External dc Field of a Long Solenoid

The external dc field of a long solenoid is investigated. An exact solution is obtained in the form of an infinite series. For three cases, the series reduces to simple analytical expressions amenable to slide-rule computation: (a) the infinitely long solenoid for which the external field is zero, (b) the field far from the ends of a long finite solenoid for radial distances close to the solenoid surface, and (c) the long solenoid at radii comparable to and greater than the solenoid half-length. The variations with radial distance of the approximate expressions derived for cases (b) and (c) are plotted parametrically over a large-range of radial distances. Exact results are also plotted for two typical solenoid length-to-radius values obtained by numerical integration. Agreement of the exact results with the approximate expression for case (c) over the entire range of radial distances is extremely good. Expression (b) is a good approximation only for small radial distances; however, for typical solenoid length-to-radius values, the two expressions (b) and (c) converge for small r.