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$9.[40$ points$]$ Pictured at right is a schematic of a cyclotron particle accelerator. It operates by applying a potential difference between the two conducting "Dees" $($They$\text{'}$re shaped like two halves of very short cylindrical cans.$)$ The potential difference creates an electric field between the Dees, but E is very nearly zero inside each Dee. Charged particles are released from source Z in one of the Dees near the center of the cyclotron and are accelerated by the potential difference towards the other Dee.

a$)$ If we choose to accelerate Alpha particles

in our cyclotron, determine the speed, $\backslash (\backslash$mathrm$\{$v$\}\_\{\backslash$mathrm$\{$a$\}\}\backslash ),$ of the Alpha particles after traversing the potential difference once, if the potential difference between the Dees is $40$ kV $.($An alpha particle is essentially a helium nucleus: $2$ protons and $2$ neutrons$)$

b$)$ If a uniform magnetic field $\backslash (\backslash$mathrm$\{$B$\}=0\mathrm{.}50\backslash$mathrm$\{$~T$\}\backslash )$ is applied downwards, the alpha particle's path will be bent into a semicircle. From the magnetic force acting on the Alpha particle, determine the radius of this semicircle. Is it oriented clockwise or counterclockwise looking down on the cyclotron? $($in the same direction as B $).$

c$)$ How much time does it take the alpha particle to travel along this semicircle? Why is it ok to assume that the Alpha particle moves with a constant speed while inside the Dee? If the Alpha particle were moving at $\backslash (\backslash$mathrm$\{$v$\}\_\{\backslash$mathrm$\{$c$\}\}=4\backslash$mathrm$\{$v$\}\_\{\backslash$mathrm$\{$a$\}\}\backslash )(4$ times faster$),$ how much time would it take to travel along the semicircle it makes then? $($Be careful, is it moving in the same semicircle as it was at speed $\backslash (\backslash$mathrm$\{$v$\}\_\{\backslash$mathrm$\{$a$\}\}\backslash )?)$

d$)$ Explain why we can't have a constant potential difference between our two Dees if we want the particle to keep speeding up as it passes the gap between the two Dees. The cyclotron frequency is the frequency that we need to switch the potential difference with so that our alpha particle is always sped up by the electric field between the Dees. It is given by $\backslash ($ f$=\backslash$frac$\{|$q$|$ B$\}\{2\backslash$pi m$\}\backslash ),$ does this agree with the relationship that $\backslash ($ f$=\backslash$frac$\{1\}\{$T$\}\backslash )$ Where the time that you calculated in part c$)$ represents half the period T for one full circular motion?

e$)$ With the cyclotron set to the cyclotron frequency, each time the Alpha particles make a crossing from one Dee to the other, they are perfectly timed to receive a boost in speed from the electric field. If our cyclotron has a radius of $0\mathrm{.}5$ m $,$ what is the speed of the Alpha particles when they exit?

f$)$ How many full cycles did the Alpha particles make around the cyclotron? $($Hint: the Alpha particles do not have the same increase in speed during each trip from one Dee to the other, but they do have the same increase in KE$...)$