Problem 2

A positively charged particle of charge

q

and mass

m

enters a uniform magnetic field of magnitude

B

with velocity

v

perpendicular to the field. The field region is wide enough for a circular orbit; neglect gravity and relativistic effects. (a) Find the orbit radius and direction of turning. (b) Find the period and cyclotron angular frequency. (c) If the particle travels a circular arc of angle

\phi

(in radians) before leaving the field, find the time and arc length and explain how the time depends on

v

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\boxed{r=\frac{mv}{qB},\;T=\frac{2\pi m}{qB},\;\omega=\frac{qB}{m},\;t_\phi=\frac{m\phi}{qB},\;s_\phi=\frac{mv\phi}{qB}}

problems.proof.analysis

The instantaneous force follows the right-hand rule for $q\mathbf v\times\mathbf B$ , setting the bend direction. The period is independent of $v$ in the nonrelativistic regime.

problems.proof.pitfall. Do not confuse radius and period:

r

depends on

v

, while the nonrelativistic cyclotron period does not.