Abstract :
With a cylindrical relativistic beam the self electric field force is cancelled by the self magnetic force within (1 − β2) and with almost the same cancellation in practical toroidal geometries. If the return current of the beam (magnetic image) in the wall and the electrostatic charge (electrostatic image) on the wall coincide, then this approximate cancellation allows very high space charge limits on current for β close to unity 0 But in a wall of finite resistivity the magnetic image decays and cancellation of forces would decrease. In fact, an iron pole outside the vacuum chamber would then create the wrong sign image and add to, not subtract from, the electrostatic force toward the wall. This is a general problem for all accelerators. It can be largely solved by placing the betatrons primary winding, which carries the same load ampere turns as the beam, at the vacuum tube wall so that when the wall image decays and the beam´s magnetic field penetrates, the properly distributed primary coil ampere-turns at the wall take over the image´s role Fig. 2a and 2b. For a rapid beam motion, wall conductivity can be good enough to provide the necessary high frequency image current; but the fixed primary wire currents act only on the long time average beam field which penetrates the wall and which must be terminated within the wall primary to decouple beam flux from the magnet coils. The magnetic cancellation of the electrostatic beam force is thus approximately maintained. Other parallel connected primary turns carrying the betatron magnetizing current would be distributed to produce the vacuum field shape and accelerating flux. Engineering compromises must be made in designing the vacuum chamber for primary distribution and for carrying the high frequency current image across the gap. This is a general design problem for which solutions are described in the case of a conventional betatron in [1].
