Problem 2: CyclotronIn the medical field, there are a variety of reasons for needing access to a particle accelerator. It sounds fancy calling it that, but even a lowly dental x-ray needs to accelerate particles in order to generate the x-rays. Another use for a particle accelerator is to generate high speed protons, which could be used directly for medical benefit (proton therapy for treating some cancers), or for colliding with certain nuclei to create “radiotracers”, such as radioactive fluorine used in PET (positron emission tomography) scans. For these latter two cases, it is most common to use a cyclotron, where an initially low energy proton moves in a circle in a magnetic field, and with each lap, is accelerated by well-timed applied voltage (electric field).a)In a uniform magnetic field B, find the number of laps per second (ie, frequency, f, in Hz) that a particle of charge qand mass mtravel in a circle in the cyclotron. (Looking for an answer in variables). Note: an interesting thing about a cyclotron is that the frequencies are independent of the particle velocity. Hint: What force does a movingchargedparticle feel in a magnetic field? That’s the force keeping it going in a circle (ie centripetal acceleration). It’s also worth remembering that 푣=휔푟for circular motion, and 휔=2휋푓.b)For proton therapy, protons are needed at an energy of 70 MeV. If a magnetic field of 1Tesla is the limit of what’s practicallyobtainable, what diameter cyclotron is needed?And what would the frequencybe (using equation from part b)
This frequency is also the frequency your applied electric field would need to oscillate at in order to accelerate the proton. Hint: Find a relationship between vand rby setting the magnetic force equal to the centripetal acceleration, and then relate vwith kinetic energy to solve for R.