Ch 36: Special Relativity

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 36: Special Relativity

Problem 59b

Chapter 36, Problem 59b

The half-life of a muon at rest is 1.5 μs. Muons that have been accelerated to a very high speed and are then held in a circular storage ring have a half-life of 7.5 μs. What is the total energy of a muon in the storage ring? The mass of a muon is 207 times the mass of an electron.

Verified step by step guidance

Understand the problem: The half-life of a muon increases due to time dilation, a relativistic effect. The goal is to find the total energy of the muon in the storage ring, which includes its rest energy and relativistic kinetic energy.

Relate the observed half-life to the time dilation factor (γ). The time dilation formula is:

t = γ t

₀, where

t

is the dilated half-life (7.5 μs),

t

₀ is the rest half-life (1.5 μs), and

γ

is the Lorentz factor. Solve for

γ

γ = \frac{t}{t}

₀.

Calculate the rest energy of the muon using Einstein's equation:

E

_rest=mc2. The mass of the muon is 207 times the mass of an electron, and the mass of an electron is approximately

9.11 \times 10 {-31}^{kg}

. Substitute the values to find the rest energy.

Relate the total energy to the Lorentz factor:

E

_total=γE_rest. Use the previously calculated

γ

and

E

_rest to express the total energy.

Combine the results: Substitute the values of

γ

and

E

_rest into the total energy formula to express the total energy of the muon in the storage ring. This will give the final expression for the total energy.

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Key Concepts

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

Half-Life

Half-life is the time required for half of a sample of a radioactive substance to decay. In the context of muons, their half-life increases when they are moving at relativistic speeds due to time dilation, a phenomenon predicted by Einstein's theory of relativity. This means that the faster a muon travels, the longer it appears to live from the perspective of an observer.

Relativistic Effects

Relativistic effects arise when objects move at speeds close to the speed of light, leading to significant changes in their physical properties. For muons in a storage ring, their increased half-life is a direct result of time dilation, which states that time passes more slowly for objects in motion compared to those at rest. This concept is crucial for understanding how the muon's behavior changes under high-speed conditions.

Total Energy of a Particle

The total energy of a particle in relativistic physics is the sum of its rest mass energy and its kinetic energy. For a muon, this can be calculated using the equation E = mc² for rest mass energy and the relativistic kinetic energy formula. In the case of the muon in the storage ring, knowing its mass and the effects of its increased half-life allows for the determination of its total energy while in motion.

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