Question
While trying to understand the answers in this question, I found out that I have some difficulty in understanding why the magnetic mirror reduces the velocity of a particle that moves towards the more dense magnetic field lines. I would like to ask: is there some governing law/underlying principle that explains why the velocity towards either end gets reduced (and then increased again towards the other end)?
Spiralling Motion
I understand the circular/spiralling/spring-like/helical motion of the particle around the field lines, because that can be explained by the Lorentz force (assuming that the particle has some initial velocity component that is tangential to a magnetic field line). The underlying principle is nicely visualised in this image from this website :
But the Lorentz force in this image is in plane, and it does not have a component in the direction of the field lines, so one could argue that this Lorentz force should not influence the velocity vector component parallel to the magnetic field lines.
Hypothesis
However, when looking at these two images:
I thought perhaps the circular motion is tilted at an angle w.r.t. the line from the centre of the two sinks/magnets, which would imply that the Lorentz force would, at some points along the circle, have a component parallel to the direction of this centreline (still not parallel to the magnetic field line, because that is tilted in the areas of increased magnetic field line density). But then I thought that that effect should cancel out. However, the image below
indicates that the spiralling motion also moves the particle a bit closer to the magnet/end/mirror at each circle. So I thought perhaps that component that is tangential to the centreline of the mirrors might be a bit bigger at the some point in the circular motion than at the next semi-circular position, causing a net force parallel to the centreline. And this net force would then be able to slow down the particle, and I imagine, accelerate it away once it has come to a stop. (Assuming the stop is unstable).
Sketch
To illustrate this hypothesis consider the following 3D image/sketch, where:
- The black line is a magnetic field line
- The pink line represents the circular motion caused by the Lorentz Force (CCW as drawn (from the side))
- The green arrow is the Lorentz force
- the purple/dashed line is the centreline connecting one centre of the magnet with the other
- The turquoise arrow is the component parallel to the centreline
- Point A is where the turquoise velocity component is larger than that it was when it was at B (half a circle ago).
- Point B is where the turquoise velocity component points in direction of the mirror, but it is smaller than that it will be in point A, causing a net difference.
