Problem 62 - College Physics

Question

A photon with wavelength of 0.1100 nm collides with a free electron that is initially at rest. After the collision, the photon's wavelength is 0.132 \(\mathrm{nm}\) . (a) What is the kinetic energy of the electron after the collision? What is its speed? (b) If the electron is suddenly stopped (for example, in a solid target), all of its kinetic energy is used to create a photon. What is the wavelength of this photon?

Step-by-Step Solution

Answer

Kinetic energy: \(0.29 \times 10^{-15} \, \text{J}\). Speed: \(3.18 \times 10^6 \, \text{m/s}\). New photon's wavelength: \(6.85 \, \text{nm}\).

Step 1: Understand the Compton Scattering

The problem refers to a photon's collision with a free electron, where the Compton Effect applies. The change in wavelength of the photon can be used to find information about the photon's and electron's energy after the collision.

Step 2: Calculate the Change in Wavelength

Calculate the change in wavelength of the photon using:\[ \Delta \lambda = \lambda' - \lambda = 0.132 \, \text{nm} - 0.110 \, \text{nm} = 0.022 \, \text{nm} \]

Step 3: Use the Compton Equation

The Compton wavelength shift is given by:\[ \Delta \lambda = \frac{h}{mc} (1 - \cos \theta) \]Solving for the angle \(\theta\) is not necessary for kinetic energy calculation as it relates only to photon energy change.

Step 4: Calculate the Increase in Photon Energy

Using the energy-wavelength relation, \(E = \frac{hc}{\lambda}\), find the initial and final energy:\[ E_i = \frac{(6.626 \times 10^{-34} \, \text{Js}) (3 \times 10^8 \, \text{m/s})}{0.110 \times 10^{-9} \, \text{m}} \approx 1.80 \times 10^{-15} \, \text{J} \]\[ E_f = \frac{(6.626 \times 10^{-34} \, \text{Js}) (3 \times 10^8 \, \text{m/s})}{0.132 \times 10^{-9} \, \text{m}} \approx 1.51 \times 10^{-15} \, \text{J} \]//The energy lost by the photon is the gain in energy of the electron:\[ \Delta E = E_i - E_f \approx 0.29 \times 10^{-15} \, \text{J} \]

Step 5: Determine the Kinetic Energy of the Electron

The kinetic energy of the electron after the collision is the photon energy loss (change in photon energy):\[ KE_e = \Delta E \approx 0.29 \times 10^{-15} \, \text{J} \]

Step 6: Calculate the Speed of the Electron

Use the kinetic energy expression to find speed:\[ KE = \frac{1}{2} mv^2 \]\[ v^2 = \frac{2 \cdot (0.29 \times 10^{-15})}{9.11 \times 10^{-31}} \]\[ v \approx \sqrt{\left( \frac{0.58 \times 10^{-15}}{9.11 \times 10^{-31}} \right)} \approx 3.18 \times 10^6 \, \text{m/s} \]

Step 7: Calculate Wavelength of Photon Created by Electron Stoppage

Use the same energy-wavelength relation:\[ \lambda_{new} = \frac{hc}{KE} \]\[ \lambda_{new} = \frac{(6.626 \times 10^{-34} \, \text{Js}) (3 \times 10^8 \, \text{m/s})}{0.29 \times 10^{-15} \, \text{J}} \approx 6.85 \times 10^{-9} \, \text{m} \] or \(6.85 \, \text{nm}\)

Step 8: Conclusion

The kinetic energy of the electron after the collision is approximately \(0.29 \times 10^{-15} \, \text{J}\), and its speed \(v\) is approximately \(3.18 \times 10^{6} \, \text{m/s}\). When the electron stops and its kinetic energy converts to a photon, the new photon's wavelength is approximately \(6.85 \, \text{nm}\).

Key Concepts

Photon-Electron Collision

When a photon collides with an electron, we encounter a fascinating phenomenon in physics known as Compton scattering. This event showcases how light, viewed as a particle, interacts with material particles such as electrons. During the collision, the photon, which can be imagined as a tiny packet of light energy, transfers some of its energy to the electron. This energy exchange causes the photon's wavelength to increase, indicating a reduction in its energy.

• The electron starts at rest before the collision.
• Energy and momentum conservation laws govern the interaction.
• The change in photon wavelength reveals information about energy transfers.

In this problem, both the initial and final wavelengths of the photon are given, allowing us to calculate changes in energy and thus predict the behavior of both photon and electron post-collision.

Kinetic Energy of Electron

Kinetic energy refers to the energy that a body possesses due to its motion. After a photon collides with an electron, the electron gains some energy from the photon, which translates into kinetic energy. We can calculate this gain using the work-energy principle and energy conservation. Here’s how:

• Calculate the initial and final photon energies using the formula:
  \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength.
• The energy transferred to the electron is the difference between the initial and final photon energy.
• This energy becomes the kinetic energy of the electron.

By understanding these steps, you can see how the electron achieves its motion and why its kinetic energy is vital to comprehend photon interactions in physics.

Energy-Wavelength Relation

The relationship between energy and wavelength is fundamental when examining photon-behavior scenarios like Compton scattering. This relation is described by the formula \( E = \frac{hc}{\lambda} \), showing energy is inversely proportional to wavelength. Therefore, when a photon's wavelength increases, its energy decreases.

• This relationship allows us to compute changes in energy from changes in wavelength.
• Allows for determination of resultant particle characteristics when energy is transferred.
• This understanding clarifies how photon energy losses directly impact the increasing electron energy post-collision.

This concept not only aids in solving problems involving photon interactions but also emphasizes the transformational aspect of energy exchanges in quantum physics.