Electronic – Making a low frequency inductor design

You have experimentally derived the equivalence that 113 turns yields one inch of coil length, yielding the conversion factor (1 inch / 113 turns). Obviously this tacitly depends on the coil diameter, but is a sound figure for the diameter under which you established the measure.

If you have 113 turns that is one inch. If you have 226 turns that is two inches. and so forth. So Length definitely equals number of turns / 113. Looks to me like you can safely substitute N/113 for l. Just keep in mind this doesn't generalize because of the implicit dependence on diameter built into the 113.

You should probably just ask the second part of your question as a new question. It's way more interesting than the part I'm answering here :-).

Air core inductors can have small capacitance. You can get an idea how small by considering the capacitance of two wires parallel to each other of length \$\pi D\$ (where \$D\$ is the coil diameter). That's like the turn to turn capacitance in the coil, and it's a small number. Of course, the total capacitance will be more than just the series combination of these turn to turn capacitances since the windings are coupled and will add in kind of a series/parallel way, but still a small number. That might make it hard to just measure by resonance. Probe capacitance of ~15pF or 20pF could be close to the coil capacitance.

Medhurst did a lot of work during the 1940's on inter-winding capacitance of air cored single layer solenoids. An equation that is an extension of Medhursts work is:

\$C_{\ell }\$ = \$\frac{4 \ell \epsilon _o \epsilon _{\text{rx}} \left(\frac{1}{2} k_C \left(\frac{\epsilon _{\text{ri}}}{\epsilon _{\text{rx}}}+1\right)+1\right)}{\pi \text{Cos}^2 \psi }\$

where
\$k_C\$ = \$\frac{0.106 D^2}{\ell ^2}+\frac{0.717439 D}{\ell }+0.933048 \left(\frac{D}{\ell }\right)^{3/2}\$

and \$D\$ is coil diameter, \$ \ell\$ is the coil length (or height), \$\psi \$ = \$\tan ^{-1}\left(\frac{p}{\pi D}\right)\$, \$p\$ is the pitch or turn spacing, \$\epsilon _o\$ = 8.854pF/m, \$\epsilon _{\text{ri}}\$ is relative permittivity of the coil form (2.56 for polystyrene, for example), and \$\epsilon _{\text{rx}}\$ is relative permittivity of space external to the coil (probably just air). For your case of \$D\$ = 0.3048m, \$ \ell\$ = 1.1811m, and \$p\$ = 1.08712mm, \$C_ \ell\$ ~ 21pF.

Reference for this is "The self-resonance and self-capacitance of solenoid coils" by David Knight. The equation is 5.3 on p 25, but there is a lot of detail of its development.

You might also be interested in the work of Gary Johnson who has written a book about Tesla coils. Here is a sample "Tesla Coil Impedance", and the rest is on his web page.