Breakdown voltages of toroids

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Original poster: "Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br> 

Hi all:

I have run some simulations of toroid breakdown voltages to see how
they compare with a sphere.
I started with a sphere with 2 cm of diameter, that has a
breakdown voltage of about 30 kV.
I kept the 2 cm as the minor diameter of the toroid and increased
the major diameter in 2 cm increments.
I calculated also the capacitances. The results were:

2 cm    sphere:    30.0 kV  1.11 pF 1.00
4x2 cm  toroid:    45.2 kV  1.94 pF 1.51
6x2 cm  toroid:    56.6 kV  2.71 pF 1.89
8x2 cm  toroid:    65.8 kV  3.44 pF 2.19
10x2 cm toroid:    73.5 kV  4.15 pF 2.45
12x2 cm toroid:    80.1 kV  4.83 pF 2.67
14x2 cm toroid:    85.6 kV  5.50 pF 2.85
16x2 cm toroid:    91.1 kV  6.15 pF 3.04
18x2 cm toroid:    95.7 kV  6.79 pF 3.19
20x2 cm toroid:    99.9 kV  7.41 pF 3.33

6.697x2 cm toroid: 60.0 kV  3.07 pF 2.00
15.56x2 cm toroid: 90.0 kV  6.01 pF 3.00

The last column is the ratio to 30 kV, the breakdown voltage due to
the minor diameter alone.
It's easy to scale these results for toroids of other dimensions.
For example, a 30x10 cm toroid would have 5 times the breakdown
voltage of the 6x2 cm toroid, or 283 kV, and 25 times (5x5) the
free-space capacitance, or 13.5 pF.
It's not difficult to interpolate an approximate curve in the
breakdown voltage data and derive an approximate formula.

I have also updated the Inca program to plot potential and
electric field graphs. An example of electric field plot,
for the 6x2 cm toroid in free space charged to 56.6 kV:
http://www.coe.ufrj.br/~acmq/programs/toroid.jpg

Antonio Carlos M. de Queiroz