A Fourier-Bessel transform method for efficiently calculating the magnetic field of solenoids
Jack Nachamkin · C. J. Maggiore
Summary and citation · read the original at the source · none found
In one page
A solenoid is a coil of wire, and running current through it makes a magnetic field. Every magnet in a plasma experiment, an ion-beam line or a confinement device is one. The hard part is that the textbook formulas give you the field on the axis, or far away, and go to pieces exactly where an engineer needs numbers most: inside the coil pack itself, in among the windings. Jack Nachamkin and C. J. Maggiore, working at Los Alamos in 1980, published a way to get the field everywhere. They rewrite the problem as a Fourier-Bessel transform — a sum over Bessel functions, the natural mathematics of anything cylindrical — and then prove that their series converges at every point, windings included. They add a trick to keep the arithmetic from losing precision, so the computation stays cheap, and an appendix that extends the convergence to the general case by cutting the winding into pieces. Unglamorous, exact, and still the kind of tool a coil designer needs.
Why it matters hereChapter 9 is about making and holding self-organising plasma, and everything in it that is hardware rather than theory begins with a coil whose field you have to know to the millimetre. Chapter 11 has the same requirement with higher stakes: every claimed rotating-superconductor gravity effect is a small residual measured against a large magnetic background, and you cannot interpret one without a field map you trust inside the windings as well as outside them.
What it claims
01A numerical procedure for calculating the magnetic field of a solenoid is derived from the properties of Bessel functions, and is shown to be convergent everywhere.Author abstract, opening sentences; Department of Energy OSTI record 7183578
Settled physics02The convergence holds within the windings of the solenoid as well as outside it — the region where the elementary axial and far-field formulas are of no use.Author abstract, first paragraph; procedure detailed in the main text
Settled physics03A simple method is used to ensure numerical significance while allowing economical computational times, so the accuracy is not bought with machine hours.Author abstract; the most critical part of the procedure is detailed in the main text
Settled physics04In the appendix the procedure is generalised to universal convergence by appropriate partitioning of the solenoid windings.Author abstract, closing sentence; appendix
Settled physics05The work was carried out at the Los Alamos Scientific Laboratory under United States Department of Energy contract W-7405-ENG-36 and published in the Journal of Computational Physics, volume 37, pages 41 to 55, in August 1980 — twelve years before the same Jack Nachamkin published the Air Force Phillips Laboratory analysis of force-free time-harmonic plasmoids that this library also holds.Crossref record for the article and OSTI record 7183578, research organisation and contract fields
Published and peer-reviewed
The way in
https://doi.org/10.1016/0021-9991(80)90003-0WHAT THIS IS. A methods paper: Journal of Computational Physics, volume 37, issue 1, pages 41 to 55, August 1980. LICENCE. Crossref carries only Elsevier’s text-and-data-mining user licence, which is not a licence to republish, and OpenAlex and Unpaywall both report the article closed with no repository copy. TEXT. ScienceDirect answers automated requests with a refusal rather than the page, and no preprint or repository copy was found through Inspire, Semantic Scholar or Fatcat, so no text of the article is reproduced here. SOURCE READ. The United States Department of Energy holds a public record of the paper, because the work was done at the Los Alamos Scientific Laboratory under contract W-7405-ENG-36: OSTI record 7183578, at osti.gov/biblio/7183578, carries the authors’ own abstract, and that abstract was read in full for this sheet. Every claim below is located against it. TITLE AND AUTHOR NAMES. Crossref carries the title in lower case and the second author as ‘C.J Maggiore’; both are set here in ordinary form. AFFILIATION NOTE FOR THE LIBRARY. This library’s directory entry for Jack Nachamkin records his affiliation as unconfirmed. The Department of Energy record confirms one: Los Alamos Scientific Laboratory, 1980. His 1992 plasmoid analysis was published through the Air Force Phillips Laboratory.
How to cite it
Jack Nachamkin, C. J. Maggiore (1980) A Fourier-Bessel transform method for efficiently calculating the magnetic field of solenoids. doi:10.1016/0021-9991(80)90003-0
Where it sits in the curriculum
Plasmoids, charge clusters and the orbsGravity control and superconductors