Ampere's Circuital Law

Ampere's law for magnetic fields can be written as $$\boxed{\oint \vec{B}\cdot\mathrm{d}\vec{s} = \mu_0 I}$$ For concentric rings around an electric wire, the infinitesimal path length $\mathrm{d}s$ can be written as $\mathrm{d}s = r\,\mathrm{d}\varphi$, where $r$ is the distance to the wire. In this case the integral can be written as $$Br\int_0^{2\pi} \mathrm{d}\varphi = \mu_0 I$$ Since $B$ and $r$ are constant along one circle, they can be drawn inf front of the integral. Calculating this integral and solving it for $B$ leads to $$\boxed{B = \frac{\mu_0 I}{2\pi r}}$$ This formula shows that the magnetic field decreases with the inverse distance to the center of the wire.