Maxwell's equations: meaning, derivation and applicability - Sci. Quote So if Coulombs law works best for you, then use coulomb's law - there's no need to re-invent the wheel.
As far as the original problem goes it is solved, but it uncovered great many things for me, so what is left now is curiosity because this solution implies Coulomb's and Biot-Savart law tell different and more complete story than Maxwell's equations and yet they are supposed to talk about the same E and B fields. There are two kinds of fields, "radial" like gravity and electric fields, and we have "rotational", like vortexes, whirlpools or magnetic fields. Uniform and constant "radial" field potentials have zero divergence and zero rotation (curl), it's a uniform magnitude distribution and inverse square law which defines topology and geometry of an electric field, not the other way around.
Electricity and Magnetism. Articles on "Electricity" Why three prongs?
Why do wall outlets have three holes? "Grounding" and safety. Electricity. Electromagnetism. Electromagnetic waves. Electricity. Magnetism.