An elementary argument for the magnetic field outside a solenoid

TL;DR

This paper presents an elementary argument explaining why the magnetic field outside a finite solenoid diminishes as its length increases, clarifying a classical problem in electromagnetism for undergraduate students.

Contribution

It offers a simple, accessible explanation for the vanishing external magnetic field of long solenoids, complementing existing detailed calculations and discussions.

Findings

Magnetic field outside a solenoid decreases with increasing length.

Elementary calculation shows the field becomes negligible at large distances.

Clarifies the behavior of magnetic fields for finite and infinite solenoids.

Abstract

The evaluation of the magnetic field inside and outside a uniform current density infinite solenoid of uniform cross-section is an elementary problem in classical electrodynamics that all undergraduate Physics students study. Symmetry properties of the cylinder and the judicious use of Ampere's circuital law leads to correct results; however it does not explain why the field is non zero for a finite length solenoid, and why it vanishes as the solenoid becomes infinitely long. An argument is provided in (American Journal of Physics 69, 751 (2001)) by Farley and Price, explaining how the magnetic field behaves outside the solenoid and not too far from it, as a function of the length of the solenoid. A calculation is also outlined for obtaining the field just outside the circular cross section solenoid, in the classic text Classical Electrodynamics by J.D.Jackson, 3rd ed, (John Wiley and Sons, INC) Problems 5.3, 5.4, 5.5. The purpose of this letter is to provide an elementary argument for why the field becomes negligible as the length of the solenoid is increased. An elementary calculation is provided for the field outside the solenoid, at radial distances large compared to the linear dimension of the solenoid cross section.