Problem 40 - College Physics

Question

\(\bullet\) Why are we bombarded by muons? Muons are unstable subatomic particles (more on them in Chapter 30 ) that decay to electrons with a mean lifetime of 2.2\(\mu \mathrm{s}\) . They are produced when cosmic rays bombard the upper atmosphere about 10 \(\mathrm{km}\) above the earth's surface, and they travel very close to the speed of light. The problem we want to address is why we see any of them at the earth's surface. (a) What is the greatest distance a muon could travel during its 2.2\(\mu\) s lifetime? (b) According to your answer in part (a), it would seem that muons could never make it to the ground. But the 2.2\(\mu\) lifetime is measured in the frame of the muon, and they are moving very fast. At a speed of \(0.999 c,\) what is the mean lifetime of a muon as measured by an observer at rest on the earth? How far could the muon travel in this time? Does this result explain why we find muons in cos- mic rays? (c) From the point of view of the muon, it still lives for only \(2.2 \mu s,\) so how does it make it to the ground? What is the thickness of the 10 \(\mathrm{km}\) of atmosphere through which the muon must travel, as measured by the muon? Is it now clear how the muon is able to reach the ground?

Step-by-Step Solution

Answer

Muons reach the Earth due to time dilation and length contraction, traveling much further in the Earth's frame.

Step 1: Calculate the greatest distance a muon could travel

The greatest distance a muon could travel in its lifetime is determined by the equation: \[d = v \cdot t\]where \(v\) is the speed of the muon, very close to the speed of light \(c = 3 \times 10^8 \, \text{m/s}\), and \(t\) is the lifetime \(2.2 \, \mu s = 2.2 \times 10^{-6} \, \text{s}\). Assuming \(v \approx c\):\[d = (3 \times 10^8 )(2.2 \times 10^{-6}) \]\[d \approx 660 \, \text{meters}\]

Step 2: Calculate the muon's mean lifetime from Earth's reference frame

Using time dilation in special relativity, when a muon moves at speed \(v = 0.999c\), its lifetime in the Earth's frame: \[t' = \gamma t = \frac{t}{\sqrt{1-v^2/c^2}}\]\(\gamma = \frac{1}{\sqrt{1-(0.999)^2}} \approx 22.4\)Thus, the mean lifetime \(t'\) is:\[t' = 22.4 \times 2.2 \times 10^{-6} \, \text{s} \approx 4.93 \times 10^{-5} \, \text{s}\]

Step 3: Determine the distance traveled from Earth's reference frame

Now, calculate the distance the muon can travel with this dilated lifetime:\[d' = v \cdot t' = 0.999c \cdot 4.93 \times 10^{-5} \]\[d' \approx 0.999 \times 3 \times 10^8 \times 4.93 \times 10^{-5} \]\[d' \approx 14,790 \, \text{meters} \text{ (or about 14.8 km)}\]This distance is more than enough for the muon to reach Earth's surface.

Step 4: Analyze the situation from the muon's perspective

In the muon's frame of reference, the atmosphere's thickness appears shorter due to length contraction:\[L = L_0 \sqrt{1-v^2/c^2}\]Given \(L_0 = 10,000 \, \text{m}\), cosmological speed \(v = 0.999c\):\[L \approx 10,000 \times \sqrt{1-(0.999)^2} \]\[L \approx 10,000 \times 0.0447 \approx 447 \, \text{meters}\]This contracted distance can be traversed by the muon within its own lifetime.

Key Concepts

Muons

Muons are fascinating subatomic particles closely related to electrons but much heavier. They are part of the lepton family and are produced when cosmic rays collide with molecules in Earth's upper atmosphere, about 10 kilometers above the ground. These particles have a very short lifespan, decaying into electrons and neutrinos in approximately 2.2 microseconds. Despite their short existence, muons are detected at the Earth's surface, which is puzzling when considering their expected travel capabilities.
This raises significant questions about how these particles can traverse vast distances in Earth’s atmosphere before decaying. Their ability to reach the surface can be explained through the framework of special relativity, which includes key concepts like time dilation and length contraction.

Time Dilation

Time dilation is a cornerstone of the theory of special relativity, described by Albert Einstein. It occurs when time, as measured in different frames of reference, varies according to the relative speed between them. When muons, traveling close to the speed of light at approximately 0.999c, are observed from Earth's reference frame:

• Their mean lifetime appears significantly extended.
• This extended time allows them to cover greater distances before decaying.

For an observer on the ground, a muon's lifetime stretches due to its high velocity. Using the formula:\[ t' = \gamma t = \frac{t}{\sqrt{1-v^2/c^2}} \]the dilated lifetime \( t' \) can be calculated. The result shows that muons live longer from the Earth's perspective, around 4.93 x 10^{-5} seconds, allowing them to reach the planet's surface.

Length Contraction

Length contraction is another significant phenomenon predicted by special relativity. While time dilation leads to longer perceived lifetimes for particles moving at relativistic speeds, length contraction affects distances. When seen from the muon's perspective, the thickness of the Earth's atmosphere, originally 10 kilometers, contracts in the direction of motion due to their incredible speed.
The distance they measure with the formula:
\[ L = L_0 \sqrt{1-v^2/c^2} \],
becomes much shorter, approximately 447 meters. This shortened distance is easily navigable within their brief lifespan of 2.2 microseconds.

• This contraction explains another part of the puzzle: why muons can reach observers on the Earth's surface despite their short lifespan.

Cosmic Rays

Cosmic rays are high-energy particles that arrive at Earth from space. These energetic particles interact with the atoms in our atmosphere, particularly at high altitudes, triggering a chain reaction that produces various particles, including muons.
This process is a fascinating aspect of particle physics and atmospheric science:

• Cosmic rays consist of protons and atomic nuclei and originate from events like supernovae.
• Upon striking atmospheric particles, they generate secondary particles, including the rapidly moving muons.

Understanding how cosmic rays trigger these reactions helps explain the presence of muons at Earth's surface, as these cosmic interactions unlock a vivid dance of particles in our upper atmosphere.