Motion of Charged Particles in a Magnetic Field
When a charged particle moves in a magnetic field, it is acted on by the magnetic force given by equation F = qv x B, and the motion is determined by Newton’s law. Figure below shows a simple example.
A particle with positive charge is at point O, moving with velocity in a uniform magnetic field directed into the plane of the figure. The vectors v and B are perpendicular, so the magnetic force F=qv x B has magnitude F=qvB and a direction as shown in the figure. The force is always perpendicular to v, so it cannot change the magnitude of the velocity, only its direction. To put it differently, the magnetic force never has a component parallel to the particle’s motion, so the magnetic force can never do work on the particle. This is even true if the magnetic field is not uniform. Motion of a charged particle under the action of a magnetic field alone is always motion with constant speed.
Using this principle, we see that in the situation shown in figure above the magnitudes of both F and v are constant. At point such as P and S the directions of force and velocity have changed as shown, but the magnitudes are the same. The particle therefore moves under the influence of a constant-magnitude force that is always at right angles to the velocity of the particle. Comparing these conditions with the circular motion, we can see that the particle’s path is a circle, traced out with constant speed. The centripetal acceleration is v2/R, and the only force acting is the magnetic force, so from Newton’s law, F = │q│vB = m v2 R where m is the mass of the particle. Solving for the radius of the circular path, p ath, we find R=mω │q│B
(radius of a circular orbit in a magnetic field)
We can also write this as R=p/q!B, where p=mv is the magnitude of the particle’s momentum. If the charge is negative, the particle moves clockwise around the orbit.
To get the angular velocity, w = v/R = v │q│B/mv = │q│B/m │q│B/m The number of revolution per unit time is f=w/2π. This frequency f is the independent of radius of the path. It is called the cyclotron frequency; in a particle accelerator called a cyclotron, particles moving in nearly circular paths are given a boost twice each revolution. Similarly, a magnetron, a common source or microwave radiation for microwave ovens and radar systems, emits radiation with a frequency equal to the frequency of circular motion of electrons in a vacuum chamber between the poles of a magnet. If the direction of the initial velocity is not perpendicular to the field, the velocity component parallel to the field is constant because there is no force parallel to the field. Then the particle moves in a helix. The radius of the helix is given using (radius of a circular orbit in a magnetic field), where now v is the component of velocity perpendicular to the B field. Motion of a charged particle in a non-uniform magnetic field is more complex. If a field is produced by two circular coils separated by some distance, particles near either coil experience a magnetic force toward the center of the region; particles with appropriate speeds spiral repeatedly from one end of the region to the other and back. Because charged particles can be trapped in such a magnetic field, it is called a magnetic bottle. This technique is used to confine very hot plasmas with temperatures of the order of 106 K. In a similar way the earth’s non-uniform magnetic field traps charged particles coming from the sun in doughnut-shaped regions around the earth. These regions, called the Van Allen radiation belts, were discovered in 1958 using data instruments aboard the Explorer 1 satellite. Example problem:
Electron motion in a microwave oven. A magnetron in a microwave oven emits electromagnetic waves with frequency f =2450 MHz. What magnetic field strength is
required for electrons to move in circular paths with this frequency?
Solution: The corresponding angular velocity is ω=2π f =(2π)(2450x10 =(2π)(2450x106s-1) = 1.54 x 1010 s-1. B = mω/q = (9.11 x 10-31 kg)(1.54 x 1010 s-1)
= 0.0877 T.
1.60 x 10-19 C
