Problem 29 - College Physics

Question

\(\bullet\) Particle annihilation. In proton-antiproton annihilation, a proton and an antiproton (a negatively charged particle with the mass of a proton) collide and disappear, producing electromagnetic radiation. If each particle has a mass of \(1.67 \times 10^{-27} \mathrm{kg}\) and they are at rest just before the annihilation, find the total energy of the radiation. Give your answers in joules and in electron volts.

Step-by-Step Solution

Answer

The energy of the radiation is \(3.006 \times 10^{-10} \) J or \(1.88 \times 10^9 \) eV.

Step 1: Understand the Problem

In proton-antiproton annihilation, a proton and an antiproton collide and convert their mass into electromagnetic radiation. We need to determine the total energy released in this process using Einstein's mass-energy equivalence principle.

Step 2: Use Mass-Energy Equivalence

According to Einstein's theory of relativity, the energy released by converting mass to energy is given by the equation \(E = mc^2\), where \(m\) is the total mass and \(c\) is the speed of light (\(3 \times 10^8 \, \text{m/s}\)).

Step 3: Calculate Total Mass

The total mass involved in the annihilation is the sum of the mass of the proton and the mass of the antiproton. Both have a mass of \(1.67 \times 10^{-27} \, \text{kg}\), so the total mass is \(2 \times 1.67 \times 10^{-27} = 3.34 \times 10^{-27} \, \text{kg}\).

Step 4: Calculate Energy in Joules

Using the mass-energy equivalence, the total energy \(E\) in joules is \(E = mc^2 = 3.34 \times 10^{-27} \, \text{kg} \times (3 \times 10^8 \, \text{m/s})^2 = 3.006 \times 10^{-10} \, \text{J}\).

Step 5: Convert Energy to Electron Volts

To convert energy from joules to electron volts (eV), use the conversion factor: 1 Joule = \(6.242 \times 10^{18} \, \text{eV}\). Therefore, \(E = 3.006 \times 10^{-10} \, \text{J} \times 6.242 \times 10^{18} \, \text{eV/J} = 1.88 \times 10^9 \, \text{eV}\).

Key Concepts

Mass-Energy Equivalence

The concept of mass-energy equivalence is a cornerstone of modern physics, famously encapsulated in Einstein's equation \(E = mc^2\). This principle suggests that mass can be converted into energy and vice versa. Here, \(E\) represents energy, \(m\) denotes mass, and \(c\) is the speed of light, approximately \(3 \times 10^8\) meters per second. This equation implies that even a small amount of mass can be transformed into a large amount of energy because \(c^2\) is a very large number.

In practical terms, mass-energy equivalence explains phenomena such as nuclear reactions and particle annihilation. In these processes, mass is not lost or destroyed but rather changed into other forms of energy, often releasing large amounts in the process.

By understanding this foundational principle, physicists can predict the behavior of particles under various conditions and calculate the energy outputs of high-energy collisions, such as those involving protons and antiprotons.

Proton-Antiproton Annihilation

Proton-antiproton annihilation is a fascinating example of particle interactions at a subatomic level. In this process, a proton and its antimatter counterpart, the antiproton, collide and demolish each other's mass. Antiprotons have the same mass as protons but carry a negative charge. When these two particles meet, they annihilate, converting their entire mass into energy.

This annihilation results in the production of electromagnetic radiation, typically gamma rays, which are a form of high-energy photons. Proton-antiproton annihilation demonstrates the practical implications of mass-energy equivalence, as the mass of the particles disappears and is replaced by energy in the form of radiation.

This phenomenon is not just theoretical but has been observed and measured in particle accelerators, providing insights into the fundamental forces of nature. Understanding this process also helps physicists explore the characteristics of antimatter and its potential applications.

Energy Conversion

Energy conversion is the process of changing one form of energy to another. In the context of particle physics, this often involves transforming mass into energy, as seen in proton-antiproton annihilation. When the mass of the particles vanishes, it's not lost but converted into electromagnetic energy, as explained by Einstein's mass-energy equivalence.

Such energy transformations are pivotal in understanding high-energy physics. They allow scientists to probe into the deep workings of atomic and subatomic structures and the interactions between particles. This conversion is not just a theoretical construct but is actively applied in technologies like nuclear reactors, where mass from nuclear fission is converted into energy, or in particle accelerators that observe particle collisions resulting in energy releases.

Grasping the concept of energy conversion helps us harness these reactions for practical purposes, ranging from generating power to investigating the universe's smallest constituents.

Electron Volts

An electron volt (eV) is a unit of energy commonly used in the field of particle physics. It reflects the amount of kinetic energy gained or lost by an electron as it moves across an electric potential difference of one volt. The electron volt is a much smaller and more convenient unit than the joule for measuring the energies involved in atomic and subatomic processes.

In the context of proton-antiproton annihilation, the energy can be measured in electron volts to provide a scale that is more intuitive for particle physicists. The conversion between joules and electron volts is generally straightforward: 1 joule is equal to \(6.242 \times 10^{18}\) electron volts.

The use of electron volts allows scientists to more easily compare and understand the energies involved in various reactions and to communicate findings effectively within the scientific community. It underscores how different measurement units can bridge the gap between theoretical physics and practical experimentation.

Einstein's Theory of Relativity

Einstein's theory of relativity revolutionized the understanding of physics by describing the interrelation of space, time, and energy. There are two parts to this theory: special relativity and general relativity. The principle of mass-energy equivalence arises from special relativity, which examines the physics of objects moving at high velocities, close to the speed of light.

Special relativity introduces concepts such as time dilation and length contraction, emphasizing that the laws of physics are the same for all non-accelerating observers. It also redefined how energy and mass are perceived, showing them as interchangeable quantities through the famous equation \(E = mc^2\).

This theory explains not only high-speed phenomena but also helps predict outcomes in subatomic particle interactions, like those seen in proton-antiproton annihilation. Understanding relativity is essential for comprehending modern physics and its application to high-energy astrophysical phenomena and the behavior of particles in accelerators.