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

Counter-ion density profile around a charged disk: from the weak to the strong association regime