Axially Symmetric Systems for Generating and Measuring Magnetic Fields. Part I

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Abstract
This is Part I of a systematic discussion of axially symmetric magnetic fields, both central and remote from the origin, search coils reporting the field or gradient at a single point, and mutual inductances. Here the central uniformity of symmetrical fields and gradients is analyzed by zonal harmonic expansion. Laplace's equation and symmetry restrict these fields to a few types, regardless of the detailed geometry of the generating system. Universal error-contour maps are derived for the central field or gradient in systems having errors of second, fourth, or sixth order, and for hybrid types combining second and fourth. One hybrid has an oblate error field suitable for cloud-chamber and orbital applications. Source systems include circular filaments, cylindrical or plane circular current sheets, and thick solenoids of rectangular or notched section. Each type of source may be designed to produce any of the field patterns. To this end, source constants derived for the particular source type are combined into a set of over-all coefficients that express the field constants for a complete system. Rapid methods are given for computing the source constants and from them all the field derivatives, using recurrence relations or tables of Legendre functions. In particular, computing time for thick-solenoid fields or gradients is greatly reduced, using a new series with a recurrence formula. The text includes tabular aids and reference formulas, and discusses the rate of convergence of series for central and remote fields, and for mutual inductances. Special systems briefly described include several infinite series of systems like that which starts with Ampere's loop, the Helm-holtz pair, and Maxwell's three-loop system. These have integral numbers of circular filaments from two to infinity. More practical are the thin solenoids. Those that produce a fourth-order (Helm-holtz) field by omission of central turns are fully tabulated for all lengths. A short solenoid with double-wound ends and sixth-order error may realize greater uniformity in actual practice than any previously described system. Still greater (theoretical) uniformity is achieved in two eighth-order combinations of a short solenoid with a loop pair; these are described and depicted, with error limits.