Problem (IIT JEE 1993): A current $I$ flows along the length of an infinitely long, straight, thin-walled pipe. Then the magnetic field
Solution:
By symmetry, the magnetic field inside the pipe is circumferential. Take a circular loop of radius $r$ with centre along the axis of the pipe. By symmetry, the magnitude of magnetic field is same throughout the loop. By Ampere's law, $\oint \vec{B}\cdot\mathrm{d}\vec{l}=B(2\pi r)=\mu_0I_\text{enc}$. Since $I_\text{enc}=0$, we get $B=0$ for all $r$. Thus, $B=0$ inside the pipe. The readers are encouraged to explain why there can not be a uniform field along the axis, similar to field of a solenoid.