Now suppose the smaller coil is located within the empty space in the center of the larger, and that the solenoidal axes of the two coils are not perfectly aligned. Then, basic electromagnetics tells us the two coils will mutually precess. That is, the axis of each coil instantaneously precesses around the axis of the other. For the smaller coil inside the larger, this is very basic electromagnetic theory, since (assuming the inner coil is sufficiently small) the magnetic field due to the larger coil is for practical purposes here a constant value aligned with the large coil axis. The smaller coil, being an ideal current-loop magnetic dipole, feels a torque in the large coil's magnetic field, and according to basic mechanics the instantaneous change in the small coil angular momentum must equal the torque exactly. (The angular momentum of the coil is that of the circulating electrons forming the current.) However, since the torque at all times is perpendicular to the angular momentum of the small coil, it can't change the angular momentum magnitude, only its direction, and so the only allowed motion is the precession of the small coil around the axis of the larger.
For similar but slightly more complicated reasons, the larger coil also precesses around the axis of the smaller. The added complication to precession of the outer coil is that the magnetic field due to the smaller coil is not everywhere the same direction relative to the motion of the the charge carriers in the larger coil. The net force must be found by averaging over small current elements of the larger coil, but the end result is similar in that the larger coil also precesses instantaneously around the instantaneous axis of the other.
That the two coils are instantaneously precessing around each others' axes may seem difficult to envision, but visualization can be made easier by invoking the principle of conservation of angular momentum. If angular momentum is conserved, and there are no other torques acting on the coils than those already noted, the total angular momentum of the two coils must be a constant. Since the total angular momentum here is simply the vector sum of the angular momenta of the two coils, the only possible motions are those where the two coils precess around the total angular momentum direction with equal angular velocity. The situation is illustrated on Figure 1. The vector angular momenta of the large and small coils are represented as arrows labeled L and s, and their vector sum is J = L + s.
Figure 1We should also consider what happens to the total magnetic moment of the system consisting of the two coils in free space. If the total magnetic moment is not stationary, energy will be lost as radiation into free space, and the precession motion will eventually cease as the angular momentum vectors come into alignment due to the radiation reactive forces. However, classical electrodynamics also tells us that the magnetic dipole moment of each coil is proportional to the angular momentum of each coil. Further, the constant of proprtionality is the same for each coil provided that the charge-to-mass ratio of the current carriers is the same for each coil. For our superconducting coils the charge carriers are electrons in each case, so it is concluded that stationarity of the total angular momentum implies in this case stationarity of the total magnetic moment as well.
If, instead of simply assuming that the total angular momentum is a constant of the motion, we had gone through the machinations of actually calculating the magnetic moments and fields and torques and rates of changes of the angular momenta of each of the coils, we would have found that indeed that angular momentum is conserved according to electrodynamics. Thus we have arrived at a pair of completely unsurprising conclusions. The total angular momentum and total magnetic moment of the isolated system of the two coils, that is not being acted on by any external mechanical or electromagnetic forces, are both constants of the motion.
Next it's worthwhile to consider the case where the charge carriers in one coil have a different charge-to-mass ratio than those in the other, and determine whether angular momentum and magnetic moment are still conserved in spite of this. So, suppose that the charge carriers in one of the coils are muons instead of electrons. The muonic charge is the same as the electron's, but the muon mass is much larger than the electron mass. If we suppose the current is kept constant in spite of switching the carriers from electrons to muons, then the magnetic field generated is the same, but the angular momentum of the muonic coil is much larger than it was for electrons. The torque on the muonic coil due to the electronic coil is unchanged, however, so the angular velocity of the precession must decrease in the muonic coil. The net effect is that although the total angular momentum remains approximately constant, in this case the total magnetic moment is no longer stationary. The moving total magnetic dipole moment radiates energy, and so the back-force of radiation reaction will cause the two coils eventually to align with each other. Although the total mechanical angular momentum of the system is no longer a constant of the motion, angular momentum is nonetheless conserved because the electromagnetic radiation also carries angular momentum, and the total of the mechanical and field angular momenta is constant.
