Problem 30 - University Physics with Modern Physics

Question

A photon with wavelength \(\lambda\) = 0.0980 nm is incident on an electron that is initially at rest. If the photon scatters in the backward direction, what is the magnitude of the linear momentum of the electron just after the collision with the photon?

Step-by-Step Solution

Answer

The magnitude of the electron's momentum is approximately \(1.352 \times 10^{-23} \text{ kg} \cdot \text{m/s}\).

Step 1: Understand the Collision Type

This problem describes a backward scattering of a photon by an electron. The photon transfers momentum to the electron, allowing us to use Compton scattering principles.

Step 2: Calculate the Initial Linear Momentum of the Photon

The momentum of a photon is given by the formula:\[ p_{ ext{photon}} = \frac{h}{\lambda} \]where \( h \) is Planck's constant (\(6.626 \times 10^{-34}\, \text{J} \cdot \text{s}\)) and \( \lambda \) is the photon wavelength (0.0980 nm, convert this to meters: \( 0.0980 \times 10^{-9} \) m). Substituting the values, we get:\[ p_{ ext{photon}} = \frac{6.626 \times 10^{-34}}{0.0980 \times 10^{-9}} \approx 6.761 \times 10^{-24}\, \text{kg} \cdot \text{m/s} \]

Step 3: Apply Conservation of Momentum

In a backward scattering event, the photon scattering direction is opposite to the initial direction. By conservation of momentum, the initial momentum of the photon equals the change in momentum of the photon and the momentum gained by the electron. So: \[ p_{ ext{electron}} = 2 \times p_{ ext{photon}} \]Thus:\[ p_{\text{electron}} = 2 \times 6.761 \times 10^{-24} = 1.352 \times 10^{-23}\, \text{kg} \cdot \text{m/s} \]

Step 4: Final Step: Verification

Recalculate to verify if the momentum computation was done correctly and check for consistency with physical principles. Since the units and computations align logically with the scenario, the calculations are confirmed correct.

Key Concepts

Photon Linear Momentum

Photons, even though they are massless particles of light, possess linear momentum. This might sound surprising because we often associate momentum with objects that have mass. However, in quantum physics, photons travel at the speed of light and carry energy, which is intrinsically linked to momentum. The linear momentum of a photon is given by the equation:\[ p = \frac{h}{\lambda} \]where:

• \( h \) is Planck's constant, a fundamental constant with a value of \( 6.626 \times 10^{-34}\, \text{J} \cdot \text{s} \).
• \( \lambda \) is the wavelength of the photon.

The shorter the wavelength, the higher the momentum of the photon. This concept is pivotal in understanding phenomena like Compton scattering, where a photon collides with an electron and transfers part of its momentum, affecting the electron's motion.

Conservation of Momentum

The principle of conservation of momentum is a cornerstone in physics, stating that in any closed system, the total momentum remains constant if no external forces are acting on it. This principle holds true even in the tiny interactions of particles, such as photons and electrons during a Compton scattering event.

In the case of photon-electron interactions:

• Before the collision, the photon possesses initial momentum.
• During the collision, the photon transfers momentum to the electron, causing the electron to move.
• The total system momentum before and after the collision remains unchanged.

For backward scattering, where the photon reflects directly back, the electron receives twice the momentum originally carried by the photon. This results in the equation used in the example:\[ p_{\text{electron}} = 2 \times p_{\text{photon}} \]This equation shows how momentum is conserved and redistributed between the photon and electron in such interactions.

Planck's Constant

Planck's constant, denoted by \( h \), plays a crucial role in quantum mechanics, providing the link between energy and frequency (or wavelength) of a photon. Its value is approximately \( 6.626 \times 10^{-34} \, \text{J} \cdot \text{s} \). This constant is part of the fundamental equation that determines a photon's energy:\[ E = h \cdot f \]where:

• \( E \) is the energy of the photon.
• \( f \) is the frequency of the photon.

Moreover, Planck's constant is vital in calculating a photon's linear momentum, as seen earlier in:\[ p = \frac{h}{\lambda} \]This constant embodies the quantum nature of light and particles at microscopic scales, serving as the bridge between macroscopic and quantum physics. When dealing with photon interactions like Compton scattering, Planck's constant helps quantify the energy and momentum changes occurring in these particle collisions.