Introducing corrugated surfaces in electromagnetism problems via perturbative approach

TL;DR

This paper reviews and extends a perturbative method for solving boundary value problems involving corrugated surfaces in electromagnetism, providing pedagogical examples for educational purposes.

Contribution

It offers a detailed review of existing perturbative solutions for corrugated surfaces and introduces a new application to a corrugated conducting cylinder in an electric field.

Findings

Extended perturbative approach to a corrugated conducting cylinder.

Pedagogical examples for electromagnetism courses.

Clarified methodology for boundary problems with corrugations.

Abstract

Problems involving boundary conditions on corrugated surfaces are relevant to understand nature, since, at some scale, surfaces manifest corrugations that have to be taken into account. In introductory level electromagnetism courses, a very common and fundamental exercise is to solve Poisson's equation for a point charge in the presence of an infinity perfectly conducting planar surface, which is usually done by image method. Clinton, Esrick and Sacks [Phys. Rev. B 31, 7540 (1985)] added corrugation to this surface, and solved the problem by a perturbative analytical calculation of the corresponding Green's function. In the present paper, we make a detailed pedagogical review of this calculation, aiming to popularize their results. We also present an original contribution, extending this perturbative approach to solve the Laplace's equation for the electrostatic potential for a corrugated neutral conducting cylinder in the presence of a uniform electric field (without corrugation, this is another very common model considered as an exercise in electromagnetism courses). All these calculations can be used as pedagogical examples of the application of the present perturbative approach in electromagnetism courses.