Ch 38: Photons: Light Waves Behaving as Particles

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 38: Photons: Light Waves Behaving as Particles

Problema 20

Capitolo 38, Problema 20

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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Understand the concept of Compton scattering, where a photon scatters off a particle, such as a proton, resulting in a change in the photon's wavelength.

Use the Compton wavelength shift formula:

Δλ = \frac{h}{m c} (1 - \cos θ)

, where

Δλ

is the change in wavelength,

h

is Planck's constant,

m

is the mass of the proton,

c

is the speed of light, and

θ

is the scattering angle.

Since the scattering angle

θ

is 180°, substitute this into the formula:

Δλ = \frac{h}{m c} (1 - \cos 180°)

. Note that

\cos 180°

is -1.

Calculate the change in wavelength

Δλ

using the formula:

Δλ = \frac{2 h}{m c}

. This represents a 10% change in wavelength, so

Δλ

is 0.1 times the initial wavelength

λ

₀.

Set up the equation

Δλ = \frac{2 h}{m c} = 0.1 λ

₀ and solve for the initial wavelength

λ

₀.

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Compton Scattering

Compton scattering is a phenomenon where a photon scatters off a target particle, such as an electron or proton, resulting in a change in the photon's wavelength. The change in wavelength depends on the angle of scattering and the mass of the target particle. This concept is crucial for understanding how the wavelength of the photon changes during the interaction.

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Law of Reflection

Wavelength Change Calculation

The change in wavelength during Compton scattering can be calculated using the Compton wavelength shift formula: Δλ = λ' - λ = (h/mc)(1 - cos θ), where h is Planck's constant, m is the mass of the target particle, c is the speed of light, and θ is the scattering angle. For a 180° backward scattering, cos θ = -1, which maximizes the wavelength change.

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Calculating Change in Velocity from Acceleration-Time Graphs

Photon Energy and Momentum

Photons have energy and momentum, which are related to their wavelength by E = hc/λ and p = h/λ, respectively. During scattering, the conservation of energy and momentum principles apply, affecting the photon's wavelength. Understanding these relationships helps in determining the initial wavelength required for a specific percentage change in wavelength.

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A photon scatters in the backward direction (ϕ=180\(\phi\)=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%10.0\% change in wavelength as a result of the scattering?

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Concetti chiave

Compton Scattering

Wavelength Change Calculation

Photon Energy and Momentum

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?