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 $\vec H$?
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