Direct current from curl in magnetic field From differential form of the Ampère-Maxwell equation,
$$\nabla\times{\bf H} = {\bf J} + \frac{\partial\bf D}{\partial t}$$
a curl in magnetic field $\bf H$ is established by a current density $\bf J$ and a time-varying displacement current $\bf D$.
It doesn't seem like the opposite should be true, that a magnetic field with non-zero curl at some point creates a current density or time-varying displacement current. If an infinitely-long wire was placed parallel to the x-axis in a field described by ${\bf H} = y\hat{k}$ would this result in some direct current in the wire? What am I failing to consider?
 A: Magnetic fields are produced by moving electric charges and the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property, their spin. You need current density or moving charges to produce a magnetic field.
A: Ampère-Maxwell equation establishes a relation between three fundamental fields in electrodynamics: ${\bf J}$, ${\bf H}$, and ${\bf D}$. The way such a relation can be interpreted depends on the details of the physical situation.
If the current ${\bf J}$ is an externally controlled physical quantity, it may be considered as a source in the full set of Maxwell equations. However, although common in many applications,  this is a simplification of a more general situation where ${\bf J}$, or part of it, may depend on the fields. In this case, a self-consistent solution of the full set of Maxwell equations and the constitutive relation ${\bf J}={\bf J}({\bf E},{\bf B})$ is required. In such situations, it isn't very meaningful either to consider ${\bf J}$ as the source of the fields or to look at the curl of ${\bf H}$ as the origin of the current.
