Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 30: Problem 13

Estimate the magnetic field strength required at the LHC to make 7.0-TeV protons travel in a circle of circumference \(27 \mathrm{km}\). Start by deriving an expression, using Newton's second law, for the field strength \(B\) in terms of the particle's momentum \(p,\) its charge \(q,\) and the radius \(r\). Even though derived using classical physics, the expression is relativistically correct. (The estimate will come out much lower than the actual value of 8.33 T. In the LHC, the protons do not travel in a constant magnetic field; they move in straight-line segments between magnets.)

Short Answer

Expert verified
Question: Estimate the magnetic field strength required at the Large Hadron Collider (LHC) for a 7.0-TeV proton traveling in a circle of circumference 27 km. Answer: The magnetic field strength required at the LHC for a 7.0-TeV proton is approximately ____ T (tesla).

Step by step solution

01

Determine the centripetal force acting on the proton

The centripetal force (\(F_c\)) acting on the proton is given by the formula: \(F_c = \frac{mv^2}{r}\), where \(m\) is the mass of the proton, \(v\) is its velocity, and \(r\) is the radius of the circle.
02

Determine the magnetic force acting on the proton

The magnetic force (\(F_B\)) acting on a charged particle moving at a velocity \(v\) in a magnetic field \(B\) is given by the formula: \(F_B = qvB\), where \(q\) is the charge of the particle.
03

Apply Newton's second law

According to Newton's second law, the net force acting on the proton is equal to the centripetal force. Therefore, \(F_B = F_c\), which gives us: \(qvB = \frac{mv^2}{r}\).
04

Derive an expression for the magnetic field strength (B)

From the previous step, we can derive an expression for \(B\) as follows: $$B = \frac{mv}{qr}$$
05

Convert the energy of the proton into momentum

The energy of the proton is given in TeV (teraelectronvolts). First, we need to convert it into electronvolts (eV) by multiplying it by \(10^ {12}\). Then, we find the momentum using the relation: \(E^2 = (pc)^2 + (mc^2)^2\), where \(E\) is the energy, \(p\) is the momentum, and \(c\) is the speed of light. Solve for \(p\) to get: \(p = \frac{\sqrt{E^2 - (mc^2)^2}}{c}\).
06

Estimate the magnetic field strength at the LHC

Using the expression derived in Step 4 and the momentum calculated in Step 5, we can now estimate the magnetic field strength at the LHC: $$B = \frac{mv}{qr} \approx \frac{(\frac{\sqrt{E^2 - (mc^2)^2}}{c})v}{q(\frac{27\mathrm{km}}{2\pi})}$$Plug in the values for charge of a proton (\(q = 1.6 \times 10^{-19}\mathrm{C}\)), mass of the proton (\(m = 1.67 \times 10^{-27}\mathrm{kg}\)), speed of light (\(c\)), and energy (given as 7 TeV). Please note that the protons are not traveling at the speed of light but in this estimation it is adequate to make this approximation to get an idea for the strength required. Then, calculate the magnetic field strength.

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

A proton in Fermilab's Tevatron is accelerated through a potential difference of 2.5 MV during each revolution around the ring of radius \(1.0 \mathrm{km} .\) In order to reach an energy of 1 TeV, how many revolutions must the proton make? How far has it traveled?
When a proton and an antiproton annihilate, the annihilation products are usually pions. (a) Suppose three pions are produced. What combination(s) of \(\pi^{+}, \pi^{-},\) and \(\pi^{0}\) are possible? (b) Suppose five pions are produced. What combination(s) of \(\pi^{+}, \pi^{-},\) and \(\pi^{0}\) are possible? (c) What is the maximum number of pions that could be produced if the kinetic energies of the proton and antiproton are negligibly small? The mass of a charged pion is \(0.140 \mathrm{GeV} / c^{2}\) and the mass of a neutral pion is \(0.135 \mathrm{GeV} / c^{2}.\)
The energy at which the fundamental forces are expected to unify is about \(10^{19}\) GeV. Find the mass (in kilograms) of a particle with rest energy \(10^{19} \mathrm{GeV}\).
Which fundamental force is responsible for each of the decays shown here? [Hint: In each case, one of the decay products reveals the interaction force.] (a) \(\pi^{+} \rightarrow\) \(\mu^{+}+v_{\mu},\) (b) $\pi^{0} \rightarrow \gamma+\gamma(\mathrm{c}) \mathrm{n} \rightarrow \mathrm{p}^{+}+\mathrm{e}^{-}+\bar{v}_{\mathrm{e}}$
In the Cornell Electron Storage Ring, electrons and positrons circulate in opposite directions with kinetic energies of 6.0 GeV each. When an electron collides with a positron and the two annihilate, one possible (though unlikely) outcome is the production of one or more proton-antiproton pairs. What is the maximum possible number of proton-antiproton pairs that could be formed?
See all solutions

Recommended explanations on Physics Textbooks

Quantum Physics

Read Explanation

Dynamics

Read Explanation

Electricity and Magnetism

Read Explanation

Nuclear Physics

Read Explanation

Translational Dynamics

Read Explanation

Electricity

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