Some aspects of Magnetotistatics
Physics 212 2010, Electricity and Magnetism
Michael Dine
Department of Physics
University of California, Santa Cruz
November 2010
Physics 212 2010, Electricity and Magnetism
Some aspects of Magnetotistatics
The complication in the case of magnetism is the vector
~ and ~J. So the multipole expansion is more
character of A
complicated to work out. We will describe a general approach
which allows a systematic multipole expansion. But we begin
~ in Cartesian
by simply expanding our expression for A
coordinates:
Z
1 ~ 0
1
~
~
(1)
d 3x 0
J(~x ).
A(x ) =
~
c
|x − ~x 0 |
This expression requires some massaging to obtain a
~ · ~J = 0. It
transparent expression. Critical is that, for statics, ∇
is helpful to write this equation with components for the different
indices.
Physics 212 2010, Electricity and Magnetism
Some aspects of Magnetotistatics
Z
1
1
d 3x 0
Ji (~x 0 )
Ai (~x ) =
c
|~x − ~x 0 |
Z
~x · ~x 0
1
≈
d 3 x 0 (1 + 2 )Ji (~x 0 ).
cr
r
The first term vanishes. This follows by writing
Z
Z
3 0
d x Ji = d 3 x 0 ∂k0 (xi0 Jk ) = 0
where the first step follows by performing the differentiation,
using current conservation, and the last follows from the
assumption that the current distribution is localized, plus
Gauss’s theorem.
Physics 212 2010, Electricity and Magnetism
Some aspects of Magnetotistatics
(2)
(3)
The second term may be rewritten in a form which involves the
dipole moment. We start by noting that
Z
d 3 x 0 (xi0 Jj + xj0 Ji ) = 0.
(4)
This follows because the integrand is, in fact, a total derivative;
it is equal to
Z
d 3 x 0 ∂k0 (xi0 xj0 Jk )
where, again, we used current conservation.
Physics 212 2010, Electricity and Magnetism
Some aspects of Magnetotistatics
(5)
~
This allows us to rewrite A:
Z
1
1
Ai (~x ) = 3 d 3 x 0 xi (xi0 Jj − xj0 Ji ).
2
cr
(6)
The last equation has a structure reminiscent of a cross
product. Indeed, whenever we have an anti symmetric tensor,
Fij , we can make a (pseudo) vector,
Vi = ijk Fjk .
(7)
~ field this way, where Fjk = ∂j Ak − ∂k Aj .
We can think of the B
Physics 212 2010, Electricity and Magnetism
Some aspects of Magnetotistatics
We can recover F from B, if we like:
Fij =
Check:
1
ijk Bk
2
1
ijk klm (∂l Am − ∂m Al )
2
= ∂i Aj − ∂j Ai .
Physics 212 2010, Electricity and Magnetism
Some aspects of Magnetotistatics
(8)
~ , where
In the present case, we define the vector, m
~ × ~x
~ =m
A
r3
where
1
~ =
m
2c
Physics 212 2010, Electricity and Magnetism
Z
d 3 x 0 × ~J.
Some aspects of Magnetotistatics
(9)
(10)
For the case of a collection of particles of mass mi , charges qi ,
located at ~xi ,
X
~J(~x ) =
(11)
qi ~vi δ (3) (~x − ~xi ),
we have
1
~ =
m
2c
Z
d 3 x 0~x 0 × ~J(~x 0 )
(12)
1
qi ~xi × ~vi
2c
X qi
~Li .
=
2mi c
This formula is not quite right for spin; in the case of the
electron, for example, the magnetic moment is approximately
ge ~
~ =
m
S.
mc
Here, the Dirac equation gives g = 2, while QED gives
corrections in a power series in α. For the muon, g is measured
so accurately that one is sensitive to corrections from the strong
and weak interactions (it is also calculated as accurately).
=
Physics 212 2010, Electricity and Magnetism
Some aspects of Magnetotistatics
Force on a Dipole; Potential of a Dipole
Because there is not an energy associated with a charge in a
magnetic field, we examine the force on a localized charge
distribution:
Z
1
~
Fi =
d 3 x ~J × B.
