2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/167971The electric field increases toward infinity in the narrow region between closely adjacent perfect conductors as they approach each other. Much attention has been devoted to the blow-up estimate, especially in two dimensions, for the practical relevance to high stress concentration in fiber-reinforced elastic composites. In this paper, we establish optimal estimates for the electric field associated with the distance between two spherical conductors in $n-${dimensional spaces for $n \geq 2$}. {The novelty of these estimates is that they explicitly describe the dependency of the blow-up rate on the geometric parameters: the radii of the conductors.}Prof. Mikyoung Lim is added as an auther to the revison, 27 pagesAnalysis of PDEsMathematical Physics15A15, 15A09, 15A23text