# Capacitance per Unit Length of Concentric Cylinders

$$\require{physics}$$ 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          0.148


We 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         64.598


Machinists 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