02/03/2014PHY 712 Spring 2014 -- Lecture 81 PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 8: Start reading Chapter 4 Multipole moment

PHY 712 Spring 2014 -- Lecture 8 102/03/2014

PHY 712 Electrodynamics10-10:50 AM MWF Olin 107

Plan for Lecture 8:

Start reading Chapter 4

Multipole moment expansion of electrostatic potential –

A. Spherical coordinates

B. Cartesian coordinates



Poisson and Laplace equation in spherical polar coordinates

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

cos

sinsin

cossin

rz

ry

rx

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


,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


Properties of spherical harmonic functions

cos!

!

4

12

:spolynomial Legendre Associated toipRelationsh

cos4

12

:spolynomial Legendre toipRelationsh

''coscos'ˆˆ''

:ssCompletene

sin

)conventionShortley -Condon (standard ,1,

0

*

mllm

ll

lm

*lmlm

mm'll'*

l'm'lm*

l'm'lm

mlm

lm

Pml

mllθ,φY

Pl

θ,φY

,φθYθ,φY

δδθ,φYθ,φYθ dθ dφ θ,φYθ,φYdΩ

YY

rr


)(

!

!1)(

)(11)(

0For

0)(1

11

:equation aldifferenti Legendre Associated

0)(11

:equation aldifferenti Legendre

2/2

2

22

2

xPml

mlxP

xPdx

dxxP

m

xPx

mll

dx

dx

dx

d

xPlldx

dx

dx

d

ml

mml

lm

mmmm

l

ml

l

Legendre and Associated Legendre functions


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


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


Example:

''''''1

12

11

that Suppose

''12

4''

4

1

'

''

4

1

:density charge isolatedfor ,

alueboundary v with potential ticelectrosta of form General

0

121

0

13

0

3

0

rdrrrrdrrr

θ,φYl

θ,φYr

,φθYθ,φYr

r

lrd

rd

r

lm

r

r

lllm

lllm

lm

lmlmlm

lm

*lmlml

l

r

r

r

rr

rr

r


a

r a

r

lm

r

r

lllm

lllm

lm

drrr

θ,φYθ,φYVa

qΦ

ar

drrrdrrr

θ,φYθ,φYVa

qΦ

ar

rdrrrrdrrr

θ,φYl

Φ

ar

arθ,φYθ,φYVa

qr

Va

qx

0

421111

0

0

421111

0

0

121

0

1111

''1

3

8

6

1

For

''''1

3

8

6

1

For

''''''1

12

11

03

8

2

1

Suppose

r

r

r

r

Example:


2

552

0

0

421111

0

2532

0

0

421111

0

1111

cossin6

''1

3

8

6

1

For

cossin6

''''1

3

8

6

1

For

03

8

2

1

Suppose

r

a

Va

q

drrr

θ,φYθ,φYVa

qΦ

ar

rarVa

q

drrrdrrr

θ,φYθ,φYVa

qΦ

ar

ar

arθ,φYθ,φYVa

qr

Va

qx

a

r a

r

r

r

r

Example -- continued:


Example -- continued:

2

552

0

2532

0

cossin6

For

cossin6

For

r

a

V

qΦ

ar

rarVa

qΦ

ar

r

r


Notion of multipole moment:

)ρ(rrrrdQ

x,y,z)(i,jQ

)ρ(rd

)ρ(rdq

q

)ρ(Yrrdq

)ρ(q

ijji

ij

lml

lm

lm

''''3 '

: componentsmoment quadrupole thedefine

'' '

:moment dipole thedefine

' '

:moment monopole thedefine

--tion representaCartesian In the

'','' '

:on distributi charge (confined) theof moment thedefine

--tion representa harmonic spherical In the

23ij

3

3

*3

r

rrp

p

r

r

r


Significance of multipole moments

lml

lmlm

*lm

l

lml

lm

lm

*lmlml

l

r

θ,φY

l

q

,φθYrrdr

θ,φY

l

r

,φθYθ,φYr

r

lrd

rd

r

10

0

31

0

13

0

3

0

12

4

4

1

'''''12

4

4

1

: ofextent theoutside For

''12

4''

