UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 17
Prof. Steven Errede
LECTURE NOTES 17
MULTIPOLE EXPANSION OF THE MAGNETIC VECTOR POTENTIAL A ( r )
As we saw in the case of electrostatics, we carried out a multipole expansion of the scalar
∞
electrostatic potential V ( r ) = ∑ Vn ( r ) that was valid for distant observation points (field points)
n=0
P ( r ) far from a localized electrostatic source charge density distribution ρTOT ( r ′ ) , which in turn
∞
enabled us to a corresponding solution for E ( r ) = ∑ En ( r ) via E ( r ) = −∇V ( r ) .
n =0
∞
1
n=0
4πε o
V ( r ) = ∑ Vn ( r ) =
∞
1
∑ r ( ) ∫ ( r ′) P ( cos Θ′) ρ ( r ′) dτ ′
n=0
n
n +1
with: cos Θ′ = rˆirˆ′ and r = r
n
v′
r′ = r′
Likewise, we can similarly/analogously carry out the same kind of multipole expansion for
∞
the magnetic vector potential A ( r ) = ∑ An ( r ) , obtaining an expression for the magnetic vector
n=0
potential that is valid for distant observation / field points P ( r ) far from a localized
magnetostatic source current density distribution – e.g. a filamenary/line current I ( r ′ ) , a surface
current density K ( r ′ ) , or a volume current density J ( r ′ ) , which Obtaining a solution for A ( r )
∞
then enables us to obtain a corresponding solution for the magnetic field B ( r ) = ∑ Bn ( r ) via
n=0
B (r ) = ∇ × A(r ) .
Thus, we carry out a power series / Taylor series / binomial expansion in r ′ r with r ′
A ( r ) {as we did in the electrostatics case for V ( r ) } where r
r for
(
r ′ ) is the distance from the
origin (located near to the charge / current source distribution). For r r ′ , the multipole
moment expansion will be dominated by the lowest-order non-vanishing multipole; higher-order
terms in the expansion can be neglected/ignored.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
1
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 17
Prof. Steven Errede
Suppose we have a filamentary/line current loop, as shown in the figure below:
P ( r ) Observation / Field Point
r′ )
(Drawing not to scale, r
r
I
r ≡ r − r′
r = r = r − r ′ (with r
x̂
ϑ
ẑ Θ′
I
r′ )
ŷ cos Θ′ = rˆirˆ′
r′
d ′ ( = dr ′ )
I
Contour of integration C ′
S ( r ′ ) Current Source Point
As we found before for the case of electrostatics, we can write a power-series expansion of 1/r
(for r r ′ ) as:
n
1
1
1 ∞ ⎛ r′ ⎞
=
= ∑ ⎜ ⎟ Pn ( cos Θ′ ) with cos Θ′ = rˆirˆ′
r
r 2 + r ′2 − 2rr ′ cos Θ′ r n =0 ⎝ r ⎠
Ordinary Legendre′
polynomial of 1st
kind , of order n
Then, for a filamentary/line current source distribution with steady current I:
⎛μ
A(r ) = ⎜ o
⎝ 4π
⎛μ
A(r ) = ⎜ o
⎝ 4π
I ( r ′ ) d ′ ⎛ μo ⎞
d ′ ( r′)
⎞
=
I
⎟ ∫C′
⎜
⎟ ∫
r
⎠
⎝ 4π ⎠ C ′ r
n
⎞ ∞ 1
⎟ I ∑ n +1 ∫ C ′ ( r ′ ) Pn ( cos Θ′ ) d ′ ( r )
⎠ n=0 r
⎛ μ ⎞ ⎧1
A(r ) = ⎜ o ⎟ I ⎨
⎝ 4π ⎠ ⎩ r
∫
C′
d ′ ( r′) +
1
r2
∫
C′
(for I ( r ′ ) = I = constant ∀ r ′ )
with cos Θ′ = rˆirˆ′
r ′ ( cos Θ′ ) d ′ ( r ′ ) +
1
r3
∫ ( r ′)
C′
2
1⎞
⎫
⎛3
2
⎜ cos Θ′ − ⎟ d ′ ( r ′ ) + ...⎬
2⎠
⎝2
⎭
The first term (~ 1/r) in the expansion is the magnetic monopole term, the 2nd term (~ 1/r2) is the
magnetic dipole term, the 3rd term (~ 1/r3) is the magnetic quadrupole term, etc. for the multipole
expansion of the magnetic vector potential A ( r ) .
