Ch 37: The Foundations of Modern Physics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 37: The Foundations of Modern Physics

Problem 27b

Chapter 37, Problem 27b

What is the velocity, as a fraction of c, of an electron with 2.0 GeV total energy? Hint: This problem uses relativity.

Verified step by step guidance

Start by recalling the total energy equation in special relativity:

E = γmc²

, where

γ

is the Lorentz factor,

m

is the rest mass of the electron, and

c

is the speed of light.

Rearrange the equation to solve for the Lorentz factor

γ

γ = E / (mc²)

. Here,

E

is the total energy (2.0 GeV), and

mc²

is the rest energy of the electron (approximately 0.511 MeV). Convert units so they are consistent (e.g., GeV to MeV).

The Lorentz factor is also related to velocity by the equation:

γ = 1 / √(1 - v²/c²)

. Substitute the value of

γ

obtained in the previous step into this equation.

Rearrange the equation to solve for

v/c

, the velocity as a fraction of the speed of light:

v/c = √(1 - 1/γ²)

. Plug in the value of

γ

to compute

v/c

Simplify the expression to find the numerical value of

v/c

. This will give the velocity of the electron as a fraction of the speed of light.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Relativistic Energy

In the context of special relativity, the total energy of an object is given by the equation E = γmc², where γ (gamma) is the Lorentz factor. This factor accounts for the effects of relativistic speeds, which become significant as an object's velocity approaches the speed of light (c). The total energy includes both rest mass energy and kinetic energy, and understanding this concept is crucial for solving problems involving high-energy particles like electrons.

Lorentz Factor

The Lorentz factor, denoted as γ, is defined as γ = 1 / √(1 - v²/c²), where v is the velocity of the object and c is the speed of light. This factor increases as the velocity of the object approaches the speed of light, leading to significant relativistic effects such as time dilation and length contraction. It is essential for calculating the relativistic momentum and energy of particles moving at high speeds.

Velocity as a Fraction of c

In relativistic physics, expressing velocity as a fraction of the speed of light (c) allows for a clearer understanding of how fast an object is moving relative to the ultimate speed limit in the universe. For example, if an electron's velocity is 0.9c, it is traveling at 90% of the speed of light. This concept is vital for analyzing the behavior of particles in high-energy physics, particularly when calculating their relativistic effects.

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Related Practice

Textbook Question

In one of Thomson’s experiments he placed a thin metal foil in the electron beam and measured its temperature rise. Consider a cathode-ray tube in which electrons are accelerated through a 2000 V potential difference, then strike a 10 mg copper foil. What is the electron-beam current if the foil temperature rises 6.0°C in 10 s? Assume no loss of energy by radiation or other means. The specific heat of copper is 385 J/kg K .

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Textbook Question

The factor γ appears in many relativistic expressions. A value γ = 1.01 implies that relativity changes the Newtonian values by approximately 1% and that relativistic effects can no longer be ignored. At what kinetic energy, in MeV, is γ = 1.01 for (a) an electron, (b) a proton, and (c) an alpha particle?

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