Chapter 16: Motion in Magnetic Fields

16.1 Introduction

In this chapter, we will explore the motion of charged particles in magnetic fields. Understanding the behavior of particles in magnetic fields is important for various applications, including particle accelerators, magnetic confinement fusion, and magnetic resonance imaging.

16.2 Motion of Charged Particles in a Uniform Magnetic Field

When a charged particle enters a uniform magnetic field perpendicular to its velocity, it undergoes circular motion due to the magnetic force acting on it. The radius of the circular path is given by:

where is the radius, is the mass of the particle, is its velocity, is its charge, and is the magnetic field strength. The time it takes for the particle to complete one full revolution is called the cyclotron period, which is given by:

16.3 Helical Motion

If a charged particle enters a magnetic field at an angle to the field lines, its motion is a combination of circular motion perpendicular to the field and linear motion parallel to the field. This results in a helical trajectory. The pitch of the helix (distance between consecutive loops) is given by:

where is the component of the particle’s velocity parallel to the magnetic field and is the cyclotron period.

16.4 Cyclotron and Synchrotron Accelerators

Cyclotron and synchrotron accelerators are devices used to accelerate charged particles to high energies using magnetic and electric fields. In a cyclotron, particles are accelerated between two D-shaped electrodes in a uniform magnetic field, causing them to move in circular paths of increasing radius. In a synchrotron, particles are accelerated in a large circular ring with varying magnetic fields that keep them moving in a constant-radius path.

16.5 Magnetic Confinement Fusion

Magnetic confinement fusion is a method to achieve nuclear fusion by confining a plasma of charged particles within a magnetic field. The magnetic field prevents charged particles from escaping, allowing the plasma to reach the high temperatures and pressures required for fusion to occur. Tokamak and stellarator devices are examples of magnetic confinement fusion machines.

16.6 Magnetic Resonance Imaging (MRI)

Magnetic resonance imaging (MRI) is a non-invasive imaging technique that utilizes a strong magnetic field and radio waves to generate images of the body’s internal structures. In MRI, the magnetic field causes hydrogen nuclei in the body to align with the field. Radio waves are then applied to flip the nuclei’s alignment, and the subsequent relaxation process produces a detectable signal used to create detailed images.

Chapter Summary

In this chapter, we covered the motion of charged particles in magnetic fields, the principles of cyclotron and synchrotron accelerators, magnetic confinement fusion, and magnetic resonance imaging. Understanding the behavior of charged particles in magnetic fields has significant implications for a wide range of applications in physics and technology.

Continue to Chapter 17: The Direct-Current Motor

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