01/31/2014phy 712 spring 2014 -- lecture 71 phy 712 electrodynamics 10-10:50 am mwf olin 107 plan...
TRANSCRIPT
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