Ch 38: Photons: Light Waves Behaving as Particles

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 38: Photons: Light Waves Behaving as Particles

Problem 15

Chapter 37, Problem 15

X rays with initial wavelength $0.0665$ nm undergo Compton scattering. What is the longest wavelength found in the scattered x rays? At which scattering angle is this wavelength observed?

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Understand the concept of Compton scattering: Compton scattering involves the change in wavelength of X-rays or gamma rays when they are scattered by electrons. The change in wavelength is given by the Compton wavelength shift formula.

Use the Compton wavelength shift formula: Δλ = λ' - λ = (h/mc) * (1 - cos(θ)), where λ is the initial wavelength, λ' is the scattered wavelength, h is Planck's constant, m is the electron mass, c is the speed of light, and θ is the scattering angle.

Identify the longest wavelength: The longest wavelength occurs when the change in wavelength Δλ is maximized. This happens when the scattering angle θ is 180 degrees, because cos(180°) = -1, which maximizes the term (1 - cos(θ)).

Calculate the longest wavelength: Substitute θ = 180° into the Compton wavelength shift formula to find λ'. Use the given initial wavelength λ = 0.0665 nm and the constants h, m, and c to compute λ'.

Determine the scattering angle for the longest wavelength: As calculated, the longest wavelength is observed at a scattering angle of 180 degrees.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Compton Scattering

Compton scattering is a phenomenon where X-rays or gamma rays are scattered by electrons, resulting in a change in wavelength. This effect is significant in quantum physics as it demonstrates the particle-like properties of light. The change in wavelength depends on the angle at which the photon is scattered.

Wavelength Shift Formula

The wavelength shift in Compton scattering is given by the formula: Δλ = (h/mc)(1 - cos θ), where h is Planck's constant, m is the electron mass, c is the speed of light, and θ is the scattering angle. This formula allows us to calculate the change in wavelength based on the scattering angle, crucial for determining the longest wavelength.

Scattering Angle

The scattering angle in Compton scattering is the angle between the incident and scattered photon directions. It plays a critical role in determining the extent of the wavelength shift. The longest wavelength occurs at the maximum scattering angle, which is 180 degrees, where the photon is scattered directly backward.

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The photoelectric work function of potassium is $2.3$ eV. If light that has a wavelength of $190$ nm falls on potassium, find the stopping potential in volts.

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A photon scatters in the backward direction ( $phi equals 180$ °) from a free proton that is initially at rest. What must the wavelength of the incident photon be if it is to undergo a $10.0 percent sign$ change in wavelength as a result of the scattering?

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Textbook Question

A horizontal beam of laser light of wavelength $585$ nm passes through a narrow slit that has width $0.0620$ mm. The intensity of the light is measured on a vertical screen that is $2.00$ m from the slit.

(a) What is the minimum uncertainty in the vertical component of the momentum of each photon in the beam after the photon has passed through the slit?

(b) Use the result of part (a) to estimate the width of the central diffraction maximum that is observed on the screen.

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Textbook Question

An electron and a positron are moving toward each other and each has speed $0.500 c$ in the lab frame. What is the kinetic energy of each particle?

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Textbook Question

The photoelectric work function of potassium is $2.3$ eV. If light that has a wavelength of $190$ nm falls on potassium, find the kinetic energy, in electron volts, of the most energetic electrons ejected.

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The cathode-ray tubes that generated the picture in early color televisions were sources of x rays. If the acceleration voltage in a television tube is $15.0$ kV, what are the shortest-wavelength x rays produced by the television?

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X rays with initial wavelength 0.06650.0665 nm undergo Compton scattering. What is the longest wavelength found in the scattered x rays? At which scattering angle is this wavelength observed?

Verified video answer for a similar problem:

Key Concepts

Compton Scattering

Wavelength Shift Formula

Scattering Angle

X rays with initial wavelength $0.0665$ nm undergo Compton scattering. What is the longest wavelength found in the scattered x rays? At which scattering angle is this wavelength observed?