∞
Thus, we see that: A ( r ) = ∑ An ( r ) where n = order of the magnetic multipole, and:
n=0
⎛μ
An ( r ) = ⎜ o
⎝ 4π
⎛μ
An ( r ) = ⎜ o
⎝ 4π
⎛μ
An ( r ) = ⎜ o
⎝ 4π
2
⎞ 1
⎟ n +1
⎠r
⎞ 1
⎟ n +1
⎠r
⎞ 1
⎟ n +1
⎠r
∫ ( r ′) P ( cos Θ′) I ( r ′) d
n
n
C′
′ for filamentary/line currents I ( r ′ )
∫ ( r ′) P ( cos Θ′) K ′ ( r ′) da′
for surface/sheet current densities K ( r ′ )
∫ ( r′) P ( cos Θ′) J ( r′) dτ ′
for volume current densities J ( r ′ )
n
n
S′
n
v′
n
⊥
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 17
Prof. Steven Errede
The reader of these lecture notes may have already realized that since (empirically) there are
no (N/S) magnetic charges / no magnetic monopoles have been (conclusively / convincingly)
ever observed in our universe, i.e. all magnetic field phenomena arises from (relative) motional
effects of electric charges that the n = 0 term in the multipole expansion of the magnetic vector
potential A ( r ) does not exist in nature. Mathematically we can also see this for the n = 0 term:
A0 ( r ) ≡ 0 because e.g.
∫
C′
d ′ ( r ′ ) ≡ 0 around a closed contour of integration
This is a consequence of Maxwell’s equation ∇i B ( r ) = 0
∞
⇒ A ( r ) = ∑ An ( r )
n =1
Thus, the dominant term for magnetostatics is the (n = 1) magnetic dipole term, e.g. for a
filamentary/line current I ( r ′ ) :
⎛μ ⎞ I
A1 ( r ) = Adipole ( r ) = ⎜ o ⎟ 2 ∫ r ′ cos Θ′d ′ ( r ′ ) with cos Θ′ = rˆirˆ′ and r = rrˆ, r ′ = r ′rˆ′
⎝ 4π ⎠ r C ′
⎛μ ⎞ I
= ⎜ o ⎟ 2 ∫ ( rˆir ′ ) d ′ ( r ′ )
⎝ 4π ⎠ r C ′
⎛μ ⎞ I
= ⎜ o ⎟ 3 ∫ ( r ir ′ ) d ′ ( r ′ )
⎝ 4π ⎠ r C ′
Now if C = any constant vector, then (see Griffiths 1.106, 7 & 8 p. 57):
∫ ( C ir ′ ) d
C′
Where: a ≡ ∫ ′ da =
S
1
2
∫
C′
′ = a × C = −C × a
r ′ × d ′ ( r ′ ) = vector area of the contour loop
a = anˆ where the unit normal n̂ associated with the vector area enclosed
by the contour loop is defined by the right hand rule.
And:
Thus (here): r = C because the observation / field-point P ( r ) (by definition) is a constant
vector, pointing from the defined origin ϑ to the observation / field point P ( r ) .
Then:
∫ ( rˆir ′) d
C′
′ = ∫ da′ × rˆ = −rˆ × ∫ da′
S′
S′
⎛ μ ⎞ m × rˆ
and thus: Adipole ( r ) = ⎜ o ⎟ 2 where: m ≡ I ∫ ′ da′ = Ia = magnetic dipole moment of loop.
S
⎝ 4π ⎠ r
(SI units of m = Amp – m2)
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
3
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 17
Prof. Steven Errede
Griffiths Example 5.13:
Determine the magnetic dipole moment m associated with a “book-end” shaped loop carrying
steady current I as shown in the figure below:
ẑ
w
I I
B
I
I
ϑ
w I
ŷ
I
w
A
I
I
x̂
Use the principle of linear superposition: Superpose two square current loops – one in the x-y
plane (of square side w) and another one in the x-z plane (also of square side w). The side in
common (line segment AB ) to both square loops have currents I flowing in opposite directions,
hence the total current along line segment AB vanishes!