4

1

'

''

4

1

:density charge isolatedfor ,

alueboundary v with potential ticelectrosta of form general Recall

rr

r

r

rr

rr

r


Relationship between spherical harmonic and Cartesian forms of multipole moments:

z

yx

pq

ippq

qq

4

3

8

3

4

1

10

11

00

zz

yzxz

yyxyxx

Qq

iQQq

QiQQq

16

5

72

15

2288

15

20

12

22


58

3

2

1

53

8

2

1 : thatNote

5

2cossin

653

8

6

1

''1

3

8

6

1

for that showed previously We

03

8

2

1

5

1111

5

11

2

5

02

5

11110

0

421111

0

1111

a

Va

qqqp

a

Va

qq

r

a

V

q

r

aθ,φYθ,φY

Va

q

drrr

θ,φYθ,φYVa

qΦ

ar

ar

arθ,φYθ,φYVa

qr

Va

qx

x

a

r

r

Consider previous example:


ji

jiij

lml

lmlm

*lm

l

lml

lm

r

rrQ

rr

q

r

θ,φY

l

q

,φθYrrdr

θ,φY

l

r

,53

0

10

31

0

2

1

4

1

:expansionCartesian of In terms

12

4

4

1

'''''12

4

4

1

: ofextent theoutside For

rp

r

rr

r

General form of electrostatic potential in terms of multipole moments:


Example of multipole expansion in evaluating energy of a very localized charge density r(r) in a electrostatic field F(r) (such as an nucleus in the field due to the electrons in an atom).

ji jiij

rr

rrQq

rd

rdW

,

2

0

2

0

3

3

0

6

100

2

10

Ep

rrrrr

rr


Simple examples of multipole distributions

x

y

z

x

y

z

qdp

ddq

z 2

ˆˆ 33

zrzrr

yyxxzz QQqdQ

ddq

21

212

333

4

2ˆˆ

rzrzrr


Another example of multipole distribution

,,

,,

,5

4

3

2,

1

4

3

2,

5

4

3

2

3

2sin

sin2

31

2

1cos

2

3,

5

4 : thatNote

sin64

20200000

20200000

2000202

2220

2/2

3

YrYr

YrYr

YYY

Y

ea

r

a

q ar

r

r

r

''''''

1

12

4

4

10

121

0

rdrrrrdrrrl lm

r

r

lllm

lllm

645

4

3

2

644

3

2 /2

320/

2

300arar e

a

r

a

qre

a

r

a

qr


Another example of multipole distribution -- continued

5

5

3

4

3

3

2

2/

3

2

020

3

3

2

2/

000

144246211

5

4

4

6

2444

3114

4

1

a

r

a

r

a

r

a

r

a

re

r

qar

a

r

a

r

a

re

r

qr

ar

ar

cos61

4

:spolynomial Legendrefor in terms ;For

23

2

0

Pr

a

r

q

r

r

cos1204

1

4

:spolynomial Legendrefor in terms ;0For

23

2

0

Pa

r

a

q

r

r



cos1204

1

4

:spolynomial Legendrefor in terms ;0For

23

2

0

Pa

r

a

q

r

r

ji ji

ij rrQW

,

2 0

6

1

:ninteractio quadrupole electricfor nsImplicatio

3

02

2

2

2

2

2

3

222

0

120

1

4

0

2

100

240

2

4

1

4

scoordinateCartesian of in terms ;0For

a

q

zyx

a

yxz

a

q

r

r



2

2

2

2

2

2

,

2 000

6

10

6

1

:ninteractio quadrupole Electric

zQ

yQ

xQ

rrQW zzyyxx

ji jiij

30

2

2404

2

1

2

1 nuclei, symmetricFor

a

QqW

QQQqQ yyxxzz

Documents

02/03/2014PHY 712 Spring 2014 -- Lecture 81 PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 8: Start reading Chapter 4 Multipole moment