phy 712 electrodynamics 10-10:50 am mwf olin 107 plan for lecture 7: start reading chapter 3
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PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 7: Start reading Chapter 3 Solution of Poisson equation in for special geometries – Cylindrical Spherical. Solution of the Poisson/Laplace equation in various geometries -- cylindrical geometry with z -dependence. f. r. - PowerPoint PPT PresentationTRANSCRIPT
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
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
frfr
frrr
rr
,,
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 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
PHY 712 Spring 2014 -- Lecture 7 501/31/2014
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
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
zVLz
mnm
mnmnmnmnmmn frfr
frfrfr
kr
)( rkJm
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
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
PHY 712 Spring 2014 -- Lecture 7 801/31/2014
mnmnmmn m
Lzn
LnIAz
zzVza
,
sinsin),,(
boundariesother allon 0),,(),(),,(
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
PHY 712 Spring 2014 -- Lecture 7 901/31/2014
ff
fff
ff
frfr
frffr
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
Lzndzmd
LanI
zVL
zndzmdA
αzV
mL
znL
nIAz
zzVza
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
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
PHY 712 Spring 2014 -- Lecture 7 1101/31/2014
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
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
)(xJaρ/a)(xρ/a)J(xJρdρ
xJx
muJum
dud
udud
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
cossinsincossin
rzryrx
PHY 712 Spring 2014 -- Lecture 7 1501/31/2014
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
PHY 712 Spring 2014 -- Lecture 7 1601/31/2014
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
PHY 712 Spring 2014 -- Lecture 7 1701/31/2014
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
PHY 712 Spring 2014 -- Lecture 7 1801/31/2014
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
PHY 712 Spring 2014 -- Lecture 7 1901/31/2014
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
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,φYrr
l''
124
'1
1
rr
PHY 712 Spring 2014 -- Lecture 7 2101/31/2014
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