Abstract: We resolve a long-standing electrostatics puzzle dating back to Maxwell: how do charge density and electric field scale near the ends of a thin conducting cylinder? Our argument is based on a local resummation of the perturbative series solution to the singular-integral equation of slender-body theory. This approach allows us to address the theory’s notorious difficulty in handling the end regions of slender cylinders and other “truncated” shapes tapered on the diameter scale (as opposed to “needle”-like shapes tapered on the longitudinal body scale, e.g. spheroids). The resulting scaling law is universal across broad classes of slender geometries and physical scenarios, ranging from diffusion to Stokes flow and elasticity. In certain problems, this scaling law dictates not just localized phenomena—relevant to critical phenomena such as dielectric breakdown, material fracture and cavitation—but also key integral quantities. Indeed, we were led to this research while attempting to address a slightly more recent problem in fluid dynamics: estimating the speed of “active” nanorods swimming in a highly viscous liquid. Building on the above general theory, we find that the scaling of the swimming speed with the slenderness typically lies “halfway" between the scaling established for needle-shaped particles and that erroneously predicted by earlier theories for cylindrical rods.   Joint work with Gunnar G. Peng