Engineering Inductance of a layered Solenoid?

AI Thread Summary
The standard formula for solenoid inductance applies only to single-layer configurations, complicating calculations for multi-layer solenoids. Historical research by mathematicians like Maxwell and E.B. Rosa has led to various inductance formulas, but many variables affect accuracy, including wire length and insulation properties. Accurate inductance measurement often requires low-frequency testing due to the behavior of inductors resembling transmission lines and antennas. Resources such as Grover's work and historical papers provide insights into multi-layer inductance calculations. Ultimately, precise inductance computation is often more reliable than laboratory measurements.
JoeBeef
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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?
 
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JoeBeef said:
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.
 
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Likes berkeman, jim mcnamara, DaveE and 3 others
Awesome thank you very much
 

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