Magnetostatic field: solution to Poisson's equation and Boundary Conditions

[wzy75]

How to derive boundary conditions for interfaces between ferromagnetic material and air?
Please see the attached figure. Any hints will be greatly appreciated!

 

[wzy75]

I can see how the second equation in (7) is derived from the continuity of normal component of B, but still cannot figure out how to derive the first equation in (7), i.e.
$$V_{in}=V_{out}$$.

From the continuity of tangential component of H, I only get
$$\frac{\partial V_{in}}{\partial t}=\frac{\partial V_{out}}{\partial t}$$,
which is different from
$$V_{in}=V_{out}$$.

I must have been missing something here. Could anybody give me some hints?