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An electron and a positron have masses of \(9.11 \times 10^{-31} \mathrm{kg}\) . They collide and both vanish, with only electromagnetic radiation appearing after the collision. If each particle is moving at a speed of 0.20 \(\mathrm{c}\) relative to the laboratory before the collision, determine the energy of the electromagnetic radiation.

Short Answer

Expert verified
The energy of the electromagnetic radiation is \(1.67 \times 10^{-13} \text{ Joules}.\)

Step by step solution

01

Use the Relativistic Energy Formula

The total energy of a moving particle in relativistic physics is given by \( E = \frac{m c^2}{\sqrt{1 - \left( \frac{v}{c} \right)^2}} \). Here, \( m \) is the rest mass, \( c \) is the speed of light, and \( v \) is the velocity of the particle. For the electron and positron, both have rest mass \( m = 9.11 \times 10^{-31} \text{ kg} \) and velocity \( v = 0.20 \cdot c \).
02

Calculate the Energy of One Particle

Calculate the energy of one electron (or positron) using the formula from Step 1:\[E = \frac{9.11 \times 10^{-31} \cdot (3 \times 10^8)^2}{\sqrt{1 - (0.20)^2}}\]Compute \(1 - (0.20)^2 = 0.96\) and its square root gives approximately \(0.98\). Substitute into the formula:\[E \approx \frac{9.11 \times 10^{-31} \times 9 \times 10^{16}}{0.98}\approx 8.37 \times 10^{-14} \text{ Joules}\]
03

Calculate Total Energy for Both Particles

Since the collision involves both an electron and a positron, each contributing the energy calculated previously, the total energy is:\[E_{\text{total}} = 2 \times 8.37 \times 10^{-14} = 1.67 \times 10^{-13} \text{ Joules}\]
04

Conclusion

The calculated total energy \(1.67 \times 10^{-13} \text{ Joules}\) represents the energy of the electromagnetic radiation produced from the annihilation of the electron and positron.

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Key Concepts

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

Electron-Positron Annihilation
When an electron and a positron collide, they undergo a fascinating process called annihilation. This event occurs because an electron, which is a negatively charged particle, meets its antimatter counterpart, the positron, which is positively charged. They cancel each other out completely.

In this annihilation, both particles' masses are converted into energy, typically in the form of electromagnetic radiation, according to Einstein's mass-energy equivalence principle. This principle is captured in the famous equation:

  • \( E = mc^2 \),
where \( E \) is energy, \( m \) is mass, and \( c \) is the speed of light.

The complete conversion of mass into energy in such microscopic interactions like electron-positron annihilation provides a spectacular demonstration of the relationship between mass and energy. This can be challenging to imagine because we don't observe mass turning into energy in everyday life.

However, in the subatomic realm, these phenomena are a crucial part of how the universe functions.
Electromagnetic Radiation
The energy released from the electron-positron annihilation appears as electromagnetic radiation. But what exactly is this radiation? Simply put, electromagnetic radiation consists of waves of electric and magnetic fields that move through space at the speed of light.

These waves vary widely in wavelength and frequency, forming what is known as the electromagnetic spectrum. It ranges from very long radio waves to very short gamma rays. In the context of annihilation, the energy typically produces photons in the gamma-ray part of this spectrum.

Photons, the elementary particles of light, carry the electromagnetic force and have no mass. This makes them the ideal carriers of energy released during annihilation. Here's what happens:

  • The total energy from the annihilated particles is converted into photon energy.
  • Photons are released in opposite directions to conserve momentum.
This transformation from matter to pure energy is a striking demonstration of how energy can manifest in different forms in the universe.
Conservation of Energy
The annihilation process showcases the fundamental principle of conservation of energy, which states that energy cannot be created or destroyed, only transformed. In this case, the mass energy of the electron and positron becomes electromagnetic energy.

In every interaction, including subatomic processes like annihilation, total energy before and after the event remains constant. Here's how this principle is applied:
  • The energy from the rest mass of the electron and positron, calculated using their mass and velocity, is transformed into electromagnetic energy.
  • Before the interaction, the system's energy includes the kinetic energy of both the electron and positron.
  • After annihilation, the system's energy is completely in the form of electromagnetic radiation.
The conservation of energy is central to numerous physical processes beyond annihilation. It provides a reliable framework to predict the behavior of systems and ensures consistency in theoretical physics equations.

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Most popular questions from this chapter

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