- #1
nagyn
- 26
- 0
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?