ẑ
I
Square Loop 2
(side w, area A = w × w = w2)
w
I
A
I
ϑ
w I I
I
w
I
B
I
I
Square Loop 1
(side w, area A = w × w = w2)
I
ŷ
I
x̂
m2 = Ia2 = Ia2 nˆ2 = Iw2 yˆ
m1 = Ia1 = Ia1nˆ1 = Iw2 zˆ
I
I
Loop #2
Loop #1
By the principle of linear superposition, the total magnetic dipole moment is:
mtot = m1 + m2 = Ia1 + Ia2
a1 = Ia1nˆ1 = w2 zˆ
mtot = Iw2 zˆ + Iw2 yˆ = Iw2 ( yˆ + zˆ )
a2 = Ia2 nˆ2 = w2 yˆ
by the right-hand rule
mtot = mtot = m12 + m22 = 2 Iw2
4
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 17
Prof. Steven Errede
mtot = Iw2 ( yˆ + zˆ )
ẑ
45o
x̂ out of page
ŷ
Note that (here) the magnetic dipole moments m1 , m2 and mtot are independent of the choice
of origin because the magnetic monopole moment of this magnetic charge distribution is zero.
Recall that the electric dipole moment p associated with an electric charge distribution is
also independent of the choice of origin, but ONLY when the electric monopole moment
(i.e. the net electric charge) associated with that electric charge distribution is zero.
The magnetic dipole moments discussed thus far are obviously for a physical magnetic dipole
– i.e. one with finite spatial extent. A pure / ideal magnetic dipole moment has NO spatial extent
– its area a → 0 while its current I → ∞, keeping the product m = Ia = constant.
For r r ′ , we asymptotically realize the case for an ideal / pure / point magnetic dipole,
e.g. magnetic moments of atoms, molecules, etc. have r ′ few Ǻngstroms (~ few x 10−10 m)
whereas r ~ 1 – few cm typically.
The Magnetic Field Associated with a Magnetic Dipole Moment
It is easiest to first calculate the magnetic vector potential A ( r ) and then calculate the
corresponding magnetic field B ( r ) = ∇ × A ( r ) associated with a magnetic dipole moment m by
choosing (without any loss of generality) to have the origin ϑ at the location of the magnetic
dipole, i.e. place m at r ′ = 0 and also orient the magnetic dipole moment such that m = mzˆ
(i.e. align m ║ to the ẑ -axis).
Then: cos Θ′ = rˆirˆ′ = cos θ (i.e. θ = the usual polar angle) and: r ≡ r − r ′ = r − 0 = r , r
P ( r ) Observation / Field Point
ϕ̂
ẑ
r′
r =r
Θ′ = θ
θ
m
S ( r ′ ) Source Point ϑ
= Local Origin
ŷ
ϕ
ϕ̂
x̂
Note: zˆ = cos θ rˆ − sin θθ
SI units of m = Amp-m2
m = Ia
From the multipole moment expansion of the magnetic vector potential A ( r ) we have:
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
5
UIUC Physics 435 EM Fields & Sources I
⎛μ
Adipole ( r ) = ⎜ o
⎝ 4π
Fall Semester, 2007
Lecture Notes 17
(
⎞ m × rˆ
⎟ 2
⎠ r
m = mzˆ = m cos θ rˆ − sin θθ
(
Prof. Steven Errede
)
)
zˆ × rˆ = cos θ rˆ − sin θθ × rˆ
m × rˆ = m sin Θ where Θ = opening angle between zˆ and rˆ
And:
r̂ × θ = ϕˆ
θ × r̂ = −ϕˆ
rˆ × rˆ = 0
θ × ϕˆ = r̂
ϕˆ × θ = − r̂
θ ×θ = 0
ϕˆ × r̂ = θ
r̂ × ϕˆ = −θ
ϕˆ × ϕˆ = 0
⇐
Very Useful
Table # 2
But Θ = θ here, and thus m × rˆ = m sin θ .
Note that m × rˆ points in the +ϕ̂ direction, because m × rˆ = mzˆ × rˆ = + m sin θϕˆ .
∴
valid for r characteristic
⎛ μ ⎞ m sin θ ˆ
ˆ
Adipole ( r ) = ⎜ o ⎟
A
fcn
r
,
ϕ
=
θ
ϕ
i.e.