(13)
c
~ in a Taylor series about the origin:
Expanding B
Z
Z
1
0
3
0
0
0
3
0
~
Fi = ijk Bk (0) Jj (~x )d x + Jj (~x )~x · ∇Bk (0)d x .
c
(14)
Physics 212 2010, Electricity and Magnetism
Some aspects of Magnetotistatics
As before, first integral vanishes; second we rewrite, again,
antisymmetrizing in indices:
Z
1
(15)
Fi = − ijk (x`0 Jj − xj0 J` )∇` Bk (0)d 3 x 0 .
2c
The expression in partenthesis is jkm mm , so using our
identities, we have:
~
~ · B)
F = ∂i (m
(16)
so we can think of a potential:
~ = −∇U
~
F
Physics 212 2010, Electricity and Magnetism
1
~
~ · B.
U=− m
c
Some aspects of Magnetotistatics
(17)
Some aspects of Magnetism in Materials
~ ~x ).
As for dielectrics, suppose a magnetization density, M(
Then, allowing also for “free" currents we have
"
#
Z
~j(~x 0 )
~ ~x 0 ) × (~x − ~x 0 )
M(
3
0
~ ~x ) = d x
A(
+
(18)
|c~x − ~x 0 |
|~x − ~x 0 |3
"
~j(~x 0 )
1
~ ~x 0 ) × ∇
~0
= d x
+ M(
|c~x − ~x 0 |
|c~x − ~x 0 |
"
#
Z
~J(~x 0 )
~ ~x 0 )
~ 0 × M(
∇
3 0
= d x
+
.
|c~x − ~x 0 |
|c~x − ~x 0 |
Z
3 0
Physics 212 2010, Electricity and Magnetism
Some aspects of Magnetotistatics
#
So the magnetization acts as an effective current density,
~JM = c ∇
~
~ × M.
(19)
~ becomes:
The equation for curl B
~ = 4π ~J + 4π ∇
~
~ ×B
~ × M.
∇
c
(20)
~ ≡B
~ − 4π M,
~
So with H
~ = 4π ~J
~ ×H
∇
c
Physics 212 2010, Electricity and Magnetism
~ = 0.
~ ·B
∇
Some aspects of Magnetotistatics
(21)
~ and H.
~
To solve these equations, we need a relation between B
For most systems, linear,
~ = µH
~
B
(22)
(the origin, in a sense, of the µ0 in Jackson, permeability of the
vacuum).
µ > 1: paramagnetic
µ < 1: diamagnetic Differences typically part in 105 difference
from one.
Ferromagnets: much more complicated; hysteresis (depends
on history of preparation of system).
Physics 212 2010, Electricity and Magnetism
Some aspects of Magnetotistatics
Boundary value problems with magnetic materials
(very brief).
~
Boundary conditions: if no currents, continuity of tangential H,
~
normal B.
~
~
∇ × H = 0, so
~ =Φ
~ M ∇2 Φ = 0
H
(23)
and use methods familiar from electrostatics (see Jackson).
Physics 212 2010, Electricity and Magnetism
Some aspects of Magnetotistatics
Magnetism in Materials (brief)
Quantum mechanics required to understand (nice introduction
in Feynman, volume 2; see also Eisenberg and Resnick).
Paramagnetism: occurs in systems with unpaired spins.
~
~ ·B
Magnetic moments tend to align with the field (−m
potential). Alignment strongest at low temperatures.
~ = χB
~
M
(24)
χ ∼ 1/T (Curie law).
Diamagnetism: associated with Lenz’s law (but also needs
quantum mechanics).
Ferromagnetism: not due to magnetic forces, but to forces in
atoms which have net effect of aligning the spins. (“Exchange"
interactions, in systems with unfilled shells (d). Occurs only in
iron, cobalt, nickel, gadolinium and dyprosium (and alloys of
these materials).
Physics 212 2010, Electricity and Magnetism
Some aspects of Magnetotistatics