When one puts a note on the refrigerator door with a magnet, one gets the impression that the attractive force is in the direction of the magnetic field. On the other hand, we said that the magnetic force is perpendicular to the field lines. Before we resolve this apparent contradiction, let’s see what the magnetic force on a particle (an ion or electron) is supposed to be. The force is called the Lorentz force,5 and it has five main features. (1) It acts only on particles with an electric charge. (2) It is propor­tional to the strength of the magnetic field, as one would expect. (3) It does not affect a particle that is stationary nor one that moves only along a field line. Only the perpendicular motion of a particle — that which takes it from one field line to an adjacent one — counts. (4) The force is perpendicular to both the particle velocity and the field line. (5) The force depends on the electric charge on the particle and is in opposite directions for positive and negative charges. This is a mouthful, but here is what it means. If a proton, say, is stationary, it feels no force. If the proton moves strictly along a field line, it also feels no force. If it moves across field lines, however, the magnetic field will push it, not backwards, but in a perpendicular direction. An ion and an electron both have the same charge, but of opposite signs, so the Lorentz force on them is in opposite directions. As we shall see, this will cause the protons and electrons to revolve in small circles around a field line. Refrigerator magnets seem to pull along the field, though. This is because permanent magnets are more complicated.6

A cartoon of the orbits of an ion and an electron in a magnetic field is shown in Fig. 4.9. The X in the center indicates that the magnetic field, labeled B, points into the page. The arrows indicate the Lorentz force, which is everywhere perpendicular to both the particle velocity and the magnetic field. If the velocity is constant, the force is inward everywhere with the same strength, so the orbits are circles. Note that the motions are in opposite directions because the charges have opposite signs. Imagine taking a yo-yo, stretching it out, and swinging it with a steady motion in a circle over your head. The string pulls the yo-yo inward with the same force at all times, so the yo-yo moves in a circle. Here, the magnetic field applies a force just like that of the string. This gyration orbit is called a cyclotron orbit, since the first cyclo­trons used this principle to keep the protons inside a circular chamber. It is also called a Larmor orbit because in science you can get something named after you without paying a huge endowment. The radius of the circle is called its Larmor radius.

Since the magnetic force is always perpendicular to the field’s direction, particles move in the parallel direction without being influenced by the magnetic field. A magnetized plasma, then, doesn’t look like Fig. 4.4, where ions and electrons are free to move in any direction. Instead, it would look like Fig. 4.10, where the charged particles gyrate in their Larmor orbits and move unimpeded in the direction of the magnetic field B. Field lines are like invisible railroad tracks that guide the motion of charged particles.

How big is a Larmor orbit? In a cyclotron, the orbit is the size of a large labora­tory because the protons have very large energies. In a fusion reactor, a deuteron has a Larmor radius of about 1 cm, when compared with a plasma radius of about a meter. An electron’s orbit is much smaller than a deuteron’s, even if it has the same energy. This is the result of two effects. With the same energy, an electron would move much faster because it is much less massive than a deuteron. So you would think that its orbit would be larger than a deuteron’s. However, the Lorentz force that curves the orbit is stronger with higher velocity. The upshot is

that the electron’s orbit is smaller by the square root of the mass ratio, or about 60 in this case. In Fig. 4.10, the electron orbit was greatly enlarged in order to be visible.

Since these gyration orbits are so much smaller than the plasma that they are immersed in, we don’t have to track the particle motions in such detail. We only have to track the motion of the centers of the circles, which are called guiding centers. In the future, when we talk about the motion of plasma particles, we will mean the motion of the guiding centers.

We can now return to the question, “How can a magnetic field hold a plasma?” We have seen that a magnetic field does not apply a force to a particle that will stop it from following field lines, so field lines that end on a boundary somewhere cannot prevent a plasma from hitting a wall.7 On the other hand, plasma cannot go across field lines because the magnetic force simply keeps charged particles spinning in small Larmor orbits around the same field line. Obviously, the solution is to make a field with lines that close on themselves and do not end. That’s the very first step in designing a magnetic bottle!