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. Cylindrical

B. Spherical

PHY 712 Spring 2014 -- Lecture 7 201/31/2014

PHY 712 Spring 2014 -- Lecture 7 301/31/2014

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

rf

zZQRz

z

,,

0 1

1

0 :equation Laplace

2

2

2

2

2

2

z

PHY 712 Spring 2014 -- Lecture 7 401/31/2014

Cylindrical geometry continued:

kρNkρJ Rm

kd

dR

d

Rd

eQQmd

Qd

ekzkzzZZkdz

Zd

mm

im

kz

, 01

)( 0

),cosh(),sinh()( 0

2

22

2

2

22

2

22

2

rf

z

:ypossibilit One

,,

0 :equation Laplace 2

zZQRz

PHY 712 Spring 2014 -- Lecture 7 501/31/2014

Cylindrical geometry continued:

kρKkρI Rm

kd

dR

d

Rd

eQQmd

Qd

ekzkzzZZkdz

Zd

mm

im

ikz

, 01

)( 0

),cos(),sin()( 0

2

22

2

2

22

2

22

2

rf

z

:ypossibilitAnother

,,

0 :equation Laplace 2

zZQRz

PHY 712 Spring 2014 -- Lecture 7 601/31/2014

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

0)( where

sin)sinh()(),,(

boundariesother allon 0),,(

),(),,(

,

akJ

mzkkJAz

z

VLz

mnm

mnmnmnmnmmn

kr

)( kJm

m=0

m=1m=2

PHY 712 Spring 2014 -- Lecture 7 701/31/2014

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

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

VkJdmd

A

V

mzkkJAz

z

VLz

PHY 712 Spring 2014 -- Lecture 7 801/31/2014

mnmnmmn m

L

zn

L

nIAz

z

zVza

,

sinsin),,(

boundariesother allon 0),,(

),(),,(

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

kr

)( kIm m=0

m=1 m=2

PHY 712 Spring 2014 -- Lecture 7 901/31/2014

2

0 0

22

2

0 0

,

sincos

),(sincos

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

sinsin),,(

boundariesother allon 0),,(

),(),,(

L

m

L

mn

mn

mnmnmmn

Lzn

dzmdL

anI

zVL

zndzmd

A

αzV

mL

zn

L

nIAz

z

zVza

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

PHY 712 Spring 2014 -- Lecture 7 1001/31/2014

Green’s function for Dirchelet boundary value inside cylindar:

LzS Lz

V

n m mnmnmmn

mnmnmnmmnmim

mnm

Vz

zzGdd

zzzGdzddz

LkakJk

zLkzkkJkJe

πa

zzG

akJ

zza

VLz

'; '

0

12

1

'

2

',''

',,',,','''

4

1

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

1),,(

sinh

sinhsinh'8

)',,',,',(

0)( :zerosfunction Bessel of in termsExpansion

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

),(),,(

PHY 712 Spring 2014 -- Lecture 7 1101/31/2014

Comments on cylindrical Bessel functions

)(),()(

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

0)(11

2

2

2

2

uKuIuF

uiNuJuHuNuJuF

uFu

m

du

d

udu

d

mmm

mmmmmm

m

m=0

J0

K0 I0/50

N0

PHY 712 Spring 2014 -- Lecture 7 1201/31/2014

m=1

J1N1

K1I1/50

PHY 712 Spring 2014 -- Lecture 7 1301/31/2014

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

)(xJa

ρ/a)(xρ/a)J(xJρdρ

xJx

muJu

m

du

d

udu

d

PHY 712 Spring 2014 -- Lecture 7 1401/31/2014

Poisson and Laplace equation in spherical polar coordinates

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

cos

sinsin

cossin

rz

ry

rx

PHY 712 Spring 2014 -- Lecture 7 1501/31/2014

,1,sin

1sin

sin

1

:functions harmonic Spherical

,)(,,

0sin

1sin

sin

111

:,, potential ticelectrostafor equation Laplace

22

2222

2

lmlm

lmlmlm

YllY

YrRr

rr

rr

r

Poisson and Laplace equation in spherical polar coordinates -- continued

PHY 712 Spring 2014 -- Lecture 7 1601/31/2014

Properties of spherical harmonic functions

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

Pl

θ,φY

,φθYθ,φY

δδθ,φYθ,φYθ dθ dφ θ,φYθ,φYdΩ

YY

rr

PHY 712 Spring 2014 -- Lecture 7 1701/31/2014

Useful identity:

lm

*lmlml

l

,φθYθ,φYr

r

l''

12

4

'

11

rr

lm

*lmlml

l

,φθYθ,φYr

r

lrd

rd

r

''12

4''

4

1

'

''

4

1

:for vanishingpotential ticelectrosta

with density charge isolatedfor Example

13

0

3

0

r

rr

rr

r

PHY 712 Spring 2014 -- Lecture 7 1801/31/2014

Example -- continued

r

arQ

ea

Q

r

rdrr

,φθYd

ea

Q

,φθYθ,φYr

r

lrd

ar

ml*

lm

ar

lm

*lmlml

l

/erf

4

''4

4

4'' '

' :Suppose

''12

4''

4

1

0

/'2/331

0

0

2

0

00

/'2/33

13

0

22

22

r

r

rr

PHY 712 Spring 2014 -- Lecture 7 1901/31/2014

Useful identity:

lm

*lmlml

l

,φθYθ,φYr

r

l''

12

4

'

11

rr

0

2

2

2/12

'ˆˆ1

2

1'ˆˆ

2

3'ˆˆ1

1

'ˆˆ21

1

'

1

:proof"" of Elements

ll

l

Pr

r

r

r

r

r

r

r

rr

rr

r

rr

rrrr

rrrr

PHY 712 Spring 2014 -- Lecture 7 2001/31/2014

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,φYr

r

l''

12

4

'

11

rr

PHY 712 Spring 2014 -- Lecture 7 2101/31/2014

Some spherical harmonic functions:

2

1cos

2

3

4

5ˆ

cos sin8

15ˆ

sin32

15ˆ

cos4

3ˆ

sin8

3ˆ

4

1ˆ

220

12

2222

10

11

00

r

r

r

r

r

r

Y

eY

eY

Y

eY

Y

i

i

i