Consider a hollow cylinder carrying a current $I$ and a wire outside the cylinder carrying a current $I'$. Let's say the cylinder is symmetrical with even current distribution etc.. so the $\mathbf{B}$ field at any point (due to current in cylinder) within the cylinder is zero by Amperes Law. However, this doesn't mean the $\mathbf{B}$ field is zero within the cylinder entirely - there is a $\mathbf{B}$ field contribution from the wire. So my question is: What is the usefulness of Amperes Law?
Does Ampere's Law only tell me something about the $\mathbf{B}$ field from a particular source?
Also say we have a solid cylinder inside a hollow cylinder with radii $a$ and $b$ respectively. They have opposite current directions. Then by Ampere, the $\mathbf{B}$ field at some point $P$ where $a < P < b$ is given as $B = \frac{\mu I}{2\pi r}, I $ the current in the solid cylinder. Is it really? The $\mathbf{B}$ field from the hollow cylinder will be in the opposite direction at $P$ and so acts to cancel the $\mathbf{B}$ field at $P$ from the solid cylinder thus resulting in zero net $\mathbf{B}$ field, no? Yet the $\mathbf{B}$ field at $P$ is in fact nonzero?
I understand how the non zero $\mathbf{B}$ field was obtained using Ampere's Law, but the Amperian loop which coincides with $P$ does not simply shield the $\mathbf{B}$ field from the hollow cylinder. So I am struggling to see why the $\mathbf{B}$ field would be nonzero.
Many thanks.