Sharp estimate of electric field from a conductive rod and application

TL;DR

This paper provides precise estimates of electric field perturbations caused by an inhomogeneous conductive rod, with applications to electrical impedance tomography, advancing understanding of boundary effects and inverse problems.

Contribution

It introduces sharp quantitative estimates of electric field perturbations considering the rod's geometry, especially near boundary curvature, using layer-potential techniques.

Findings

Accurate characterization of electric field localization near boundary curvature.

Development of layer-potential methods for anisotropic geometries.

Application to single-measurement detection in electrical impedance tomography.

Abstract

We are concerned with the quantitative study of the electric field perturbation due to the presence of an inhomogeneous conductive rod embedded in a homogenous conductivity. We sharply quantify the dependence of the perturbed electric field on the geometry of the conductive rod. In particular, we accurately characterise the localisation of the gradient field (i.e. the electric current) near the boundary of the rod where the curvature is sufficiently large. We develop layer-potential techniques in deriving the quantitative estimates and the major difficulty comes from the anisotropic geometry of the rod.The result complements and sharpens several existing studies in the literature. It also generates an interesting application in EIT (electrical impedance tomography) in determining the conductive rod by a single measurement, which is also known as the Calderon's inverse inclusion problem in the literature.