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


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


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


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:


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:


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


'

'''

'

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:


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




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


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


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


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

02/05/2014phy 712 spring 2014 -- lecture 91 phy 712 electrodynamics 11-11:50 am mwf olin 107 plan...

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