reading Chapter 3