Ch 37: The Foundations of Modern Physics

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 37: The Foundations of Modern Physics

문제 27b

37장, 문제 27b

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

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.