Confusion in the guiding center of gyrating particles

Question

In Chen's Introduction to Plasma Physics and Controlled Fusion, while discussing magnetic mirrors, it's considered a magnetic field pointing in the $\hat{z}$ direction as shown below:

Going further, it's said that, if we assume that a gyrating particle has a guiding center on $\hat{z}$, then we can get the following: \begin{equation} (I)\space v_\theta \space is\space a\space constant\\ (II)\space F_{\parallel}= - \mu\nabla_{\parallel}B \end{equation} Whereas $(II)$ depends on $(I)$ to be possible, since the calculation of the magnetic moment relies on it.

I do not fully understand why that's the case. That's a pretty important thing to understand, since you can't derive the invariance of the magnetic moment without understanding these, especially the first one.

Maybe that's because my understanding of the guiding center is rather too shallow, because I just see it as the point on which the particle rotates around.

Besides, I didn't quite understand how valid that is. We are in a region of variable magnetic field, and on this convention for the gyration of the particles, we make a development that is only valid for uniform magnetic fields (Larmor radius, to be specific). How valid is that? Because it seems to me that all further development is based on a fragile premise.

So, summarizing, I actually have two questions:

1. How can we properly argue that $(I)$ and $(II)$ are valid?
2. How valid is it to assume, under the proper conditions, a uniform magnetic field on a region of variable magnetic field, supporting the later arguments in the results that depart from this statement?

I have a slight feeling that I may be making a serious conceptual mistake. Nevertheless, I'm not identifying it. If anyone can help me, I'll be grateful.

Answer 1

How can we properly argue that (I) and (II) are valid?

I will start by referencing my answer at https://physics.stackexchange.com/a/670591/59023.

So long as the magnetic field is not time-varying and the spatial variation is slow compared to the gyration of the particle (i.e., inverse cyclotron frequency), the field can do no net work on the particle. That means the particle kinetic energy will be conserved, i.e., the total particle speed is constant. If the variation is slow (relative to above constraints), we can also assume that the following is a constant $p_{\perp}^{2}/B$, where $p_{\perp}$ is the momentum orthogonal the quasi-static magnetic field, $\mathbf{B}$. Thus, so far we have: $$ \begin{align} v_{\parallel}^{2} + v_{\perp}^{2} & = v_{o}^{2} \tag{0a} \\ \frac{ v_{\perp}^{2} }{ B } & = \frac{ v_{\perp o}^{2} }{ B_{o} } \tag{0b} \end{align} $$ where the subscript $o$ implies the initial conditions.

The key part is that the spatial variation happens slower than the particle gyrates, i.e., the spatial variation gradient scale length is longer than the particle gyroradius so that it can gyrate many times before reaching the final field strength region. This tends to put constraints on the parallel vs perpendicular momentum magnitudes because if you move too fast along $\mathbf{B}$ the second constraint (Equation 0b) no longer holds. Similarly, if the perpendicular momentum is too large, the particle gyroradius is much larger than the thickness of the region where $\mathbf{B}$ changes in magnitude. If this is the case, the particle can basically ballistically propagate through the region and/or start to gradient drift transverse to the magnetic field gradient direction (e.g., see https://physics.stackexchange.com/a/556682/59023).

How valid is it to assume, under the proper conditions, a uniform magnetic field on a region of variable magnetic field, supporting the later arguments in the results that depart from this statement?

I really am not sure how to answer this as I am not sure what you are asking.