This is a question from my high school physics class years ago. I'm embarrassed to say I still don't understand the answer.
Say we have a non-conducting (like garnet) spherical shell, with -1 C charge spread evenly over the outer surface. At the center of the shell is a +2 C charge. What is the direction of the electric field just outside the outer surface?
Now, I know how to answer this in terms of Gauss's law: net positive charge inside the sphere, the field points outwards. I also know how to answer it in terms of integrating Coulumb's law: same answer. Or, you could use the fact that electric charge is the divergence of the electric field; the field should reduce in strength by half as you pass from inside the sphere to outside the sphere, so, again, same answer.
But I also know that "field lines flow out of positive charges and into negative charges". So what's going on? Are there really supposed to be field lines flowing out of the negatively charged surface?