To this point we have been considering only classical electrodynamics. However, when it was first recognized that electrons must possess an "intrinsic", or non-orbital, angular momentum and an associated intrinsic magnetic dipole moment, it was also immediately recognized that the constant of proportionality, or gyromagnetic ratio, between intrinsic angular momentum and magnetic moment must be twice the classically-expected value. This doubling of the gyromagnetic ratio is conventionally expressed by including in the expression for the magnetic moment in terms of the intrinsic angular momentum (or "spin"), an additional "g-factor" of two. This factor of two has never been derived from classical electrodynamic electron models, if it is assumed that the charge-to-mass ratio of the electron is a constant throughout its volume.
Now let us consider what happens when one of the two coils is allowed to have a nonunity g-factor. For an electron-like g-factor of two, the magnetic moment is doubled compared to the classical g-factor of unity, for a fixed value of angular momentum. If we suppose the g-factor of the small coil is changed to two, while keeping the large coil classical g-factor of unity (and returning again to electron currents in both coils), then it's clear that the rate of precession of the small coil is doubled. The magnetic field at the small coil due to the large coil is unchanged, but the torque on the small coil has doubled due to its magnetic moment doubling, whilst the small coil angular momentum is unchanged. This is a net doubling of the rate of precession of the small coil. However, the rate of precession of the large coil also doubles, since the magnetic field of the small coil that drives its precession has doubled. So, the total angular momentum is a constant of the motion, if there are no radiative effects. However, evaluating the total magnetic moment, it's plain that it cannot be stationary if the total angular momentum is stationary (and both coils are precessing), since the magnetic moments are no longer both in equal proportion to the coils' angular momenta. Thus, seen from afar, the magnetic moment of the system is precessing, and this type of motion must radiate electromagnetic energy and momentum, and decay due to radiation reactive force.
Finally, there is an important additional effect that needs to be added to our system of two electronic-current-carrying coils, if it is to be a useful toy representation of atomic spin-orbit coupling, and that is the Thomas precession. The Thomas precession (also sometimes called Wigner rotation) is a relativistic effect wherein a reference frame that is accelerating and translating relative to another must also rotate compared to the other. The effect of the Thomas precession is to approximately halve the rate of precession of the intrinsic spin of a particle with a g-factor of two such as an electron. If the small coil with the g-factor of two also undergoes Thomas precession, while the large coil does not, plainly the angular momentum will no longer be a constant of the motion, even neglecting radiation reaction, if there is no change in the rate of the precession of the large coil due to the Thomas precession acting on the small coil. But, the nature of the Thomas precession is such that it only acts on the electron intrinsic spin and not on the orbital angular momentum vector, represented here by the angular momentum of the current-carrying electrons of the large coil. Also, the Thomas precession has only reduced the rate of precession of the small coil, not its magnetic field as experienced by the large coil. So the rate of precession of the large coil cannot change as would be necessary to maintain total angular momentum constancy, and thus it's clear that the total mechanical angular momentum is not a constant of the motion. But what about the field angular momentum? Can it restore angular momentum constancy? It turns out it cannot. Calculating the total magnetic moment, we find that the factor of one-half due to the Thomas precession (the celebrated Thomas factor that resolved the spin-orbit coupling anomaly of the 1920s) has compensated the g-factor of two that previously caused the total magnetic moment to move in spite of stationarity of total angular momentum. Present the Thomas precession, we now have the situation of a stationary magnetic moment in spite of moving total angular momentum, so that there is no need for the motion of the total angular momentum to lead to radiative decay, nor require an external torque.
The precession of the total angular momentum, even absent an externally-applied magnetic field, due to the spin-orbit coupling is a feature of quantum theory, but it has been seen here to be simply the classical electrodynamic consequence of the electron possessing a g-factor of two. This raises a further question. Will this same sort of behavior, of motion of the total angular momentum but stationarity of total magnetic moment, occur for other g-factor values than two? It turns out, g=2 is the unique g factor where the total magnetic moment can be stationary in the presence of spin-orbit interaction. (The proof can be found in my paper, "Regarding L. H. Thomas's paper of 1927 ....," linked at right.) For any other g-factor, including the classical value of unity, in the presence of Thomas precession, neither the total angular momentum nor the total magnetic moment is a constant of the motion. This suggests to me that the electron g-factor being two may be related to or even a consequence of the Thomas precession.
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