Determine the magnetostatic field in the continuum hyperboloid setup from boundary conditions

Determine the complete magnetostatic field H(r) with rotational symmetry about the z-axis for the continuum limit (N → ∞) of straight equidistant currents arranged on the hyperboloid x^2 + y^2 − (z cot φ)^2 = R^2 with total current J, by solving the divergence-free equation ∇ · H(r) = 0 for H(r) = H_z(r,z) e_z + H_r(r,z) e_r subject to the following boundary conditions on the xy-plane: outside the hyperboloid (r > R), H(r, z=0) = (J sin φ)/(2π r) e_z × e_r; inside the hyperboloid (r < R), H(r, z=0) points along e_z with magnitude J cos φ/(2π R). Ascertain whether these boundary conditions are sufficient to determine H(r,z) everywhere and derive the explicit solution.

Background

The paper studies the static magnetic field produced by straight currents arranged equidistantly on the surface of a hyperboloid defined by x^2 + y^2 − (z cot φ)^2 = R^2. After deriving expressions for the field along the central axis and analyzing the continuum limit (N → ∞) with fixed total current J, the author uses rotational symmetry about the z-axis to reduce the problem.

On the xy-plane, the author derives explicit boundary conditions: outside the hyperboloid (r > R), the field is azimuthal with magnitude (J sin φ)/(2π r); inside (r < R), the field points in the z-direction with constant magnitude J cos φ/(2π R). The author then proposes solving the divergence-free condition ∇ * H = 0 for a field of the form H(r) = H_z(r,z) e_z + H_r(r,z) e_r using these boundary conditions, but reports not having obtained the complete solution and leaves it as an exercise.