Inductance of a layered Solenoid?

[JoeBeef]

Homework Statement

For a final laboratory project, I am trying to find an experimental value for the permeability of free space by determining the Inductance of a solenoid.
I have found a good value of inductance for my solenoid, but the problem is that it is that my solenoid has multiple layers of coil.

Relevant Equations

L = (μ₀ * μᵣ * N² * A) / l

I realize that the standard formula for the inductance of a solenoid (L = (μ₀ * N² * A) / l, where A is the cross sectional area, N is the number of turns and l is the length) is for solenoids with only one layer of wire. I thought I could find μ₀ using this simple formula, but it does not work.
Is there a standard formula for coils with more layers? I looked around online but I could not find anything, and me TA did not know either. How can I go about calculating the permeability constant given my experimental inductance?

 

[Baluncore]

Relevant Equations: L = (μ₀ * μᵣ * N² * A) / l

I thought I could find μ₀ using this simple formula, but it does not work.

Formulas for the self inductance of almost any winding configuration had been developed a century ago, by a number of brilliant applied mathematicians. Maxwell in England, and then E.B. Rosa at the Bureau of Standards in the USA, dominated the competitive field. It was then summed up in a book by F.W. Grover, who was at the NBS during the later period.

At that time, they were all engaged in a futile search for a standard inductor, but with inductors there are just too many confounding variables. The length of the wire measured in half-wavelengths, the dielectric constant of the wire insulation, or the conductive screening that reflects the field, all play a part. All inductors, to some extent, morph into transmission lines and antennas. That necessitates measuring the inductance at very low frequencies, to get any reliable value of self-inductance.

See for example:
"The self-inductance of a solenoid of any number of layers"
Volume: Bulletin of the Bureau of Standards, Vol. 4, 383-390 (1908)
Scientific Paper 84 (S84). By: Cohen, L. Published: 1907.
https://archive.org/details/selfin438339019088484cohe

Most of those reports can be found here:
http://archive.org/search.php?query...llection:NBSBulletin AND subject:"Inductance"

Inductance Calculations. Working Formulas and Tables. By Frederick Warren Grover. 1946,
has been reprinted a number of times, including by Dover.
https://www.google.com.au/books/edition/Inductance_Calculations/K3KHi9lIltsC?hl=en&gbpv=0

I will warn you not to get lost in that labyrinth of formulas, as I almost did for a year. With the work of the NBS, you can always compute the inductance more accurately than you can possibly measure self-inductance in the laboratory.

 

[JoeBeef]

Awesome thank you very much