What is the role of surface currents in Ampere's Law with H fields?

[McLaren Rulez]

I'm using Griffiths Electrodynamics and I can't figure out something. For reference, this is on page 269 of the third edition.

In the derivation for Ampere's law as it applies to the auxillary field H (i.e integral of H over a loop = free current passing through), there seems to be no word on the surface currents. If I had an Amperian loop that cut through the boundary between two surfaces, shouldn't we also include the bound surface current K_b along with whatever free current passes through that loop?

Thank you, in advance.

 

[Meir Achuz]

H is introduced so that Ampere's law with H includes only free currents.

 

[netheril96]

You said, free currents. So why include bound currents?

 

[yungman]

Bounded or not, surface current is on the surface, it does not cut through the surface, right? So the surface current whether it is bounded or free don't cut through the surface ( or the current don't string the Amperian loop.).