Problem 35 - College Physics

Question

\(\cdot\) An x ray with a wavelength of 0.100 nm collides with a electron that is initially at rest. The \(x\) ray's final wavelength 0.110 nm. What is the final kinetic energy of the electron?

Step-by-Step Solution

Answer

The final kinetic energy of the electron is equal to the energy lost by the photon.

Step 1: Understanding the Problem

In this problem, we have an X-ray photon with an initial wavelength of 0.100 nm colliding with a stationary electron. After the collision, the wavelength of the X-ray increases to 0.110 nm. We need to determine the final kinetic energy of the electron after the collision.

Step 2: Utilize Compton Wavelength Shift Equation

The change in wavelength (abla \(\Delta \lambda\)) when a photon collides with a stationary electron is described by the Compton wavelength shift equation: \(\Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos \theta)\), where \(h\) is Planck's constant, \(m_e\) is the electron mass, and \(c\) is the speed of light. Here, \(\theta\) is the scattering angle. From the problem, \(\Delta \lambda = 0.110 \text{ nm} - 0.100 \text{ nm} = 0.010 \text{ nm}\).

Step 3: Calculate the Scattering Angle

Since the specific angle \(\theta\) isn't provided, it's initially more practical to consider the energy changes involved rather than finding \(\theta\) directly. This will simplify to calculating the changes in photon energy, as the angle can be deduced if required later using the energy calculations and electron's momentum.

Step 4: Calculate Energy of Photons

The energy of a photon is given by \(E = \frac{hc}{\lambda}\). Calculate initial and final energies. Initial energy of photon, \(E_i = \frac{hc}{0.100 \text{ nm}}\). Final energy of photon, \(E_f = \frac{hc}{0.110 \text{ nm}}\).

Step 5: Calculate the Energy Lost by the Photon

The change in the photon's energy is the energy it lost (or transferred to the electron), \(\Delta E = E_i - E_f\). Substitute the values from the previous step to find this energy change.

Step 6: Calculate Final Kinetic Energy of the Electron

The energy lost by the photon is gained by the electron as kinetic energy. Therefore, the final kinetic energy of the electron is \(\Delta E = E_i - E_f\). Utilize the computed photon energy values to find the kinetic energy of the electron.

Key Concepts

Wavelength Shift

The concept of "wavelength shift" is central to understanding Compton scattering. In this phenomenon, a photon—often an X-ray—collides with a stationary electron, resulting in a change in the photon's wavelength. This shift occurs because energy is transferred from the photon to the electron during the collision.
In the provided exercise, the initial wavelength of the X-ray photon is 0.100 nm, and after the collision, it increases to 0.110 nm. The change in wavelength, or wavelength shift, \( \Delta \lambda \), is calculated as:

• Initial wavelength: \( \lambda_i = 0.100 \text{ nm} \)
• Final wavelength: \( \lambda_f = 0.110 \text{ nm} \)
• Wavelength shift: \( \Delta \lambda = \lambda_f - \lambda_i = 0.110 \text{ nm} - 0.100 \text{ nm} = 0.010 \text{ nm} \)

Such a shift is indicative of energy transfer, where the photon loses some energy, dispersing its wavelength longer, and thereby influencing the resultant kinetic energy of the electron.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. In the context of Compton scattering, this concept comes into play after the photon and the electron collision occurs. The energy lost by the photon during this interaction is gained by the electron, resulting in the electron acquiring kinetic energy.
To illustrate, the energy transferred from the photon to the electron can be computed by first finding the energy of the photon before and after the collision. The kinetic energy of the electron turns out to be equal to the energy difference of the photon between its states:

• Photon energy loss = \( E_i - E_f \)
• Kinetic energy of electron = \( E_i - E_f \)

This approach is key to understanding how the energy distribution affects the appearance of the wavelength shift in the overall interaction.

Photon Energy

Photon energy is a crucial component in understanding Compton scattering, as it directly relates to the wavelength of the photon. The energy of a photon ties its wavelength via the formula \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant and \( c \) is the speed of light.
In this exercise, the initial and final energy of the X-ray photon are computed using their respective wavelengths:

• Initial photon energy: \( E_i = \frac{hc}{0.100 \text{ nm}} \)
• Final photon energy: \( E_f = \frac{hc}{0.110 \text{ nm}} \)

The difference, \( \Delta E = E_i - E_f \), represents the energy transferred to the electron, highlighting the fundamental role of photon energy in driving the changes observed in the scattering process.

Electron Collision

An electron collision in the context of Compton scattering describes the interaction between the photon and the initially stationary electron. Upon colliding, the photon transfers part of its energy to the electron, imparting kinetic energy.
This energy transfer is what accounts for the post-collision behavior, namely the increase in wavelength of the photon and the newfound energy in the electron:

• The photon slows down (wavelength increases), which implies energy loss.
• The electron "absorbs" the lost energy and accelerates, gaining kinetic energy.

The concept behind electron collision is fundamentally based on the conservation of energy, where no energy is created or destroyed, only transferred. Understanding this allows us to gauge the energy relationships within the scattering event.