Magnetic Mirrors: Exploring Conservation & Torque in Plasma Physics

[Loro]

I'm taking a short course in plasma physics, and we've covered quickly the magnetic mirror.

So the idea is that an ion, say a proton, moves along a helical path along parallel B-field lines - say in z-direction. Then the lines converge towards the z-axis, forming a shape like a bottleneck, with the B-field strength increasing in the direction of the narrow part; and on that converging bit, if the proton is not too fast, it gets slowed down, and reflected back.

So the way our lecturer explained it to us is that the z-velocity $v_{z}$ of the proton decreases, and that this kinetic energy has to go somewhere, because energy is conserved (B-field can't do work) so it gets converted into the energy of the rotational component of its motion (around the B-field lines) and so the tangential velocity $v_{\bot}$ increases.

Then it comes to a stop and this process reverses.

With all that I agree. I'm rarely satisfied with these sorts of explanations, but I drew it, etc. and found where the forces responsible for these changes in motion come from.

So from $v_{\bot} × B$ , there follows a force, that on average has got a component in z-direction, and that's what's stopping it.

Then from $v_{z} × B$, there follows a force which is always tangential and is speeding the rotations up. And if we calculate the energy loss due to these forces, we indeed get 0.

Then the lecturer says, that angular momentum wrt the z-axis is conserved too - from that he derives $v_{z}$ as a function of B (because B increases along the z-axis), and calculates that the B, at which the "stop" occurs, is:

$B=B_{0} (\frac{v_{\bot 0}^2 + v_{z 0}^2}{v_{\bot 0}^2} )$

where these quantities with zero are meant to be: before it entered the bottleneck.

And I know this formula is correct, however I can't agree that the angular momentum is conserved. There is this force that's speeding the rotations up - it clearly has got a torque. I told it to the lecturer and he said, that the angular momentum is conserved, "because it can't go anywhere"...

What's the true explanation, or my mistake?

 

[M Quack]

Angular momentum is v x r. As you increase B, the force v x B increases, forcing a smaller radius. At the same time, the torque you mention increases the speed, so that in the end the angular momentum v x r remains constant.

In the adiabatic approximation, the kinetic energy of the particle (mostly electrons!) is conserved. If you assume that both energy and angular momentum are conserved, then you end up with the exchange between v_parallel and v_perp that eventually gives v_parallel=0 at the reflection point.

Keep in mind that this is only an approximation. The higher order terms beyond this "adiabatic approximation" are called drifts, and there are grad B x B drifts, E x B drifts, ...

 

[Loro]

Thanks,

We did cover drifts, and apparently the "braking" force here might me explained by a $\nabla B \times B$ drift, caused by the radial gradient of the field. This gives the same force, as the one we get from Lorentz.

When you say it's just an approximation - how can the energy not be conserved? - if it wasn't then the B-field would have to do work!

The interesting thing is that the two effects that you said about, just cancel each other out to conserve momentum. So I guess my question is - is there any deeper explanation of this conspiracy?

 

[M Quack]

I would have to dig out my plasma physics book to give a more detailed answer...

The approximation is that angular momentum is conserved. In a static magnetic field, energy should be conserved exactly - just as you point out.

I think grad B x B is pointing the wrong way. It should cause drift around the axis of the bottleneck.

 

[haruspex]

There is this force that's speeding the rotations up - it clearly has got a torque.

Isn't the force at right angles to the motion, i.e. centripetal? That would mean there is no torque.

 

[Loro]

haruspex, the $v_{\bot} × B$ is perpendicular to the rotational motion, and doesn't have a torque.

But $v_{z} × B$ does point tangentially and causes the $\nabla B × B$ drift mentioned by M Quack.

So yeah thanks a lot M Quack, I think the explanation that it's an approximation, helps me.

But still it's quite mysterious, because if it's an approximation, then the angular momentum is at least very close to being conserved, while there's a tangential force changing the motion quite dramatically, and I guess there must be some reason for that, that I don't understand.

...Or is it just because the angle between $v_{z}$ and $B$ is so small, that the approximation works, and if the bottleneck was steeper - it wouldn't?