Chapter 10: Problem 21

A high-energy photon can collide with a free electron and give it some energy. (This is called the Compton effect.) How are the photon's energy, frequency, and wavelength affected by the collision?

Short Answer

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Answer: After a high-energy photon collides with a free electron, its energy, frequency, and wavelength change. The conservation of energy and momentum principles, as well as the Compton scattering formula, are used to determine these changes. The photon's new energy, frequency, and wavelength can be calculated using equations derived from these principles and the relationship between energy, frequency, and wavelength (E = hf and λ = c/f).

Step by step solution

1. Initial Conditions

Before the collision, we have a high-energy photon with energy E1, frequency f1, and wavelength λ1, and a free electron at rest with mass m_e and zero initial momentum.

2. After Collision Conditions

After the collision, the photon has a new energy E2, frequency f2, and wavelength λ2, while the electron has gained some kinetic energy and momentum. We will call the electron's final velocity v and its final momentum p_e.

3. Conservation of Energy

Using the conservation of energy, the sum of the initial energies (photon's energy and electron's energy) should be equal to the sum of the final energies (new photon's energy and electron's kinetic energy). We can write this equation as: E1 = E2 + (1/2)m_e * v^2

4. Conservation of Momentum

Using the conservation of momentum, the initial momentum (zero, as the electron was at rest) should be equal to the final momentum (sum of photon's and electron's momentum). We can write this equation as: 0 = h/λ2 - h/λ1 + m_e * v where h is the Planck's constant.

5. Compton Scattering Formula

Compton scattering formula relates the change in photon's wavelength (Δλ) to the scattering angle (θ): Δλ = λ2 - λ1 = (h/(m_e * c)) * (1 - cos(θ)) Here, Δλ is the change in wavelength, m_e is the electron's mass, c is the speed of light, and θ is the angle between the initial and final directions of the photon.

6. Finding the Photon's New Energy, Frequency, and Wavelength

Using the Compton scattering formula and the relationship between energy, frequency, and wavelength (E = h * f and λ = c/f), we can calculate the new energy E2, frequency f2, and wavelength λ2 of the photon after the collision. For example, to find the new energy: E2 = E1 - (1/2)m_e * v^2 To find the new frequency and the wavelength, we can use: f2 = E2/h λ2 = c/f2

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A high-energy photon can collide with a free electron and give it some energy. (This is called the Compton effect.) How are the photon's energy, frequency, and wavelength affected by the collision?

Short Answer

Step by step solution

1. Initial Conditions

2. After Collision Conditions

3. Conservation of Energy

4. Conservation of Momentum

5. Compton Scattering Formula

6. Finding the Photon's New Energy, Frequency, and Wavelength

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