An elementary argument for the magnetic field outside a solenoid

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 Farley and Price (2001 Am. J. Phys. 69 751), 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 Jackson, 3rd edn (John Wiley and Sons, Inc.), problems 5.3–5.5. The purpose of this paper is to provide an elementary argument for why the field becomes negligible as the length of the solenoid is increased. A quantitative analysis is provided for the field outside the solenoid, at radial distances large compared to the linear dimension of the solenoid cross section.