An approach to the magnetic field of a finite solenoid with a circular cross-section

We present a simple approach, different from approaches in the literature, for calculating the magnetic field of a finite solenoid with a circular cross section, carrying an ideal surface current with an ordinary azimuthal component and a small axial component due to nonideal winding. The results are given as an infinite series whose terms are elementary functions or simple polynomials rather than other infinite series. It is easy to deal with the results because the polynomials involved are partial binomial expansions. These series converge rapidly everywhere, except near the edges of the solenoid. From these results, it is very easy to derive some general properties of the magnetic field, to reduce the field to greatly simplified forms in various special regions, and to address the special cases of infinite and semi-infinite solenoids. The azimuthal field due to nonideal winding is shown to be small everywhere. A solenoid with a small but finite thickness in the radial direction is investigated in detail, and we present simple analytic approximate results for its magnetic field. A comparison with an exact numerical evaluation shows that these are very good approximations.