A cyclotron designed to accelerate protons has a magnetic field of magnitude $0\mathrm{.}365$ T over a region of radius $1\mathrm{.}90$ m$.$

$($a$)$ What is the cyclotron frequency?

$($b$)$ What is the maximum speed acquired by the protons?

Part $1$ of $4-$ Conceptualize

Cyclotrons can be used to produce radioactive substances used in diagnosis and treatment. We expect the angular frequency to be on the order of millions of radians per second. We expect the maximum speed of the proton to be a substantial fraction of the speed of light.

Part $2$ of $4-$ Categorize

We apply the equations of angular frequency to the proton that accelerates in the magnetic field of the cyclotron.

Part $3$ of $4-$ Analyze

$($a$)$ Cyclotron frequency refers to the angular frequency or angular speed of the accelerated particle. For the angular speed of a proton, we have the following.

$=$

qBm

$$

$=$

$1\mathrm{.}601019$ C

T

$1\mathrm{.}671027$ kg

$=$