$$\require{physics}$$ Capacitance per unit length of concentric cylinders: $$ \frac{C}{L} = \frac{2 \pi \epsilon_0}{\ln{(b/a)}} $$ Where "b" is the radius of the outer cylinder, and "a" is the radius of the inner cylinder. For our purposes, (b/a) is always 1.61803... Let's call this alpha. $$ \alpha = \frac{1 + \sqrt{5}}{2} = 1.61803... $$ So then we have: $$ \frac{C}{L} = \frac{2 \pi \epsilon_0}{\ln{\alpha}} $$ Let's solve this numerically. $$ 2 \pi = 6.28318530718 $$ $$ \epsilon_0 = 8.85418782 \times 10^{-12} \; Farads / meter $$ $$ \ln{ \left( \frac{1 + \sqrt{5}}{2} \right)} = 0.48121182505960347 $$ $$ \frac{C}{L} = \frac{2 \pi \epsilon_0}{\ln{\alpha}} $$ $$ \frac{C}{L} = \frac{6.28318530718 \times 8.85418782 \times 10^{-12}}{0.48121182505960347} $$ $$ \frac{C}{L} = 1.1560917650089283 \times 10^{-10} $$ $$ \frac{C}{L} = 115.60917650089283 \times 10^{-12} $$ $$ C = 115.609 pF / meter $$ To find the capacitance between each pair of concentric cylinders, we will multiply the expression above by the length of the shorter (outer) cylinder. This neglects the "fringing fields" which splay out around the ends of the cylinders, but I don't know if anyone knows how to account for those in their calculatoins. Lengths are in meters. Capacitance is in pF.

## LENGTH CAPACITANCE -------------------------------------------- 1 1.014 72.451 2 0.627 44.777 3 0.387 27.674 4 0.239 17.103 5 0.148We should measure 72.451 pF between the first and second cylinders. And we should measure 44.777 pF between the second and third. You get the idea :) These values are for the prototype, which has the following physical dimensions, in *inches*.

## DIAMETER LENGTH CIRCUMF ------------------------------------------------ 1 3.000 39.924 9.425 2 4.854 24.674 15.250 3 7.854 15.250 24.674 4 12.708 9.425 39.924 5 20.562 5.825 64.598Machinists and Fabricators in the US use inches. But the permittivity of free space is measured in Farads per meter. Here's another way of writing it, if you really like looking at tables.

## DIAMETER LENGTH CIRCUMF pF -------------------------------------------------------------- 1 3.000 39.924 9.425 72.451 2 4.854 24.674 15.250 44.777 3 7.854 15.250 24.674 27.674 4 12.708 9.425 39.924 17.103 5 20.562 5.825 64.598