A slender dielectric body embedded in an arbitrary external potential

This paper determines the electrostatic potential and field taking place both inside and outside a slender dielectric body embedded in a given potential ø₀. This task is actually reduced to the determination of the occurring polarization surface-charge density q which depends on ø₀, on the body shape but also on the ratio δ = ε₂/ε₁ of the dielectric constan (ε₂ outside and ε₁ inside the body). The adopted procedure consists in asymptotically expanding and inverting (with respect to the small slenderness ratio of the body) the well-known Fredholm boundary integral equation of the second kind governing the function q. The technical difficulties such an approach encounters are bypassed by employing a systematic formula in getting the asymptotic estimate of certain integrals depending upon a small parameter. Contrary to other works in the field, this method authorizes us to handle the case of non-axisymmetric slender bodies. As an illustration the theory is applied to a body of elliptical cross-section and comparisons are presented for a slender dielectric ellipsoid embedded in a special potential ø₀ for which the exact density q is obtained in a closed form.