Problem 37 - University Physics with Modern Physics

Question

CP A photon with wavelength \(\lambda=0.1050 \mathrm{nm}\) is incident on an electron that is initially at rest. If the photon scatters at an angle of \(60.0^{\circ}\) from its original direction, what are the magnitude and direction of the linear momentum of the electron just after the collision with the photon?

Step-by-Step Solution

Answer

The electron's momentum is increased by collision. Calculate all variables to verify the exact direction and magnitude.

Step 1: Understand the Problem

We have a photon with a given wavelength (\lambda = 0.1050 \, \mathrm{nm}) hitting a stationary electron. After the collision, the photon is scattered at an angle of \(60.0^{\circ}\). We need to find the magnitude and direction of the electron's momentum after the collision.

Step 2: Calculate the Initial Momentum of the Photon

We start by calculating the initial momentum of the photon using the formula:\[ p_{\text{photon}} = \frac{h}{\lambda} \]where \(h\) is Planck's constant \(6.626 \times 10^{-34} \, \mathrm{m^2 \, kg/s}\), and \(\lambda\) is the wavelength of the photon given as \(0.1050 \, \mathrm{nm} = 0.1050 \times 10^{-9} \, \mathrm{m}\). Calculate \(p_{\text{photon}}\).

Step 3: Use Compton Scattering

Apply the Compton scattering formula to find the change in wavelength of the photon:\[ \Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c}(1 - \cos \theta) \]where \(m_e\) is the electron mass \(9.109 \times 10^{-31} \, \mathrm{kg}\), and \(c\) is the speed of light \(3 \times 10^{8} \, \mathrm{m/s}\). \(\theta\) is the scattering angle \(60.0^{\circ}\). Solve for \(\Delta \lambda\) and find the new wavelength \(\lambda'\).

Step 4: Calculate Final Momentum of the Photon

With the new wavelength \(\lambda'\) obtained from the previous step, calculate the photon's final momentum after scattering:\[ p_{\text{photon, final}} = \frac{h}{\lambda'} \]

Step 5: Apply Conservation of Momentum

Using momentum conservation in the x and y directions:- In the x-direction: \(p_{\text{photon, initial}} = p_{\text{photon, final}} \cos(\theta) + p_{\text{electron, x}}\)- In the y-direction: \(0 = p_{\text{photon, final}} \sin(\theta) - p_{\text{electron, y}}\)Solve these equations to find \(p_{\text{electron, x}}\) and \(p_{\text{electron, y}}\).

Step 6: Calculate the Magnitude and Direction of Electron's Final Momentum

The magnitude of the electron's momentum \(p_{\text{electron}}\) is given by:\[ p_{\text{electron}} = \sqrt{(p_{\text{electron, x}})^2 + (p_{\text{electron, y}})^2} \]The direction \(\phi\) (angle with respect to the original direction of the photon) is given by:\[ \phi = \tan^{-1} \left( \frac{p_{\text{electron, y}}}{p_{\text{electron, x}}} \right) \]Calculate the magnitude and direction.

Key Concepts

Photon Momentum

When studying Compton Scattering, understanding photon momentum is crucial. Photons, the particles that make up light, possess momentum despite having no mass. This is a fascinating concept because momentum is typically associated with objects that have mass. The formula to calculate the momentum of a photon is given by:

• \( p_{\text{photon}} = \frac{h}{\lambda} \)

where \( h \) is Planck's constant, and \( \lambda \) represents the photon's wavelength. This equation shows that a photon’s momentum is inversely proportional to its wavelength, meaning shorter wavelengths correspond to higher momentum.
This relationship between momentum and wavelength is key to understanding how photons interact with electrons during Compton scattering.

Wavelength Change

During Compton scattering, an essential concept to grasp is the change in wavelength of the photon. When a photon collides with an electron, its wavelength increases. This change in wavelength, \( \Delta \lambda \), can be calculated using the Compton wavelength shift formula:

• \( \Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos \theta) \)

Here, \( \lambda' \) is the photon's new wavelength after scattering, \( m_e \) is the electron's rest mass, \( c \) is the speed of light, and \( \theta \) represents the scattering angle.
This formula quantitatively describes how the photon's energy and wavelength are altered due to the interaction. Understanding this relationship helps us predict how light changes as it bounces off particles, an important foundation in quantum mechanics.

Conservation of Momentum

The conservation of momentum is a fundamental law of physics that allows us to understand the outcome of the collision between the photon and the electron in Compton scattering. According to this principle, the total momentum of the system must remain constant before and after the collision.
In the context of Compton scattering, this means the initial momentum of the photon must equal the total momentum of the scattered photon and electron combined. The exercise break down includes conserving momentum in both horizontal and vertical directions:

• X-direction: \( p_{\text{photon, initial}} = p_{\text{photon, final}} \cos(\theta) + p_{\text{electron, x}} \)
• Y-direction: \( 0 = p_{\text{photon, final}} \sin(\theta) - p_{\text{electron, y}} \)

Solving these equations allows us to find the components of the electron's momentum after the collision. By applying the Pythagorean theorem, we can also determine the magnitude, and trigonometry allows us to find the direction of the electron’s momentum. This thorough understanding of momentum conservation helps us predict and analyze the results of particle interactions.