Problem 3: A proton in a cyclotron gains AK = 2eAV of kinetic energy per revolution, where AV is th potential between the dees. Although the energy gain comes in small pulses, the proton makes so man ΔΚ revolutions that it is reasonable to model the energy as increasing at the constant rate P = dk = AK, when T is the period of the cyclotron motion. This is power input because it is a rate of increase of energy. Fin an expression for r(t), the radius of a proton's orbit in a cyclotron, in terms of m, e, B, P, and t. Assum 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 kinet energy K of the proton in terms of m, e, B, and r. qB¹ b) Show that equation d dk = P results in the following differential equation for r as a function of Make sure to show all the steps that lead to this equation. rdr = Pm e² B2- Pm e2 B2 c) This differential equation can be solved by separating the variables. For that, rewrite it as r dr Pdt and integrate both sides. The expression on the left side should be integrated from r = 0 (cente to r(t) (an intermediate radius at the moment of time t), while the expression on the right side should b integrated from 0 to t. From here, obtain the expression for r(t) in terms of m, e, B, P, and t.
Problem 3: A proton in a cyclotron gains AK = 2eAV of kinetic energy per revolution, where AV is th potential between the dees. Although the energy gain comes in small pulses, the proton makes so man ΔΚ revolutions that it is reasonable to model the energy as increasing at the constant rate P = dk = AK, when T is the period of the cyclotron motion. This is power input because it is a rate of increase of energy. Fin an expression for r(t), the radius of a proton's orbit in a cyclotron, in terms of m, e, B, P, and t. Assum 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 kinet energy K of the proton in terms of m, e, B, and r. qB¹ b) Show that equation d dk = P results in the following differential equation for r as a function of Make sure to show all the steps that lead to this equation. rdr = Pm e² B2- Pm e2 B2 c) This differential equation can be solved by separating the variables. For that, rewrite it as r dr Pdt and integrate both sides. The expression on the left side should be integrated from r = 0 (cente to r(t) (an intermediate radius at the moment of time t), while the expression on the right side should b integrated from 0 to t. From here, obtain the expression for r(t) in terms of m, e, B, P, and t.
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
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Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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I really need help with all of the problems because I am having a difficult time with this problem I did the problem four times and I don't get it is there a chance you can help me with all of the problems and label them
![**Problem 3: Cyclotron Dynamics and Kinetic Energy**
In a cyclotron, a proton gains an increase in kinetic energy, \( \Delta K = 2e \Delta V \), per revolution, where \( \Delta V \) is the potential difference between the dees. Despite the energy gain occurring in small pulses, the proton undergoes numerous revolutions, justifying the model of energy increasing at a constant rate \( P = \frac{dK}{dt} = \frac{\Delta K}{T} \), where \( T \) is the period of the cyclotron motion. This represents a power input as it signifies a rate of increase of energy. Your task is to find an expression for \( r(t) \), the radius of a proton’s orbit in a cyclotron, using \( m, e, B, P, \) and \( t \). Assume \( r = 0 \) at \( t = 0 \).
---
**Part (a)**
Using the formula for the cyclotron orbit radius, \( r = \frac{mv}{qB} \), derive a formula for the kinetic energy \( K \) of the proton in terms of \( m, e, B, \) and \( r \).
---
**Part (b)**
Show that the equation \( \frac{dK}{dt} = P \) results in the differential equation:
\[ r \frac{dr}{dt} = \frac{Pm}{e^2 B^2} \]
Ensure all steps leading to this equation are clear.
---
**Part (c)**
Solve this differential equation by separating the variables. Rewrite it as \( r \, dr = \frac{Pm}{e^2 B^2} \, dt \), then integrate both sides. Integrate the left side from \( r = 0 \) to \( r(t) \). Integrate the right side from \( 0 \) to \( t \). From here, derive the expression for \( r(t) \) in terms of \( m, e, B, P, \) and \( t \).
---
**Part (d)**
Express the power input of the cyclotron, \( P = \frac{\Delta K}{T} \), using \( m, e, B, \) and \( \Delta V \).
---
**Part (e)**
A relatively small cyclotron, 2.0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe03336ee-221b-42fe-a81a-096801f8da53%2F3b645f3e-25bb-4527-b0c2-48b39fa1ed23%2F8s5iu6d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 3: Cyclotron Dynamics and Kinetic Energy**
In a cyclotron, a proton gains an increase in kinetic energy, \( \Delta K = 2e \Delta V \), per revolution, where \( \Delta V \) is the potential difference between the dees. Despite the energy gain occurring in small pulses, the proton undergoes numerous revolutions, justifying the model of energy increasing at a constant rate \( P = \frac{dK}{dt} = \frac{\Delta K}{T} \), where \( T \) is the period of the cyclotron motion. This represents a power input as it signifies a rate of increase of energy. Your task is to find an expression for \( r(t) \), the radius of a proton’s orbit in a cyclotron, using \( m, e, B, P, \) and \( t \). Assume \( r = 0 \) at \( t = 0 \).
---
**Part (a)**
Using the formula for the cyclotron orbit radius, \( r = \frac{mv}{qB} \), derive a formula for the kinetic energy \( K \) of the proton in terms of \( m, e, B, \) and \( r \).
---
**Part (b)**
Show that the equation \( \frac{dK}{dt} = P \) results in the differential equation:
\[ r \frac{dr}{dt} = \frac{Pm}{e^2 B^2} \]
Ensure all steps leading to this equation are clear.
---
**Part (c)**
Solve this differential equation by separating the variables. Rewrite it as \( r \, dr = \frac{Pm}{e^2 B^2} \, dt \), then integrate both sides. Integrate the left side from \( r = 0 \) to \( r(t) \). Integrate the right side from \( 0 \) to \( t \). From here, derive the expression for \( r(t) \) in terms of \( m, e, B, P, \) and \( t \).
---
**Part (d)**
Express the power input of the cyclotron, \( P = \frac{\Delta K}{T} \), using \( m, e, B, \) and \( \Delta V \).
---
**Part (e)**
A relatively small cyclotron, 2.0
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