Problem 79 - Fundamentals of Physics

Question

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

Step-by-Step Solution

Answer

The momentum is approximately 2.457 MeV/c.

Step 1: Understand the Kinetic Energy Formula for Relativistic Electron

In relativistic physics, the relationship between total energy \( E \), rest energy \( E_0 \), and kinetic energy \( K \) is given by: \( E = E_0 + K \), where \( E_0 = m_0c^2 \) is the rest energy of the electron. Here, \( m_0 \) is the rest mass of the electron and \( c \) is the speed of light.

Step 2: Introduce the Relativistic Energy-Momentum Relation

The relation between energy, momentum, and mass in relativistic settings is given by: \( E^2 = (pc)^2 + (m_0c^2)^2 \). We need to find the momentum \( p \), with given kinetic energy \( K = 2.00 \text{ MeV} \).

Step 3: Calculate the Total Energy

The rest mass energy of an electron is \( m_0c^2 = 0.511 \text{ MeV} \). With \( K = 2.00 \text{ MeV} \), the total energy \( E \) is \( E = 0.511 + 2.00 = 2.511 \text{ MeV} \).

Step 4: Apply the Relativistic Energy-Momentum Relation

Using the equation \( E^2 = (pc)^2 + (m_0c^2)^2 \), plug in the values: \( (2.511)^2 = (pc)^2 + (0.511)^2 \).

Step 5: Solve for Momentum (p)

Rearrange the equation to solve for \((pc)^2\) and then for \( p \): \[ (pc)^2 = (2.511)^2 - (0.511)^2. \] Calculate \((2.511)^2 = 6.3008\) and \((0.511)^2 = 0.2611\). Therefore, \((pc)^2 = 6.3008 - 0.2611 = 6.0397\). So, \( pc = \sqrt{6.0397} = 2.457 \text{ MeV} \). Thus, \( p = 2.457 \text{ MeV} / c \).

Key Concepts

Kinetic Energy

Kinetic energy, often abbreviated as KE, is the energy that an object possesses due to its motion. In the context of relativistic physics, kinetic energy is part of the total energy equation for particles moving at speeds close to the speed of light. Here’s why this is important:

• The classical formula for kinetic energy (\( K = \frac{1}{2}mv^2 \)) becomes inadequate at high velocities.
• For high-speed particles, kinetic energy is calculated using the relativistic energy formula: \( E = E_0 + K \), where \( E_0 \) is the rest energy of the particle.
• This shows how energy accumulates differently at relativistic speeds compared to everyday speeds.

Knowing the kinetic energy of high-speed particles helps in calculating other important quantities like momentum. For the exercise, the kinetic energy given is \(2.00 \text{ MeV}\) which fits into this framework for an electron.

Relativistic Energy-Momentum Relation

The relativistic energy-momentum relation is a cornerstone of modern physics. It connects energy, momentum, and rest mass in the realm of special relativity. Understanding this relationship is crucial:

• It is expressed as: \( E^2 = (pc)^2 + (m_0 c^2)^2 \). This equation shows the interplay between momentum \( p \), speed of light \( c \), and rest mass energy \( m_0c^2 \).
• The formula unifies concepts of energy and momentum, applying to all particles no matter their speed.
• It provides a method to calculate momentum when energy is known, which is essential in understanding particle behavior in high-energy physics.

In the exercise, by knowing the total energy \( E \) and the rest energy \( m_0c^2 \), we can compute the electron's momentum using this relation. Such calculations are typically done by squaring the known values, rearranging the equation, and solving for the desired attribute, which in this case is momentum \( p \).

Rest Energy

Rest energy is the energy an object contains due to its rest mass, as per the famous equation \( E_0 = m_0c^2 \). Here’s what makes rest energy significant:

• It represents the "energy at rest," meaning it is the intrinsic energy a particle has even without movement.
• Rest energy is fundamental in calculations involving high-energy and relativistic particles.
• This form of energy explains how mass could theoretically be transformed into energy, essential for concepts like nuclear energy.

In the problem, the rest energy of an electron is given as \(0.511 \text{ MeV}\). This value is consistently used to compute other energies like total and kinetic energy. Since rest energy is integral to finding the values deduced in the solution, it's clearly a vital component in any relativistic physics calculation.