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

t_\phi=\frac{\phi}{\omega}=\frac{m\phi}{qB},\qquad s_\phi=r\phi=\frac{mv\phi}{qB}

problems.proof.analysis

For radians, $\phi=\omega t$ , while arc length is $r\phi$ . Time is independent of speed because faster particles also follow proportionally larger orbits.