A proton in a cyclotron gains K = 2eV of kinetic energy per revolution, where V is

Question:

A proton in a cyclotron gains ΔK = 2eΔV of kinetic energy per revolution, where ΔV is the potential between the dees. Although the energy gain comes in small pulses, the proton makes so many revolutions that it is reasonable to model the energy as increasing at the constant rate P = dK/ dt = ΔK/T, where T is the period of the cyclotron motion. This is power input because it is a rate of increase of energy.
a. Find an expression for r(t), the radius of a proton’s orbit in a cyclotron, in terms of m, e, B, P, and t. Assume that r = 0 at t = 0.

b. A relatively small cyclotron is 2.0 m in diameter, uses a 0.55 T magnetic field, and has a 400 V potential difference between the dees. What is the power input to a proton, in W?
c. How long does it take a proton to spiral from the center out to the edge?