<< Chapter < Page Chapter >> Page >

Learning objectives

By the end of this section, you will be able to:

  • Describe the effects of a magnetic field on a moving charge.
  • Calculate the radius of curvature of the path of a charge that is moving in a magnetic field.

The information presented in this section supports the following AP® learning objectives and science practices:

  • 3.C.3.1 The student is able to use right-hand rules to analyze a situation involving a current-carrying conductor and a moving electrically charged object to determine the direction of the magnetic force exerted on the charged object due to the magnetic field created by the current-carrying conductor. (S.P. 1.4)

Magnetic force can cause a charged particle to move in a circular or spiral path. Cosmic rays are energetic charged particles in outer space, some of which approach the Earth. They can be forced into spiral paths by the Earth’s magnetic field. Protons in giant accelerators are kept in a circular path by magnetic force. The bubble chamber photograph in [link] shows charged particles moving in such curved paths. The curved paths of charged particles in magnetic fields are the basis of a number of phenomena and can even be used analytically, such as in a mass spectrometer.

A drawing representing trails of bubbles in a bubble chamber.
Trails of bubbles are produced by high-energy charged particles moving through the superheated liquid hydrogen in this artist’s rendition of a bubble chamber. There is a strong magnetic field perpendicular to the page that causes the curved paths of the particles. The radius of the path can be used to find the mass, charge, and energy of the particle.

So does the magnetic force cause circular motion? Magnetic force is always perpendicular to velocity, so that it does no work on the charged particle. The particle’s kinetic energy and speed thus remain constant. The direction of motion is affected, but not the speed. This is typical of uniform circular motion. The simplest case occurs when a charged particle moves perpendicular to a uniform B size 12{B} {} -field, such as shown in [link] . (If this takes place in a vacuum, the magnetic field is the dominant factor determining the motion.) Here, the magnetic force supplies the centripetal force F c = mv 2 / r size 12{F rSub { size 8{c} } = ital "mv" rSup { size 8{2} } /r} {} . Noting that sin θ = 1 size 12{"sin"θ=1} {} , we see that F = qvB size 12{F= ital "qvB"} {} .

Diagram showing an electrical charge moving clockwise in the plane of the page. Velocity vectors are tangent to the circular path. The magnetic field B is oriented into the page. Force vectors show that the force on the charge is toward the center of the charge’s circular path as the charge moves.
A negatively charged particle moves in the plane of the page in a region where the magnetic field is perpendicular into the page (represented by the small circles with x’s—like the tails of arrows). The magnetic force is perpendicular to the velocity, and so velocity changes in direction but not magnitude. Uniform circular motion results.

Because the magnetic force F size 12{F} {} supplies the centripetal force F c size 12{F rSub { size 8{c} } } {} , we have

qvB = mv 2 r . size 12{ ital "qvB"= { { ital "mv" rSup { size 8{2} } } over {r} } "." } {}

Solving for r size 12{r} {} yields

r = mv qB . size 12{r= { { ital "mv"} over { ital "qB"} } "." } {}

Here, r size 12{r} {} is the radius of curvature of the path of a charged particle with mass m size 12{m} {} and charge q size 12{q} {} , moving at a speed v size 12{v} {} perpendicular to a magnetic field of strength B size 12{B} {} . If the velocity is not perpendicular to the magnetic field, then v size 12{v} {} is the component of the velocity perpendicular to the field. The component of the velocity parallel to the field is unaffected, since the magnetic force is zero for motion parallel to the field. This produces a spiral motion rather than a circular one.

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics for ap® courses' conversation and receive update notifications?

Ask