Question

Calculate the speed parameter \(\beta\) of a particle with a momentum of \(12.5 \mathrm{MeV} / c\) if the particle is \((a)\) an electron and \((b)\) a proton.

Short answer

The speed parameter \( \beta \) is calculated to be approximately 0.930 and 0.014 for the electron and proton, respectively.

Step-by-step solution

Identify Given Information and Formulas

Firstly, it's important to gather the necessary parameters. The problem gives that the momentum of the particle, denoted by \( p \), is 12.5 MeV/c. The formula to be used is the one for relativistic momentum (\( p = \gamma m v \)), and the definition of speed parameter (\( \beta = \frac{v}{c} \)).

Calculate β for the Electron

Using the relativistic momentum formula and the definition of speed parameter, rewrite p in terms of m and β to get the formula for β. For an electron, the mass (m) is approximately 0.511 MeV/c². Substitute the given momentum and the mass of the electron into the equation and solve for β.

Calculate β for the Proton

For a proton, the mass (m) is approximately 938 MeV/c². Just like with the electron, substitute these values into the equation derived in step 2 and solve for β. Using a calculator, make sure to apply the correct order of operations (multiplication and division before addition and subtraction).

Key concepts

Speed Parameter (Beta)

Understanding the speed parameter, often symbolized by the Greek letter \( \beta \), is crucial when delving into the realm of relativistic physics. \( \beta \) is defined as the ratio of the velocity \( v \) of a particle to the speed of light \( c \). Mathematically, this is represented as \( \beta = \frac{v}{c} \). By expressing speed as a fraction of the speed of light, \( \beta \) provides an intuitive sense of how "relativistic" a particle's velocity is.

If \( \beta \) is close to 1, the particle is moving close to the speed of light, entering the relativistic regime where classical mechanics no longer suffice. For lower values of \( \beta \), classical approximations might still apply, but accuracy in high speeds demands Einstein's theory of relativity.

In problems, calculating \( \beta \) typically involves the momentum \( p \), the mass \( m \), and a relativistic factor known as \( \gamma \) (gamma). Since \( p = \gamma m v \), you can see \( \beta = \frac{p}{\gamma m c} \), which is used when \( \gamma \) is part of the solution.

Momentum

Momentum is a fundamental concept in physics representing the product of mass and velocity. However, in relativistic physics, momentum behaves differently than in classical contexts. The relativistic momentum is expressed as \( p = \gamma m v \), where \( \gamma \), the Lorentz factor, accounts for the effects of relativity.

The Lorentz factor \( \gamma = \frac{1}{\sqrt{1 - \beta^2}} \) grows significantly as the speed of an object approaches the speed of light. This adjustment is necessary because, according to Einstein's theory of relativity, as an object's speed increases towards the speed of light, its momentum increases significantly more than predicted by Newtonian mechanics alone.

Relativistic momentum can further be understood as the limiting behavior of classical momentum at high speeds. Therefore, understanding how to manipulate the equation \( p = \gamma m v \) is essential for solving problems involving high-speed particles, such as electrons and protons in the realm of particle physics.

Relativity

Relativity, a cornerstone of modern physics introduced by Albert Einstein, redefines our understanding of concepts like time and space. Special relativity, in particular, addresses the physics of objects moving at a significant fraction of the speed of light.

A key result of special relativity is the realization that measurements of time and space are not absolute but depend on the relative motion between the observer and what is being observed. This idea is captured in equations such as \( E = mc^2 \), which reflect the interplay between energy (\( E \)) and mass (\( m \)).

Relativistic effects become significant when speeds approach that of light, requiring modifications to previously accepted Newtonian notions. For example, time dilation explains how a moving clock ticks slower compared to a stationary one, while length contraction describes the shortening of objects in the direction of motion. Understanding these relativistic effects is crucial when working with high-speed particles like electrons and protons, which behave differently than they would at lower, non-relativistic speeds.