I have a three dimensional current carrying coil. This coil has rectangular cross section, with a rectangular hole cut in the middle. The aim is then to assign the current density at points inside the coil, such the magnitude of current density, J = sqrt(Jx^2 + Jy^2 + Jz^2) = Constant and the direction of current flow is rotational and continuous around the coil. Mathematically, the value J = IN / A, where A is the cross sectional area normal to the current flow, I is the current and N the number of turns in the coil. The figure below illustrates this:
What I'm seeking is expressions of the form Jx(x,y,z) = f(x,y,z), Jy(x,y,z) = g(x,y,z), Jz(x,y,z) = h(x,y,z) which achieve the rotational, continuous current flow I'm interested in. For the coil as seen here, Jz = 0, I would also expect the general functional f(x,y,z) to involve four terms - 2 zero terms for arms of the coil where Jx = 0 and two non-zero terms with equal magnitude but opposite direction.
I have been searching for an extended period to try and find/set up these mathematical expressions but haven't yet been successful. A starting point, say how to do this for a single segment of my coil, any thoughts, ideas or suggestions are greatly appreciated!