Problem 48 - Physics for Scientists and Engineers with Modern Physics

Question

(a) How much energy is released when an electron and a positron annihilate each other? (b) How much energy is released when a proton and an antiproton annihilate each other? (All particles have \(K \approx 0 .)\)

Step-by-Step Solution

Answer

Electron-positron annihilation releases \(1.64 \times 10^{-13}\) J, proton-antiproton \(3.00 \times 10^{-10}\) J.

Step 1: Understand Particle Annihilation

Annihilation occurs when a particle meets its corresponding antiparticle. The masses of these particles are converted into energy. The most famous equation relating mass and energy is Einstein's mass-energy equivalence equation: \[ E = mc^2 \] where \( E \) is the energy, \( m \) is the mass, and \( c \) is the speed of light in a vacuum, \( c \approx 3 \times 10^8 \text{ m/s} \).

Step 2: Calculate Energy from Electron-Positron Annihilation

The mass of an electron (and a positron since it has the same mass) is approximately \( 9.11 \times 10^{-31} \text{ kg} \). When they annihilate, their total mass is converted into energy:\[ E_{e^-e^+} = 2m_ec^2 = 2 \times 9.11 \times 10^{-31} \times (3 \times 10^8)^2 \]Calculating this, we get:\[ E_{e^-e^+} \approx 1.64 \times 10^{-13} \text{ J} \].

Step 3: Calculate Energy from Proton-Antiproton Annihilation

The mass of a proton (and an antiproton) is approximately \( 1.67 \times 10^{-27} \text{ kg} \). Similar to the electron-positron case, the energy released when a proton and antiproton annihilate is:\[ E_{p\bar{p}} = 2m_pc^2 = 2 \times 1.67 \times 10^{-27} \times (3 \times 10^8)^2 \]Calculating this, we find:\[ E_{p\bar{p}} \approx 3.00 \times 10^{-10} \text{ J} \].

Step 4: Interpret the Results

For both annihilation scenarios, the entire mass is converted into energy. The amount of energy released is proportional to the mass of the particles involved. Given the significantly higher mass of protons compared to electrons, more energy is released in proton-antiproton annihilation.

Key Concepts

Mass-Energy Equivalence

In the world of physics, a groundbreaking concept introduced by Albert Einstein is known as mass-energy equivalence. The essence of this concept is captured in the equation \( E = mc^2 \), where \( E \) represents energy, \( m \) symbolizes mass, and \( c \) is the speed of light in a vacuum, approximately \( 3 \times 10^8 \text{ m/s} \). This simple yet profound equation tells us that mass can be converted into energy and vice versa.

When we talk about mass-energy equivalence, we're essentially discussing the idea that energy and mass are two forms of the same thing. This principle allows us to understand phenomena like particle annihilation, where particles and their corresponding antiparticles can entirely convert their mass into energy. It's a core principle behind nuclear reactions and particle physics experiments.

Key points to grasp, include that:

• The mass of both particles contributes to the total energy output.
• Greater mass will yield a greater amount of energy upon conversion.
• This principle is crucial in understanding both theoretical physics and practical applications.

Understanding mass-energy equivalence helps explain how massive amounts of energy can be produced from relatively small particles, as showcased in high-energy physics experiments.

Electron-Positron Annihilation

Electron-positron annihilation is an interesting phenomenon in which an electron, a negatively charged particle, meets its antiparticle, the positron, which is positively charged. Both particles are identical in mass, around \( 9.11 \times 10^{-31} \text{ kg} \), but have opposite charges.

Upon their meeting, these particles annihilate, meaning their entire mass is transformed into energy. This conversion is beautifully illustrated using Einstein's equation: \( E = mc^2 \). In this case, the total mass from both particles converts into energy:

• The total energy released is calculated as \( 1.64 \times 10^{-13} \text{ J} \).
• This process often results in the emission of two gamma-ray photons.

The energy release during this type of annihilation is relatively small compared to larger particles, but is still significant in research and technology, particularly in medical imaging technologies like PET scans (Positron Emission Tomography).

Electron-positron annihilation is essential in our understanding of fundamental particles and their interactions, serving as a model for understanding larger-scale particle and antiparticle dynamics.

Proton-Antiproton Annihilation

Proton-antiproton annihilation is similar in concept to electron-positron annihilation but involves heavier particles. A proton is a positively charged particle with a mass of about \( 1.67 \times 10^{-27} \text{ kg} \). The antiproton, its antiparticle, has the same mass but a negative charge.

When protons and antiprotons meet, they annihilate each other, resulting in a conversion of their entire mass into energy. As per the mass-energy equivalence principle:

• The energy released is calculated as \( 3.00 \times 10^{-10} \text{ J} \).
• This much larger energy release, compared to electron-positron annihilation, is due to the greater mass of the proton and antiproton.
• The annihilation often produces several pions, which are lighter particles, instead of just photons.

Proton-antiproton annihilation is essential for understanding high-energy physics, often explored in particle accelerators like the Large Hadron Collider. The energy and new particles produced can reveal insights into the forces that hold atomic nuclei together, helping us unlock mysteries of the universe at the smallest scales. Understanding this process is crucial for both theoretical physics and practical innovations.