by wendy » Tue Oct 04, 2016 7:56 am
The general form of the ampere law in higher dimensions does not suppose current, but 'circulation'.
A ring here is a division-boundary on a solid, such as the outline of a section against one axis. But the division does not have to be coplanar. Such a ring, when spanned by a surface (N-1 space), gives rise to a 'vector area', which is independent of the shape of the surface. This is the general reflex of the rule m = IA (magnetic moment = current * area).
This means that the ring has a parity, in the form of a circulation, (I), which gives a sense of direction to A (ie the normals to A, which are added vectorially, acquire their sense from the perimeter). As with the ordinary partition of the area into a mesh, and each little mesh-cell has the same I and sense, this replicates in every dimension.