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Magnetic Mirror

Intuition

Squeeze magnetic field lines together at two ends of a tube and you've built a bottle for charged particles. A particle spiraling toward the pinch feels the field strengthen; its gyration steals energy from its forward motion until — if the pinch is strong enough — it stops, turns, and bounces back. Two mirrors face-to-face trap particles indefinitely. Nature builds them too: Earth's Van Allen belts are mirror machines.

The field

The course models a mirror through its flux function in cylindrical coordinates \((r, \theta, z)\):

\[\psi = B_0\,\pi r^2\left(1 + \frac{z^2}{L^2}\right), \qquad \mathbf{B} = \frac{1}{2\pi}\nabla\psi\times\nabla\theta\]

which evaluates to

\[B_r = -\frac{B_0\, r z}{L^2}, \qquad B_z = B_0\left(1 + \frac{z^2}{L^2}\right)\]

\(|\mathbf{B}|\) is weakest at the midplane (\(z = 0\)) and grows toward the mirror points. Field lines are contours of \(\psi\) — and a well-magnetized particle's guiding center stays on its contour.

Why particles reflect: the adiabatic invariant

The magnetic moment

\[\boxed{\,\mu = \frac{m v_\perp^2}{2B}\,}\]

is an adiabatic invariant: nearly constant when the field changes slowly over a gyration. Since \(B\) does no work, \(v^2 = v_\parallel^2 + v_\perp^2\) is also constant. As the particle rides into stronger \(B\), constant \(\mu\) forces \(v_\perp^2\) to grow — so \(v_\parallel^2\) must shrink. If \(B\) reaches \(B_\text{reflect} = B_\text{min}\, v^2/v_{\perp,0}^2\) before the throat, the particle reflects.

The loss cone

Particles with too much parallel velocity escape. With mirror ratio \(R_m = B_\text{max}/B_\text{min}\), the trapping condition on the midplane pitch angle \(\theta\) is

\[\sin^2\theta > \frac{B_\text{min}}{B_\text{max}} = \frac{1}{R_m}\]

Velocities inside the loss cone (\(\theta < \theta_\text{trap}\), \(\sin\theta_\text{trap} = R_m^{-1/2}\)) stream straight through. This leak is why mirror fusion machines struggled — collisions constantly scatter particles into the cone.

Numerical experiment

Integrate orbits with the 3-D leapfrog solver in the field above (worked example):

  • Launch with various \(v_\parallel/v_\perp\): watch trapped orbits bounce and passing orbits escape
  • Verify \(\mu\) stays constant while \(v_\parallel\) oscillates
  • Verify the guiding center hugs a \(\psi\)-contour

Common mistakes

  • Thinking the magnetic force decelerates the particle. \(\mathbf{B}\) does no work; it redirects kinetic energy from parallel to perpendicular. Total speed never changes.
  • Treating \(\mu\) as exactly conserved. It's adiabatic — conserved to exponential accuracy while fields vary slowly; fast field changes (or sharp field gradients) break it.
  • Forgetting the loss cone. A mirror is a leaky bottle by construction; only pitch angles outside the cone are held.

Knowledge graph position

Prerequisites: Guiding center, Leapfrog. Leads to: Fermi acceleration.

Quiz

Q1 (computational). A mirror has \(R_m = 4\). What fraction of an isotropic particle population is trapped?

Answer

Loss cone half-angle: \(\sin\theta_c = 1/2 \Rightarrow \theta_c = 30°\). Solid-angle fraction lost (both cones): \(1 - \cos\theta_c = 1 - \frac{\sqrt3}{2} \approx 0.134\) each side ⇒ trapped fraction \(= \cos\theta_c \approx 87\%\).

Q2 (conceptual). A trapped particle's \(v_\perp\) is largest at which point of its bounce?

Answer

At the mirror (turning) points, where \(B\) is largest: \(\mu = mv_\perp^2/2B\) const forces \(v_\perp^2 \propto B\). There \(v_\parallel = 0\) — all energy is perpendicular.

Q3 (MCQ). Earth's radiation belts trap particles because the geomagnetic field:

  • (a) is uniform (b) strengthens toward the poles (c) vanishes at the equator (d) is purely radial
Answer

(b). Field lines converge near the magnetic poles — natural mirror points; particles bounce pole-to-pole.