The Magnetic field for an n-cusped Epi-and Hypo-Cycloids loop current

Abstract

We calculate the magnetic field generated by a steady current that takes the shape of two types of special curves: hypocycloids and epicycloids with n numbers of sides. The computation was performed in the center of the referred curves. For this purpose, we use the Biot-Savart law which is studied in every introductory-level electricity and magnetism course. The result is quite general because it is obtained as a function of the number of sides of the curve and in terms of a parameter that identifies the type of curve considered (= −1 hypocycloids and = +1 epicycloids).

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

“…where a and b are the radii of the circumferences necessary to generate the hypocycloid curve [20]. To obtain a closed and non-intersecting curve, we set the relationship a/b = n, where n is an integer number.…”