 Solution

For a steady thin-wire current in vacuum, the Biot-Savart law isThe line element follows conventional current direction. More generally replace $Id{\mathbf{\ell }}^{\prime }$ with $\mathbf{J}({\mathbf{r}}^{\prime })d^3{r}^{\prime }$. The cross product gives the circulating field direction and superposition allows contributions from separate wire segments to be added.

Integration gives $B={\mu }_{0}I/(2\pi r)$ around a long straight wire. On the axis of a circular loop of radius a, $B_z={\mu }_{0}I{a}^2/[2(a^2+z^2{)}^{3/2}]$, including the centre value ${\mu }_{0}I/(2a)$. Coil turns multiply the field when their geometries coincide; distributed coils and solenoids follow by integration. This is a magnetostatic or quasistatic calculation: rapidly changing currents require retarded electrodynamics and appropriate displacement-current effects, not an instantaneous wire formula. ^[ref]