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Chapter 37: Problem 79

What is the momentum in MeV/c of an electron with a kinetic energy of \(2.00 \mathrm{MeV} ?\)

Short Answer

Expert verified
The momentum of the electron is approximately 2.456 MeV/c.

Step by step solution

01

Understand the Given Information

We are given the kinetic energy of an electron as 2.00 MeV. We need to find its momentum in MeV/c.
02

Use the Formula for Relativistic Energy

The total energy of an electron is given by the formula: \[ E^2 = (pc)^2 + (m_0c^2)^2 \] where \( E \) is the total energy, \( p \) is the momentum, \( c \) is the speed of light, and \( m_0c^2 \) is the rest energy of the electron (0.511 MeV).
03

Calculate the Total Energy

The total energy \( E \) is the sum of the rest energy and kinetic energy. Therefore, \[ E = KE + m_0c^2 = 2.00 \, \text{MeV} + 0.511 \, \text{MeV} = 2.511 \, \text{MeV} \]
04

Solve for Momentum

Rearrange the formula \( E^2 = (pc)^2 + (m_0c^2)^2 \) to find \( pc \):\[ (pc)^2 = E^2 - (m_0c^2)^2 \]Substitute \( E = 2.511 \, \text{MeV} \) and \( m_0c^2 = 0.511 \, \text{MeV} \):\[ (pc)^2 = (2.511)^2 - (0.511)^2 \]\[ (pc)^2 = 6.3001 - 0.2611 = 6.039 \]\[ pc = \sqrt{6.039} \approx 2.456 \, \text{MeV/c} \]
05

Result

The momentum of the electron is approximately 2.456 MeV/c.

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Key Concepts

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

Kinetic Energy
Kinetic energy is a concept that helps us understand the energy that an object possesses due to its motion. For particles moving at speeds comparable to the speed of light, like electrons in this problem, we use a relativistic approach to determine kinetic energy.
This is because ordinary Newtonian mechanics doesn't work well at high speeds. Instead of using the classical formula for kinetic energy, \( KE = \frac{1}{2} mv^2 \), we consider how energy behaves under Einstein's theory of relativity. In relativistic terms, kinetic energy, combined with rest energy, allows us to determine the total energy of a particle.
Total Energy
Total energy for a moving particle in relativity is more comprehensive. It includes both the kinetic energy and the rest energy, i.e., the energy a particle has even when at rest. The formula used to find the total energy in high velocity scenarios is:- \( E = KE + m_0c^2 \) Here, \( KE \) is the kinetic energy and \( m_0c^2 \) is the rest energy of the particle. For the example of the electron, the kinetic energy is 2.00 MeV, and the rest energy is 0.511 MeV.
Therefore, the total energy becomes the sum of these, totaling 2.511 MeV. This total energy can then be substituted into the relativistic energy-momentum equation to find the momentum.
Rest Energy
Rest energy is the energy inherent in an object when it is not moving. It's depicted by the famous equation \( E = m_0c^2 \) from Einstein's relativity theory. Even when a particle is stationary, it contains energy due to its rest mass.
For electrons, the rest energy is a constant value of approximately 0.511 MeV. This concept helps to understand that energy doesn鈥檛 solely come from movement or interaction; every particle holds intrinsic energy simply by having mass. When combined with kinetic energy, rest energy provides the complete energy picture of a particle.
Relativity
Relativity, introduced by Albert Einstein, revolutionized our understanding of space, time, and energy. In scenarios where particles move at high speeds close to the speed of light, classical mechanics fails to accurately describe motion.
  • Relativistic concepts like time dilation and length contraction become significant, affecting how we evaluate particles鈥� behavior.
  • Momentum and energy are not simply additive components but are intertwined in the space-time fabric. This relationship is captured by the energy-momentum equation \( E^2 = (pc)^2 + (m_0c^2)^2 \), which blends energy and momentum into a single entity.
This equation allows us to deduce one quantity if we know the others, illustrating the interconnected nature of relativistic physics.

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Most popular questions from this chapter

A certain particle of mass \(m\) has momentum of magnitude \(m c\). What are (a) \(\beta\), (b) \(\gamma\), and (c) the ratio \(K / E_{0}\) ?

A rod lies parallel to the \(x\) axis of reference frame \(S\), moving along this axis at a speed of \(0.630 c\). Its rest length is \(1.70 \mathrm{~m}\). What will be its measured length in frame \(S\) ?

(Come) back to the future. Suppose that a father is \(20.00 \mathrm{y}\) older than his daughter. He wants to travel outward from Earth for \(2.000 \mathrm{y}\) and then back for another \(2.000 \mathrm{y}\) (both intervals as he measures them) such that he is then \(20.00 \mathrm{y}\) younger than his daughter. What constant speed parameter \(\beta\) (relative to Earth) is required?

Certain wavelengths in the light from a galaxy in the constellation Virgo are observed to be \(0.4 \%\) longer than the corresponding light from Earth sources. (a) What is the radial speed of this galaxy with respect to Earth? (b) Is the galaxy approaching on receding from Earth?

An alpha particle with kinetic energy \(7.70 \mathrm{MeV}\) collides with an \({ }^{14} \mathrm{~N}\) nucleus at rest, and the two transform into an \({ }^{17} \mathrm{O}\) nucleus and a proton. The proton is emitted at \(90^{\circ}\) to the direction of the incident alpha particle and has a kinetic energy of \(4.44 \mathrm{MeV}\). The masses of the various particles are alpha particle, \(4.00260 \mathrm{u} ;{ }^{14} \mathrm{~N}\). \(14.00307\) u; proton, \(1.007825 \mathrm{u}\); and \({ }^{17} \mathrm{O}, 16.99914 \mathrm{u}\). In \(\mathrm{MeV}\), what are (a) the kinetic energy of the oxygen nucleus and (b) the \(Q\) of the reaction? (Hint: The speeds of the particles are much less than \(c .\) )
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