= Solution
For steady currents in vacuum, the <Biot-Savart law> sums the contributions of current elements:
$$
\boxed{\mathbf B(\mathbf r)=\frac{\mu_0I}{4\pi}\int
\frac{d\boldsymbol\ell'\times(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3}.}
$$
It is useful for known wire geometries; a circular loop of radius $a$ gives $B=\mu_0I/(2a)$ at its centre.
<Ampère's circuital law> in <magnetostatics> is
$$
\boxed{\oint\mathbf B\cdot d\boldsymbol\ell=\mu_0I_{\rm enc}.}
$$
With sufficient symmetry it finds fields directly: a long straight wire gives $B=\mu_0I/(2\pi r)$, while a long <solenoid> has interior $B\simeq\mu_0nI$ and a small exterior field away from its ends. Without symmetry, a circulation value alone does not determine the local field. For time-dependent <electric fields>, use the <Ampère-Maxwell equation> with the displacement-current term $\mu_0\epsilon_0\,d\Phi_E/dt$ as well as conduction current.
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