Hello, please I really need help with part D and part E I really don't understand understand those two problems I have done all of the other parts I just need help with that one, please I need help with part D and part E and can you label them ΔΚProblem 3: A proton in a cyclotron gains AK = 2eAV of kinetic energy per revolution, where AV is thepotential between the dees. Although the energy gain comes in small pulses, the proton makes so manyrevolutions that it is reasonable to model the energy as increasing at the constant rate P = K = AK, whereT is the period of the cyclotron motion. This is power input because it is a rate of increase of energy. Findan expression for r(t), the radius of a proton's orbit in a cyclotron, in terms of m, e, B, P, and t. Assumethat r = 0 at t = 0.a) Using the formula for the radius of the cyclotron orbit, r = mg, work out a formula for the kineticenergy K of the proton in terms of m, e, B, and r.b) Show that equation d dK = P results in the following differential equation for r as a function of t:Make sure to show all the steps that lead to this equation.rder t=Pme² B².3Pme²B²c) 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 beintegrated 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.e) A relatively small cyclotron is 2.0 m in diameter, uses a 0.55 T magnetic field, and has a 400 V potentialdifference between the dees. What is the power input to a proton, in W?f) How long does it take a proton to spiral from the center out to the edge? (Answer: 2.17 ms.)

Answered: Image /qna-images/answer/3409f5f9-b67b-4444-8d15-396b4641da51.jpg

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.

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.

University Physics Volume 2

18th Edition

ISBN:9781938168161

Author:OpenStax

Publisher:OpenStax

Chapter11: Magnetic Forces And Fields

Section: Chapter Questions

Problem 55P: The magnetic field in a cyclotron is 1.25 T, and the maximum orbital radius of the circulating...

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Ferromagnetism

Ferromagnetism is a mechanism by which some materials show magnetic properties. Iron and magnetite, an oxide of iron, are two naturally occurring materials that show the property of ferromagnetism. The word ferromagnetism is derived from the Latin name of iron, "Ferrum" as it was the first observed material to show such property. The property of ferromagnetism is shown by only a few substances such as iron, nickel, cobalt, and their alloys. Some transition metals and alloys of rare-earth metals also show ferromagnetism.

Question

Hello, please I really need help with part D and part E I really don't understand understand those two problems I have done all of the other parts I just need help with that one, please I need help with part D and part E and can you label them

ΔΚ
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 = 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.
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 dK = P results in the following differential equation for r as a function of t:
Make sure to show all the steps that lead to this equation.
rder t
=
Pm
e² B².
3
Pm
e²B²
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.
e) 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?
f) How long does it take a proton to spiral from the center out to the edge? (Answer: 2.17 ms.)

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