The Van de Graaff Generator is a favourite project for the amateur science constructor. However, many designs are flawed by incorrect assumptions about the profile of the "sphere", or high voltage terminal. The maximum potential possible for the terminal is determined by its smallest radius of curvature. This will be where the supporting structure and belts enter the terminal.
A large diameter sphere with a small reentrant radius will charge up only to the potential where corona initiates at this small radius. The rest of the sphere adds only capacitance, giving a fatter spark, but not increased potential.
The following example of a generator is an excellent example of the correct design of a Van de Graaff high voltage terminal. Instead of a sphere, the terminal is a flat topped toroid. This design also has a correctly configured charge collection system, with the belt being oppositely charged for its return journey. Equipotential rings are used on the supporting column for smoothing the voltage distribution along the column.
Half a million volts at around 200uA is generated by this 1938 design by Trump, Merrill & Safford.
1) High-voltage terminal, 30in. diameter and 12in. high made by two aluminium spinnings, joined together at the equator by a brass ring.
2) Lower grounded terminal, consisting of an aluminium spinning identical with the lower half of the h.t. terminal. A heater is placed under this spinning to prevent condensation of moisture inside the generator on insulating surfaces.
3) Insulating column, 38in. long, 12in. outside diameter with 1/8in. wall thickness, made of Textolite or Micarta.
4) Upper aluminium casting used to attach the high-voltage terminal to the top of the insulating column. The lower surface of this casting serves as a smooth continuation of the spinning surface to the column.
5) Aluminium ring shaped casting, carrying the upper pulley frame.
6) 4 rubber supports, 1 1/4in. diameter, 1 1/2in. long for mechanical and electrical insulation of the upper pulley frame from the terminal.
7) Pillow blocks with self-aligning bearings.
8) Pillow block mounting for adjusting the tension and running position of the main belt.
9) Insulating supports for upper corona spray rod.
10) Upper corona spray rod, 5/8in. brass with gramophone needles spaced 1/2in. and projecting 3/8in.
11) Upper collector comb, as (10)
12) 1/2 h.p., 3,450 r.p.m. driving motor. For heavier belts a more powerful motor would be necessary.
13) Adjustable hinged motor mounting for setting the belt tension.
14) Generator base made of a standard flanged steel head 3/16in. thick and 30in. diameter.
15) Dynamically balanced hollow steel pulleys 4in. diameter, 10in. long with 1/8in. (in radius) per foot taper extending 3in. at each end of the pulley.
Full Article (large graphics files) Trump, J.G., Merrill, F.H., and Safford, F.J. Rev. Sci. Instrum., 9 (1938) 398