Magnetic field problem with axial symmetry and its solution in terms of Fourier-Bessel expansions

A method for reaching implicit analytical solutions is explained, demonstrated and verified for certain problems of potential theory, characterized by having boundaries which are parts of co-ordinate surfaces. Different Fourier expansions for the potential are matched over an interface of the problem by the solution of linear simultaneous equations in their coefficients. Two problems are completed by numerical computations and the construction of equipotentials. Comparisons are made with known results for the easier problem, and with experimental results for the other. The method is considered as a computational alternative to relaxation and finite differences, and the axisymmetric problem is shown to be aimed at applications in the design of electrical machines.