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02/05/2014PHY 712 Spring 2014 -- Lecture 91 PHY 712 Electrodynamics 11-11:50 AM MWF Olin 107 Plan...
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Transcript of 02/05/2014PHY 712 Spring 2014 -- Lecture 91 PHY 712 Electrodynamics 11-11:50 AM MWF Olin 107 Plan...
PHY 712 Spring 2014 -- Lecture 9 102/05/2014
PHY 712 Electrodynamics11-11:50 AM MWF Olin 107
Plan for Lecture 9:
Continue reading Chapter 4
Dipolar fields and dielectrics
A. Electric field due to a dipole
B. Electric polarization P
C. Electric displacement D and dielectric functions
PHY 712 Spring 2014 -- Lecture 9 202/05/2014
PHY 712 Spring 2014 -- Lecture 9 302/05/2014
PHY 712 Spring 2014 -- Lecture 9 402/05/2014
PHY 712 Spring 2014 -- Lecture 9 502/05/2014
General results for a multipole analysis of the electrostatic potential due to an isolated charge distribution:
''''''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
PHY 712 Spring 2014 -- Lecture 9 602/05/2014
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
PHY 712 Spring 2014 -- Lecture 9 702/05/2014
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:
PHY 712 Spring 2014 -- Lecture 9 802/05/2014
rpprpr
rE
rpr
r
35
2
0
30
3
4 3
4
1
:field ticElectrosta
4
1
:potential ticElectrosta
: ofextent theoutside For
r
r
r
r
Focus on dipolar contributions:
PHY 712 Spring 2014 -- Lecture 9 902/05/2014
Coarse grain representation of macroscopic distribution of dipoles:
iii
iii
q rrr
r
rrprP
rP
3mono
mono
3
:density charge electric Mono
:on polarizati Electric
3
04
1
: dipole single a and
charge monopole single afor potential ticElectrosta
rr
q
q
rpr
p
PHY 712 Spring 2014 -- Lecture 9 1002/05/2014
'
'''
'
1'' '
'
''' :Note
'
'''
'
''
4
1
: dipoles and
charges monopole of scollectionfor potential ticElectrosta
4
1
: dipole single a and
charge monopole single afor potential ticElectrosta
333
3
333
0
30
rr
rP
rrrP
rr
rrrP
rr
rrrP
rr
rr
p
rpr
p
rdrdrd
rdrd
q
rr
q
q
mono
i
i
Coarse grain representation of macroscopic distribution of dipoles -- continued:
PHY 712 Spring 2014 -- Lecture 9 1102/05/2014
rrD
rPrErD
rrPrE
rPrrEr
rr
rP
rr
rr
p
mono
mono
mono
mono
i
i
rdrd
q
:law sGauss' of form cMacroscopi
:fieldnt Displaceme Define
1
'
'''
'
''
4
1
: dipoles and
charges monopole of scollectionfor potential ticElectrosta
0
0
0
2
33
0
Coarse grain representation of macroscopic distribution of dipoles -- continued:
PHY 712 Spring 2014 -- Lecture 9 1202/05/2014
Coarse grain representation of macroscopic distribution of dipoles -- continued:
material theoffunction dielectric therepresents
1
:constant ality proportion a with field electrican of presence
in the fieldon polarizati a produce and epolarizabl are materialsMany
e00
e0
e
rErErPrErD
rErP
Boundary value problems in the presence of dielectrics
rErrDr
r
rErD
r
ˆ continuousˆ
:ˆ normal surface of in terms s,dielectric obetween tw surface aAt
0 and 0
0For mono
PHY 712 Spring 2014 -- Lecture 9 1302/05/2014
Boundary value problems in the presence of dielectrics – example:
a
e e0
z
E0Eq
rrD
rrD
rErD
0 For
For
0 and 0
ar
ar
rr
rr
:At 0 rrar
PHY 712 Spring 2014 -- Lecture 9 1402/05/2014
Boundary value problems in the presence of dielectrics – example -- continued:
cos For
:At
0
0
θrEr
rrar
r
rr
rr
01
0
cos
cos
lll
lll
ll
ll
Pr
CrB
PrA
r
r
/2
1/
/2
3
scontribute 1only --Solution
03
0
010
01
01
EaCEA
EB
l
PHY 712 Spring 2014 -- Lecture 9 1502/05/2014
Boundary value problems in the presence of dielectrics – example -- continued:
cos /2
1/
cos /2
3
02
3
0
0
00
Er
ar
rE
r
r
Ea
r/a
/e e0=
10 2 1