Cyclotron Motion

When a charged particle moves in a magnetic field, the Lorentz force $\mathbf{F} = q\mathbf{v} \times \mathbf{B}$ acts perpendicular to the velocity, causing the particle to gyrate in a circle. This circular motion is called cyclotron motion (or gyromotion).

Gyrofrequency (Cyclotron Frequency)

The angular frequency of the circular orbit:

$\omega_c = \frac{|q| B}{m}$

This is independent of the particle's speed — all particles of the same charge-to-mass ratio gyrate at the same frequency in a given field.

Gyroperiod

The time for one complete orbit:

$T_c = \frac{2\pi}{\omega_c} = \frac{2\pi m}{|q| B}$

Larmor Radius (Gyroradius)

The radius of the circular orbit depends on the perpendicular velocity $v_\perp$:

$r_L = \frac{m v_\perp}{|q| B}$

Key parameters:

• $e = 1.602 \times 10^{-19}$ C
• $m_e = 9.109 \times 10^{-31}$ kg (electron)
• $m_p = 1.673 \times 10^{-27}$ kg (proton)

Physical Intuition

• Electrons gyrate at much higher frequencies than ions (smaller mass → higher $\omega_c$)
• Higher energy particles have larger Larmor radii
• In fusion devices, small Larmor radii confine particles to magnetic field lines
• Cyclotron resonance heating (ICRH, ECRH) drives plasma heating by injecting waves at $\omega_c$

Implement the three gyration functions below.