 Figure 1
 Dipole as 2 charges:

A
magnetic dipole is typically modeled as a current loop. If
the area enclosed in the loop is
A and the current
in the loop is
I, then the magnitude of the dipole
is
IA. However, we can also model it as
two monopoles: A + and  magnetic charge, each charge of
magnitude
b, separated by a
length
l. The dipole vector points from
the "" charge to the "+" charge, and its magnitude is
bl
(
figure 1).
Of
course, in classical E&M, magnetic monopoles are explicitly
ruled out by the equation
.
On this page we'll pretend we can relax that, and so allow
monopoles into the theory. This actually leads to some
significant problems in the foundation of the relativistic formulation
of E&M, but we need not concern ourselves with that here.
At
the level of the classical formulation of Maxwell's equations, allowing
nonzero divergence of
B yields a
perfectly consistent
theory, and a fine model; the only apparent problem with it is that
there we've never found any magnetic monopoles in the "real world"!
Of course to have a reasonable theory we couldn't really just
stop with allowing
to be nonzero; we'd also need to expand the rest of Maxwell's equations
to include magnetic currents and the effect of an electric field on
magnetic charges in motion. But, since we're only concerned
here
with stationary magnetic charges, we can safely ignore those issues as
well. We will now go directly to our main concern,
which is
to show that a dipole made of two monopoles would produce the same
field as a currentloop dipole.
We could just derive
the fields
for both dipole models and compare them. However, much of our
purpose here is to avoid the need to find the field of a currentloop
dipole directly. So, we're going to take an indirect approach.
We
observe that, if two objects
feel the same forces
in the same situation, they must also
exert
the same forces in the same situation. Otherwise, if momentum
were conserved in the interactions with one of the objects, it wouldn't
be conserved in interactions involving the other. So, if our
two
dipole models feel the same forces due to an external magnetic field,
then they must both exert the same forces on a magnet  otherwise,
with one or the other of our models, momentum would not be conserved.
(You may feel that, before we can use this argument, we
should
show that our "expanded" version of electrodynamics doesn't lead to a
violation of conservation of momentum. For now, we're just
going
to
assume the expanded model still conserves
momentum.)
We will now proceed to find the torque and force on a
2charge
dipole due to a nonuniform magnetic field. We'll then compare
the
values with those we found for a
current
loop dipole.
Torque on a Two Charge Dipole
 Figure 2
 torque on dipole:

In
figure 2 we show the dipole
in an external magnetic field, with the
B
field oriented parallel to the
z
axis. The dipole lies in the
yz
plane; we haven't drawn the
x
axis. The dipole is rotated at angle θ relative to
the
B field.
The
B
field exerts forces on the charges as shown in figure 2; to zeroth
order in the strength of the field, the net force is zero.
However, the
torque depends directly
on the field strength, and its magnitude is:
(1)
In figure
1, we can see that the force on the dipole would be such that it would
rotate
toward the
z
axis. Using the right hand rule, we see that the torque
vector must point out of the page, along the
x
axis. So the torque vector must be,
(2)
where we
have used μ to represent the vector of the dipole, with
magnitude
bl.
This matches equation (
DipoleForce:7).
Force
on a Two Charge Dipole
 Figure 3
 force on a dipole:

To
find the forces, we'll rotate the axes in figure 2 so that the dipole
lies on the
z axis (
figure 3).
From
figure 3, we see that the net
z
component of the
force on the dipole is zero if the
field is uniform. If the field varies, however, the
z
component of the force must be
(3a)
Similarly
we can see that the
x and
y
components of the force must be
(3b)
(3c)
With the
dipole lying on the
z axis, the
dipole's
z component is
bl
and its other two components are zero. So, using the fact
that μ
_{z} = μ, we can rewrite (3ac) as
(4)
and,
using the fact that μ
_{x} = μ
_{y}
= 0, we can write (4) as
(5)
Since μ
is not a function of position, we can move the dot product under the
gradient sign, and obtain
(6)
This
value is independent of the orientation of the coordinates, and it
matches equation (
DipoleForce:17).
Conclusion
The
torque and force on a twocharge dipole due to an external magnetic
field matches the torque and force on a current loop dipole in the same
field. So, as we concluded above, the fields generated by the
two
must be identical; if we can find the field of a twocharge dipole
we'll also know the field of a current loop dipole.
Furthermore,
if we wish to determine how a (permanent) current loop dipole (or spin
dipole) will behave in a particular situation we can examine a
twocharge dipole model instead, if that simplifies the problem for us.
Page
created on 12/27/06