Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 39: Problem 10

Consider a hypothetical force mediated by the exchange of bosons that have the same mass as protons. Approximately what would be the maximum range of such a force? You may assume that the total energy of these particles is simply the rest-mass energy and that they travel close to the speed of light. If you do not make these assumptions and instead use the relativistic expression for total energy, what happens to your estimate of the maximum range of the force?

Short Answer

Expert verified
Answer: The maximum range of the force is approximately 1.32 * 10^-15 m.

Step by step solution

01

Identifying given values

We are given the following information: - Mass of the boson (m_boson) is the same as the mass of a proton (m_proton). - The total energy of the particles is equal to their rest-mass energy. - The particles travel close to the speed of light.
02

Calculate rest-mass energy

First, we need to calculate the rest-mass energy of the boson (E_boson) using the mass-energy equivalence formula: E_boson = m_boson * c^2 where c is the speed of light, around 3 * 10^8 m/s. Since the mass of a proton (m_proton) is approximately 1.67 * 10^-27 kg, the rest-mass energy of a boson is: E_boson = m_proton * c^2 = (1.67 * 10^-27 kg) * (3 * 10^8 m/s)^2 E_boson ≈ 1.50 * 10^-10 J
03

Calculate the maximum range of the force

We can find the maximum range of the force (R_max) by dividing the rest-mass energy of the boson by Planck's constant (h) and the speed of light, which gives the wavelength of the boson. R_max = h * c / E_boson Planck's constant (h) is approximately 6.626 * 10^-34 Js. Using the calculated rest-mass energy, we can find the maximum range of the force: R_max = (6.626 * 10^-34 Js) * (3 * 10^8 m/s) / (1.50 * 10^-10 J) R_max ≈ 1.32 * 10^-15 m
04

Estimate the maximum range using relativistic expression

We can now estimate the maximum range when using the relativistic expression for total energy, which is given by: E_total = m_boson * c^2 / sqrt(1 - (v^2 / c^2)) As v approaches c, the denominator sqrt(1 - (v^2 / c^2)) approaches zero, which causes the total energy (E_total) to become infinite. Therefore, using the relativistic expression for total energy does not provide a meaningful estimate of the maximum range of the force, as it results in an infinite range.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Draw a Feynman diagram for an electron-proton scattering, \(e^{-}+p \rightarrow e^{-}+p\), mediated by photon exchange.

In a positron annihilation experiment, positrons are directed toward a material such as a metal. What are we likely to observe in such an experiment, and how might it provide information about the momentum of electrons in the metal?

Within three years after it begins operation, the proton beam at the Large Hadron Collider at CERN is expected to reach a luminosity of \(10^{34} \mathrm{~cm}^{-2} \mathrm{~s}^{-1}\) (this means that in a \(1-\mathrm{cm}^{2}\) area, \(10^{34}\) protons encounter each other every second). The cross section for collisions, which could lead to direct evidence of the Higgs boson, is approximately \(1 \mathrm{pb}\) (picobarn). [These numbers were obtained from "Introduction to LHC physics," by G. Polesello, Journal of Physics: Conference Series \(53(2006), 107-116 .]\) If the accelerator runs without interruption, approximately how many of these Higgs events can one expect in one year at the LHC?

A proton and neutron interact via the strong nuclear force. Their interaction is mediated by a particle called a meson, much like the interaction between charged particles is mediated by photons-the particles of the electromagnetic field. a) Perform a rough estimate of the mass of the meson from the uncertainty principle and the known dimensions of a nucleus \(\left(\sim 10^{-15} \mathrm{~m}\right)\). Assume the meson travels at relativistic speed. b) Use a line of reasoning similar to the one in part (a) to prove that the theoretically expected rest mass of the photon is zero. \(.\)

Which of the following particles does not have integer spins? a) photon b) \(\pi\) meson c) \(\omega\) meson d) \(\nu_{\mathrm{e}}\) lepton
See all solutions

Recommended explanations on Physics Textbooks

Turning Points in Physics

Read Explanation

Dynamics

Read Explanation

Fields in Physics

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Thermodynamics

Read Explanation

Physical Quantities And Units

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free