The electric potential of an infinite conducting cylinder with an n-cusped hypocycloidal cross-section

Abstract

Generally, during a course in electromagnetism, boundary conditions are used in conjunction with the Laplace equation to determine the electric potential of a system of objects in regions of space free of electric charges. However, for objects with unconventional geometries such as the hypocycloid, this is not an easy task. In the case where the problem can be reduced to two dimensions, there are simpler approximations such as complex-variable with conformal transformation. In this work, we use the last approach, to calculate analytically the electric potential of an infinite conducting cylinder with an n-cusped hypocycloidal cross-section and charge Q per unit length. In addition, we verify some of the results using numerical methods.