Current geometry and Ampere's law

Question

Under the right circumstances, Ampere's law $\oint \vec H\cdot d\vec \ell=I_{encl}$ can be used to deduce the field $\vec H$ at a point from the current enclosed by the circuit which produces $\vec H$. This can be done when one can find a current-enclosing contour on which the field is constant in magnitude, something that can occur only in highly symmetrical situations: the symmetries of the current distribution are reflected in the symmetries of $\vec H$, meaning that the geometry of the Amperian loop enclosing the current is usually closely related to the symmetry of the source current distribution.

All textbook examples use cylindrical or planar current distributions (or modifications thereof, such as the infinite solenoid or the toroid, or even semi-infinite cylinders), resulting in circular or rectangular loops.

Can people provide examples of other non-trivial current distributions, coordinate systems and contours for which one can put Ampere's law to good use to find the field $\vec H$?

Answer 1

The question behind your question may be: is there a 3D version of the relevant law and its associated theorem? Yes, there is. The equivalences go like this: $$\int_{∂S} d(⋯) = \int_S d×∇(⋯), \quad \int_{∂V} d(⋯) = \int_V dV∇(⋯),$$ where $S$ and $V$ are, respectively, a 2D surface and 3D volume, where $∂S$ and $∂V$ are their respective 1D and 2D boundaries, where $d = (dx, dy, dz)$ is a line element on $∂S$, $d = (dy∧dz, dz∧dx, dx∧dy)$ is surface element on $S$ and on $∂V$, and $dV = dx∧dy∧dz$ a volume element in $V$.

Thus, for the magnetic field $$, one has the usual 2D version of the law: $$\int_{∂S} d· = \int_S d×∇· = \int_S d·∇× = \int_S d· = I_S,$$ as the integral version of $∇× = $, where $$ = + \frac{∂}{∂t}$$ is the "total" current density; but also a 3D version: $$\quad \int_{∂V} d× = \int_V dV∇× = \int_V dV.$$

As to the question, per se: there are no restrictions on the shapes of the 2D region $S$, just so long as it has a well-defined boundary $∂S$. (That means: no fractal surfaces or the like). So the surface $S$ for the current $I_S$ is passing through can be other shapes, not just solid circles or rectangles. The 3D volume $V$ doesn't have to be a cylinder. It can be any shape, as well, just so long as it has a well-defined boundary $∂V$.