Three-dimensional field computation of charged sectors: A semianalytical method

An analytical representation of the potential created by charged sectors of a ring has been investigated. The surface charge density of the ring is supposed to be constant along a radius, assuming that a Fourier series development allows simplification into a linear superposition of sinusoidal surface densities. In addition to the obvious electrostatic devices, this approach also applies-because of the similarity of the magnetic equations-to permanent magnets with a uniform magnetization. Laplace's equation is solved by means of separation of variables. A Hankel transform, followed by a numerical integration of a Laplace transform, leads to the identification of the coefficients. Most generally, this integration involves elliptic functions which allow a fast and precise numerical treatment. At any point in space, the potentials are computed with the same simplicity and nearly the same accuracy. The proposed method introduces no systematic error and requires only a small memory-size desk computer. It is also much faster than purely numerical methods such as FEM or FDM; any change in the geometrical dimensions can also be achieved without previous reconditioning. The results of this method compare favorably with those already obtained for homopolar distributions and also with previous experiments on an eight-pole-pair distribution of magnets.