Boundary Conditions in Time-Varying Electromagnetics

I just finished delivering my lecture on time-varying fields. It is very essential to discuss on the boundary conditions. In fact, I have covered in 4 occasions including today: in electrostatic case, steady current case, magnetostatic case, and time-varying case.

Surprisingly, there is no difference between time-varying as compared to the time-invariant case. The method remains the same, so does the result. I will go through the tangential components from Faraday's Law of Induction and Ampere Circuital Law.
 
From Eq. (1-1), we then draw a reactangle at the boundary between medium 1 and 2 as shown in Figure 1 below:

As goes to zero, we will have the following tangential component:

                           (1-2)