/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 The subatomic particles called m... [FREE SOLUTION] | 91Ó°ÊÓ

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The subatomic particles called muons behave exactly like electrons, except that a muon's mass is greater by a factor of \(206.77\). Muons are continually bombarding the Earth as part of the stream of particles from space known as cosmic rays. When a muon strikes an atom, it can displace one of its electrons. If the atom happens to be a hydrogen atom, then the muon takes up an orbit that is on the average \(206.77\) times closer to the proton than the orbit of the ejected electron. How many times greater is the electric force experienced by the muon than that previously felt by the electron?

Short Answer

Expert verified
The muon experiences approximately 42758.5 times greater electric force than the electron.

Step by step solution

01

Understanding the Relationship Between Force and Distance

The electric force between two charges is governed by Coulomb's law, given by the formula \( F = \frac{k \, |q_1 q_2|}{r^2} \), where \( F \) is the force, \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges. In this problem, the charge interaction is between the muon (or electron) and the proton.
02

Determine the Distance Ratio

We are told that the muon orbits at an average distance that is \( 206.77 \) times closer to the proton than the electron does. Therefore, the distance of the muon to the proton can be expressed as \( r_{\text{muon}} = \frac{r_{\text{electron}}}{206.77} \), where \( r_{\text{electron}} \) is the original orbit radius for the electron.
03

Compute the Force Ratio

Using Coulomb's law, the electric force for the electron is \( F_{\text{electron}} = \frac{k \, |e \cdot e|}{r_{\text{electron}}^2} \), and for the muon, it is \( F_{\text{muon}} = \frac{k \, |e \cdot e|}{(\frac{r_{\text{electron}}}{206.77})^2} \). The force ratio is then \( \frac{F_{\text{muon}}}{F_{\text{electron}}} = \left(\frac{r_{\text{electron}}}{\frac{r_{\text{electron}}}{206.77}}\right)^2 = (206.77)^2 \).
04

Calculate the Final Result

Calculating \( (206.77)^2 \) gives approximately \( 42758.5 \). Therefore, the electric force experienced by the muon is approximately \( 42758.5 \) times greater than that experienced by the electron.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Muon
Muons are intriguing particles that share many characteristics with electrons. However, a significant difference is their mass. A muon's mass is approximately 206.77 times greater than that of an electron. Despite this larger mass, muons act just like electrons in many ways. They are also part of what we call lepton family in subatomic particles. Despite being heavier, muons have a short-lived existence. They typically last only a few millionths of a second before decaying into other particles. This makes them fascinating yet elusive targets of study in particle physics. In cosmic ray experiments, muons are quite significant since they reach the Earth after being created in the upper atmosphere and can penetrate deep into materials due to their high energy.
Electric Force
The idea of electric force is essential in understanding interactions at an atomic and subatomic level. Coulomb's Law gives us the equation to calculate this force between two charged objects:
  • \( F = \frac{k \cdot |q_1 q_2|}{r^2} \)
Here, \( F \) represents the electric force, \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges. In the specific case of muons and electrons interacting with protons, when the distance between the charges is reduced, the force increases. Because a muon orbits much closer to the proton compared to an electron in a hydrogen atom, the electric force it experiences is significantly greater. As the orbit distance declines by a factor of 206.77, the force felt by the muon increases by about 42758.5 times.
Cosmic Rays
Cosmic rays have an awe-inspiring and mysterious origin, typically originating from outer space phenomena such as solar flares and distant supernovas. These rays are composed mainly of high-energy particles that constantly bombard the Earth. Within this mix of particles are muons, which are a byproduct of cosmic rays interacting with particles in Earth's atmosphere. As cosmic rays strike atoms high above us, they create showers of subatomic particles, including muons, that cascade towards the Earth's surface. These muons are highly penetrating and can reach areas even deep below the surface, carrying valuable information about their cosmic origins and the conditions they traveled through on their journey to Earth.
Subatomic Particles
Subatomic particles are the fundamental constituents of matter, with electrons, protons, and muons being some of the most well-known types.
Matter at its most basic level is composed of these tiny, invisible particles.
They interact with each other through fundamental forces, like the electromagnetic force described through Coulomb's Law.
Subatomic particles come in various flavors, including fermions and bosons.
Electrons and muons are classified as fermions; they are known as leptons, a category of particles with half-integer spin.
  • They do not participate in strong nuclear interactions, unlike protons and neutrons which are baryons.
The study of subatomic particles reveals much about the fundamental nature of our universe, explaining how different forces come into play to create the complex world we see today.

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