On the electrostatic equilibrium of charges and cavities in a conductor

TL;DR

This paper rigorously proves that in electrostatic equilibrium, the electric field within each cavity of a conductor depends only on charges inside that cavity and on the surface charges of that cavity, clarifying an elementary but often overlooked property.

Contribution

It provides two complete proofs of the property that the field in each cavity depends solely on internal and surface charges, filling a gap in standard electrostatics literature.

Findings

Confirmed the independence of cavity fields from external charges

Provided rigorous proofs using electrostatic uniqueness theorems

Clarified an elementary property often not fully justified in textbooks

Abstract

We consider a charged conductor of arbitrary shape, in electrostatic equilibrium, with one or more cavities inside it, and with fixed charges placed outside the conductors and inside the cavities. The field inside a particular cavity is then only due to charges within that cavity itself and to the surface charge induced on the surface of the same cavity. A similar statement holds for the exterior of the conductor. Although this is an elementary property of conductors, it is not a trivial statement, as explained in this article. Undergraduate texts in electrodynamics do not discuss at length or provide a complete argument for an important problem such as this. Two simple and complete proofs are provided in this note with the help of the standard electrostatic uniqueness theorems.