Looking at a globe, we see lines that do not end. The latitude lines go in circles around the earth and end on themselves (Fig. 4.11). The longitude lines go north and south until they reach the poles, where they continue over to the other side of the earth (Fig. 4.12). Why can’t we make a magnetic bottle shaped like a sphere with magnetic field lines that go either north-south or east-west? Here’s why. If we look down at the north pole, say, in Fig. 4.11, we see that the field lines go around in smaller and smaller circles. As one gets closer to the pole, the magnetic field must get weaker and weaker, since the fields on opposite sides of the circle are in opposite directions and tend to cancel each other. Exactly at the pole, the field must be zero, since it cannot be in two directions at the same time. This is called an

O-type null. The plasma will leak out at the poles, since there is no magnetic field there to confine it. If we now look at a configuration in which the field lines are like longitude lines, Fig. 4.12 shows that the field lines point toward (or away from) one another at the poles, or cross one another at an angle. Again, the field at the pole must be zero, since it cannot be in two directions at once. This is called an X-type null. A simple shape that is topologically equivalent to a sphere cannot be made into a magnetic bottle. It will have a big leak at the poles, where there is no mag­netic field to hold the plasma.

The simplest shape that will work is a torus, a three-dimensional volume like a tire or a doughnut, with a hole in it, as shown in Fig. 4.13. Mathematicians would call it a doubly connected space. Field lines that have no ends can be imbedded in such a chamber in such a way that ions and electrons cannot find a way out by moving along the field lines. Such closed field lines are of two types. Toroidal field lines, of which one is shown in Fig. 4.13a, go around the torus in the long way, encircling the hole. Poloidal field lines, shown in Fig. 4.13b, go around the short way and do not encircle the hole. Remember that field lines are just a graphic way to show the direction of the magnetic field. There is an infinite number of field lines. The torus is entirely filled with magnetic field, so that plasma placed inside will not, in principle, escape. The ion and electron guiding centers simply move along the field lines and never hit the wall, as long as the field lines they’re on do not wander out of the torus.

Now imagine combining toroidal and poloidal fields into the same torus. A toroidal field line going around the long way will also bend the short way, like an old-fashioned barber pole or the stripes on a candy cane. The field line will look like a Slinky® toy stretched around a lamppost; it is a helix bent into a circle. The generic toroidal and poloidal types of magnetic field will not work. Combining them into a helix is the beginning of the art of making magnetic bottles. All this is necessary because magnetic fields do not stop particles from moving longitudinally, and therefore they must not end on a material wall.