Problem 32 - University Physics with Modern Physics

Question

In proton-antiproton annihilation a proton and an antiproton (a negatively charged proton) collide and disappear, producing electromagnetic radiation. If cach 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 total energy is \(3.00 \times 10^{-10}\) J or \(1.87 \times 10^{9}\) eV.

Step 1: Identify Given Values

In this problem, we know the mass of the proton and antiproton as \( m = 1.67 \times 10^{-27} \mathrm{kg} \) each. Since they are at rest before annihilation, their combined initial kinetic energy is zero.

Step 2: Apply Mass-Energy Equivalence

According to Einstein's mass-energy equivalence principle, the energy equivalent of a mass \( m \) is given by \( E = mc^2 \). Here \( c = 3 \times 10^8 \mathrm{m/s} \) is the speed of light. For two particles, the total mass is \( 2m \).

Step 3: Calculate Total Energy in Joules

Substitute the values into the formula: \( E = 2m c^2 = 2 \times 1.67 \times 10^{-27} \mathrm{kg} \times (3 \times 10^8 \mathrm{m/s})^2 \). Simplify it to get the energy in joules.

Step 4: Convert Joules to Electron Volts

The energy conversion from joules to electron volts can be performed using the conversion factor \(1 \, ext{eV} = 1.602 \times 10^{-19} \, ext{J}\). So, divide the energy in joules by \( 1.602 \times 10^{-19} \) to find the energy in electron volts.

Key Concepts

Proton-Antiproton Annihilation

When a proton encounters an antiproton, something fascinating happens. They annihilate each other. This isn't just a dramatic word; it means the particles literally disappear and convert into energy. This transformation occurs because protons and antiprotons are matter and antimatter counterparts. In annihilation events, the rest mass energy of these particles converts into other forms of energy, such as electromagnetic radiation. What started as two particles now becomes pure energy in a new form. Before annihilation, the proton and antiproton are typically at rest, having no kinetic energy. Their inherent rest mass is what fuels the conversion to energy. The total mass in a proton-antiproton annihilation reaction is crucial, as it determines how much energy will be released during the process.

Energy Conversion

Energy conversion is a key concept in physics, especially in annihilation reactions. With mass-energy equivalence, mass can transform into energy and vice versa. The famous equation from Einstein, \( E = mc^2 \), plays an important role here. It tells us how much energy equivalent corresponds to a given mass. In the context of proton-antiproton annihilation, both particles have a mass of \(1.67 \times 10^{-27} \mathrm{kg}\). When these particles annihilate, their combined mass converts into energy. The speed of light \((3 \times 10^8 \mathrm{m/s})\) acts as a conversion factor, emphasizing how a small amount of mass can translate into a significant amount of energy. This process highlights the core idea that energy and mass are two forms of the same thing. It also reminds us that conversion factors are important, allowing us to express energy in different units, like joules or electron volts.

Einstein's Theory of Relativity

Einstein's theory of relativity is groundbreaking in understanding how mass and energy interplay. One key aspect of this theory is the mass-energy equivalence principle, expressed as \( E = mc^2 \). This simple yet profound equation explains that mass can be converted into energy. In practical terms, it revolutionizes how we think about the universe and reactions like proton-antiproton annihilation. Relativity extended our understanding of physics by showing that time, space, and energy are interconnected. It goes beyond the classical views that treated mass and energy as distinct entities. By linking mass and energy, Einstein opened up new pathways for understanding the immense energy potential stored in tiny amounts of matter. This has profound implications for technology, physics, and our grasp of cosmic events.

Electromagnetic Radiation

Electromagnetic radiation is the form of energy produced in proton-antiproton annihilation. When the particles annihilate, their mass converts entirely into energy. This energy manifests primarily as electromagnetic waves. Electromagnetic radiation spans a broad spectrum that includes visible light, radio waves, X-rays, and more. In the context of annihilation, the energy is typically released as gamma rays—high-frequency electromagnetic waves. Gamma rays carry significant energy, reflective of the mass-energy conversion that occurs during the reaction. Understanding electromagnetic radiation is essential to grasp how energy propagates through space as waves. This form of radiation is quite versatile, responsible for various technological and natural phenomena, from powering communications to contributing to our understanding of the universe. Electromagnetic waves produced in such reactions travel at the speed of light, showcasing once again the principle of mass-energy equivalence and emphasizing the dynamic interplay between mass disappearance and energy emergence.