  Vacuum force is an interesting low energy test for brane worlds due to its
dependence on field's modes and its role in submillimeter gravity experiments.
In this work we generalize a previous model example: the scalar field vacuum
force between two parallel plates lying in the brane of a Randall-Sundrum
scenario extended by $p$ compact dimensions (RSII-$p$). Upon use of Green's
function technique, for the massless scalar field, the 4D force is obtained
from a zero mode while corrections turn out attractive and depend on the
separation between plates as $l^{-(6+p)}$. For the massive scalar field a
quasilocalized mode yields the 4D force with attractive corrections behaving
like $l^{-(10+p)}$. Corrections are negligible w.r.t. 4D force for
$AdS_{(5+p)}$ radius less than $\sim 10^{-6}$m. Although the $p=0$ case is not
physically viable due to the different behavior in regard to localization for
the massless scalar and electromagnetic fields it yields an useful comparison
between the dimensional regularization and Green's function techniques as we
describe in the discussion.