(
)
dipole
2
a.
size of m , i.e. r
⎝ 4π ⎠ r
Then: Bdipole ( r ) = ∇ × Adipole ( r ) =
μo
4π
(
⎛m⎞
⎜ 3 ⎟ 2 cos θ rˆ + sin θθ
⎝r ⎠
)
valid for r characteristic
a.
size of m , i.e. r
Compare this result to the electric field of an electric dipole with electric dipole moment p = qd :
1 ⎛ p⎞
valid for r characteristic
Edipole ( r ) =
⎜ 3 ⎟ 2 cos θ rˆ + sin θθ
4πε o ⎝ r ⎠
size of p = qd , i.e. r d .
(
)
They have the same form!!
We can also write Bdipole ( r ) in coordinate-free form by using:
( )
m = mzˆ = m ⎡ cos θ rˆ − m sin θθ ⎤ = ( mirˆ ) rˆ + miθ θ { cos θ = zˆ irˆ and sin θ = zˆ iθˆ }
⎣
⎦
Then:
3 ( mirˆ ) rˆ − m = 3m cos θ rˆ + m sin θθ − m cos θ rˆ
= 2m cos θ rˆ + m sin θθ
Then, in coordinate-free form the magnetic field associated with a physical magnetic dipole
moment, m = Ia is:
μ ⎛1⎞
Bdipole ( r ) = o ⎜ 3 ⎟ ⎡⎣3 ( mirˆ ) rˆ − m ⎤⎦ valid for r characteristic
4π ⎝ r ⎠
a.
size of m , i.e. r
Compare this result to the coordinate-free form of the electric field associated with a physical
electric dipole with dipole moment, p :
Edipole ( r ) =
6
1 ⎛1⎞
valid for r characteristic
⎜ 3 ⎟ ⎡⎣3 ( p irˆ ) rˆ − p ⎤⎦
4πε o ⎝ r ⎠
size of p = qd , i.e. r d .
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 17
Prof. Steven Errede
For completeness’ sake, we give the coordinate-free form of the magnetic field associated
with a point magnetic dipole moment, m :
point
Bdipole
(r ) =
μo ⎛ 1 ⎞
8μ o
mδ 3 ( r )
⎜ 3 ⎟ ⎡⎣3 ( mirˆ ) rˆ − m ⎤⎦ +
4π ⎝ r ⎠
3
n.b. valid for all r.
Note that the δ-function term compensates for the singularity at r = 0 associated with the first
term, and arises from calculating the average magnetic field over an infinitesimally small sphere
of infinitesimal radius ε that entirely contains the current density associated with the magnetic
dipole moment m (See Griffiths Problem 5.59, p. 254). In quantum mechanics, this δ-function
term is responsible for hyperfine splitting of bound electron energy levels in atoms!
Compare this result to the coordinate-free form of the electric field associated with a point
electric dipole moment, p (See P435 Lect. Notes 8, p. 8, and/or Griffiths Problem 3.42, p.157):
point
Edipole
(r ) =
1 ⎛1⎞
1
pδ 3 ( r )
⎜ 3 ⎟ ⎡⎣3 ( p irˆ ) rˆ − p ⎤⎦ −
4πε o ⎝ r ⎠
3ε o
n.b. valid for all r.
where again the δ-function term compensates for the singularity at r = 0 associated with the first
term, and arises from calculating the average electric field over an infinitesimally small sphere of
infinitesimal radius ε that entirely contains the charge densities associated with the electric
dipole moment p .
