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

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 ϵ that identifies the type of curve considered (ϵ = −1 hypocycloids and ϵ = + 1 epicycloids).我们计算了呈两类特殊曲线形态的稳恒电流所产生的磁场：内摆线（hypocycloid）与外摆线（epicycloid），二者均具有n个侧边。本次计算在上述曲线的中心位置展开。为此，我们采用了所有入门级电磁学课程中都会讲授的毕奥-萨伐尔定律（Biot-Savart law）。所得结果具有普适性，因为其表达式以曲线的侧边数为变量，同时通过参数ϵ来区分所考虑的曲线类型：当ϵ = −1时为内摆线，ϵ = +1时为外摆线。