Explanation 1) Concept: We can equate the centripetal force and Coulomb force between the muon and the H e + nucleus to find the equation for square of velocity. By using quantization condition, we can find the radius. By substituting these values in the equation for total energy, we can prove the given equation. 2) Formula: i) F = - m v 2 r ii) F = - k Z e 2 r 2 iii) L = n ℏ = m v r iv) K = 1 2 m v 2 v) U = - k Z e 2 r 3) Given: i) Charge of muon is - e ii) Mass of muon is m = 207 m e = 207 × 9.11 × 10 - 31 k g 4) Calculations: Suppose the muon of mass m is orbiting in circular uniform motion around the H e + nucleus with velocity v .It, thus, experiences a centripetal force given by F = - m v 2 r This force must be balanced by the Coulomb force between the muon and the H e + nucleus, which is given as F = - k Z e 2 r 2 Where, k = 1 / 4 π ε 0 Therefore, - m v 2 r = - k Z e 2 r 2 v 2 = k Z e 2 m r … … … … … … … … … … … … … … … … … … … … … … ( 1 ) Since the magnitude of the angular momentum is quantized as L = n ℏ where ℏ = h / 2 π And as L = m v r , we get, m v r = n ℏ Rearranging this equation for v , v = n ℏ m r Substituting this in equation ( 1 ) , n ℏ m r 2 = k Z e 2 m r On simplifying, r = n 2 ℏ 2 k Z e 2 m … … … … … … … … … … … … … … … … … … … … … … ( 2 ) Now, the kinetic energy of muon is K = 1 2 m v 2

10th Edition

ISBN: 9781118230725

Author: David Halliday, Robert Resnick, Jearl Walker

Publisher: Wiley, John & Sons, Incorporated

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Chapter 39, Problem 68P

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The energy levels of the system are given by

E=-11.3 keVn2

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Chapter 39, Problem 67P

Chapter 39, Problem 69P

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Chapter 39 Solutions

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The energy levels of the system are given by E = - 11.3 k e V n 2

Solution Summary: The author explains that the energy levels of the system are given by E=-11.3keVn2.

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To show:

The energy levels of the system are given by

E=-11.3 keVn2

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Chapter 39 Solutions

The energy levels of the system are given by E = - 11.3 k e V n 2

Solution Summary: The author explains that the energy levels of the system are given by E=-11.3keVn2.

Concept explainers

To show: The energy levels of the system are given by E=-11.3 keVn2

Want to see the full answer?

Chapter 39 Solutions

To show:

The energy levels of the system are given by

E=-11.3 keVn2