Creation of a Magnetic Dipole Moment from N & S Magnetic Charges:
We can create a magnetic dipole moment m (at least conceptually) in a manner completely
analogous to that associated with making an electric dipole moment p = qd from two opposite
electric charges +q and −q, but instead using N and S magnetic charges ±g for m = gd :
n.b. SI units of: Ampere-m2 ⇒ SI units of magnetic charge, g: Ampere-meters
ẑ
+g (N pole)
m = mzˆ = gd
d
−g (S pole)
We summarize below the magnetic dipole moments associated with filamentary/line,
surface/sheet and volume current densities:
1
2
1
m=
2
1
m=
2
m=
1
I r′ × d ′
C′
2 ∫C′
1
ˆ ′
⊥
∫ S ′ r ′ × K ( r ′) da⊥′ = 2 K ∫ S ′ r ′ × Kda
r′ × I ( r ′) d ′ =
∫
∫
v′
If I = I = constant ∀ r ′
If K = K = constant ∀ r ′
r ′ × J ( r ′ ) dτ ′
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
7
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 17
Prof. Steven Errede
A comparison of the magnetic dipole fields associated with a pure / ideal / point magnetic
dipole moment, m point versus a physical / finite spatial extent magnetic dipole moment
m phys = Ia = I π R 2 zˆ (e.g. a = a = π R 2 for a magnetic dipole loop of radius R)
Pure vs. Physical
Magnetic Dipole
A comparison of the electric dipole fields associated with a pure / ideal / point electric dipole
moment, p point versus a physical / finite spatial extent electric dipole moment p phys = qd
Pure vs. Physical
Electric Dipole
8
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 17
Prof. Steven Errede
The Magnetic Vector Potential Aquad ( r ) and Magnetic Field Bquad ( r )
Associated with a Magnetic Quadrupole Moment Qm
We can create / build a magnetic quadrupole moment using two back-to-back magnetic
dipoles in analogy to how an electric quadrupole was generated from two back-to-back electric
dipoles - i.e. use the principle of linear superposition, e.g. using magnetic charges +gm = N,
−gm = S poles, or using two identical current loops back-to-back to make a linear magnetic
quadrupole:
m1 = + gdzˆ
Magnetic
Dipole #1
“up”
m2 = − gdzˆ
Magnetic
Dipole #2
“down”
ẑ
+gm N
ẑ
+ gm
d
−gm
S
−gm
S
d
(shrink)
−2gm
2S
d
+ gm
d
+gm
N
N
N
SI units of magnetic charge: gm = Ampere-meters
ẑ
m1 = Ia1 = I π R 2 zˆ
I
a1 = π R 2 zˆ
Two identical magnetic dipole loops
carrying opposing equal currents I, each
of radius R and separation distance d = R.
R
I
I
R=d
a2 = π R zˆ
2
R
I
m2 = Ia2 = − I π R 2 zˆ
Then (for r
⎛μ
r ′ ): Aquad ( r ) = ⎜ o
⎝ 4π
⎛μ
=⎜ o
⎝ 4π
⎞ I
⎟ 3
⎠r
⎞ I
⎟ 3
⎠r
∫ ( r ′) P ( cos Θ′) d ′ ( r ′) for line currents,
2
2
C′
∫ ( r′)
C′
2
1⎞
⎛3
2
⎜ cos Θ′ − ⎟ d ′ ( r ′ ) with cos Θ′ = rˆirˆ′
2⎠
⎝2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
9
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 17
Prof. Steven Errede
The Magnetic Quadrupole Moment Tensor (in terms of discrete magnetic charges):
# discrete magnetic charges
Qm ≡
r1 = + dzˆ
g m1 = + g m
r2 = 0 zˆ
g m2 = −2 g m
r3 = − dzˆ
g m3 = + g m
n =3
(
)
1
3ri ri − 1ri 2 g mi
∑
2 i =1
ˆˆ 0 0
xx
ˆˆ 0
Unit Dyadic: 1 = 0 yy
ˆˆ
0 0 zz
2
ri = ri iri (i = 1,2,3)
in
Cartesian
coordinates
(
)
=0 ⎞
ˆˆ − 1
3 zz
1
1
2 ⎛ =0
2
2
ˆˆ
ˆˆ − d 21 = 2 g m d 2
ˆˆ
Thus: Q m = g m 3d zz − d 1 − g m ⎜ 3i0 zz − 01 ⎟ + g m 3d 2 zz
⎟ 2
2
2
2 ⎜⎝
⎠
(
)
charge #1 + g m
at r1 =+ dzˆ
Then for r
r′ :
⎛μ
Aquad ( r ) = ⎜ o
⎝ 4π
(
charge #3 + g m
at r3 =− dzˆ
charge # 2 − 2g m
at r2 = 0 zˆ
3cos 2 Θ′ − 1) ⎛ μo
⎞
2⎛ 1 ⎞(
=⎜
2
g
d
⎟ m ⎜ 3⎟
2
⎝r ⎠
⎠
⎝ 4π
)
⎞
2⎛ 1 ⎞
⎟ 2 g m d ⎜ 3 ⎟ P2 ( cos Θ′ ) cos Θ′ = rˆirˆ′
⎝r ⎠
⎠
Qm = Q m = 2 g m d 2 Amp-meters*meters2 = Amp-meters3
In terms of current loops:
Qm = 2md = 2 Iad = 2π R 2 dI = 2π R 3 I
Qm = 2 g m d 2 or: Qm = 2md = 2 Iad = 2 I (π R 2 ) d = 2 I (π R 3 )
( R = d ) Amp-meters3
{ d = R here}.
