/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 68 What is the energy of each of th... [FREE SOLUTION] | 91Ó°ÊÓ

91Ó°ÊÓ

What is the energy of each of the photons resulting from an annihilation event? A. \(\frac{1}{2} m_{\mathrm{e}} v^{2},\) where \(v\) is the speed of the positron. B. \(m_{\mathrm{e}} v^{2}\) C. \(\frac{1}{2} m_{\mathrm{e}} c^{2}\) D. \(m_{\mathrm{e}} c^{2}\)

Short Answer

Expert verified
Each photon has energy \(m_e c^2\) (Option D).

Step by step solution

01

Understanding Annihilation

Annihilation is a process where a particle and its antiparticle, such as an electron and a positron, collide and are converted entirely into energy. This energy is typically released as two photons to conserve both energy and momentum.
02

Applying the Energy-Mass Equivalence Principle

According to Einstein's theory of relativity, the energy of a particle can be represented by the formula \(E=mc^2\), where \(m\) is the mass of the particle and \(c\) is the speed of light. In the annihilation event, both the electron and positron's rest mass are converted to energy.
03

Calculating Total Energy Released

In an annihilation event, the total energy released is the sum of the rest energies of the electron and positron. Each has a rest mass \(m_e\). Therefore, the total energy released is \(2 \times m_e c^2\).
04

Dividing Energy Between Photons

The total energy released is equally divided between two photons to conserve both energy and momentum. Thus, each photon carries energy \(m_e c^2\).
05

Identifying the Correct Option

According to the calculation, each photon has the energy of \(m_e c^2\). This matches with option D.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Annihilation Event
An annihilation event is a fascinating process in physics involving a particle and its antiparticle. When these two collide, such as an electron meeting a positron, they are entirely transformed into energy. This event is remarkable because it's not only about destruction but also about transformation. Interestingly, the energy released is often in the form of two photons. This pair of photons ensures the conservation of energy and momentum in the system.
  • Annihilation involves converting mass into energy, following the principles of relativity.
  • Photon pairs are generated to maintain equilibrium in the physical laws.
Understanding annihilation events bridges the gap between matter and energy, demonstrating one of the universe's intriguing principles of symmetry and conservation.
Energy-Mass Equivalence
The concept of energy-mass equivalence is central to understanding annihilation events. Introduced by Albert Einstein, this principle is encapsulated in the equation \(E=mc^2\). Here, \(E\) stands for energy, \(m\) is mass, and \(c\) is the speed of light in a vacuum, a universal constant. This equation signifies that mass can be converted into energy and vice versa. Energy-mass equivalence explains how even small particles like electrons and positrons possess rest mass energy that can be transformed into a significant amount of photon energy upon annihilation.
  • The speed of light (\(c\)) is extraordinarily high, making even tiny masses capable of converting into huge energy outputs.
  • This principle also lays the groundwork for understanding how nuclear reactions operate and the potentials within atomic particles.
By realizing that mass is a concentrated form of energy, we grasp how energy liberation occurs in atomic level interactions.
Einstein's Theory of Relativity
Einstein's theory of relativity is a cornerstone of modern physics, transforming our understanding of concepts like time, space, mass, and energy. Within this theory, both the special and general theories of relativity play crucial roles. The special theory of relativity, in particular, introduces the principle of mass-energy equivalence through the iconic equation \(E=mc^2\). This principle shows the deep connection between mass and energy, as seen in annihilation events where electron-positron pairs convert entirely into energy.
  • Relativity challenges the classical notions of absolute time and space, proposing instead that these are relative and interlinked.
  • It offers a new perspective on how natural laws such as energy conservation apply even at high velocities close to the speed of light.
Grasping relativity not only enhances comprehension of atomic and subatomic phenomena, but it also enlightens our perception of the cosmos.
Electron-Positron Pair
An electron-positron pair is fundamental to understanding particle interactions and annihilation events. Electrons are negatively charged particles that, along with protons and neutrons, make up atoms. In contrast, positrons are the antimatter counterparts of electrons, possessing the same mass but a positive charge. When these two particles encounter each other, they form an electron-positron pair. Upon collision, they annihilate, converting their mass into energy, typically released as two photons, as required by conservation laws.
  • This transformation highlights the delicate balance between matter and antimatter.
  • Electron-positron pairs serve as a significant area of study within particle physics, offering insights into the universe's symmetry and forces.
The behavior and interactions of such pairs deepen our understanding of fundamental physics, contributing to advancements in fields like quantum mechanics and cosmology.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

How many protons and how many neutrons are there in a nucleus of (a) neon, \(_{10}^{21} \mathrm{Ne}(\mathrm{b})\) zinc, \(_{30}^{65} \mathrm{Zn}(\mathrm{c})\) silver, \(_{47}^{108} \mathrm{Ag}\)

Pair annilation. Consider the case where an electron \(\mathrm{e}^{-}\) and a positron \(\mathrm{e}^{+}\) annihilate each other and produce photons. Assume that these two particles collide head-on with equal, but small, speeds. (a) Show that it is not possible for only one photon to be produced. (Hint: Consider the conservation law that must be true in any collision.) (b) Show that if only two photons are produced, they must travel in opposite directions and have equal energy. (c) Calculate the wavelength of each of the photons in part (b). In what part of the electromagnetic spectrum do they lie?

Show that the net result of the proton-proton fusion chain that occurs inside our sun can be summarized as $$6 \mathrm{p}^{+} \rightarrow_{2}^{4} \mathrm{He}+2 \mathrm{p}^{+}+2 \beta^{+}+2 \gamma+2 \nu_{\mathrm{e}}$$

The critical density of the universe is \(5.8 \times 10^{-27} \mathrm{kg} / \mathrm{m}^{3}\) . (a) Assuming that the universe is all hydrogen, express the critical density in the number of \(\mathrm{H}\) atoms per cubic meter. (b) If the density of the universe is equal to the critical density, how many atoms, on the average, would you expect to find in a room of dimensions 4 \(\mathrm{m} \times 7 \mathrm{m} \times 3 \mathrm{m} ?\) (c) Compare your answer in part (b) with the number of atoms you would find in this room under normal conditions on the earth.

Radioactive fallout. One of the problems of in-air testing of nuclear weapons (or, even worse, the use of such weapons!) is the danger of radioactive fallout. One of the most problematic nuclides in such fallout is strontium-90 \(\left(^{90} \mathrm{Sr}\right),\) which breaks down by \(\beta^{-}\) decay with a half-life of 28 years. It is chemically similar to calcium and therefore can be incorporated into bones and teeth, where, due to its rather long half- life, it remains for years as an internal source of radiation. (a) What is the daughter nucleus of the \(^{90}\) Sr decay? (b) What percent of the original level of 90 \(\mathrm{Sr}\) is left after 56 years? (c) How long would you have to wait for the original level to be reduced to 6.25\(\%\) of its original value?

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.