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

02/06/2013 PHY 712 Spring 2013 -- Lecture 9 2

02/06/2013 PHY 712 Spring 2013 -- Lecture 9 3

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

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

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:

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:

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

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:

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:

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

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

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

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