PHY 712 Electrodynamics 11-11:50 AM MWF Olin 107 Plan for...
Transcript of PHY 712 Electrodynamics 11-11:50 AM MWF Olin 107 Plan for...
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02/06/2013 PHY 712 Spring 2013 -- Lecture 9 1
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
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02/06/2013 PHY 712 Spring 2013 -- Lecture 9 2
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02/06/2013 PHY 712 Spring 2013 -- Lecture 9 3
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02/06/2013 PHY 712 Spring 2013 -- Lecture 9 4
General results for a multipole analysis of the electrostatic potential due to an isolated charge distribution:
''''''112
11
that Suppose
''12
4''4
1
'''
41
:density chargeisolatedfor ,alueboundary v with potentialticelectrostaofformGeneral
0
121
0
13
0
3
0
rdrrrrdrrr
θ,φYl
θ,φYr
,φθYθ,φYrr
lrd
rd
r
lm
r
r
lllm
lllm
lm
lmlmlm
lm
*lmlml
l
r
r
r
rrrr
r
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02/06/2013 PHY 712 Spring 2013 -- Lecture 9 5
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 distributicharge(confined) theofmoment thedefine--tion representaharmonicsphericalIn the
23ij
3
3
*3
r
rrp
p
r
r
r
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02/06/2013 PHY 712 Spring 2013 -- Lecture 9 6
ji
jiij
lml
lmlm
*lm
l
lml
lm
rrr
Qrr
q
rθ,φY
lq
,φθYrrdrθ,φY
l
r
,53
0
10
31
0
21
41
:expansionCartesian ofIn terms
124
41
'''''12
44
1:ofextent theoutsideFor
rpr
rr
r
General form of electrostatic potential in terms of multipole moments:
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02/06/2013 PHY 712 Spring 2013 -- Lecture 9 7
rpprprrE
rpr
r
35
2
0
30
343
41
:fieldticElectrosta4
1:potentialticElectrosta
:ofextent theoutsideFor
rr
r
r
Focus on dipolar contributions:
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02/06/2013 PHY 712 Spring 2013 -- Lecture 9 8
Coarse grain representation of macroscopic distribution of dipoles:
iii
iii
q rrrr
rrprPrP
3mono
mono
3
:density chargeelectricMono
:on polarizatiElectric
304
1:dipolesingleaand
chargemonopolesingleafor potentialticElectrosta
rrq
q
rpr
p
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02/06/2013 PHY 712 Spring 2013 -- Lecture 9 9
''''
'1'''
'''':Note
''''
'''
41
:dipolesandchargesmonopoleofscollectionfor potentialticElectrosta
41
:dipolesingleaandchargemonopolesingleafor potentialticElectrosta
333
3
333
0
30
rrrP
rrrP
rrrrrP
rrrrrP
rrrr
p
rpr
p
rdrdrd
rdrd
qrr
q
q
mono
i
i
Coarse grain representation of macroscopic distribution of dipoles -- continued:
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02/06/2013 PHY 712 Spring 2013 -- Lecture 9 10
rrDrPrErD
rrPrE
rPrrEr
rrrP
rrrr
p
mono
mono
mono
mono
i
i
rdrd
q
:lawsGauss'offormcMacroscopi:fieldnt DisplacemeDefine
1
''''
'''
41
:dipolesandchargesmonopoleofscollectionfor potentialticElectrosta
0
0
0
2
33
0
Coarse grain representation of macroscopic distribution of dipoles -- continued:
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02/06/2013 PHY 712 Spring 2013 -- Lecture 9 11
Coarse grain representation of macroscopic distribution of dipoles -- continued:
material theoffunction dielectric therepresents1
:constant ality proportiona with fieldelectrican ofpresencein thefieldon polarizatiaproduceandepolarizablarematerialsMany
e00
e0
e
rErErPrErDrErP
Boundary value problems in the presence of dielectrics
rErrDrr
rErDr
ˆcontinuousˆ:ˆnormalsurfaceofin termss,dielectricobetween twsurfaceaAt
0and00For mono
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02/06/2013 PHY 712 Spring 2013 -- Lecture 9 12
Boundary value problems in the presence of dielectrics – example:
a
0
z
E0E
rrD
rrDrErD
0For For
0and0
arar
rr
rr:At 0 rrar
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02/06/2013 PHY 712 Spring 2013 -- Lecture 9 13
Boundary value problems in the presence of dielectrics – example -- continued:
cosFor
:At
0
0
θrEr
rrar
r
rr
rr
01
0
cos
cos
lll
lll
ll
ll
PrCrB
PrA
r
r
/21/
/23
scontribute1only --Solution
03
0
010
01
01
EaCEA
EBl
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02/06/2013 PHY 712 Spring 2013 -- Lecture 9 14
Boundary value problems in the presence of dielectrics – example -- continued:
cos/2
1/
cos/2
3
02
3
0
0
00
Erar
rE
r
r