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The "Two Charge" Model of a Magnetic Dipole

Figure 1 -- Dipole as 2 charges:
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 file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_fApZVq.png.  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 file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_iqBSRN.png 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 current-loop 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 current-loop 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 2-charge 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:
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)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_QvvvF7.png

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)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_hbZh7J.png

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:
force on 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)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_fwFNyK.png

Similarly we can see that the x and y components of the force must be

(3b)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_7QT672.png

(3c)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_FWHEUy.png

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 μz = |μ|, we can rewrite (3a-c) as

(4)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_TdRzI5.png

and, using the fact that μx = μy = 0, we can write (4) as

(5)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_0xV9mc.png

Since μ is not a function of position, we can move the dot product under the gradient sign, and obtain

(6)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_3gkkOQ.png

This value is independent of the orientation of the coordinates, and it matches equation (DipoleForce:17).


The torque and force on a two-charge 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 two-charge 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 two-charge dipole model instead, if that simplifies the problem for us.

Page created on 12/27/06