Motion of a charged particle in a uniform magnetic field
We know that the magnetic force acting on a charged particle moving in a magnetic field is perpendicular to the velocity of the particle and that consequently the work done on the particle by the magnetic force is zero. Let us now consider the special case of a positively charged particle moving in a uniform magnetic field with the initial velocity vector of the particle perpendicular to the field. Let us assume that the direction of the magnetic field is into the page. Figure 29.17 shows that the particle moves in a circle in a plane perpendicular to the magnetic field.
The particle moves in this way because the magnetic force FB is at right angles to v and B and has a constant magnitude qvB. As the force deflects the particle, the directions of v and FB change continuously, as Figure 29.17 shows. Because FB always points toward the center of the circle, it changes only the direction of v and not its magnitude. As Figure 29.17 illustrates, the rotation is counter-clockwise for a positive charge. If q were negative, the rotation would be clockwise. We can use Equation 6.1 to equate this magnetic force to the radial force required to keep the charge moving in a circle:
That is, the radius of the path is proportional to the linear momentum mv of the particle and inversely proportional to the magnitude of the charge on the particle and to the magnitude of the magnetic field. The angular speed of the particle (from Eq. 10.10) is
The period of the motion (the time that the particle takes to complete one revolution) is equal to the circumference of the circle divided by the linear speed of the particle:
These results show that the angular speed of the particle and the period of the circular motion do not depend on the linear speed of the particle or on the radius of the orbit. The angular speed ω is often referred to as the cyclotron frequency because charged particles circulate at this angular speed in the type of accelerator called a cyclotron.
If a charged particle moves in a uniform magnetic field with its velocity at some arbitrary angle with respect to B, its path is a helix. For example, if the field is directed in the x direction, as shown in Figure 29.18, there is no component of force in the x direction. As a result, ax = 0, and the x component of velocity remains constant. However, the magnetic force qv x B causes the components νy and νz to change in time, and the resulting motion is a helix whose axis is parallel to the magnetic field. The projection of the path onto the yz plane (viewed along the x axis) is a circle. (The projections of the path onto the xy and xz planes are sinusoids!) Equations 29.13 to 29.15 still apply provided that ν is replaced by ν⊥ = √(νy2 + νz2).
When charged particles move in a non-uniform magnetic field, the motion is complex. For example, in a magnetic field that is strong at the ends and weak in the middle, such as that shown in Figure 29.20, the particles can oscillate back and forth between the end points. A charged particle starting at one end spirals along the field lines until it reaches the other end, where it reverses its path and spirals back. This configuration is known as a magnetic bottle because charged particles can be trapped within it. The magnetic bottle has been used to confine a plasma, a gas consisting of ions and electrons. Such a plasma-confinement scheme could fulfill a crucial role in the control of nuclear fusion, a process that could supply us with an almost endless source of energy. Unfortunately, the magnetic bottle has its problems. If a large number of particles are trapped, collisions between them cause the particles to eventually leak from the system.
The Van Allen radiation belts consist of charged particles (mostly electrons and protons) surrounding the Earth in doughnut-shaped regions (Fig. 29.21). The particles, trapped by the Earth’s nonuniform magnetic field, spiral around the field lines from pole to pole, covering the distance in just a few seconds. These particles originate mainly from the Sun, but some come from stars and other heavenly objects. For this reason, the particles are called cosmic rays. Most cosmic rays are deflected by the Earth’s magnetic field and never reach the atmosphere. However, some of the particles become trapped; it is these particles that make up the Van Allen belts. When the particles are located over the poles, they sometimes collide with atoms in the atmosphere, causing the atoms to emit visible light. Such collisions are the origin of the beautiful Aurora Borealis, or Northern Lights, in the northern hemisphere and the Aurora Australis in the southern hemisphere.
Auroras are usually confined to the polar regions because it is here that the Van Allen belts are nearest the Earth’s surface. Occasionally, though, solar activity causes larger numbers of charged particles to enter the belts and significantly distort the normal magnetic field lines associated with the Earth. In these situations an aurora can sometimes be seen at lower latitudes.