Magnetic field in a rotating uniformly-charged infinite cylinder

[Delta2]

Yes you can do that, if you carefully apply Ampere's law for a symmetrical rectangular loop inside the cylinder you ll end up with an equation of 0(path integral)=0(total current enclosed).

Now that I think of it , you get 0=0 even if the symmetrical loop has its two parallel to the axis of the cylinder, sides outside the cylinder. So sorry it seems that my proof at post #34 doesn't work.

 

[Delta2]

The only argument I see now is based on symmetry. Due to cylindrical symmetry we can argue that the field outside the cylinder will be in the z-direction. But again due to symmetry we can't decide if it will point in the up or down direction. So the only symmetrical option is to be zero.