Ch 44: Particle Physics and Cosmology

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 44: Particle Physics and Cosmology

Problem 4a

Chapter 44, Problem 4a

A proton and an antiproton annihilate, producing two photons. Find the energy, frequency, and wavelength of each photon if the $p$ and $\overline{p}$ are initially at rest.

Verified step by step guidance

Start by understanding the annihilation process: When a proton and an antiproton annihilate, their entire rest energy is converted into the energy of the two photons. The rest energy of a particle is given by Einstein's equation:

E_{rest} = mc

², where

m

is the mass of the particle and

c

is the speed of light.

Calculate the total rest energy of the proton and antiproton system. Since the proton and antiproton have the same mass, the total rest energy is:

E_{total} = 2mc

², where

m

is the mass of a proton.

Since the proton and antiproton are initially at rest, their total momentum is zero. By conservation of momentum, the two photons produced must have equal and opposite momenta. This ensures that the system's total momentum remains zero. Therefore, the energy of each photon is equal to half of the total rest energy:

E_{photon} = mc

².

Relate the energy of each photon to its frequency using the Planck-Einstein relation:

E_{photon} = hν

, where

h

is Planck's constant and

ν

is the frequency of the photon. Solve for the frequency:

ν = E_{photon} / h

Relate the frequency of each photon to its wavelength using the wave equation:

c = λν

, where

λ

is the wavelength and

c

is the speed of light. Solve for the wavelength:

λ = c / ν

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Key Concepts

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

Mass-Energy Equivalence

Mass-energy equivalence, expressed by Einstein's equation E=mc², states that mass can be converted into energy and vice versa. In the context of particle annihilation, the rest mass of the proton and antiproton is converted into energy, which is then emitted as photons. This principle is fundamental for calculating the total energy produced in the annihilation process.

Photon Properties

Photons are massless particles of light that carry energy and momentum. The energy of a photon is directly related to its frequency (E=hf) and inversely related to its wavelength (λ=c/f), where h is Planck's constant and c is the speed of light. Understanding these relationships is crucial for determining the energy, frequency, and wavelength of the photons produced in the annihilation.

Conservation of Energy

The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In the case of proton-antiproton annihilation, the total energy before the event (the rest mass energy of the particles) equals the total energy after the event (the energy of the resulting photons). This principle allows us to calculate the energy of each photon produced in the annihilation.

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Related Practice

Textbook Question

The starship Enterprise, of television and movie fame, is powered by combining matter and antimatter. If the entire $400$ -kg antimatter fuel supply of the Enterprise combines with matter, how much energy is released? How does this compare to the U.S. yearly energy use, which is roughly $1.0 \times 10^{20}$ J?

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Textbook Question

A proton and an antiproton annihilate, producing two photons. Find the energy, frequency, and wavelength of each photon if the $p$ and $\overline{p}$ collide head-on, each with an initial kinetic energy of $620$ MeV.

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Textbook Question

Two equal-energy photons collide head-on and annihilate each other, producing a $µ^{+} µ^{-}$ pair. The muon mass is given in terms of the electron mass in Section $44.1$ . Calculate the maximum wavelength of the photons for this to occur. If the photons have this wavelength, describe the motion of the $µ^{+}$ and immediately after they are produced.

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Textbook Question

An electron with a total energy of $30.0$ GeV collides with a stationary positron. What is the available energy?

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Textbook Question

A neutral pion at rest decays into two photons. Find the energy, frequency, and wavelength of each photon. In which part of the electromagnetic spectrum does each photon lie? (Use the pion mass given in terms of the electron mass in Section $44.1$ .)

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A proton and an antiproton annihilate, producing two photons. Find the energy, frequency, and wavelength of each photon if the pp and p‾\(\overline{p}\) are initially at rest.

Verified video answer for a similar problem:

Key Concepts

Mass-Energy Equivalence

Photon Properties

Conservation of Energy

A proton and an antiproton annihilate, producing two photons. Find the energy, frequency, and wavelength of each photon if the $p$ and $\overline{p}$ are initially at rest.