Current geometry and Ampere's law

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 $\vec H$?

• You can use the differential form $\vec\nabla\times{}\vec{B}=\mu_0\vec{J}$ in any geometry, and you probably do every time you use a FEM method to solve a magnetics problem. – The Photon Mar 12 '17 at 16:44
• @ThePhoton I can't see your suggestion being correct. For an infinitely long wire carrying current $I$ the $\vec B$ field at distance $\rho$ is just $\vec B=\frac{mu_0I}{2\pi \rho}\hat \phi$ and one easily verifies that $\nabla\times \vec B=0$ near that point, consistent with $\vec J=0$ near that point. Of course this is expected since $\vec\nabla\times B$ is local whereas the integral form of Ampere's law is global. – ZeroTheHero Mar 12 '17 at 17:45
• The two forms are mathematically equivalent according to Stokes' theorem. – The Photon Mar 13 '17 at 15:39
• The differential form works at every point not on contours and surfaces. So if you are using the differential form, you will have to work out the curl of B at every point. – PhyEnthusiast Mar 16 '17 at 11:57
• Ampere's law is used to derive the continuity conditions for H. Hence Ampere's law is used extensively whenever there is an interface between different media. – Rob Jeffries Nov 25 '18 at 23:03