A charged particle experiences a force when moving through a magnetic field. What happens if this field is uniform over the motion of the charged particle? What path does the particle follow? In this section, we discuss the circular motion of the charged particle as well as other motion that results from a charged particle entering a magnetic field.

The simplest case occurs when a charged particle moves perpendicular to a uniformB-field ((Figure)). If the field is in a vacuum, the magnetic field is the dominant factor determining the motion. Since the magnetic force is perpendicular to the direction of travel, a charged particle follows a curved path in a magnetic field. The particle continues to follow this curved path until it forms a complete circle. Another way to look at this is that the 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.A negatively charged particle moves in the plane of the paper in a region where the magnetic field is perpendicular to the paper (represented by the small [×] ’s—like the tails of arrows). The magnetic force is perpendicular to the velocity, so velocity changes in direction but not magnitude. The result is uniform circular motion. (Note that because the charge is negative, the force is opposite in direction to the prediction of the right-hand rule.)[An illustration of the motion of a charged particle in a uniform magnetic field. The magnetic field points into the page. The particle is negative and moves in a clockwise circle. Its velocity is tangent to the circle, and the force points toward the center of the circle at all times.]

In this situation, the magnetic force supplies the centripetal force [{F}_{\text{c}}=\frac{m{v}^{2}}{r}.] Noting that the velocity is perpendicular to the magnetic field, the magnitude of the magnetic force is reduced to [F=qvB.] Because the magnetic forceFsupplies the centripetal force [{F}_{c},] we have[qvB=\frac{m{v}^{2}}{r}.]Solving forryields[r=\frac{mv}{qB}.]Here,ris the radius of curvature of the path of a charged particle with massmand chargeq, moving at a speedvthat is perpendicular to a magnetic field of strengthB. The time for the charged particle to go around the circular path is defined as the period, which is the same as the distance traveled (the circumference) divided by the speed. Based on this and(Figure), we can derive the period of motion as[T=\frac{2\pi r}{v}=\frac{2\pi }{v}\phantom{\rule{0.2em}{0ex}}\frac{mv}{qB}=\frac{2\pi m}{qB}.]If the velocity is not perpendicular to the magnetic field, then we can compare each component of the velocity separately with the magnetic field. The component of the velocity perpendicular to the magnetic field produces a magnetic force perpendicular to both this velocity and the field:[{v}_{\text{perp}}=v\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}\theta ,\phantom{\rule{0.5em}{0ex}}{v}_{\text{para}}=v\text{cos}\phantom{\rule{0.1em}{0ex}}\theta .]where [\theta] is the angle betweenvandB. The component parallel to the magnetic field creates constant motion along the same direction as the magnetic field, also shown in(Figure). The parallel motion determines thepitchpof the helix, which is the distance between adjacent turns. This distance equals the parallel component of the velocity times the period:[p={v}_{\text{para}}T.]The result is ahelical motion, as shown in the following figure.A charged particle moving with a velocity not in the same direction as the magnetic field. The velocity component perpendicular to the magnetic field creates circular motion, whereas the component of the velocity parallel to the field moves the particle along a straight line. The pitch is the horizontal distance between two consecutive circles. The resulting motion is helical.[An illustration of a positively charged particle moving in a uniform magnetic field. The field is in the positive x direction. The initial velocity is shown as having a component, v sub para, in the positive x direction and another component, v sub perp, in the positive y direction. The particle moves in a helix that loops in the y z plane (counterclockwise from the particle’s perspective) and advances in the positive x direction.]

While the charged particle travels in a helical path, it may enter a region where the magnetic field is not uniform. In particular, suppose a particle travels from a region of strong magnetic field to a region of weaker field, then back to a region of stronger field. The particle may reflect back before entering the stronger magnetic field region. This is similar to a wave on a string traveling from a very light, thin string to a hard wall and reflecting backward. If the reflection happens at both ends, the particle is trapped in a so-called magnetic bottle.

Trapped particles in magnetic fields are found in theVan Allen radiation beltsaround Earth, which are part of Earth’s magnetic field. These belts were discovered by James Van Allen while trying to measure the flux ofcosmic rayson Earth (high-energy particles that come from outside the solar system) to see whether this was similar to the flux measured on Earth. Van Allen found that due to the contribution of particles trapped in Earth’s magnetic field, the flux was much higher on Earth than in outer space.Aurorae, like the famous aurora borealis (northern lights) in the Northern Hemisphere ((Figure)), are beautiful displays of light emitted as ions recombine with electrons entering the atmosphere as they spiral along magnetic field lines. (The ions are primarily oxygen and nitrogen atoms that are initially ionized by collisions with energetic particles in Earth’s atmosphere.) Aurorae have also been observed on other planets, such as Jupiter and Saturn.(a) The Van Allen radiation belts around Earth trap ions produced by cosmic rays striking Earth’s atmosphere. (b) The magnificent spectacle of the aurora borealis, or northern lights, glows in the northern sky above Bear Lake near Eielson Air Force Base, Alaska. Shaped by Earth’s magnetic field, this light is produced by glowing molecules and ions of oxygen and nitrogen.