I Stern-Gerlach experiment with a classical object

[Killtech]

TL;DR Summary

looking for help calculating Stern-Gerlach experiment for classical objects like a compass used instead of a particle

I want to have a make/calculate a classical analog of the SG experiment within classical physics to understand all the relevant forces at play here. Within this context i would like to stick to classical physics only (yeah, I want it to compare to QM later but that is besides the discussion) and I hope to get some help in getting this right.

So the general setup of the experiment (the SG device) is macroscopic magnet setup which as such can be used identically. To make it practical everything should happen in zero-G and vacuum (no friction nor gravity). Instead of a quantum particle I want take identically build compasses with a frictionless mounted magnetic needle. Let the initial orientation and angular momentum of the needle be random/unknown.

Okay, now my understanding is that once a compass enters the magnetic field this will apply a torque through the magnetic moment of the needle – I want to setup the parameters such that the change of the angular momentum becomes significant compared to the magnetic gradient dipole force – so the needle needs to have extremely low mass and enough flight time in the SG device while the magnetic field needs to be low and the compass itself heavy to reduce the deflection of the entire body.

Anyhow the torque will however change the orientation of the needle and thus that of its magnetic moment which again should change its torque. The combined ODE for the needles angle should look just like for a pendulum in a gravity field (making some simplifying assumptions like a negligible deflection angle of the compass to keep the $B$ field near constant along the path), right?

Now assuming the needles oscillates fast enough couldn’t I just simplify things for the effective deflecting force acting on the compass by just taking the average magnetic moment vector of the needle over an entire oscillation period? Now I am not sure what the best method is to calculating these averages for a no-small-angle approximation pendulum. Any ideas?

Without calculation I would expect that in the case a) where the needles moment is exactly aligned along the magnetic field already it shouldn’t move so the effective moment should stay the same. In case where it a different orientation b) it should be a reduced value but always parallel to the magnetic field because the oscillation must be symmetric around the pivot point. c) If the needle is close to making a full circle or slightly faster than that (i.e. around the unstable equilibrium) it should spend most time upside down resulting in having an average magnetic momentum antiparallel to $B$ and if it spins too fast d) the total deflecting force should even net out to zero.

Now if I wanted to get discrete deflection angles shouldn’t it be sufficient to add a little friction to the needle to get a dampened oscillation that will converge the needle towards the only stable configuration? Unfortunately this makes only the pivot point parallel to $B$ stable while the unstable equilibrium at $\phi=\pi$ would swiftly decay. Then again one could cheat a little with a less trivial friction model e.g. by also adding static friction to make it stable.

Assuming everything is scaled up to dimensions that allow the oscillation to settle at an equilibrium long before leaving the SG device, shouldn’t this result in the deviation angles of the outgoing compasses amass around two points only and where the proportions of those two populations would depend on the initial condition/angular distribution/friction model?

Also what would happen if instead of considering friction I would take energy loss through EM-radiation into account? Given that I have a spinning dipole shouldn’t this oscillation classically lose energy (and therefore angular momentum)? Is there an analogue Larmor formula for that particular case?

Okay, before i get into a thorough calculation i would like to now if that setup/assumptions are reasonable first or if there are any logical errors i made.

 

[jambaugh]

The appropriate analog would be a spinning sphere magnetized along its axis of rotation. Shooting such a projectile through an SG magnet will cause it to deflect as a smooth function of its component of spin in the direction the SG magnet is set up. This was the expected outcome for elementary particles and the surprise was that the respective spin component was quantized with only the two values.

The spinning allows the object to keep its angle with respect to the magnetic field even with the torque the field applies. It instead precesses. The asymmetric non-linearity of the SG magnet's field causes the deflecting force.

 

[Killtech]

The appropriate analog would be a spinning sphere magnetized along its axis of rotation. Shooting such a projectile through an SG magnet will cause it to deflect as a smooth function of its component of spin in the direction the SG magnet is set up. This was the expected outcome for elementary particles and the surprise was that the respective spin component was quantized with only the two values.

The spinning allows the object to keep its angle with respect to the magnetic field even with the torque the field applies. It instead precesses. The asymmetric non-linearity of the SG magnet's field causes the deflecting force.

Well, my intention here was specifically to understand the role the torque can play, which is why I set it up to give it a greater importance. In particular i do want to see what happens if the object can change its angle with respect to the $B$.

A spinning sphere would exhibit a different behavior as you noted, therefore it isn't what I was looking for. Also a homogeneous sphere shouldn't be able to lose any energy to radiation like an spinning rod magnet (the compass needle) would - therefore it cannot use that effect to dampen its oscillation either.