Problem 50 - Fundamentals of Physics

Question

To four significant figures, find the following when the kinetic energy is \(10.00 \mathrm{MeV}:\) (a) \(\gamma\) and (b) \(\beta\) for an electron \(\left(E_{0}=\right.\) \(0.510998 \mathrm{MeV}\) ), (c) \(\gamma\) and (d) \(\beta\) for a proton \(\left(E_{0}=938.272 \mathrm{MeV}\right)\) and (e) \(\gamma\) and (f) \(\beta\) for an \(\alpha\) particle \(\left(E_{0}=3727.40 \mathrm{MeV}\right)\).

Step-by-Step Solution

Answer

Electron: \(\gamma = 20.57\), \(\beta = 0.9990\); Proton: \(\gamma = 1.0107\), \(\beta = 0.1474\); Alpha: \(\gamma = 1.0027\), \(\beta = 0.07357\).

Step 1: Understanding the Variables

Identify the variables involved. The kinetic energy (\(KE\)) given is 10.00 MeV. The rest energies (\(E_0\)) are provided for the electron, proton, and alpha particle as 0.510998 MeV, 938.272 MeV, and 3727.40 MeV, respectively.

Step 2: Compute Total Energy (Rest + Kinetic)

The total energy (\(E\)) for each particle is the sum of rest energy \(E_0\) and kinetic energy \(KE\). Therefore, \(E = E_0 + KE\). For the electron, \(E = 0.510998 + 10 = 10.510998 \ \text{MeV}\). Similarly, compute \(E\) for the proton and alpha particle.

Step 3: Calculate Lorentz Factor \(\gamma\)

The Lorentz factor \(\gamma\) is given by \(\gamma = \frac{E}{E_0}\). Substitute \(E\) and \(E_0\) for each particle. For the electron, \(\gamma = \frac{10.510998}{0.510998}\). Calculate \(\gamma\) to four significant figures and repeat for the proton and alpha particle.

Step 4: Determine \(\beta\)

\(\beta\), the ratio of speed to the speed of light, is calculated using \(\beta = \sqrt{1 - \frac{1}{\gamma^2}}\). Use the \(\gamma\) values obtained in Step 3 to compute \(\beta\) for each particle to four significant figures.

Step 5: Compile Results

Record the calculated values of \(\gamma\) and \(\beta\) for each particle.- Electron: \(\gamma = 20.57\), \(\beta = 0.9990\)- Proton: \(\gamma = 1.0107\), \(\beta = 0.1474\)- Alpha particle: \(\gamma = 1.0027\), \(\beta = 0.07357\)

Key Concepts

Kinetic Energy

Kinetic energy is a fundamental concept in physics that describes the energy an object has due to its motion. In this exercise, we're dealing with kinetic energy as it applies to subatomic particles like electrons, protons, and alpha particles. But unlike the classical mechanics scenario, where kinetic energy is given by the simple formula \[ KE = \frac{1}{2}mv^2 \] (where \(m\) is the mass and \(v\) is the velocity), in relativistic mechanics, things work differently.

When particles travel at speeds close to the speed of light, we must use a different formula to account for relativistic effects. The kinetic energy is a part of the total energy \(E\), which includes the rest energy \(E_0\). The relationship used in the exercise demonstrates the impact of kinetic energy by adding it to the rest energy to determine the total energy:

• Total Energy \(E = E_0 + KE\)

Here, we see how adding the kinetic energy increases the total energy of the particle, altering its effective mass and behavior according to relativity theory. This approach prevents inconsistencies that occur when speeds near the speed of light.

Relativistic Mechanics

Relativistic mechanics is the extension of classical mechanics, developed by Albert Einstein, that accurately describes the motion of objects traveling at speeds close to the speed of light.

In the problem, the Lorentz factor \(\gamma\) is key to understanding relativistic effects. It transforms the equations of classic physics into those compatible with special relativity. The Lorentz factor is given by:

• \( \gamma = \frac{E}{E_0} \)

where \(E\) represents total energy and \(E_0\) is the rest energy of the particle. Another important parameter is \(\beta\), which represents the velocity of the particle as a fraction of the speed of light \(c\):

• \( \beta = \sqrt{1 - \frac{1}{\gamma^2}} \)

These equations showcase how significantly the particle's behavior diverges from predictions based on Newtonian physics. As \(\gamma\) increases, so does the particle's relativistic mass and energy. The closer \(\beta\) gets to 1, the more evident are the relativistic effects.

This understanding is crucial for studies in particle physics and for technologies like particle accelerators, where energies are so high that classical physics is not sufficient for precise calculations.

Rest Energy

Rest energy is the inherent energy contained within a particle solely due to its mass, even when the particle isn't moving. This concept is pivotal in Einstein's theory of relativity and is described by the famous equation \(E_0 = mc^2\), where \(E_0\) is the rest energy, \(m\) is the rest mass, and \(c\) is the speed of light.

In the context of the exercise, rest energy serves as the baseline energy level from which any motion creates additional kinetic energy. For instance, in the case of an electron, its rest energy is \(0.510998 \ \text{MeV}\). Rest energy accounts for the mass-energy equivalence principle, demonstrating that mass can be converted into energy and vice versa.

This principle highlights why particles have an intrinsic energy value even at rest, which drastically influences their total energy calculations in relativistic contexts. The impact of rest energy becomes more pronounced when comparing particles like protons and alpha particles, which have much higher rest energies than electrons due to their larger masses. Knowing how to work with rest energy is essential for understanding the behavior of particles in high-energy environments and underlies many applications in nuclear physics and cosmology.