Why, when rotation reverses the current, should the field also be reversed?

Rotation of the source around the radial axis reverses the source (the current is then in the -z direction) and hence must reverse the field.

Says the book. And for this reason, the radial component has to be zero. I couldn't see the relationship between the two.

Source:

http://web.mit.edu/6.013_book/www/chapter1/1.4.html

• can you include a picture, or describe "the source"? – user363165 Sep 15 '16 at 4:00
• sure, i included – kihlaj Sep 15 '16 at 11:22

the text is referring to the axis marked "rotation axis", with a rotation symbol. The text really should say "rotation of the source around the radial axis by 180 degrees", since that is what it means and is clearer. This just means flipping the cylinder end-on-end. This operation flips the source current from pointing up to pointing down, which is the same as a switch $I \to -I$ and should flip the directions of all magnetic fields. This is true for any magnetic field component pointing up or down (flipped) or pointing on the $\phi$ direction (also flipped). However, a magnetic field component $B_r$ in the radial $r$ direction is not flipped by this operation. The only number $B_r$ such that $-B_r = B_r$, as required by the symmetry of the current source, is $B_r = 0$.