Introducing corrugated surfaces in electromagnetism problems via perturbative approach.

In physics, problems involving boundary conditions on corrugated surfaces are relevant to understanding nature, since, at some scale, the surfaces manifest corrugations that may 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 planar conducting surface, which is usually done by image method, but also via Green's function. Clinton, Esrick and Sacks [Phys. Rev. B 31, 7540 (1985)], using a perturbative analytical calculation of the Green function, solved this problem introducing corrugation to the surface. In the present paper, we make a detailed pedagogical review of the calculations of these authors. Moreover, we present an original result, applying this perturbative approach to investigate the introduction of corrugation in another very common exercise in electromagnetism courses. Specifically, we solve the Laplace equation for the electrostatic potential to a corrugated neutral conducting cylinder in the presence of an uniform electric field, obtaining how the potential, electric field, and surface charge density are affected by the corrugation. These calculations can be used as pedagogical examples of the application of this perturbative approach in electromagnetism courses.