Magnetic Bottle

Vary the initial pitch angle to compare reflection with escape. Pitch α₀ is the angle between the electron velocity and the field at the centre, z = 0. The particle starts toward +z. The examples use mirror ratio Rm = 4 with pitch 60° or 10°.

The prescribed field magnitude along the axis is B(z) = B₀[1 + (Rm − 1)(z/L)²], with L = 1 m and open ends at ±L. It is defined here only inside that region. The adiabatic guiding-centre model conserves μ = m v⊥²/(2B) and E = m v∥²/2 + μB. The mirror force −μ dB/dz gives an analytic harmonic orbit: z = A sin(ωt), A = L cot(α₀)/√(Rm − 1), ω = v₀ sin(α₀)√(Rm − 1)/L.

Reflection inside the region requires sin²(α₀) > 1/Rm. The loss-cone half-angle is αc = asin(1/√Rm). At the mirror points v∥ = 0 and B = B₀/sin²(α₀). Smaller pitch angles reach an end before reflecting. At equality the turning point touches an open end; this tool stops there, with v∥ = 0. Roundoff within 32 machine epsilons of Rm sin²(α₀) = 1 counts as equality.

At Rm = 4 and pitch 60°, the mirror points are ±⅓ m. With v₀ = 10⁶ m/s, the full bounce period is about 4.189 µs. Changing B₀ changes μ, gyro radius and gyro frequency, but not the mirror positions or bounce period in this prescribed model. Frequency is |q|B/(2πm), radius is m v⊥/(|q|B).

The axial positions and time readouts follow the model. The field-tube width is a schematic 1/√B outline, not a calculated coil geometry. The dot marks the guiding centre; no full Lorentz trajectory or gyromotion is drawn. Trapped playback takes four seconds per full bounce; escaping playback takes two seconds to the end. The displayed conversion gives model time per playback second. Hidden tabs suspend playback; delayed frames are capped at 0.1 s. Reduced-motion preferences start paused. The step button advances to the next turning point or the open end.

This is a nonrelativistic, collisionless, adiabatic illustration with electron mass 9.1093837×10⁻³¹ kg and charge −1.602176634×10⁻¹⁹ C. Adiabatic conservation assumes slow field variation over a gyro orbit; it is imposed here, not checked by integrating the Lorentz equation. No electric field, scattering, azimuthal drift, self-consistent plasma or Earth radiation-belt geometry is calculated. Supported inputs remain B₀ = 0.0001–10 T, Rm = 1.2–20, pitch = 5–85°, speed = 10³–10⁷ m/s.

Reference: Richard Fitzpatrick, Magnetic Mirrors and Adiabatic Invariants.