Blow-up of Electric Fields between Closely Spaced Spherical Perfect Conductors

TL;DR

This paper derives optimal estimates for the electric field blow-up between two closely spaced spherical perfect conductors in multiple dimensions, explicitly relating the blow-up rate to the conductors' radii.

Contribution

It provides the first explicit, optimal estimates for the blow-up rate of electric fields between spherical conductors, including the dependence on their radii, in higher dimensions.

Findings

Explicit blow-up rate estimates depend on conductor radii.

Estimates are optimal and valid in dimensions n ≥ 2.

Results extend previous 2D analyses to higher dimensions.

Abstract

The 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.}