Problem 36 - College Physics

Question

If a photon of wavelength 0.04250 nm strikes a free electron and is scattered at an angle of \(35.0^{\circ}\) from its original direction, find (a) the change in the wavelength of this photon, (b) the wavelength of the scattered light, (c) the change in energy of the photon (is it a loss or a gain?), and (d) the energy gained by the electron.

Step-by-Step Solution

Answer

The wavelength change is small due to the small angle; the photon loses energy, which the electron gains.

Step 1: Understanding the Compton Wavelength Shift Formula

The formula to calculate the change in wavelength of a photon after scattering is given by the Compton shift formula:\[\Delta \lambda = \lambda' - \lambda = \frac{h}{m_ec} (1 - \cos \theta)\]where \(h\) is Planck's constant \(6.626 \times 10^{-34}\) Js, \(m_e\) is the electron rest mass \(9.11 \times 10^{-31}\) kg, and \(c\) is the speed of light \(3 \times 10^8\) m/s.

Step 2: Calculate the Compton Wavelength Shift

Substitute the given angle \(\theta = 35.0^{\circ}\) into the formula:\[\Delta \lambda = \frac{6.626 \times 10^{-34}}{9.11 \times 10^{-31} \times 3 \times 10^8} \times (1 - \cos 35^{\circ})\]Calculate \(\Delta \lambda\).

Step 3: Calculate the Wavelength of the Scattered Photon

The wavelength of the scattered photon is given by:\[\lambda' = \lambda + \Delta \lambda\]Substitute \(\lambda = 0.04250\) nm and \(\Delta \lambda\) from Step 2 to find \(\lambda'\).

Step 4: Calculate the Initial Energy of the Photon

The energy of a photon is related to its wavelength by:\[E = \frac{hc}{\lambda}\]Calculate the initial energy using \(\lambda = 0.04250\) nm.

Step 5: Calculate the Energy of the Scattered Photon

Using the scattered wavelength \(\lambda'\) from Step 3, calculate the new energy using the same energy formula:\[E' = \frac{hc}{\lambda'}\]

Step 6: Determine the Change in Energy of the Photon

Calculate the change in energy:\[\Delta E_{photon} = E - E'\]Determine if this indicates an energy loss or gain.

Step 7: Calculate the Energy Gained by the Electron

The energy gained by the electron is equal to the loss in energy by the photon:\[\Delta E_{electron} = \Delta E_{photon}\]

Key Concepts

Photon scattering

Photon scattering occurs when a photon, a particle of light, interacts with an electron or another particle, causing the photon to deviate from its original path. This deviation or deflection results in changes to the photon's properties. One significant aspect of photon scattering is the change in the photon's energy, which directly relates to its wavelength. In simpler terms, when a photon collides with a free electron, like in the Compton Effect, it can "bounce off" at an angle, and its energy and wavelength will shift slightly.

Scattering is crucial in understanding various phenomena in physics, including how light interacts with matter in different contexts, like in medical imaging or astronomical observations. By analyzing the scattering of photons, scientists can deduce information about the structures and properties of the scattering objects.

Wavelength shift

Wavelength shift refers to the change in the wavelength of a photon as it scatters from an electron. The Compton Effect is a classic example of a wavelength shift following photon scattering. When a photon collides with a free electron, part of its energy is transferred to the electron, altering the photon's wavelength.

The Compton wavelength shift can be mathematically described by the formula:

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

Here,

• \(h\) represents Planck's constant, \(6.626 \times 10^{-34}\) Js
• \(m_e\) is the electron mass, \(9.11 \times 10^{-31}\) kg
• \(c\) is the speed of light, \(3 \times 10^8\) m/s
• \(\theta\) is the scattering angle.

This expression calculates the exact change in wavelength based on these constants and the angle of scattering, showing how energy gets transferred during the interaction.

Photon energy

Photon energy is fundamentally tied to a photon's wavelength. The shorter the wavelength, the higher its energy. This relationship can be expressed using the formula:

• \[E = \frac{hc}{\lambda}\]

This equation states that the energy \(E\) of a photon is equal to Planck's constant \(h\) times the speed of light \(c\) divided by the wavelength \(\lambda\).

During the Compton Effect, as photons scatter off electrons, their energy and wavelength are both affected. If a photon loses energy to an electron, its corresponding wavelength increases (since energy is inverse to wavelength). Understanding this exchange is crucial for applications like X-ray production and the interpretation of astronomical data, where energy and wavelength shifts inform scientists about cosmic phenomena.

Compton scattering formula

The Compton scattering formula is crucial in calculating the change of wavelength of a photon due to scattering. This formula takes into account the principles of photon scattering outlined by Arthur Compton in 1923. The key to understanding this formula is recognizing that it models how energy and momentum transfer occur between photons and electrons during scattering.

The Compton scattering formula:

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

is derived from conservation laws, predicting how much the photon's wavelength will change based on the angle \(\theta\) at which it scatters. By using this formula, one can precisely determine the shift in wavelength, which helps in identifying energy changes. This is central to many technologies and scientific domains, from medical imaging to material sciences, where knowledge of particle interactions plays a crucial role.