Explaining Magnetic Bottle Force: Why Does It Point Inward?

[Cyrus]

hi, my physics book has a little picture of a magnetic bottle but does not really explain why the force always points to the center. Can you explain it to me. The magnitude of the velocity can never change when in a magnetic field, but the particle will be in Simple harmonic motion inside the bottle because it goes back and fourth. This means the velocity along the axis of the bottle has to change between -v to 0 to v. So it has to be gained somewhere else. Is this gain in how big the rings become, or is it gained in BOTH going faster along BIGGER rings. I am having trouble drawing a diagram to see that the force does point to the center. Every time I try it, I find that the as the magnetic field starts to curve, the component of velocity along the axis of the bottle goes down. But the gain in velocity is along the spiral path. This does not make much sense, because this would not mean a force pointing to the center of the bottle. It would just always point to the center of circular spiral like a centripital force does. Thanks for any help you give me.

 

[NoTime]

The force generated by a magnetic field on a charged particle is always perpendicular to the direction of travel.
This, for the single magnet case, means particles go in a circle.
The momentum of the particle does not change.
The center of particle motion relative to the center of the bottle depends on how/where it was injected.
Does that help?

 

[Tide]

Energy and angular momentum are conserved so that as a charged particle moves along magnetic flux lines towards stronger magnetic field its perpendicular (tangential) velocity must increase because the gyroradius is getting smaller. Therefore, the component of velocity parallel to the magneitic field must decrease. Eventually, the parallel component goes to zero if the magnetic field gets strong enough and the particle "reflects" back toward the region of weaker field. This is the same principle that allows for the formation of the auroras near the poles.

 

[Cyrus]

What happens when it reaches the other end and comes to a stop. At that instant, how come it does not just sit there and go around in a circle at a much faster rate, (since it converted its forward speed into rotational speed in order for speed to be constant). Whats causing it to move back after that instant, the magnetic field should be nearly straight by the time it comes to a stop at the other end. Unless I am incorrect in assuming it stops at some point at which the field is "uniform." Would it be more sensible to say that it stops somewhere like, 3/4 of the way down, at which point its clearly not uniform.

If it were uniform when it stopped, then there would be no way for it to be pulled back in would there? It would sit there going around and around at a rapid speed.