Magnetic field from a ‘thick coil/solenoid’ or ‘annular conductor’ or &c.
Posted: February 8, 2011 Filed under: Uncategorized Comments Off on Magnetic field from a ‘thick coil/solenoid’ or ‘annular conductor’ or &c.It happens that the integrals for calculating the magnetic field from all but the simplest objects (cuboid and cylindrical magnets) quickly become very difficult to solve, and in most cases closedform solutions are not known. But reducing the integral required to a simpler form makes a huge difference in whether the result can be efficiently calculated on a computer.
I was amused recently when looking into the literature on the magnet field produced by ‘thick coils’ — i.e., a coil winding with many turns both long and deep — to find at least one new paper every decade for the last five consecutive decades on essentially the same work (by different authors). Here’s a probably incomplete list:

1971: Magnetic field of thick finite DC solenoids by Danilov, V.I. and Ianovici, M.

1982: Vector potential and magnetic field of currentcarrying finite arc segment in analytical form, Part III: Exact computation for rectangular cross section by L. Urankar.

1988: Analytical calculation of the 3D magnetostatic field of a toroidal conductor with rectangular cross section by Babic et al.

1993: Magnetic field evaluation for thick annular conductors by B. Azzerboni et al.

2006: Magnetic field of a cylindrical coil by V. Labinac et al.

2010: Calculation of the Magnetic Field Created by a Thick Coil by Romain Ravaud et al.
I haven’t implemented any of them to evaluate whether they give consistent results; my testing of Ravaud’s 2010 integral seems to give good results so I’m happy enough with his work for now. (I’m using his integral to — attempt to — derive the force produced on a permanent magnet in the vicinity of the coil, so I haven’t written code to reproduce his work exactly.)
What I find interesting is that although the work has been generalised to some extent over the years (see, e.g., the 1982 paper in which only arc segments are considered) in each case an equation is presented to calculate the magnetic field and presumably they all give identical results (they’re just solutions to a certain integral, after all). But in each case the papers were deemed worthy of publication. Hopefully new researchers looking for such works don’t miss the newer — and hopefully more useful — ones!
It’s worth noting that the 2010 paper links to a location with the claim that you can download their Mathematica code containing their equations. (Although I can’t see it there right now—hurrumph.) I think that the proliferation of code for the reproduction of such works is one of the best things we can do to accelerate the progress of the academic world. I’ll have more to say about this sort of thing in the future.