= Solution
For a steady thin-wire current in vacuum, the <Biot-Savart law> is
$$
\boxed{\mathbf B(\mathbf r)=\frac{\mu_0I}{4\pi}\int\frac{d\boldsymbol\ell'\times(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3}.}
$$
The line element follows conventional current direction. More generally replace $I\,d\boldsymbol\ell'$ with $\mathbf J(\mathbf r')d^3r'$. The cross product gives the circulating field direction and superposition allows contributions from separate wire segments to be added.
Integration gives $B=\mu_0I/(2\pi r)$ around a long straight wire. On the axis of a circular loop of radius a, $B_z=\mu_0Ia^2/[2(a^2+z^2)^{3/2}]$, including the centre value $\mu_0I/(2a)$. 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. <University Physics Volume 2 by Samuel J. Ling and others>[Chapter12, steady-current fields and Biot–Savart applications]{ref}
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