Thank you all for the comments. They were very helpful to clear up my confusions.

@Jason Starr: Thanks for the remark. I do require that $G$ is of finite type over $Z_p$. I fixed the question.

Yes, that is what I meant. I corrected the question to make the statement clearer. Thank you for for the remark.

There is a conjecture due to Bjorn Poonen (www-math.mit.edu/~poonen/papers/leopoldt.pdf) which implies that if $J$ is simple, then $r'=r$. So, if you believe it you will have to pick $J$ to be a product of two elliptic curves, with ranks 2 and 0 respectively.For any Jacobian of this type $r'$ will be 1.