Electrostatics of finite-thickness conducting cylindrical shells: coupling elliptic-kernel integral equations

We formulate and solve a reduced electrostatic model of an open finite-length cylindrical conductor represented by two coaxial lateral surfaces of inner radius a, outer radius b=a+t, and common length L, maintained at the same prescribed potential V₀. The annular end surfaces are omitted, and the regions adjacent to the inner and outer cylindrical surfaces are characterized by permittivities ɛᵢₙ and ɛₒᵤₜ, respectively. Exploiting axial symmetry, the surface-integral formulation is reduced analytically to a coupled 2×2 system of one-dimensional singular integral equations for the two surface-charge densities. The self- and mutual-interaction kernels are obtained in closed form in terms of complete elliptic integrals. The equations are solved using a Chebyshev-weighted Nyström discretization that incorporates the square-root endpoint behavior of the charge densities and treats the weak logarithmic self-interaction singularities through an explicit local diagonal prescription. Surface-charge profiles and capacitance are computed as functions of the aspect ratio α=a/L, radial ratio δ=b/a, and dielectric contrast κ=ɛₒᵤₜ/ɛᵢₙ. The zero-thickness limit δ→1⁺ recovers the single-surface open-cylinder formulation, whereas increasing radial separation progressively shifts the charge toward the outer cylindrical surface. In the slender-cylinder regime, finite radial separation produces only subleading corrections to the dominant logarithmic behavior. In the opposite short-cylinder regime, the capacitance does not approach a finite plateau; instead, for fixed δ>1, it retains a slow logarithmic decay described by a coupled-ring asymptotic expression derived from the large-α kernel structure. The numerical results are further validated against an independent graded-panel boundary-element discretization, providing benchmark data for coupled axisymmetric electrostatic solvers.