The External Magnetic Field Created by the Superposition of Identical Parallel Finite Solenoids

TL;DR

This paper demonstrates that the external magnetic field of parallel finite solenoids can be nearly uniform and significant, with analysis of arrangements and novel insights into energy distribution, relevant for educational purposes.

Contribution

It introduces a numerical approach to analyze external magnetic fields of finite solenoids and reveals new findings about energy storage and field uniformity.

Findings

External fields can be nearly uniform and substantial.

Energy stored outside exceeds internal energy for short solenoids.

Field uniformity depends on solenoid arrangement and dimensions.

Abstract

Using superposition and numerical approximations of a published analytical expression for the magnetic field generated by a finite solenoid, we show that the magnetic field external to parallel identical solenoids can be nearly uniform and substantial, even when the solenoids have lengths that are large compared to their radii. We study two arrangements of solenoids---a ring of parallel solenoids whose surfaces are tangent to a common cylindrical surface and to nearest neighbours, and a large finite hexagonal array of parallel solenoids---and summarize how the magnitude and uniformity of the resultant external field depend on the solenoid length and distances between solenoids. We also report some novel results about single solenoids, e.g., that the energy stored in the internal magnetic field exceeds the energy stored in the spatially infinite external magnetic field for even short solenoids. These results should be broadly interesting to undergraduates learning about electricity and magnetism as novel examples of superposition based on a familiar source of magnetic fields.