Back

Cyclotron motion in a uniform magnetic field

What you are seeing: a charged particle (q = m = 1) in a uniform out-of-page magnetic field B. The Lorentz force F=qv×B\mathbf{F} = q \mathbf{v} \times \mathbf{B} is always perpendicular to v\mathbf{v}, so it does no work; the speed is constant. The trajectory is a circle of radius r=mv/(qB)r = m v / (q B) traced at angular frequency ωc=qB/m\omega_c = q B / m. Increasing BB tightens the orbit but speeds up the rotation, leaving the period T=2πm/(qB)T = 2\pi m / (q B) shorter.

Vary BB: smaller BB gives bigger circles. Vary the initial speed: at fixed BB, larger vv gives bigger circles but the same period. The field-strength readout shows the resulting radius and period.

Figure 1. Cyclotron motion. Method: RK4 integration of v˙=q/mv×B\dot{\mathbf{v}} = q/m \, \mathbf{v} \times \mathbf{B}.
B1.00
|v|1.00
charge q1.00
mass m1.00
speed2

WHAT TO TRY

  • Vary each control and watch the rail readouts respond.
  • Compare the diagnostic plot against the live scene.