Magnetic Moment.
Ian R. Gatland, Georgia Institute of Technology
Physics 3122 Notes, July 22, 2010
Preliminaries.
Consider a vector field,
J
(
r
)
, that is localized (zero on and outside a surface
S
surrounding
a volume
Γ
) and divergence free (
∇
⋅
J
=
0
) together with two scalar fields,
f
(
r
) and
g
(
r
)
.
Then
[
f
J
⋅
∇
g
+
Γ
∫
g
J
⋅
∇
f
]
d
τ
=
[
f
J
⋅
∇
g
+
∇
⋅
(
fg
J
)
−
Γ
∫
f
∇
⋅
(
g
J
)]
d
τ
=
∇
⋅
(
fg
J
)
Γ
∫
d
τ
+
[
f
J
⋅
∇
g
−
Γ
∫
fg
∇
⋅
J
−
f
J
⋅
∇
g
]
d
τ
=
fg
J
⋅
S
∫
d
a
−
fg
∇
⋅
J d
τ
=
0
Γ
∫
(1)
using the localization and divergence conditions.
Case 1: Equation (1) with
f
= 1 and
g
=
x
provides
[
J
⋅
∇
x
+
Γ
∫
x
J
⋅
∇
1]
d
τ
=
[
J
⋅
ˆ
x
+
Γ
∫
0]
d
τ
=
J
x
Γ
∫
d
τ
=
0
(2)
and similar results with
y
and
z
in place of
x
.
Case 2: Equation (1) with
f
=
x
and
g
=
x
provides
[
x
J
⋅
∇
x
+
Γ
∫
x
J
⋅
∇
x
]
d
τ
=
2
xJ
x
Γ
∫
d
τ
=
0
(3)
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 Fall '08
 Staff
 Physics

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