MULTIPOLE EXPANSION

Class Activities: Multipole

Dipole moment - off center

x

r1

r2

d

+q

-q

p = qiri= ?

i

å

= +qd

MD6.1

The dipole moment,

Dipole moment - off center

x

y

r1

r2

d

+q

-q i i

i

q r ?=å

A) + qd

B) - qd

C) zero

D) None of these, it's more complicated now!

MD6.1

Which of the following is correct (and "coordinate

free")?

A) B)

C) D)

E) None of these

A small dipole (dipole moment p=qd) points

in the z direction.

We have derived V(r ) »1

4pe0

qd cosq

r2=

1

4pe0

qd z

r3

 

V (

r ) =1

4pe0

p × ˆ r

r2

 

V (

r ) =1

4pe0

p ×

r

r2

 

V (

r ) =1

4pe0

p ́ ˆ r

r2

3.22

a

 

V (

r ) =1

4pe0

p × ˆ r

r3

An ideal dipole (tiny dipole moment p=qd)

points in the z direction.

We have derived

 

E (

r ) =

p

4pe0r3

2cosq ˆ r + sinq

q ( )

Sketch this E field...

(What would change if the dipole

separation d was not so tiny?)

3.22

b

E(r ) =p

4pe0r3

2cosq r̂ + sinq q̂( )

Sketch this E field…

3.22

b

For a dipole at the origin pointing in the z-direction, we

have derived

( )dip 3

0

p ˆˆ(r) 2cos sin4 r

= q + q qpe

E r

For the dipole p = q d shown, what does the

formula predict for the direction of E(r=0)?

MD6 - 2

x

z

+

-

d

A)Down B) Up C) some other direction

D) The formula doesn't apply.

An ideal dipole (tiny dipole moment p=qd)

points in the z direction.

We have derived

 

E (

r ) =

p

4pe0r3

2cosq ˆ r + sinq

q ( )

(What would change if the dipole

separation d was not so tiny?)

3.22

b

StreamPlot[

{(3*x*y)/(Sqrt[x^2 + y^2])^5, (2*y*y – x*x)/(Sqrt[x^2 + y^2])^5},

{x, -2, 2}, {y, -2, 2}]

E(r ) =p

4pe0r3

2cosq r̂ + sinq q̂( )

What is the magnitude of the dipole

moment of this charge distribution?

A) qd

B) 2qd

C) 3qd

D) 4qd

E) It's not determined

(To think about: How does V(r) behave as |r| gets large?)

3.27 p = qiri

i

å

Dipole moment - off center

d

+2q

-q

p = qiri

i

åMD6.1

What is the dipole moment of this system?

(Note: it is NOT overall neutral!)

A) qd

B) 2qd

C)3

2qd

D) 3 qd

E) Something else

(or, not defined)!

x 0

Dipole moment - off center

d/2

d

+2q

-q

p = qiri

i

åMD6.1

What is the dipole moment of this system?

(Note: same as last question, just shifted in z!)

A) qd

B) 2qd

C)3

2qd

D) 3 qd

E) Something else

(or, not defined)!

d/2

0 x

r1

r2

You have a physical dipole, +q and -q

a finite distance d apart.

When can you use the expression:

A) This is an exact expression everywhere.

B) It's valid for large r

C) It's valid for small r

D) ?

 

V (

r ) =1

4pe0

p × ˆ r

r2

3.22

c

You have a physical dipole, +q and -q,

a finite distance d apart.

When can you use the expression

A) This is an exact expression everywhere.

B) It's valid for large r

C) It's valid for small r

D) ?

3.22

d

Which charge distributions below produce

a potential which looks like C/r2 when you

are far away?

E) None of these, or more than one of these!

(Note: for any which you did not select, how

DO they behave at large r?)

3.25

Which charge distributions below produce

a potential which looks like C/r2 when you

are far away?

E) None of these, or more than one of these!

(Note: for any which you did not select, how

DO they behave at large r?)

3.26

In which situation is the dipole term

the leading non-zero contribution to

the potential?

 

r(r)

A) A and C

B) B and D

C) only E

D) A and E

E) Some other combo

 

+r0

s =s 0 cos(q)

3.28

Electric properties of matter

No flies were harmed in the process

V(r) =1

4peo

r(r ')dt '

Âòòò

x

z

-d l

x

z

-d l

V(z) =1

4peo

l dz '

Âò

V(z) =1

4peo

l dz '

Âò

x

z

-d

dz’ z’

l

Â

V(z) =1

4peo

l dz '

Âò

=if z>0

1

4pe0

ldz '

(z - z ')z'=-d

z'=0

ò

=l

4pe0

(-)ln(z - z ') |-d

0

=l

4pe0

ln(z + d

z)

x

z

-d l

dz’ z’

Â

Let me know how far along you are:

A) DONE with page 1

B) DONE with page 2

C) DONE with page 3!

On paper (don’t forget your name!) in your

own words (by yourself):

What is the idea behind the multipole

expansion? What does it accomplish?

In what limits/cases is it useful?