Controlled Thermonuclear Fusion — Question 7

Question 7

Magnetic-mirror reflection follows from the adiabatic invariant

A particle moves from field $B_0$ toward $B_{\max}=4B_0$. Assume no electric field, conserved kinetic energy and conserved magnetic moment $\mu=mv_\perp^2/(2B)$. Initial pitch angle $\theta_0$ is between velocity and magnetic field.

1. Derive the condition for reflection before reaching the maximum field.

2. Find the loss-cone half-angle and decide whether initial angles $20^\circ$ and $45^\circ$ reflect.

Question 7 - Solution

Magnetic-mirror reflection follows from the adiabatic invariant

(a) Method and calculation

$$\frac{v_\perp^2}{B}=\frac{v^2\sin^2\theta_0}{B_0},\quad v_\parallel^2=v^2-v_\perp^2.$$$$v_\parallel=0\Rightarrow B_m=B_0/\sin^2\theta_0\le B_{\max}.$$

(b) Interpretation and further calculation

$$\boxed{\sin^2\theta_0\ge B_0/B_{\max}=1/4,\quad\theta_{\mathrm{lc}}=30^\circ}.$$$$\boxed{20^\circ:\text{ passes the mirror};\qquad45^\circ:\text{ reflects}}.$$

Physical check

Equality gives a turning point at the field maximum. The invariant requires gradual field variation over an orbit and negligible scattering; the opposite end has a corresponding loss cone.