Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
9th Edition
ISBN: 9781305372337
Author: Raymond A. Serway | John W. Jewett
Publisher: Cengage Learning
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Chapter 40, Problem 31P

(a)

To determine

The scattering angle of the photon and the electron.

(a)

Expert Solution
Check Mark

Answer to Problem 31P

The scattering angle of the photon and the electron is 43.0° .

Explanation of Solution

Write the equation for the momentum of the photon before scattering.

  p0=E0c                                                                                                                     (I)

Here, p0 is the momentum of the photon before scattering, E0 is the energy of the photon before scattering and c is the speed of light in vacuum.

Write the equation for the energy of the incident photon.

  E0=hcλ0                                                                                                                    (II)

Here, h is the Plank’s constant and λ0 is the wavelength of the incident photon.

Put equation (II) in equation (I).

  p0=hc/λ0c=hλ0                                                                                                              (III)

Write the equation for the momentum of the photon after scattering.

  p=Ec                                                                                                                   (IV)

Here, p is the momentum of the photon after scattering and E is the energy of the photon after scattering.

Write the equation for the energy of the scattered photon.

  E=hcλ                                                                                                                   (V)

Here, E is the energy of the scattered photon and λ is the wavelength of the scattered photon.

Put equation (V) in equation (IV).

  p=hc/λc=hλ                                                                                                             (VI)

Refer to figure P40.31,and write the equation for the conservation of momentum in x direction.

  p0=pcosθ+pecosθ

Here, θ is the angle of scattering and pe is the momentum of the electron after the scattering.

Put equations (III) and (VI) in the above equation.

  hλ0=hλcosθ+pecosθ=(hλ+pe)cosθ                                                                                         (VII)

Refer to figure P40.31,and write the equation for the conservation of momentum in y direction.

  0=psinθpesinθ

Neglect the trivial solution θ=0 for the above equation.

  psinθ=pesinθp=pe                                                                                                (VIII)

Put equation (VI) in the above equation.

  hλ=pe                                                                                                                 (IX)

Put equation (IX) in equation (VII) and rewrite it for λ .

  hλ0=(hλ+hλ)cosθ=2hλcosθλ=2hλ0hcosθ=2λ0cosθ                                                                                              (X)

Write the equation for the Compton shift.

  λλ0=hmec(1cosθ)                                                                                          (XI)

Here, me is the mass of the electron.

Put equation (X) in equation (XI).

  2λ0cosθλ0=hmec(1cosθ)

Solve the above equation.

  2λ0cosθλ0=hmec(1cosθ)=hmechmeccosθ(2λ0+hmec)cosθ=λ0+hmec                                                                     (XII)

Rewrite equation (II) for λ0 .

  λ0=hcE0

Put the above equation in equation (XII).

  (2hcE0+hmec)cosθ=hcE0+hmec2hc2me+hE0mecE0cosθ=hc2me+hE0mecE01mecE0(2c2me+E0)cosθ=1mecE0(c2me+E0)cosθ=mec2+E02mec2+E0

Rewrite the above equation for θ .

  θ=cos1mec2+E02mec2+E0                                                                                     (XIII)

Conclusion:

It is given that the energy of the incident photon is 0.880 MeV .

The value of the mec2 is 0.511 MeV .

Substitute 0.511 MeV for mec2 and 0.880 MeV for E0 in equation (XIII) to find θ .

  θ=cos10.511 MeV+0.880 MeV2(0.511 MeV)+0.880 MeV=43.0°

Therefore, the scattering angle of the photon and the electron is 43.0° .

(b)

To determine

The energy and momentum of the scattered photon.

(b)

Expert Solution
Check Mark

Answer to Problem 31P

The energy of the scattered photon is 0.602 MeV and the momentum is 3.21×1022 kgm/s .

Explanation of Solution

Put equation (X) in equation (V).

  E=hc2λ0cosθ=hcλ0(2cosθ)

Put equation (II) in the above equation.

  E=E02cosθ                                                                                                          (XIV)

Conclusion:

Substitute 0.880 MeV for E0 and 43.0° for θ in equation (XIV) to find E .

  E=0.880 MeV2cos43.0°=0.602 MeV

Substitute 0.602 MeV for E and 3.00×108 m/s for c in equation (IV) to find p .

  p=0.602 MeV1.60×1013 J1 MeV3.00×108 m/s=3.21×1022 kgm/s

Therefore, the energy of the scattered photon is 0.602 MeV and the momentum is 3.21×1022 kgm/s .

(c)

To determine

The kinetic energy and momentum of the scattered electron.

(c)

Expert Solution
Check Mark

Answer to Problem 31P

The kinetic energy of the scattered electron is 0.278 MeV and its momentum is 3.21×1022 kgm/s .

Explanation of Solution

Write the equation for the kinetic energy of the scattered electron.

  Ke=E0E                                                                                                        (XV)

Here, Ke is the kinetic energy of the scattered electron.

Conclusion:

Substitute 0.880 MeV for E0 and 0.602 MeV for E in equation (XV) to find Ke .

  Ke=0.880 MeV0.602 MeV=0.278 MeV

Substitute 3.21×1022 kgm/s for p in equation (VIII) to find pe .

  pe=3.21×1022 kgm/s

Therefore, the kinetic energy of the scattered electron is 0.278 MeV and its momentum is 3.21×1022 kgm/s .

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Chapter 40 Solutions

Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University

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