The pitch is given by Equation 11.8, the period is given by Equation 11.6, and the radius of circular motion is given by Equation 11.5. After setting the radius and the pitch equal to each other, solve for the angle between the magnetic field and velocity or \theta. The pitch of the motion relates to the parallel velocity times the period of the circular motion, whereas the radius relates to the perpendicular velocity component. (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.Įxample Helical Motion in a Magnetic FieldĪ proton enters a uniform magnetic field of 1.0\phantom At what angle must the magnetic field be from the velocity so that the pitch of the resulting helical motion is equal to the radius of the helix? Strategy Aurorae, like the famous aurora borealis (northern lights) in the Northern Hemisphere ( Figure 11.9), are beautiful displays of light emitted as ions recombine with electrons entering the atmosphere as they spiral along magnetic field lines. 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. These belts were discovered by James Van Allen while trying to measure the flux of cosmic rays on Earth (high-energy particles that come from outside the solar system) to see whether this was similar to the flux measured on Earth. Trapped particles in magnetic fields are found in the Van Allen radiation belts around Earth, which are part of Earth’s magnetic field. If the reflection happens at both ends, the particle is trapped in a so-called magnetic bottle. This is similar to a wave on a string traveling from a very light, thin string to a hard wall and reflecting backward. The particle may reflect back before entering the stronger magnetic field region. 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. While the charged particle travels in a helical path, it may enter a region where the magnetic field is not uniform. The pitch is the horizontal distance between two consecutive circles. 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 direction of motion is affected but not the speed.įigure 11.8 A charged particle moving with a velocity not in the same direction as the magnetic field. The particle’s kinetic energy and speed thus remain constant. 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 continues to follow this curved path until it forms a complete circle. Since the magnetic force is perpendicular to the direction of travel, a charged particle follows a curved path in a magnetic field. If the field is in a vacuum, the magnetic field is the dominant factor determining the motion. The simplest case occurs when a charged particle moves perpendicular to a uniform B-field ( Figure 11.7). 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. Describe how to determine the radius of the circular motion of a charged particle in a magnetic fieldĪ charged particle experiences a force when moving through a magnetic field.Explain how a charged particle in an external magnetic field undergoes circular motion.By the end of this section, you will be able to:
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