Cylindrical Magnets and Ideal Solenoids

TL;DR

This paper provides an exact, computationally efficient formula for the magnetic field of an ideal solenoid, facilitating accessible and accurate calculations for educational and experimental purposes.

Contribution

It introduces a new exact solution for the ideal solenoid's magnetic field expressed in a simple form, enabling rapid computation and practical application in teaching and experiments.

Findings

The formula is highly accurate compared to approximate methods.

The computational algorithm is fast enough for real-time calculations.

Experimental data from dropping a magnet through a conducting tube matches the simulation.

Abstract

Both wire-wound solenoids and cylindrical magnets can be approximately modeled as ideal, azimuthally symmetric solenoids. We present here an exact solution for the magnetic field of an ideal solenoid in an especially easy to use form. The field is expressed in terms of a single function that can be rapidly computed by means of a compact, highly efficient algorithm, which can be coded as an add-in function to a spreadsheet, making field calculations accessible even to introductory students. In computational work these expressions are not only accurate but also just as fast as most approximate expressions. We demonstrate their utility by numerically simulating the experiment of dropping a cylindrical magnet through a nonmagnetic conducting tube and then comparing the calculation with data obtained from experiments suitable for an undergraduate laboratory.