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International audienceThis 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 d = e2/e1 of the dielectric constan (e2 outside and e1 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

A slender dielectric body embedded in an arbitrary external potential