Magnetic dipole current loop separation distance, d
Thus for r
⎛ μ ⎞Q
r ′ : Aquad ( r ) = ⎜ o ⎟ 3m P2 ( cos Θ′ ) where cos Θ′ = rˆirˆ′
⎝ 4π ⎠ r
We can also write this in coordinate-free form {valid for r
r ′ , d (R = d here)} as:
ˆˆ − 1 ⎤ ⎫⎪ ⎛ μo ⎞ Qm ⎡ 3cos 2 Θ′ − 1 ⎤ ⎛ μo ⎞ Qm
⎛ μo ⎞ Qm ⎧⎪ ⎡ 3zz
Aquad ( r ) = ⎜
⎥ × rˆ ⎬ = ⎜
⎟ 3 ⎨rˆ × ⎢
⎟ 3 ⎢
⎥ = ⎜ 4π ⎟ r 3 P2 ( cos Θ′ )
2
⎝ 4π ⎠ r ⎪⎩ ⎣ 2 ⎦ ⎭⎪ ⎝ 4π ⎠ r ⎣
⎠
⎦ ⎝
We can obtain Bquad ( r ) from: Bquad ( r ) = ∇ × Aquad ( r ) (…an exercise for the energetic student...)
10
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 17
Prof. Steven Errede
Thus, we can write the multipole expansion of the magnetic vector potential A ( r ) as:
∞
⎛ μ ⎞ mˆ × rˆ ⎛ μ
A ( r ) = ∑ An ( r ) = ⎜ o ⎟ 2 + ⎜ o
⎝ 4π ⎠ r
⎝ 4π
n =1
magnetic dipole
term n =1
⎞ rˆ × Q m × rˆ
+ ....
⎟
r3
⎠
magnetic quadrupole
term n = 2
Once A ( r ) is determined, we can obtain B ( r ) from B ( r ) = ∇ × A ( r ) .
We can write the magnetic quadrupole tensor Q m as:
⎛ Qm
⎜ xx
Q m = ⎜ Qmxy
⎜
⎜ Qm
⎝ xz
Qmyx
Qmyy
Qmyz
Qmzx ⎞
⎟
Qmzy ⎟
⎟
Qmzz ⎟
⎠
Note that (as for the electric quadrupole moment tensor Q e ) the magnetic quadrupole moment
tensor Q m has only six independent components because Qmij = Qm ji and also note that
Qmxx + Qmyy + Qmzz = 0 i.e. Q m (like Q e ) is traceless.
For a linear magnetic quadrupole (oriented along the ẑ -axis:
Qmxx = Qmyy and thus: Qmzz = −2Qmxx = −2Qmyy = 2 g m d 2
Thus, the magnetic quadrupole tensor for a linear magnetic quadrupole is of the form:
linear
Qm
or:
linear
Qm
⎛ −1 0 0 ⎞
⎜
⎟
= 2 g m d ⎜ 0 −1 0 ⎟ ⇐
⎜ 0 0 2⎟
⎝
⎠
⎛ −1 0 0 ⎞
⎜
⎟
= 2 I ( π R 3 ) ⎜ 0 −1 0 ⎟ ⇐
⎜ 0 0 2⎟
⎝
⎠
2
For a linear magnetic quadrupole
oriented along the ẑ -axis
consisting of magnetic charges g m .
For two back-to-back magnetic
dipole loops carrying steady current
I separated by a vertical distance d.
Compare these results for the magnetic quadrupole to that for the electric quadrupole
(P435 Lecture Notes 8, p. 13-15)
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
11
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 17
Prof. Steven Errede
Another Kind of Magnetic Quadrupole Using Four Bar Magnets:
12
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.