  Using a first principles linear response approach, we study the magnetic
exchange interactions J for a series of superconducting cuprates. We reproduce
the observed spin-wave dispersions together with other experimental trends, and
show that different cuprates have similar J's regardless their T$_{c}$. The
nearest neighbor J is not sensitive to the hole-doping, which agrees with
recent experiments. For the undoped cuprates, the second nearest neighbor J is
ferromagnetic, but changes its sign with hole-doping. We also find that, in
contrast to the hopping integral, the exchange interaction is not sensitive to
the position of apical oxygen. To see the effect of the long--range nature of
the exchange on the superconducting T$_{c}$, we study the dynamical spin
susceptibility $\chi (q,\omega)$ within the t--J model using a dynamical
cluster approximation.
