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The Biot-Savart law isHere $r^{\prime }$ is the current-element position, $d{\ell }^{\prime }$ follows the electric current, and $r$ is the observation point. For a circular current loop, transverse components cancel on the axis. Each element contributes axial factor $a/\sqrt{a^2+x^2}$ in addition to the inverse-square distance. Integrating around length $2\pi a$ gives the on-axis magnetic field of a circular current loopwith axial direction set by the right-hand rule.

For a finite straight wire parametrized by $z\in [-b,b]$, at perpendicular distance $a$, the Biot-Savart law givesThe direction is azimuthal about the current axis by the right-hand rule. In the hairpin, the semicircle contributes ${\mu }_{0}I/(4a)$ and each semi-infinite straight segment contributes ${\mu }_{0}I/(4\pi a)$. For the indicated current, all three contributions point out of the page. Thus the magnetic field at the center of a hairpin wire is