A proper integral for the electric field at the surface of a conducting sphere

F M S Lima

doi:10.1590/1806-9126-rbef-2020-0182

Abstract

Abstract In a recent paper, I have shown that the electric field at the surface of a charged conducting sphere evaluates to half the field discontinuity across the surface of the sphere. This exact result has been found by solving an improper integral, which has been criticized by Assad in a very recent paper in this journal [6]. In this note, I present a simpler approach that yields a proper integral for that field. This integral also evaluates to exactly half the field discontinuity, contrarily to Assad's objections.

Highlights

For a conducting sphere with radius R and net charge Q, in electrostatic equilibrium, the electric field in any point P on its surface is given by
We have excluded that point from the domain, but we have properly included it there in Eq (3), which has led to an elementary, proper integral in Eq (4). This can be viewed as a regularization of the improper Riemann integral solved in Ref. [5], which is a valid procedure since the value of the integral of a bounded function that is continuous in an interval of integration, except at a point belonging to this interval, does not change when we remove this point from the integration domain, as follows from a well-known theorem by Cauchy.4. Speaking, this regularization is justified by noting that the charge is distributed uniformly on the surface of the sphere, with a constant density σ, so the charge dQ = σ dA = 2π σ R2 sin θ dθ in each ring will tend to zero as r → 0, which means that our point P, which encloses a null area, does not enclose a finite amount of charge
Even an older argument by Assad, as found at the end of Ref. [9], that the electric field is undefined at the surface of a conducting sphere because there is electric charge there, is incorrect

Summary

Introduction

For a conducting sphere with radius R and net charge Q, in electrostatic equilibrium, the electric field in any point P on its surface is given by [ r2 + (R − z)2] 2 according to the well-known formula for the field of a uniformly-charged ring, as given e.g. in Eq (22-16) of Ref.

Results

Conclusion