You are making the problem unnecessarily complicated. Use Ampere's Law:
B . dl = Mu0 N I, or
B . 2 Pi R = Mu0 N I, or
B = Mu0 N I / (2 Pi R)
Decompose the field from each of the two wires into its Cartesian components, add these components into the total field, and...
If your coils are extremely thin, you can use the analytic formulas published by Milan Wayne Garrett in 1963 in the Journal of Applied Physics. (These formulas use Elliptic Integrals of all three kinds. You can evaluate these elliptic integrals using Carlson's algorithms.) If your coils are...
Two of the important applications of elliptic integrals are:
1. The magnetic fields, forces, and inductances generated by current loops (circles) and solenoids (circular tubes).
2. The electric fields, forces, and capacitances generated by current loops (circles) and solenoids (circular...