Consider a proton undergoing 2d cyclotron motion (in the xy-plane) with initial velocity inv0 a magnetic field of magnitudeB pointing out of the plane. Larmor's formula for the power radiated by an accelerating charge is:P=60c3q2a2 A. What is the power radiated by a proton undergoing cyclotron motion with velocityv in a magnetic field B? Note that the power radiated is related to the kinetic energyKE of the electron by: dtdKE=P.

B. Show that the kinetic energy, KE, of the proton satisfies a differential equation of the form:dtdKE=KE , which has solution KE(t)=KE0etwhereKE0=21mv02 is the initial kinetic energy. Determine the decay rate in terms of v,B and fundamental constants.

C. Assuming that the decay constant is much less than the cyclotron frequency,fc:fc , we can approximate that the decay rate is constant over one cyclotron orbit. What is the change inKE of the proton per cyclotron period as a function of the velocity v?

D. Cyclotron accelerator: A cyclotron accelerator applies an alternating voltage with amplitude V, and with a frequency that matches the cyclotron frequency of the proton, such that the proton gains energy: eVduring each cyclotron period. The maximum speed attainable by a cyclotron occurs when the KEgain from the accelerating voltage is equal to theKE loss due to the cyclotron radiation. What is the maximum speed, vmax, to which we can accelerate a proton using a cyclotron accelerator

E. How would the maximum speed compare to the answer in Part D if we accelerated an electron with the same cyclotron?