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. dt 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. qB¹ 3 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 c) This differential equation can be solved by separating the variables. For that, rewrite it as r dr = Pm dt and integrate both sides. The expression on the left side should be integrated from r = 0 (center) e² B² 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 = A, 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.)
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. dt 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. qB¹ 3 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 c) This differential equation can be solved by separating the variables. For that, rewrite it as r dr = Pm dt and integrate both sides. The expression on the left side should be integrated from r = 0 (center) e² B² 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 = A, 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.)
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
Related questions
Question
Hello, I really need help with all of these problems since they one problem after the other is there any chance that you label these problems because I kept getting the wrong answer
Transcribed Image Text: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.)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.Recommended textbooks for you
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
Physics for Scientists and Engineers
Physics
ISBN:
9781337553278
Author:
Raymond A. Serway, John W. Jewett
Publisher:
Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:
9780321820464
Author:
Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:
Addison-Wesley
College Physics: A Strategic Approach (4th Editio…
Physics
ISBN:
9780134609034
Author:
Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:
PEARSON