  We present a comprehensive study of the two dimensional one component plasma
in the cell model with charged boundaries. Departing from weak couplings
through a convenient approximation of the interacting potential we were able to
obtain an analytic formulation to the problem deriving the partition function,
density profile, contact densities and integrated profiles that compared well
with the numerical data from Monte-Carlo simulations. Additionally, we derived
the exact solution for the special cases of $\Xi = 1, 2, 3, ...$ finding a
correspondence between the results from weak couplings and exact results.
Furthermore, we investigated the strong coupling regime taking into
consideration the Wigner formulation. Departing from this, we procured the
profile to leading order, computed the contact density values as compared to
those derived in a work on the contact theorem. We formulated adequately the
strong coupling regime for this system that differed from previous
formulations. Ultimately, we computed the first order corrections and compared
those results against numerical results from our simulations obtaining very
good agreement; this results compared equally well in the planar limit, whose
results are known.
