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Electrostatics of a finite-thickness conducting cylindrical shell: coupled elliptic-kernel integral equations

We develop an exact electrostatic formulation for a finite-length conducting cylindrical shell of finite thickness separating two dielectric media with arbitrary permittivity contrast. The boundary-value problem is reduced to a coupled system of singular integral equations with elliptic kernels governing the induced surface-charge densities on the inner and outer faces. High-accuracy numerical solutions are combined with a systematic asymptotic analysis that elucidates the interplay between geometry, thickness, and dielectric contrast. All classical limiting regimes are recovered, including the slender-body limit, the short-cylinder (ring-like) asymptote, and the thick-shell regime dominated by the outer surface. We demonstrate that the logarithmic short-cylinder behavior of zero-thickness models is a singular feature, which is regularized for any finite thickness, giving rise instead to a finite capacitance plateau. The asymptotic structure of the coupled equations explains both the electrostatic decoupling of the inner cavity in the thick-shell limit and the redistribution of charge between the two surfaces. The results provide exact benchmarks for finite cylindrical conductors, bridging classical analytical treatments and modern numerical approaches, and furnish a high-accuracy reference solution for the validation of axisymmetric electrostatic solvers.