# 11.7 Applications of magnetic forces and fields  (Page 3/11)

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Particles are accelerated to very high energies with either linear accelerators or synchrotrons. The linear accelerator accelerates particles continuously with the electric field of an electromagnetic wave that travels down a long evacuated tube. The Stanford Linear Accelerator (SLAC) is about 3.3 km long and accelerates electrons and positrons (positively charged electrons) to energies of 50 GeV. The synchrotron is constructed so that its bending magnetic field increases with particle speed in such a way that the particles stay in an orbit of fixed radius. The world’s highest-energy synchrotron is located at CERN, which is on the Swiss-French border near Geneva. CERN has been of recent interest with the verified discovery of the Higgs Boson (see Particle Physics and Cosmology ). This synchrotron can accelerate beams of approximately ${10}^{13}$ protons to energies of about ${10}^{3}$ GeV.

## Accelerating alpha-particles in a cyclotron

A cyclotron used to accelerate alpha-particles ( $m=6.64\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-27}\text{kg},q=3.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-19}\text{C}$ ) has a radius of 0.50 m and a magnetic field of 1.8 T. (a) What is the period of revolution of the alpha-particles? (b) What is their maximum kinetic energy?

## Strategy

1. The period of revolution is approximately the distance traveled in a circle divided by the speed. Identifying that the magnetic force applied is the centripetal force, we can derive the period formula.
2. The kinetic energy can be found from the maximum speed of the beam, corresponding to the maximum radius within the cyclotron.

## Solution

1. By identifying the mass, charge, and magnetic field in the problem, we can calculate the period:
$T=\frac{2\pi m}{qB}=\frac{2\pi \left(6.64\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-27}\text{kg}\right)}{\left(3.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-19}\text{C}\right)\left(1.8\text{T}\right)}=7.3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-8}\text{s.}$
2. By identifying the charge, magnetic field, radius of path, and the mass, we can calculate the maximum kinetic energy:
$\frac{1}{2}m{v}_{\text{max}}{}^{2}=\frac{{q}^{2}{B}^{2}{R}^{2}}{2m}=\frac{{\left(3.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-19}\text{C}\right)}^{2}{\left(1.8\text{T}\right)}^{2}{\left(0.50\text{m}\right)}^{2}}{2\left(6.65\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-27}\text{kg}\right)}=6.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-12}\text{J}=39\text{MeV.}$

Check Your Understanding A cyclotron is to be designed to accelerate protons to kinetic energies of 20 MeV using a magnetic field of 2.0 T. What is the required radius of the cyclotron?

0.32 m

## Summary

• A mass spectrometer is a device that separates ions according to their charge-to-mass ratios by first sending them through a velocity selector, then a uniform magnetic field.
• Cyclotrons are used to accelerate charged particles to large kinetic energies through applied electric and magnetic fields.

## Key equations

 Force on a charge in a magnetic field $\stackrel{\to }{F}=q\stackrel{\to }{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{B}$ Magnitude of magnetic force $F=qvB\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}\theta$ Radius of a particle’s path in a magnetic field $r=\frac{mv}{qB}$ Period of a particle’s motion in a magnetic field $T=\frac{2\pi m}{qB}$ Force on a current-carrying wire in a uniform magnetic field $\stackrel{\to }{F}=I\stackrel{\to }{l}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{B}$ Magnetic dipole moment $\stackrel{\to }{\mu }=NIA\stackrel{^}{n}$ Torque on a current loop $\stackrel{\to }{\tau }=\stackrel{\to }{\mu }\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{B}$ Energy of a magnetic dipole $U=\text{−}\stackrel{\to }{\mu }·\stackrel{\to }{B}$ Drift velocity in crossed electric and magnetic fields ${v}_{d}=\frac{E}{B}$ Hall potential $V=\frac{IBl}{neA}$ Hall potential in terms of drift velocity $V=Bl{v}_{d}$ Charge-to-mass ratio in a mass spectrometer $\frac{q}{m}=\frac{E}{B{B}_{0}R}$ Maximum speed of a particle in a cyclotron ${v}_{\text{max}}=\frac{qBR}{m}$

## Conceptual questions

Describe the primary function of the electric field and the magnetic field in a cyclotron.

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