Magnetic Mirror

Source lecture(s): pc368_lec08_adiabatic_invariant

Intuition

A magnetic mirror is a region where the magnetic field strength increases along a field line. Charged particles spiraling into this region slow their parallel motion and can reflect back, creating a magnetic “bottle.”

Formal Definition

Reflection occurs when the pitch angle \(\\theta\) satisfies

\[\\sin^2\\theta \\geq \\frac{B_{\\text{min}}}{B_{\\text{max}}}\]

Equivalently, the loss cone angle is

\[\\theta_{\\text{lc}} = \\arcsin\\sqrt{\\frac{B_{\\text{min}}}{B_{\\text{max}}}}\]

Mathematical Formulation

The magnetic moment is an adiabatic invariant:

\[\\mu = \\frac{m v_\\perp^2}{2 B} = \\text{const}\]

As \(B\) increases, \(v_\\perp\) grows and \(v_\\parallel\) must decrease to conserve energy. Reflection happens when \(v_\\parallel = 0\).

Derivation

Start from \(\\mu = m v_\\perp^2/(2B) =\) const.
Total energy \(E = m v_\\parallel^2/2 + \\mu B =\) const.
At the mirror point \(v_\\parallel = 0\), so \(E = \\mu B_{\\text{max}}\).
At the minimum field, \(E = \\mu B_{\\text{min}} + m v_{\\parallel,\\text{min}}^2/2\).
Equating gives the loss-cone condition.

Worked Example

For a magnetic mirror with \(B_{\\text{min}} = 0.1\) T and \(B_{\\text{max}} = 1\) T:

\[\\theta_{\\text{lc}} = \\arcsin\\sqrt{0.1} \\approx 18°\]

Only particles with \(\\theta < 18°\) escape; the rest are trapped.

Common Mistakes

Believing mirrors reflect all particles. Only those outside the loss cone are trapped.
Confusing mirror ratio \(R_m = B_{\\text{max}}/B_{\\text{min}}\) with field strength.

Quiz Questions

Why are magnetic mirrors not perfect traps?
How does increasing \(B_{\\text{max}}\) affect the loss cone?