If $r=2$ then $r'>0$. For an example where $r'=1$, take a curve such that the jacobian has a nontrivial endomorphism $f$
And such that the group of points in the jacobian is generated by $P,f(P)$ for some point $P$
Now find a prime $p$ splitting in $\mathbb{Q}(f)$ so that $f(P)=\alphaP$ for some $p$-adic number $\alpha$.
Then $r'=1$.