  We consider two-dimensional Coulomb systems confined in a disk with ideal
dielectric boundaries. In particular we study the two-component plasma in
detail. When the coulombic coupling constant $\Gamma=2$ the model is exactly
solvable. We compute the grand-potential, densities and correlations. We show
that the grand-potential has a universal logarithmic finite-size correction as
predicted in previous works. This logarithmic finite-size correction is also
checked in another solvable model: the one-component plasma.
