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 of the electron is approximately 2.456 MeV/c.

Step 1: 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.

Step 2: 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).

Step 3: Calculate the Total Energy

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

Step 4: 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 \( m_0c^2 = 0.511 \, \text{MeV} \):\[ (pc)^2 = (2.511)^2 - (0.511)^2 \]\[ (pc)^2 = 6.3001 - 0.2611 = 6.039 \]\[ pc = \sqrt{6.039} \approx 2.456 \, \text{MeV/c} \]

Step 5: Result

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

Key Concepts

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’t 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’ 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.