Problem 2

A proton of charge

q=+1.60\times10^{-19}

C and mass

m=1.67\times10^{-27}

kg moves in a uniform magnetic field

B=0.50

T. Its initial velocity has magnitude

v=2.0\times10^6

m/s and is perpendicular to

\vec B

; neglect gravity. (a) Find the magnetic-force magnitude and orbit radius. (b) Find the angular frequency, ordinary frequency, and period of the motion. (c) The proton is accelerated until its kinetic energy quadruples in the same field; find the new speed, radius, and period.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\omega=\frac{v}{r}=\frac{qB}{m}=4.79\times10^7\ \mathrm{rad/s},\quad f_c=\frac{\omega}{2\pi}=7.62\times10^6\ \mathrm{Hz}

problems.proof.analysis

Eliminate $v$ between $\omega=v/r$ and $r=mv/(qB)$ to obtain the cyclotron angular frequency $\omega=qB/m$ . The ordinary frequency is $\omega/(2\pi)$ and depends only on $q$ , $B$ , and $m$ .