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.

The propagation of electromagnetic waves in a medium with electrical and magnetic anisotropy is a subject that is not usually handled in conventional optics and electromagnetism books. During this work, we try to give a pedagogical approach to the subject, using tools that are accessible to an average physics student. In this article, we obtain the Fresnel relation in a media with electromagnetic anisotropy, which corresponds to a quartic equation in the refraction index, assuming only that the principal axes of the electric and magnetic tensors coincide. Additionally, we find the geometric location related to the different situations the discriminant of the quartic equation provides. In order to illustrate our findings, we determine the refractive index together with the plane wave equations for certain values of the parameters that meet the established conditions. The target readers of the paper are students pursuing physics at the intermediate undergraduate level.