dk dt Problem 3: A proton in a cyclotron gains AK = 2eAV of kinetic energy per revolution, where AV 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 = = AK, where T is the period of the cyclotron motion. This is power input because it is a rate of increase of energy. 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.
dk dt Problem 3: A proton in a cyclotron gains AK = 2eAV of kinetic energy per revolution, where AV 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 = = AK, where T is the period of the cyclotron motion. This is power input because it is a rate of increase of energy. 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.
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Hello I really need help because I don't understand the problem because I keep getting the wrong answers I have done all of the problems five times and I still get wrong. Is there any chance that you can help me with part A,Part B, Part C and Part D and can you label them as well thank you so so so much
Transcribed Image Text:AK
Problem 3: A proton in a cyclotron gains AK = 2eAV of kinetic energy per revolution, where AV 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 = k=, where
T is the period of the cyclotron motion. This is power input because it is a rate of increase of energy. 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.
3
a) Using the formula for the radius of the cyclotron orbit, r = mg, work out a formula for the kinetic
energy K of the proton in terms of m, e, B, and r.
b) Show that equation d P results in the following differential equation for r as a function of t:
rdr = Pm Make sure to show all the steps that lead to this equation.
=
e² B².
Pm
e² B2
c) This differential equation can be solved by separating the variables. For that, rewrite it as r dr =
dt and integrate both sides. The expression on the left side should be integrated from r = 0 (center)
to r(t) (an intermediate radius at the moment of time t), while the expression on the right side should be
integrated from 0 to t. From here, obtain the expression for r(t) in terms of m, e, B, P, and t.
d) Express the power input of the cyclotron, P = AK, in terms of m, e, B, and AV.
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