Magnetic field homogeneity of a conical coaxial coil pair

An analytical study of the magnetic field created by a double-conical conducting sheet is presented. The analysis is based on the expansion of the magnetic field in terms of Legendre polynomials. It is demonstrated analytically that the angle of the conical surface that produces a nearly homogeneous magnetic field coincides with that of a pair of loops that fulfills the Helmholtz condition. From the results obtained, we propose an electric circuit formed by pairs of isolated conducting loops tightly wound around a pair of conical surfaces, calculating numerically the magnetic field produced by this system and its heterogeneity. An experimental setup of the proposed circuit was constructed and its magnetic field was measured. The results were compared with those obtained by numerical calculation, finding a good agreement. The numerical results demonstrate a significant improvement in homogeneity in the field of the proposed pair of conical coils compared with that achieved with a simple pair of Helmholtz loops or with a double solenoid. Moreover, a new design of a double pair of conical coils based on Braunbek's four loops is also proposed to achieve greater homogeneity. Regarding homogeneity, the rating of the analyzed configurations from best to worst is as follows: (1) double pair of conical coils, (2) pair of conical coils, (3) Braunbek's four loops, (4) Helmholtz pair, and (5) solenoid pair.