Chapter 26: Problem 59
What minimum amount of electromagnetic energy is needed to produce an electron and a positron together? A positron is a particle with the same mass as an electron, but has the opposite charge. (Note that electric charge is conserved in this process. See Section 27\(-\)6.)
Short Answer
Expert verified
1.64 × 10^{-13} J is needed to produce an electron and a positron.
Step by step solution
01
Understand the Process
We need to calculate the minimum energy required to produce an electron and a positron. The positron is essentially an 'anti-electron' with the same mass as the electron but opposite charge.
02
Apply the Mass-Energy Equivalence
According to Einstein's mass-energy equivalence, the energy required to create a particle is given by the formula \(E = mc^2\), where \(m\) is the mass of the particle and \(c\) is the speed of light in a vacuum.
03
Calculate the Energy for One Particle
The mass of an electron (or positron) is approximately \(9.11 \times 10^{-31}\) kg. Substituting this into the formula \(E = mc^2\), we calculate the energy needed for one particle: \[ E = (9.11 \times 10^{-31} \text{ kg}) \times (3 \times 10^8 \text{ m/s})^2 \approx 8.19 \times 10^{-14} \text{ J} \]
04
Calculate the Total Energy for Two Particles
Since we need to create both an electron and a positron, we need double the energy calculated for one particle. Therefore, multiply the energy for one particle by 2: \[ E_{total} = 2 \times 8.19 \times 10^{-14} \text{ J} = 1.64 \times 10^{-13} \text{ J} \]
05
Conclusion - Minimum Energy Required
The minimum electromagnetic energy needed to produce an electron and a positron together is \(1.64 \times 10^{-13} \text{ J}\). This calculation assumes no additional energy is lost in the process.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Electron
An electron is a fundamental particle with a negative electric charge, often represented by the symbol \( e^- \). Electrons are a type of elementary particle belonging to a group called leptons, which means they do not experience the strong nuclear force like protons and neutrons do. Instead, they are involved in electromagnetic processes and interactions. Electrons play a critical role in many areas of physics, including electricity, magnetism, and quantum mechanics.
Here are some other important points about electrons:
Here are some other important points about electrons:
- An electron has a very small mass of approximately \( 9.11 \times 10^{-31} \text{ kg} \), which is about 1/1836 the mass of a proton.
- Electrons orbit the nucleus of an atom in various energy levels, which are distinct distances corresponding to specific energies.
- They are responsible for electricity because they move through conductive materials, generating electric currents.
Positron
A positron is an antiparticle of the electron, symbolized as \( e^+ \). It has the same mass as an electron but carries a positive charge, making it the mirror image of an electron in terms of electric charge.
Here are key attributes of positrons:
Here are key attributes of positrons:
- Positrons are part of the antimatter group of particles, highlighting the symmetry in physics known as matter-antimatter symmetry.
- When positrons and electrons meet, they annihilate each other, converting their mass into energy in the form of photons, showing mass-energy equivalence.
- Positrons play a significant role in medical imaging technologies like Positron Emission Tomography (PET) scans.
Einstein's Equation
Einstein's equation, represented by \( E = mc^2 \), is one of the most famous equations in physics. It demonstrates the mass-energy equivalence principle, which means mass can be converted into energy and vice versa. Here, \( E \) is energy, \( m \) is mass, and \( c \) is the speed of light in a vacuum, approximately \( 3 \times 10^8 \text{ m/s} \).
This equation tells us that:
This equation tells us that:
- Even a tiny amount of mass can be converted into a large amount of energy, which is fundamental in nuclear reactions.
- The speed of light \( c \) is a constant, emphasizing that energy in this context is tremendous due to the high value of \( c^2 \).
- Energy and mass are different forms of the same thing and can be transformed under certain conditions, such as in nuclear fission or fusion.