Physics for Scientists and Engineers with Modern Physics
Physics for Scientists and Engineers with Modern Physics
10th Edition
ISBN: 9781337553292
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 39, Problem 16P

(a)

To determine

The wavelengths of photons scattered at angles 30.0°, 60.0°, 90.0° , 120.0°, 150.0°, 180.0°.

(a)

Expert Solution
Check Mark

Answer to Problem 16P

The wavelengths of photons scattered at angles 30.0°, 60.0°, 90.0° , 120.0°, 150.0°, 180.0° are 120.3×1012m, 121.2×1012m, 122.4×1012m, 123.6×1012m, 124.5×1012m and 124.8×1012m respectively.

Explanation of Solution

Write the expression to find the change in wavelength.

  Δλ=hmec(1cosθ)                                                                                          (I)

Here, h is the Plank’s constant, me is the mass of electron, c is the speed of light, Δλ is the change in wavelength, θ is the scattering angle.

Write the expression to find the wavelengths of scattered photons.

  λ=λ+Δλ                                                                                                      (II)

Here, λ is the wavelengths of scattered photons, λ is the wavelength of X-rays before scattering.

Substitute equation (I) in (II) to find the wavelength of scattered photons.

  λ=λ+hmec(1cosθ)

Conclusion:

Substitute 120.0pm for λ, 6.626×1034Js for h, 9.11×1031kg for me, 2.998×108m/s for c and 30.0° for θ to find the wavelength of scattered photon at 30.0°.

  λ=120.0pm(1012m1.0pm)+6.626×1034Js(9.11×1031kg)(2.998×108m/s)(1cos(30.0°))=120.0×1012m+0.3×1012m=120.3×1012m

Substitute 120.0pm for λ, 6.626×1034Js for h, 9.11×1031kg for me, 2.998×108m/s for c and 60.0° for θ to find the wavelength of scattered photon at 60.0°.

  λ=120.0pm(1012m1.0pm)+6.626×1034Js(9.11×1031kg)(2.998×108m/s)(1cos(60.0°))=120.0×1012m+1.2×1012m=121.2×1012m

Substitute 120.0pm for λ, 6.626×1034Js for h, 9.11×1031kg for me, 2.998×108m/s for c and 90.0° for θ to find the wavelength of scattered photon at 90.0°.

  λ=120.0pm(1012m1.0pm)+6.626×1034Js(9.11×1031kg)(2.998×108m/s)(1cos(90.0°))=120.0×1012m+2.4×1012m=122.4×1012m

Substitute 120.0pm for λ, 6.626×1034Js for h, 9.11×1031kg for me, 2.998×108m/s for c and 120.0° for θ to find the wavelength of scattered photon at 120.0°.

  λ=120.0pm(1012m1.0pm)+6.626×1034Js(9.11×1031kg)(2.998×108m/s)(1cos(120.0°))=120.0×1012m+3.6×1012m=123.6×1012m

Substitute 120.0pm for λ, 6.626×1034Js for h, 9.11×1031kg for me, 2.998×108m/s for c and 150.0° for θ to find the wavelength of scattered photon at 150.0°.

  λ=120.0pm(1012m1.0pm)+6.626×1034Js(9.11×1031kg)(2.998×108m/s)(1cos(150.0°))=120.0×1012m+4.5×1012m=124.5×1012m

Substitute 120.0pm for λ, 6.626×1034Js for h, 9.11×1031kg for me, 2.998×108m/s for c and 180.0° for θ to find the wavelength of scattered photon at 180.0°.

  λ=120.0pm(1012m1.0pm)+6.626×1034Js(9.11×1031kg)(2.998×108m/s)(1cos(180.0°))=120.0×1012m+4.8×1012m=124.8×1012m

Therefore, The wavelengths of photons scattered at angles 30.0°, 60.0°, 90.0° , 120.0°, 150.0°, 180.0° are 120.3×1012m, 121.2×1012m, 122.4×1012m, 123.6×1012m, 124.5×1012m and 124.8×1012m respectively.

(b)

To determine

The energy of electrons at scattering angles 30.0°, 60.0°, 90.0° , 120.0°, 150.0°, 180.0°.

(b)

Expert Solution
Check Mark

Answer to Problem 16P

The energy of electrons at scattering angles 30.0°, 60.0°, 90.0° , 120.0°, 150.0°, 180.0° are 27.9eV, 104eV, 205eV, 305eV, 376eV, 402eV respectively.

Explanation of Solution

The energy of the electrons will be the energy lost from the photons.

Write the expression to find the energy of electrons.

  Ke=hcλ0hcλ=hc(1λ1λ)

Here, Ke is the energy of electrons, λ is the wavelength before scattering, λ is the wavelength of scattered photon.

Conclusion:

Substitute 120.0×1012m for λ, 6.626×1034Js for h, 2.998×108m/s for c and 120.3×1012m for λ to find the energy of scattered electron when scattering angle is 30.0°.

  Ke=(6.626×1034Js)(2.998×108m/s)(1120.0×1012m1120.3×1012m)(1.0J1.602×1019eV)=27.9eV

Substitute 120.0×1012m for λ, 6.626×1034Js for h, 2.998×108m/s for c and 121.2×1012m for λ to find the energy of scattered electron when scattering angle at 60.0°.

  Ke=(6.626×1034Js)(2.998×108m/s)(1120.0×1012m1121.2×1012m)(1.0J1.602×1019eV)=104eV

Substitute 120.0×1012m for λ, 6.626×1034Js for h, 2.998×108m/s for c and 122.4×1012m for λ to find the energy of scattered electron when scattering angle at 90.0°.

  Ke=(6.626×1034Js)(2.998×108m/s)(1120.0×1012m1122.4×1012m)(1.0J1.602×1019eV)=205eV

Substitute 120.0×1012m for λ, 6.626×1034Js for h, 2.998×108m/s for c and 123.6×1012m for λ to find the energy of scattered electron when scattering angle at 120.0°.

  Ke=(6.626×1034Js)(2.998×108m/s)(1120.0×1012m1123.6×1012m)(1.0J1.602×1019eV)=305eV

Substitute 120.0×1012m for λ, 6.626×1034Js for h, 2.998×108m/s for c and 124.5×1012m for λ to find the energy of scattered electron when scattering angle at 150.0°.

  Ke=(6.626×1034Js)(2.998×108m/s)(1120.0×1012m1124.5×1012m)(1.0J1.602×1019eV)=376eV

Substitute 120.0×1012m for λ, 6.626×1034Js for h, 2.998×108m/s for c and 124.8×1012m for λ to find the energy of scattered electron when scattering angle at 180.0°.

  Ke=(6.626×1034Js)(2.998×108m/s)(1120.0×1012m1124.8×1012m)(1.0J1.602×1019eV)=402eV

Therefore, The energy of scattered electron when scattering angle is 30.0°, 60.0°, 90.0°, 120.0°, 150.0°, 180.0° are 27.9eV, 104eV, 205eV, 305eV, 376eV, 402eV respectively.

(c)

To determine

The scattering angle with which the electron gets the greatest energy.

(c)

Expert Solution
Check Mark

Answer to Problem 16P

The scattering angle with which the electron gets the greatest energy is 180°.

Explanation of Solution

When the scattering angle is 180.0°, the photon strikes on the electron which is at rest with a head on collision. In the head on collision, the photon imparts its momentum to the electron.

After the head on collision the photon scattered straight back and the kinetic energy gained by the initially stationary electron will be the maximum. The kinetic energy gained by the electron at scattering angle 180° is 402eV.

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