The fundamental vector calculus definition of a force-free, field-aligned, Birkeland current is expanded in cylindrical coordinates to obtain the partial differential equations (DEs) that yield the magnetic field created by such a current. The resulting equations are put into state-variable form and an Euler, step-wise, approximation incorporating a 4th order Runge-Kutta algorithm is applied. In single-variable form, the DEs are identified as Bessel equations. J0(r) and J1(r) Bessel function solutions confirm the Euler results thus yielding closed form solutions for both the linking (azimuthal) and collinear (axial) components of the force-free field. Results show that both of these magnetic components reverse their directions and vary in magnitude in a way that aids the formation of concentric cylindrical shells of matter as has been observed in Marklund convection. Another result is the finding that magnetic fields extend relatively farther from Birkeland currents than they would from a straight-line current. They slowly decay as 1/SQRT(r) for large r.