Magnetic Field on the Axis of Fixed-Perimeter Polygonal Loops: From Local Geometry to the

In this work, an analytical formulation is developed for the magnetic field along the axis of regular polygonal loops with $N$ sides and fixed perimeter $L$, using the Biot–Savart law. A general expression valid for any $N$ is obtained, including the classical square and circular loop cases. The analysis reveals four distinct regimes of the axial magnetic field: local, extended-local, intermediate, and dipolar. In the local regime, loops with fewer sides exhibit a stronger on-axis field due to their smaller apothem; in the intermediate regime, loops with larger $N$ dominate owing to the more homogeneous current distribution; and in the dipolar regime, all geometries converge to the $\propto$ 1/z³ behavior. The normalized variables adopted ($B_z/\left({\mu }_{0}I/L\right)$ and $z/L$) enable direct comparison among all geometries with the same perimeter and clearly illustrate the continuous transition between the physical–geometrical regimes. The results provide a unified treatment that complements and extends the classical solutions for regular polygons, offering substantial didactic value for the teaching of Electromagnetism.

Keywords:
magnetic field; polygonal loops; Biot–Savart law; dipole regime; electromagnetism education