Bound Surface current across boundary of Amperian loop

Question

For a Bound surface current $\vec{K}_{b}$ 'flowing' across through the boundary of an Amperian loop is reflected by the expression $\int \vec{K}_{b}dl_{\perp}$

What does $\int \vec{K}_{b}dl_{\perp}$ means?

Thanks in advance.

Answer 1

Consider a narrow strip parallel to $K$ of width $\Delta l$. The current on the strip is $K \Delta l$. The integral represents the current passing through a line on the surface. It can be thought of as a sum of infinitesmal strips, with only the components of $\Delta l$ perpendicular to $K$ contributing.