A method for generating a high uniform magnetic field along a solenoid longitudinal axis

Abstract

In this paper we present an analytic approach to the problem of finding how should be the turn density function of a solenoid of radius R and length L in order that the axial magnetic field along the axis of the solenoid is as uniform as possible. The mathematical problem involved is to solve a Fredholm integral equation of the first kind. As in our case we can't find a solution of the integral equation, we will look for an approximate solution, shifting the problem to a minimum norm problem in an appropriate set of functions. In doing that, we will follow unintentionally a way suggested but not developed by Ewing in [4] where he refers to this problem as a problem "out of the beaten path".
We also make some considerations that show that a good behavior of the magnetic field along the axis can guarantee a good behavior also around, but near the axis itself. In the Appendix we also discuss, from a different point of view, a classical result about uniform magnetic fields: Helmholtz Coils.