A few paradoxes of Gauss' law and how to avoid them

TL;DR

This paper examines paradoxes arising from the application of Gauss' law in certain charge distributions, identifies their root causes, and proposes solutions to avoid these contradictions in standard textbook applications.

Contribution

It reveals specific charge configurations that cause contradictions in Gauss' law and offers a method to resolve these issues for accurate application.

Findings

Identified charge distributions causing contradictions

Root causes of paradoxes in Gauss' law application

Proposed solutions to avoid these paradoxes

Abstract

We present a few charge distributions for which the application of Gauss' law in its integral form, as typically outlined in standard textbooks, results in a contradiction. We identify the root cause of such contradictions and put forward a solution to resolve them.