Ampere's Law, Interface conditions for magnetic field

I'm failing to understand the derivation of the interface conditions for the tangential components of the magnetic field given her (based on d.j,griffiths)

Ampere's law in integral form is given as $$\oint_C \mathbf{H}\cdot d\mathbf{l}=\int_S(\mathbf{j_f}+\frac{\partial \mathbf{D}}{\partial t})\cdot d\mathbf{a}$$ where $$\mathbf{j}_f$$ is the current-density of free charge carriers. Now considering the gaussian rectangle below, in the limit $$h\rightarrow 0$$ gives $$H^\parallel_1 l-H^\parallel_2 l=I_f$$,where $$I_f$$ represents the enclosed current of free charges $$I_f=\int_S \mathbf{j}_f\cdot d\mathbf{a}$$

fine.

Then the author says

...The free surface current is the product of a surface current density $$\mathbf{K}_f$$ and the width of the loop;...$$H^\parallel_1 l-H^\parallel_2 l=K_f l$$

This is what confuses me:Where is $$\mathbf{K}_f$$ coming from? We already have a free surface current density- and it was called $$\mathbf{j}_f$$. What's the difference between these two? And how can a surface current density $$\mathbf{K}_f$$ have the same units as a magnetic field (A/m)? This doens't look like a density to me.

• $J_f$ is volume current, ie. the free charges if they were to stop for an instant would have nonzero volume density. $K_f$ is surface current ie., so the corresponding moving free charges would have zero volume density but not zero surface density. – hyportnex Feb 12 at 16:51
• Are you sure? S.I. units of D are C/m^2 so the time-derivative of D has units of A/m^2 and J_f has to match these units..isn't that a surface-current-density? – OD IUM Feb 12 at 16:58
• $J_f$ represents current (charges per second) passing through a unti surface, but if the charges that are moving at speed $v$ stopped for an instant ($\delta t$ seconds) then they would represent a volume of charges $v\delta t dA$ passing through that unit surface $dA$. Similarly for $K_f$ but now charges confined to a surface passing through a unit length of a line – hyportnex Feb 12 at 17:10
• ok,that helps me – OD IUM Feb 12 at 17:13
• Note: because of the skin effect this concept of surface current is especially useful for high frequency currents in metals. – hyportnex Feb 12 at 17:18