Time Period of Circular Motion in B

Period is independent of speed — a particle moving faster traces a larger circle in the same time. This independence is the principle underlying the cyclotron.

Derivation

From mc03, the radius of circular orbit: $r = mv/qB$.

The circumference of the orbit: $2\pi r = 2\pi mv/qB$.

Time to complete one full circle:

$$T = \frac{2\pi r}{v} = \frac{2\pi m v}{qBv}$$ $$\boxed{T = \frac{2\pi m}{qB}}$$

Speed Independence

$v$ cancels out completely. The period depends only on $m$, $q$, and $B$ — not on speed.

A slow particle traces a small circle; a fast particle traces a large circle — but both take the same time to complete one revolution.

Physical Consequence — The Cyclotron Principle

Since $T$ is independent of speed, a particle being accelerated in a cyclotron stays in sync with a fixed-frequency alternating electric field throughout acceleration. The resonance condition holds at every stage.

Angular Frequency

$$\omega_c = \frac{2\pi}{T} = \frac{qB}{m}$$

This is the cyclotron angular frequency. At relativistic speeds, $m$ increases and $T$ grows — the resonance breaks, limiting the classical cyclotron.