I have this simple yet intriguing question that struck me through an introductory Electromagnetism. Being familiar with Gauss' Law and Coulomb's Law, the "Charged Spherical Shell" excercise is something very common to me. A particular thing for this exercise is that if you use any method (at least as far as I know), be it a simple Gaussian or a die-hard spherical coord. Coulomb Law integration, you get the same result. Zero electrical field at any point inside the shell. But one day I thought: "What about charged rings?".
To my surprise, and if my calculations aren't mistaken, the electrical field inside the ring was zero only in the center, but non-zero everywhere else (the ring's plane). An even more intriguing fact was that now Gauss' Law and Coulomb's Law disagreed for this particular case. I took this question to a professor and he said he didn't know what to tell me, but he did state that I should trust on the Coulomb's Law result, this being an experimental law.
Why is this so? Couldn't I make a "spherical" arrangement of charged rings to make a charged spherical shell? What am I missing? Please do remember I've only gone through an introductory course.