Physics for Scientists and Engineers With Modern Physics
Physics for Scientists and Engineers With Modern Physics
9th Edition
ISBN: 9781133953982
Author: SERWAY, Raymond A./
Publisher: Cengage Learning
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Chapter 40, Problem 47P

(a)

To determine

The ratio of the wavelength of the photon to the wavelength of the electron.

(a)

Expert Solution
Check Mark

Answer to Problem 47P

The ratio of the wavelength of the photon to the wavelength of the electron is γγ1uc .

Explanation of Solution

It is given that the speed of the electron is close to the speed of light. This implies relativistic expression for energy has to be used.

Write the equation for the kinetic energy of the electron.

  K=(γ1)mec2                                                                                                          (I)

Here, K is the kinetic energy of the electron, γ is the Lorentz factor, me is the mass of the electron and c is the speed of light in vacuum.

Write the equation for the de Broglie wavelength of the electron.

  λ=hp                                                                                                                      (II)

Here, λ is the de Broglie wavelength of the electron, h is the Planck’s constant and p is the momentum of the electron.

Write the equation for the momentum of the electron.

  p=γmeu

Here, me is the mass of the electron and u is the speed of the electron.

  λ=hγmeu                                                                                                               (III)

It is given that energy of the photon is equal to the kinetic energy of the electron.

Write the relationship between the energy of the photon and the kinetic energy of the electron.

  E=K                                                                                                                    (IV)

Here, E is the energy of the photon.

Write the equation for the energy of the photon in terms of its frequency.

  E=hf                                                                                                                      (V)

Here, f is the frequency of the photon.

Write the equation for the wavelength of the photon.

  λph=cf

Here, λph is the wavelength of the photon.

Multiply the numerator and denominator of the right hand side of the above equation by h .

  λph=chhf

Replace denominator of the right hand side of the above equation by equation (V).

  λph=chE

Put equation (IV) in the above equation.

  λph=chK

Put equation (I) in the above equation.

  λph=ch(γ1)mec2                                                                                                    (VI)

Conclusion:

Take the ratio of equation (VI) to equation (III).

  λphλ=ch(γ1)mec2hγmeu=ch(γ1)mec2γmeuh=γγ1uc                                                                                        (VII)

Therefore, the ratio of the wavelength of the photon to the wavelength of the electron is γγ1uc .

(b)

To determine

The value of the ratio of the wavelength of the photon to the wavelength of the electron for the particle speed u=0.900c .

(b)

Expert Solution
Check Mark

Answer to Problem 47P

The value of the ratio of the wavelength of the photon to the wavelength of the electron for the particle speed u=0.900c is 1.60 .

Explanation of Solution

Write the expression for the Lorentz factor.

  γ=11u2/c2

Put the above equation in equation (VII).

  λphλ=11u2/c21[(1/1u2/c2)1]uc                                                             (VIII)

Conclusion:

Substitute 0.900c for u in equation (VIII) to find the value of the ratio.

  λphλ=11(0.900c)2/c21(1/1(0.900c)2/c2)10.900cc=11(0.900)20.900(1/1(0.900)2)1=1.60

Therefore, the value of the ratio of the wavelength of the photon to the wavelength of the electron for the particle speed u=0.900c is 1.60 .

(c)

To determine

The change in answer of part (b) if the particle were a proton instead of electron.

(c)

Expert Solution
Check Mark

Answer to Problem 47P

There will be no change in the answer to part (b) since the ration does not depend on mass.

Explanation of Solution

The expression for the ration of the wavelength of the photon to the wavelength of the material particle is given by equation (VIII). The equation contains only the speed of the material particle and the speed of light in vacuum. The equation is independent of the mass of the particle.

Since the ratio is independent of the mass of the material particle, the value of the ratio will be the same if the electron is replaced with a proton of same speed. This implies there will be no change in the answer to part (b) even when the electron is replaced with a proton.

Conclusion:

Thus, there will be no change in the answer to part (b) since the ration does not depend on mass.

(d)

To determine

The value of the ratio of the wavelength of the photon to the wavelength of the electron for the particle speed u=0.00100c .

(d)

Expert Solution
Check Mark

Answer to Problem 47P

The value of the ratio of the wavelength of the photon to the wavelength of the electron for the particle speed u=0.00100c is 2.00×103 .

Explanation of Solution

Equation (VIII) can be used to find the value of the ratio.

Conclusion:

Substitute 0.00100c for u in equation (VIII) to find the value of the ratio.

  λphλ=11(0.00100c)2/c21(1/1(0.00100c)2/c2)10.00100cc=11(0.00100)20.00100(1/1(0.00100)2)1=2.00×103

Therefore, the value of the ratio of the wavelength of the photon to the wavelength of the electron for the particle speed u=0.00100c is 2.00×103 .

(e)

To determine

The value the ratio of the wavelengths approaches at high particle speeds.

(e)

Expert Solution
Check Mark

Answer to Problem 47P

The value the ratio of the wavelengths approaches at high particle speeds is 1 .

Explanation of Solution

At high particles speeds, uc . As uc, uc1 .

Substitute this in the expression for Lorentz factor.

  γ111γ

As γ,γ1γ .

Conclusion:

Replace γ1 by γ and uc by 1 in equation (VII) to find the value of the ratio at high particle speeds.

  λphλγγ(1)=1

Therefore, the value the ratio of the wavelengths approaches at high particle speeds is 1 .

(f)

To determine

The value the ratio of the wavelengths approaches at low particle speeds.

(f)

Expert Solution
Check Mark

Answer to Problem 47P

The value the ratio of the wavelengths approaches at low particle speeds is .

Explanation of Solution

At low particles speeds, uc0 .

As uc0,

  γ=(1u2c2)1/2(10)1/2γ1                                                                                  (IX)

Also,

  γ1=(1u2c2)1/211(12)u2c21=12u2c2                                                                 (X)

Conclusion:

Put approximations (IX) and (X) in equation (VII) to find the value of the ratio at low particle speeds.

  λphλ1u/c(1/2)(u2/c2)=2cu

For low values of u , 2cu .

Replace 2cu by this approximation in the above equation.

  λphλ

Therefore, the value the ratio of the wavelengths approaches at low particle speeds is .

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Chapter 40 Solutions

Physics for Scientists and Engineers With Modern Physics

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