  Macromolecules, when immersed in a polar solvent like water, become charged
by a fixed surface charge density which is compensated by ``counter-ions''
moving out of the surface. Such classical particle systems exhibit poor
screening properties at any temperature and the trivial bulk regime (far away
from the charged surface) with no particles, so the validity of standard
Coulomb sum rules is questionable. In the present paper, we concentrate on the
two-dimensional version of the model with the logarithmic interaction
potential. We go from the finite disc to the semi-infinite planar geometry. The
system is exactly solvable for two values of the coupling constant $\Gamma$: in
the Poisson-Boltzmann mean-field limit $\Gamma\to 0$ and at the free-fermion
point $\Gamma=2$. We show that the finite-size expansion of the free energy
does not contain universal term as is usual for Coulomb fluids. For the
coupling constant being an arbitrary positive even integer, using an
anticommuting representation of the partition function and many-body densities
we derive a sequence of sum rules. As a result, the contact density of
counter-ions at the wall is available for the disc. The amplitude function,
which characterizes the asymptotic inverse-power law behavior of the two-body
density along the wall, is found to be related to the particle density profile.
The dielectric susceptibility tensor, calculated exactly for an arbitrary
coupling and the particle number, exhibits the anticipated disc value in the
thermodynamic limit, in spite of zero contribution from the bulk region. Some
of the results obtained in the Poisson-Boltzmann limit are generalized to an
arbitrary Euclidean dimension.
