Motion in crossed electric and magnetic fields
So far we have talked about particles in electric fields only or in magnetic fields only. There are some interesting effects when there are both kinds of fields at the same time. Suppose we have a uniform magnetic field $\FLPB$ and an electric field $\FLPE$ at right angles. Particles that start out perpendicular to $\FLPB$ will move in a curve like the one in Fig. 29–20. (The figure is a plane curve, not a helix!) We can understand this motion qualitatively. When the particle (assumed positive) moves in the direction of $\FLPE$, it picks up speed, and so it is bent less by the magnetic field. When it is going against the $\FLPE$-field, it loses speed and is continually bent more by the magnetic field. The net effect is that it has an average “drift” in the direction of $\FLPE\times\FLPB$. We can, in fact, show that the motion is a uniform circular motion superimposed on a uniform sidewise motion at the speed $v_d=E/B$—the trajectory in Fig. 29–20 is a cycloid. Imagine an observer who is moving to the right at a constant speed. In his frame our magnetic field gets transformed to a new magnetic field plus an electric field in the downward direction. If he has just the right speed, his total electric field will be zero, and he will see the electron going in a circle. So the motion we see is a circular motion, plus a translation at the drift speed $v_d=E/B$. The motion of electrons in crossed electric and magnetic fields is the basis of the magnetron tubes, i.e., oscillators used for generating microwave energy. There are many other interesting examples of particle motions in electric and magnetic fields—such as the orbits of the electrons and protons trapped in the Van Allen belts—but we do not, unfortunately, have the time to deal with them here.