Measurement of Two-Dimensional Fields. Part I: Theory

Abstract

An arbitrary two‐dimensional magnetic field may be uniquely specified by listing all coefficients of the power‐series expansion of its complex potential function. An experimental procedure is described in which the first N coefficients are derived from an N‐point harmonic analysis of the radial component of field or gradient, as measured with a dipole or quadrupole search coil at N equally spaced angular positions about an arbitrary longitudinal axis. The method may be refined by mounting two coils coaxially on a radial arm, so connected as to balance out the strongest component present and permit more accurate determination of the higher harmonics. With four coils, odd or even harmonics also may be suppressed. Analysis of the symmetry of various magnet pole structures shows that many of the coefficients vanish identically. Procedures are outlined for measuring with a quadrupole search coil the five independent elements of the magnetic gradient dyadic at a point in an unrestricted field. It is shown that in the fringing region at the ends of a long magnet gap the axial average of the transverse field can be treated as a two‐dimensional field. It is correctly measured using line dipole or quadrupole search coils, whose design is prescribed. The procedure is identical with that for the two‐dimensional case. A further extension of the same method to measurements made with the electrolytic tray analog increases greatly their speed and accuracy.