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.
University Physics Volume 3
17th Edition
ISBN:9781938168185
Author:William Moebs, Jeff Sanny
Publisher:William Moebs, Jeff Sanny
Chapter5: Relativity
Section: Chapter Questions
Problem 36P: Describe the following physical occurrences as events, that is, in the form (x, y, z, t): (a) A...
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