= Solution
The <Biot-Savart law> gives the <magnetic field> of a steady thin-wire <electric current> in vacuum:
$$
\mathbf B(\mathbf r)=\frac{\mu_0I}{4\pi}
\int\frac{d\boldsymbol\ell'\times(\mathbf r-\mathbf r')}
{|\mathbf r-\mathbf r'|^3}.
$$
The cross product fixes the circulation direction by the <right-hand rule>; fields from different <electric current> elements superpose. Integration gives $B=\mu_0I/(2\pi r)$ outside a straight infinite wire and $B=\mu_0I/(2a)$ at the centre of a circular loop of radius $a$. A distributed <electric current> replaces $I\,d\boldsymbol\ell'$ by $\mathbf J\,dV'$. \b[This is a magnetostatic law:] arbitrary time-dependent <electric currents> require retarded electromagnetic fields, not instantaneous substitution.
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