1. The problem statement, all variables and given/known data Per Maxwell's equation and my H. H. Skilling EM textbook, curl H = 0 in the absence of current density. But consider a long, thin wire along the z axis carrying time-invariant current I. By Ampere's law, at a point (x,0) outside the wire, H = I/2πx j . But curl H computed by Cramer's rule from the curl determinant is not zero. 2. Relevant equations curl H = J + ∂D /∂t 2πxH = I 3. The attempt at a solution Since I is time-invariant there is no time-varying electric field so ∂D/∂t = 0. Since H = H j = I/2πx j at (x,0), curl H = ∂H/∂x k - ∂H/∂z i but for a long wire, ∂H/∂z = 0 so we wind up with curl H = dH/dx k = -I/2π x2 k ≠ 0. So how is the paradox resolved, please? If you're familiar with the concept of little "paddle wheels" (Skilling uses) indicating curl, even there it seems curl H ≠ 0 outside the wire. All glory and thanks to the resolver!