Kelvin transformations in arbitrary dimensions

Following the initial idea presented in Amaral et al (2017 Eur. J. Phys 38 025206), we address Kelvin's inversion of coordinates, which exchanges the radial coordinate for its inverse, in arbitrary dimensional electrostatics. For two- and three-dimensional configurations (planar and cylindrical symmetries) detailed analyses are provided. Using this symmetry, which is rather overlooked in usual electromagnetism textbooks, a wide range of problems is correlated showing that some complicated situations are much simpler in a dual description. We also show that the Lagrangian changes by a total derivative term under the studied inversion leading to action invariance.