Spin of a charged sphere in a non-uniform magnetic field

[nagyn]

Homework Statement

Consider a solid spinning sphere of negative charge in a non-uniform magnetic field:

http://d2vlcm61l7u1fs.cloudfront.net/media/2f8/2f840122-3d7f-4af3-98a7-efa9b2e7d26a/phpBOhfaK.png

(the rotation illustrated at the top is supposed to be counter-clockwise).

What motion does the sphere undergo as time evolves?

Would the motion you described above affect the component of the magnetic moment measured by the Stern-Gerlach experiment?

Homework Equations

For a negative charge, magnetic moment and angular momentum point in opposite directions.

Net torque = dL/dt
torque = u X B

The Attempt at a Solution

Initially, the magnetic moment points downward perpendicular to the line drawn through the center of the sphere. (Two ways to justify this: angular momentum points in the opposite direction (right hand rule for rotation), or the net negative charge rotating in the given direction gives a positive current in the opposite direction (right hand rule for magnetic moment and current).

Using torque = u X B, torque is pointing into the page. Therefore, after a time dt, the angular momentum L will change in the direction of net torque, so L moves into the page. Consequently, magnetic moment moves out of the page and net torque starts to point more downward.

Over time the resulting motion is that the sphere precesses in and out of the page in a circular motion.

The part I'm not sure about is how the fact that the field is non-uniform affects this motion?

And to answer the Stern-Gerlach question, I believe that the answer is yes, because the components depend on the instant at which they are measured, which is why we can only talk about spin orientations as a probability distribution. Is my reasoning for this answer correct?

 

[kuruman]

Does the magnetic moment experience only a torque or is there a force also acting on it?

 

[nagyn]

Okay, so there's a magnetic force acting on the sphere

F = iL X B

At the point along the line through the center of the sphere that's higher up, the current is into the page, and therefore F points right and down (into the sphere). At the other side of the sphere F points left and up, also into the sphere. Because the first point is higher up in the magnetic field (ie the magnitude of B is greater), there is a net rightward/downward magnetic force acting on the magnetic moment at this point in time.

Therefore the sphere will also rotate so that the magnetic moment aligns with the magnetic field (makes sense as this is the state of lowest potential energy).

Consequently when u and B align, torque = u X B = 0, so precession ceases?

 

[kuruman]

F = iL X B

That's not what you should be considering. It's the Lorentz force and has nothing to do with the magnetic moment of the sphere.

What happens to a magnetic dipole in a non-uniform magnetic field? Hint: $~\vec F=-\vec{\nabla}U$ and $U=-\vec{\mu}\cdot \vec{B}$.

For the precession to cease, there must be some exchange of energy.

 

[nagyn]

It's been a while since I took EM, but hopefully I'm following you correctly:

F = +u*dB/dz

Moving in the upward direction, dB/dz is positive so F>0, so the force acting on the magnetic moment is upward.

I know that magnetic dipoles tend to align so that the moment (south-to-north poles) are in the direction of the field, but I don't remember how a magnetic force affects a dipole. Will the sphere accelerate upwards due to the net upward force, without any magnetic moment rotation?

 

[kuruman]

F = +u*dB/dz

That's it.

Will the sphere accelerate upwards due to the net upward force, without any magnetic moment rotation?

If there is no mechanism to damp the precession, the sphere will keep on precessing around the field lines while being accelerated by the force.

 

[nagyn]

So if the Lorentz force has no effect on the magnetic moment of the sphere, is the answer to the second question:

"Would the motion you described above affect the component of the magnetic moment measured by the Stern-Gerlach experiment?"

a no, because the motion the sphere experiences is the same as the motion a neutrally-charged sphere would experience in the same field?

 

[kuruman]

What is the Stern-Gerlach experiment all about?

 

[nagyn]

What is the Stern-Gerlach experiment all about?

Sending neutral atoms with random initial orientation through a non-uniform magnetic field causes discrete ouputs (spin orientations) along the same axis as the magnetic field gradient.

 

[kuruman]

Sending neutral atoms with random initial orientation through a non-uniform magnetic field causes discrete ouputs (spin orientations) along the same axis as the magnetic field gradient.

Close enough. Now if you model these atoms as spinning, negatively-charged spheres of the kind described in the question, what must be true for discrete outputs to be caused? Assume the simplest case of two outputs as was the case with the Stern-Gerlach experiment.

 

[nagyn]

Close enough. Now if you model these atoms as spinning, negatively-charged spheres of the kind described in the question, what must be true for discrete outputs to be caused? Assume the simplest case of two outputs as was the case with the Stern-Gerlach experiment.

It needs an intrinsic angular momentum, which is determined by the electrons in the outmost shell?

 

[kuruman]

Yes. What are the possible values of this intrinsic angular momentum if the atom is in an external magnetic field?

 

[nagyn]

Yes. What are the possible values of this intrinsic angular momentum if the atom is in an external magnetic field?

Assuming we’re still treating this as a spin-1/2 problem, the component of intrinsic angular momentum in the direction of the magnetic field gradient can only be either + or - h/(4*pi).

 

[kuruman]

And what will happen to a beam of particles some of which have + and some - component when they are sent through a non-uniform magnetic field? Ting in terms of the model of the spinning charged sphere.

 

[nagyn]

And what will happen to a beam of particles some of which have + and some - component when they are sent through a non-uniform magnetic field? Ting in terms of the model of the spinning charged sphere.

Particles will be accelerated either upward or downward when sent through a non uniform magnetic field depending on their orientation?

 

[kuruman]

Yes. Do you see now how to answer the questions?

 

[nagyn]

Yes. Do you see now how to answer the questions?

I think so.

The answer to question 2 would be no: as the sphere precesses the magnetic moment alternates between up and down orientation (this is due to angular momentum and torque, not the magnetic field itself). The magnetic field causes a magnetic force that will either accelerate the sphere downward or upward (with respect to the gradient) depending on the orientation of the magnetic moment at that time, so the components measured by the Stern-Gerlarch experiment (again, components measured with respect to the gradient) will only take on two discrete values.

 

[kuruman]

In relation to the S-G experiment, I think the question wants you to consider a collection of spheres, some with "up" moments and some with "down moments" and not a single sphere flipping between up and down. What would be the motion of the "up" spheres be and of the "down" spheres?

 

[nagyn]

In relation to the S-G experiment, I think the question wants you to consider a collection of spheres, some with "up" moments and some with "down moments" and not a single sphere flipping between up and down. What would be the motion of the "up" spheres be and of the "down" spheres?

(Negatively charged) spheres with “up” moments will accelerate downward and vice-versa.

 

[kuruman]

Yes, and they will still precess about the magnetic field lines.