  The current density ${\bf j}^{\rm{\bf B}}$ induced in a clean metal by a
slowly-varying magnetic field ${\bf B}$ is formulated as the low-frequency
limit of natural optical activity, or natural gyrotropy. Working with a
multiband Pauli Hamiltonian, we obtain from the Kubo formula a simple
expression for $\alpha^{\rm gme}_{ij}=j^{\rm{\bf B}}_i/B_j$ in terms of the
intrinsic magnetic moment (orbital plus spin) of the Bloch electrons on the
Fermi surface. An alternate semiclassical derivation provides an intuitive
picture of the effect, and takes into account the influence of scattering
processes in dirty metals. This "gyrotropic magnetic effect" is fundamentally
different from the chiral magnetic effect driven by the chiral anomaly and
governed by the Berry curvature on the Fermi surface, and the two effects are
compared for a minimal model of a Weyl semimetal. Like the Berry curvature, the
intrinsic magnetic moment should be regarded as a basic ingredient in the
Fermi-liquid description of transport in broken-symmetry metals.
