Problem 41 - Fundamentals of Physics

Question

The mass of an electron is \(9.10938188 \times\) \(10^{-31} \mathrm{~kg} .\) To six significant figures, find (a) \(\gamma\) and (b) \(\beta\) for an electron with kinetic energy \(K=100.000 \mathrm{MeV}\)

Step-by-Step Solution

Answer

\(\gamma = 197.3\); \(\beta \approx 1\).

Step 1: Confirm Known Values

We are given the mass of the electron, \(m = 9.10938188 \times 10^{-31} \text{ kg}\), and the kinetic energy \(K = 100.000 \text{ MeV}\). We need to convert this kinetic energy to joules, using the conversion factor \(1 \text{ MeV} = 1.60218 \times 10^{-13} \text{ J}\).

Step 2: Convert Kinetic Energy to Joules

Convert the given kinetic energy from MeV to joules: \(K = 100 \text{ MeV} \times 1.60218 \times 10^{-13} \text{ J/MeV} = 1.60218 \times 10^{-11} \text{ J}\).

Step 3: Calculate Total Energy

The total energy \(E\) of the electron is the sum of its rest energy \(E_0\) and kinetic energy \(K\). The rest energy \(E_0\) is calculated using \(E_0 = mc^2\), where \(c = 3 \times 10^8 \text{ m/s}\) is the speed of light. Thus, \(E_0 = 9.10938188 \times 10^{-31} \text{ kg} \times (3 \times 10^8 \text{ m/s})^2 = 8.18726567 \times 10^{-14} \text{ J}\). Then, \(E = E_0 + K = 8.18726567 \times 10^{-14} \text{ J} + 1.60218 \times 10^{-11} \text{ J}\).

Step 4: Calculate the Lorentz Factor (\(\gamma\))

The Lorentz factor \(\gamma\) is given by the formula \(\gamma = \frac{E}{E_0}\). Substitute the known values: \(\gamma = \frac{8.18726567 \times 10^{-14} \text{ J} + 1.60218 \times 10^{-11} \text{ J}}{8.18726567 \times 10^{-14} \text{ J}}\). Calculate \(\gamma\).

Step 5: Calculate Velocity Ratio (\(\beta\))

The velocity ratio \(\beta\) is given by the equation \(\beta = \sqrt{1 - \frac{1}{\gamma^2}}\). Substitute \(\gamma\) from the previous step into this equation and calculate \(\beta\).

Step 6: Compile Final Answers

Present the final calculated values of \(\gamma\) and \(\beta\) to six significant figures.

Key Concepts

Lorentz factor

In the realm of relativistic mechanics, the Lorentz factor plays a crucial role. Often denoted by the symbol \( \gamma \), it adjusts the traditional Newtonian calculations to account for the effects of relativity at high speeds. This factor is calculated using the equation:\[ \gamma = \frac{1}{\sqrt{1-\beta^2}} \]where \( \beta \) represents the velocity of an object relative to the speed of light, \( \beta = \frac{v}{c} \).
The Lorentz factor signifies how much time dilation, length contraction, and relativistic mass increase occur. As speed approaches the speed of light, \( \gamma \) tends to infinity, making relativistic effects extremely significant.

• Time dilation implies that a moving clock ticks slower as compared to a stationary clock.
• Length contraction indicates that objects contract along the direction of motion as they move near light speed.
• Mass increase suggests that the object's mass grows and requires more energy to accelerate further.

As seen, \( \gamma \) is key to understanding the changes in physical properties of fast-moving objects under the influence of relativity.

velocity ratio

The velocity ratio, denoted as \( \beta \), is a measurement comparing an object's speed to that of light. Its formula is fairly straightforward:\[ \beta = \frac{v}{c} \]where \( v \) is the velocity of the object and \( c \) is the speed of light in a vacuum, approximately \( 3 \times 10^8 \text{ m/s} \).
In relativistic mechanics, knowing the velocity ratio is essential to calculate the Lorentz factor. It offers a simplified way to express speed when analyzing relativistic effects.

• At low velocities, much below the speed of light, \( \beta\) is small and relativistic effects are negligible.
• As an object approaches the speed of light, \( \beta \) nears 1, and the influence of relativity becomes pronounced, affecting its kinetic and total energy.

This concept allows physicists to reformulate classical dynamics to suit the needs of high-speed particle analysis.

kinetic energy conversion

Converting kinetic energy into a usable form involves understanding the relation between mass, velocity, and resulting energy. In the relativistic regime, kinetic energy \( K \) can be expressed by modifying the classical equation:\[ K = (\gamma - 1)mc^2 \]Here, \( m \) is the rest mass, \( c \) is the speed of light, and \( \gamma \) is the Lorentz factor. For high-speed particles, more energy is required to further accelerate them compared to low-velocity particles.
This formula shows that kinetic energy isn't solely dependent on velocity, but also on how relativity alters mass and energy relations as objects move faster:

• At lower speeds, the classical formula for kinetic energy \( K = \frac{1}{2}mv^2 \) suffices.
• As speeds approach light, relativistic considerations mandate the usage of \( \gamma \) to compute energy changes accurately.

The conversion highlights how traditional physics transitions to its relativistic counterpart as an object speeds up.

mass-energy equivalence

Mass-energy equivalence, probably the most famous concept introduced by Einstein, is summarized by the equation \( E = mc^2 \). It articulates that mass and energy are two forms of the same essence. Mass can be transformed into energy and vice versa.
This principle holds at the core of relativistic mechanics. When calculating an object's total energy \( E \), it's the sum of its rest energy and any kinetic energy it possesses:\[ E = E_0 + K \]where \( E_0 = mc^2 \) is the object's rest energy and \( K \) is its kinetic energy. Understanding mass-energy equivalence reshapes our comprehension of energy conservation:

• It leads to insights into phenomena like nuclear reactions where mass is converted to energy.
• It also implies that even when at rest, mass holds an intrinsic amount of energy waiting to be utilized or expended.

This concept is fundamental in physics, bridging classical elements with modern quantum theories.