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For a Coulomb system contained in a domain Λ, the dielectric susceptibility tensor χΛ is defined as relating the average polarization in the system to a constant applied electric field, in the linear limit. According to the phenomenological laws of macroscopic electrostatics, χΛ depends on the specific shape of the domain Λ. In this paper we derive, using the methods of equilibrium statistical mechanics in both canonical and grand-canonical ensembles, the shape dependence of χΛ and the corresponding finite-size corrections to the thermodynamic limit, for a class of general ν-dimensional (ν≥2) Coulomb systems, of ellipsoidal shape, being in the conducting state. The microscopic derivation is based on a general principle: the total force acting on a system in thermal equilibrium is zero. The results are checked in the Debye–Hückel limit. The paper is a generalization of a previous one [L. Šamaj, J. Stat. Phys.100:949 (2000)], dealing with the special case of a one-component plasma in two dimensions. In that case, the validity of the presented formalism has already been verified at the exactly solvable (dimensionless) coupling Γ = 2.