MULTIPOLE EXPANSION
Class Activities: Multipole
MD6.1
The dipole moment,
p = å q i ri = ?
Dipole moment - off center
i
+q
= +q d
r1
d
r2
-q
x
MD6.1
+q
Dipole
moment - off center
d
-q
r1
y
åq
r =?
i i
i
r2
x
A) + q d
B) - q d
C) zero
D) None of these, it's more complicated now!
3.22
a
A small dipole (dipole moment p=qd) points
in the z direction.
1 qd cosq
1 qd z
=
We have derived V(r ) »
2
3
4pe0
4pe0 r
r
Which of the following is correct (and "coordinate
free")?
p × rˆ
A) V ( r ) =
4 pe0 r 2
1
C) V ( r ) =
1
4 pe0
p× r
r
2
E) None of these
B) V ( r ) =
1
p × rˆ
4 pe0
3
D) V ( r ) =
r
1
p ´ rˆ
4 pe0
2
r
3.22
b
An ideal dipole (tiny dipole moment p=qd)
points in the z direction.
We have derived E( r ) =
p
4pe0 r
3
(2cosq rˆ + sinq q )
Sketch this E field...
(What would change if the dipole
separation d was not so tiny?)
3.22
b
E(r ) =
p
4pe0 r
3
(
2 cosq r̂ + sinq qˆ
Sketch this E field…
)
MD6 - 2
For a dipole at the origin pointing in the z-direction, we
have derived
(
p
ˆ
ˆ
Edip (r) =
2cos
q
r
+
sin
q
q
4pe0 r 3
)
z
For the dipole p = q d shown, what does the
formula predict for the direction of E(r=0)?
A) Down
B) Up
+
d
C) some other direction
D) The formula doesn't apply.
x
-
3.22
b
An ideal dipole (tiny dipole moment p=qd)
points in the z direction.
We have derived E( r ) =
p
4pe0 r
3
(2cosq rˆ + sinq q )
(What would change if the dipole
separation d was not so tiny?)
E(r ) =
p
4pe0 r 3
(
2 cosq r̂ + sinq qˆ
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}]
)
3.27
p = å q i ri
i
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?)
p = å q i ri
MD6.1
Dipole moment - off
i center
What is the dipole moment of this system?
(Note: it is NOT overall neutral!)
+2q
A) q d
d
0
-q
x
B) 2q d
3
C) q d
2
D) 3 q d
E) Something else
(or, not defined)!
p = å q i ri
MD6.1
Dipole moment - off
i center
What is the dipole moment of this system?
(Note: same as last question, just shifted in z!)
+2q
d/2
r1
0
r2
d/2
-q
A) q d
d
x
B) 2q d
3
C) q d
2
D) 3 q d
E) Something else
(or, not defined)!
3.22
c
You have a physical dipole, +q and -q
a finite distance d apart.
When can you use the expression:
1 p × rˆ
V (r ) =
2
4 pe0 r
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
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) ?
3.25
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
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?)
3.28
In which situation is the dipole term
the leading non-zero contribution to
the potential?
A) A and C
B) B and D
+ r0 r(r)
C) only E
D) A and E
E) Some other combo
s = s 0 cos(q )
Electric properties of matter
No flies were harmed in the process
V(r) =
1
4peo
òòò
r (r ')dt '
z
Â
x
l
-d
V(z) =
1
4peo
ò
l dz'
z
Â
x
l
-d
V(z) =
1
4peo
ò
l dz'
z
Â
Â
dz’
z’
l
-d
x
V(z) =
=
if z>0
1
4peo
1
ò
l dz'
z
Â
z'=0
ò
Â
l dz'
4pe 0 z'=-d (z - z')
l
0
=
(-)ln(z - z') |-d
4pe0
l
z+d
=
ln(
)
4pe0
z
dz’
z’
l
-d
x
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?