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Chapter 34: Problem 51

A 150 -pm X-ray photon Compton-scatters off an electron and emerges at \(135^{\circ}\) to its original direction. Find (a) the wavelength of the scattered photon and (b) the electron's kinetic energy.

Short Answer

Expert verified
The wavelength of the scattered photon is approximately \(165\) pm and the kinetic energy of the electron is approximately \(478\) eV.

Step by step solution

01

Apply the Compton Wavelength Shift Formula

Use the Compton wavelength shift equation: \(\Delta \lambda = \lambda' - \lambda = h/(m_e c)(1 - cos(\Theta))\), where \(\Delta \lambda\) is the change in wavelength, \(\lambda\) is the initial wavelength of the photon, \(m_e\) is the electron mass, \(c\) is the speed of light, and \(\theta\) is the scattering angle. Plug in the known values: \(h = 6.62607004 × 10^{-34}\) m² kg / s, \(m_e= 9.11 × 10^{-31}\) kg, \(c\) = \(3 × 10^8\) m/s and \(\theta = 135^\circ\).
02

Calculate the scattered wavelength

To calculate the final scattered wavelength \( \lambda' \), we need to add the original wavelength \( \lambda \) to the change in wavelength \( \Delta \lambda \). Since we know the energy of the incident photon (we can calculate it from its given wavelength using the relation \( E = \frac{hc}{\lambda} \)), we can get the initial wavelength \( \lambda = \frac{hc}{E} \).
03

Calculate the electron's kinetic energy

The kinetic energy of the electron can be calculated using the conservation of energy. The difference in energy of the photon before and after the collision is the kinetic energy of the electron. This can be expressed as \( \Delta E = E_{initial} - E_{final} = K.E_{electron} \). The energy of the photon can be calculated from their wavelengths: \( E = \frac{hc}{\lambda} \). Find the initial and final energies, calculate the difference, and that will give us the electron’s kinetic energy.

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Most popular questions from this chapter

Suppose the Compton effect were significant at radio wavelengths. What problems might this present for radio and TV broadcasting?

Ultraviolet light with wavelength 75 nm shines on hydrogen atoms in their ground states, ionizing some of the atoms. What's the energy of the electrons freed in this process?

(a) Find the Compton wavelength for a proton. (b) Find the energy in electronvolts of a gamma ray whose wavelength equals the proton's Compton wavelength.

Why is the immediate ejection of electrons in the photoelectric effect surprising from a classical viewpoint?

A photon's wavelength is equal to the Compton wavelength of a particle with mass \(m .\) Show that the photon's energy is equal to the particle's rest energy.
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