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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?
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