If we had straight field lines in a cylinder, there would be no problem; but when we bend the cylinder into a torus so that the field lines do not strike a wall, the first of several toroidal effects comes into play. In Fig. 4.8b, we saw that an electromagnet generates a magnetic field by driving electric current in a coiled wire. The field lines are then formed inside the wire coil. When we bend the cylinder into a torus, as in Fig. 4.13a, the wire coil will also have to be bent to surround the torus, resulting in the configuration shown in Fig. 4.14. Each turn of the coil carries current in the direction of the arrows, generating a magnetic field purely in the toroidal direction.

Two of the field lines have been drawn. Notice how the coils crowd together as they go through the hole in the torus. The bunched current then creates a larger field at point A than at point B, which is farther from the doughnut hole. The magnetic field is always larger on the inside of a torus than on the outside. This is a toroidal effect that does not happen in a cylinder. The consequence of this effect is that charged particles no longer gyrate in perfect circles. Let’s look at the orbit of an ion in the right-hand cross section of the torus in Fig. 4.15. Normally, it will gyrate clockwise in a circular orbit, but here its orbit has been distorted into a spiral. Remember that it is the Lorentz force that makes the ion gyrate, and this force is proportional to the magnetic field strength. On the left-hand side of the orbit, the ion will feel a stronger force than it does on the right-hand side, where the field is weaker, so it will turn more tightly on the inside. The result is that the ion’s guiding center drifts downwards in this diagram. Observe that an electron drifts upwards because it has negative charge, and therefore gyrates in the opposite sense to that of the ion. This drift has been greatly exaggerated here, but nonetheless it has a huge effect on the plasma, collecting the positive charges on the bottom and the

negative charges on the top, as shown. These charge bunches will create an electric field going from the positive charge bunch at the bottom to the negative one at the top. Such a vertical electric field, as we shall see, will blow the whole plasma out toward the outer wall. Such a simple magnetic bottle will not work!

This problem was recognized very early in the game. The famous astronomer Lyman Spitzer, Jr., was riding up the long ski lift at Garmisch-Partenkirchen when he thought of a solution. Incidentally, Spitzer was the prime mover in getting the Hubble telescope built. It was not named after him, but after his death a Spitzer telescope was finally put into orbit. Spitzer’s solution was to twist the torus into a pretzel shape, as in Fig. 4.16. If you were a particle traveling along the depicted field line starting at B, you would feel a stronger magnetic field on the left than on the right. When you reach A, the strong field is on the right, now that the torus has been twisted. This is different from the circular torus of Fig. 4.14, where the strong field is always on the same side. Let’s look at the two cross sections in Fig. 4.16 in more detail. These are shown larger in Fig. 4.17. Cross section A is the same as that in Fig. 4.15, with the magnetic field pointing out of the page and with the ions drifting downwards. In cross section B, on the opposite side, the field also points out of the page instead of into the page, as in a circular torus. The fat arrows are

supposed to show this. With the field in the same direction, the ion gyrates clockwise in B, as it does in A. In B, however, the strong field is on the right side of the orbit, so the ions drift upwards instead of downwards. The vertical drifts then cancel as the ion moves along the figure-8 along a field line, and the cata­strophic separation of charges, in principle, does not occur.

This type of magnetic bottle was named a stellarator by Spitzer because it was intended to reproduce the conditions inside stars which allow them to generate fusion energy. A series of a half-dozen figure-8 stellarators was built at the Plasma Physics Laboratory in Princeton University in the 1950s to test this confinement idea. A model of a figure-8 stellarator (Fig. 4.18) was shown at the 1958 Atoms for-Peace conference in Geneva, in which thermonuclear fusion was declassified and different nations showed off their inventions. The individual coils carrying the current to generate the magnetic field can clearly be seen in this model. There was also an electron gun inside the chamber that could emit electrons that visibly traced the magnetic field lines. In addition to this model, an entire real, working stellarator was shipped to Geneva and reassembled there in the US exhibit. The Russians proudly displayed their Sputnik satellite, but their fusion exhibit was an unim­pressive, unintelligible black hunk of iron called a tokamak. It was only many years later that the world realized that that was the real star of the show.