Do all parts

Consider the situation illustrated below. Specifically, a particle with charge $\backslash ($ q $\backslash )$ and mass $\backslash ($ m $\backslash )$ enters a medium with a uniform magnetic field $\backslash (\backslash$vec$\{$B$\}\backslash ).$ The initial velocity of the particle $\backslash (\backslash$vec$\{$u$\}\backslash )$ is perpendicular to $\backslash (\backslash$vec$\{$B$\}\backslash ),$ and the magnetic force exerted on the particle causes it to move in a circle of radius $\backslash ($ a $\backslash ).($Note: the figure below uses slightly different notation than we typically use. In the figure, the velocity, $\backslash (\backslash$vec$\{$u$\}\backslash ),$ is written as $\backslash (\backslash$mathbf$\{$u$\}\backslash )$; the magnetic field, $\backslash (\backslash$vec$\{$B$\}\backslash ),$ is written as $\backslash (\backslash$mathbf$\{$B$\}\backslash )$; and the magnetic force, $\backslash (\backslash$vec$\{$F$\}\_\{$m$\}\backslash ),$ is written as $\backslash (\backslash$mathbf$\{$F$\}\_\{\backslash$mathrm$\{$m$\}\}\backslash ).)$

$($a$)$ What is the kinetic energy of the particle? Does this energy vary with time? Explain your answer. $($HINT: Magnetic fields do no work!$)$

$($b$)$ Equate the magnetic force $\backslash (\backslash$overrightarrow$\{$F$\_\{$m$\}\}\backslash )$ to the centripetal force required to rotate the particle and determine $\backslash ($ a $\backslash )$ in terms of $\backslash ($ q$,$ m$,\backslash$vec$\{$u$\}\backslash ),$ and $\backslash (\backslash$vec$\{$B$\}\backslash ).$

$($c$)$ Use your answer from part $($b$)$ to calculate how long it takes the charged particle to make a trip around its circular trajectory. Call this duration $\backslash ($ T $\backslash )$ and calculate the frequency that corresponds to this period. Does this frequency depend on the charge's initial velocity?

The frequency you calculated in part $($c$)$ is called the cyclotron frequency, and it is an important parameter for describing charged$-$particle motion in magnetic fields. Also, the basic arrangement described in this problem forms the core of one of the earliest particle accelerators; these accelerators are called cyclotrons and were developed nearby in Berkeley, CA$.$