Current charge density of the $H$-field

Question

In this Wikipedia article: https://en.wikipedia.org/wiki/Magnetic_field#:~:text=south%20to%20north.-,H%2Dfield%20and%20magnetic%20materials,-Comparison%20of%20B

we see how the volume free current charge density and the surface free current charge density are found. While I am able to understand the derivation of the volume free current charge density, I cannot understand how was the surface free current charge density was derived:

$$\vec H^{||}_1 - H^{||}_2=\vec K_f \times \vec n.$$

I have to questions regarding this equation:

1. How was the part in the right side derived?
2. How can I understand (geometrically) how is that cross product happening? I don't know how the surface free current charge density $\vec K_f$ flows in the surface to even understand to do the cross product and also know in which direction does the resulting vector points at?

Answer 1

I found a neat derivation of the above equation using a boundary condition motivated by Ampere's Law. See this.

So $K_f$ is the surface current density and $\hat n$ is the normal vector pointing in the direction of the separation from medium 1 to medium 2, normal to the boundary of separation. So whatever $H_1-H_2$ is, it must be normal to the direction of the normal vector. Try to imagine possible orientations and use the right hand rule to get the direction of $H_1-H_2$. Hope this helps.