Characterization of Electric Fields for Perfect Conductivity Problems in 3D

TL;DR

This paper derives an asymptotic formula for the electric field concentration between two spherical perfect conductors with different radii in 3D, revealing how their sizes influence the blowup behavior.

Contribution

It provides an explicit asymptotic expression for the electric field near closely spaced spherical inclusions with different radii, incorporating special functions.

Findings

Explicit blowup factor involving radii, digamma function, and Euler-Mascheroni constant

Identification of the role of inclusion radii in electric field concentration

Asymptotic formula applicable to 3D composite materials with closely spaced conductors

Abstract

In composite materials, the inclusions are frequently spaced very closely. The electric field concentrated in the narrow regions between two adjacent perfectly conducting inclusions will always become arbitrarily large. In this paper, we establish an asymptotic formula of the electric field in the zone between two spherical inclusions with different radii in three dimensions. An explicit blowup factor relying on radii is obtained, which also involves the digamma function and Euler-Mascheroni constant, and so the role of inclusions' radii played in such blowup analysis is identified.