Magnetic Mirror

Magnetic Mirror

A magnetic mirror is a region where the magnetic field strength increases along field lines, trapping charged particles through a magnetic force.

The Mirror Effect

As a charged particle moves into a region of stronger B, its perpendicular velocity (v_{\perp}) increases (to conserve the magnetic moment (\mu = mv_{\perp}^2 / 2B)). By conservation of kinetic energy, the parallel velocity (v_{\parallel}) must decrease. If (v_{\parallel}) reaches zero, the particle is reflected.

Mirror Ratio

The mirror ratio characterizes how strong the mirror is:

$R_m = \frac{B_{max}}{B_{min}}$

A higher mirror ratio means a stronger mirror that traps more particles.

Loss Cone

Particles whose pitch angle (\theta) (angle between velocity and B) satisfies:

$\sin^2(\theta_{lc}) = \frac{B_{min}}{B_{max}} = \frac{1}{R_m}$

are on the loss cone boundary. Particles with (\theta < \theta_{lc}) (too much parallel velocity) escape the mirror. Those with (\theta > \theta_{lc}) are trapped.

Fraction Trapped

The fraction of an isotropic distribution that is trapped:

$f_{trapped} = 1 - \frac{1}{\sqrt{R_m}}$

Pitch Angle

The pitch angle of a particle with velocities (v_{\perp}) and (v_{\parallel}):

$\theta = \arctan\left(\frac{v_{\perp}}{v_{\parallel}}\right)$

Applications

Magnetic mirrors appear in:

• Earth's Van Allen belts — natural magnetic mirrors trapping high-energy particles
• Fusion devices — mirror machines attempt to confine plasma for energy production
• Solar corona — particles are trapped between magnetic loops