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 $\mathbf{F} = q \mathbf{v} \times \mathbf{B}$ is always perpendicular to $\mathbf{v}$ , so it does no work; the speed is constant. The trajectory is a circle of radius $r = m v / (q B)$ traced at angular frequency $\omega_c = q B / m$ . Increasing $B$ tightens the orbit but speeds up the rotation, leaving the period $T = 2\pi m / (q B)$ shorter.

Vary $B$ : smaller $B$ gives bigger circles. Vary the initial speed: at fixed $B$ , larger $v$ 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

\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

The magnetic force stays perpendicular to the velocity, so it does no work: the speed never changes and the particle circles forever at radius r = m v / q B.
Raise B to tighten the circle or the speed to widen it, yet the cyclotron frequency q B / m holds fixed, independent of speed. That constancy is what makes a cyclotron work.
Flip the charge sign and the orbit reverses, the handedness that separates positive from negative carriers in a magnetic field.