Ampere's Law The original static law of Ampere (in integral form) was that the "circulation" (i.e., 1-D closed loop line integral of the scalar product of the magnetic field vector B with the infinitesimal tangential line element dl) is proportional to the electric current passing through any interior 2-D surface which has the closed loop as its boundary.

Note that this law is essentially topological because one can distort the shape of the surface without tearing it without changing the result. In fact, as shown by John Wheeler, all of Maxwell's electromagnetic field equations have the same profound topological foundation that "the boundary of a boundary is zero".

Maxwell's stroke of genius was to add the displacement current term which is the rate of change of the electric flux through the same surface. This implies there can be an effective electric current even in empty space where there are no actual charges. It is this displacement current which explains how transverse electromagnetic waves of radiation can propagate to infinity.

In local differential form, this Maxwell equation says that the vector curl of the magnetic field at a point in space at a given time equals the material vector electric current density plus the partial time derivative of the electric vector at that same point in space and time.