Boundary conditions of steady current

The first equation $$vec{nabla}cdot vec{J}(vec{r})=0 tag 1$$ due to Gauss's theorem is equivalent to $$int_{partial S} vec J(vec r)dvec a=0 tag 2$$ This means that there is no time change of charge enclosed in the volume. The derivation of the normal current density boundary condition follows the same path as the for the electrical field boundary conditions for the normal component of the dielectric displacement $vec D$.

The continuity of the normal component of the stationary current density is obtained using eq. (2) with a Gaussian pill box surface enclosing the interface of the media and letting the pillbox height go to zero. In the stationary case, there is no time change of a possible interface charge. This yields $$vec{J}(vec{r})_{1n}=vec{J}(vec{r})_{2n} tag 3$$ The continuity of the tangential component of the electric field follows from $nabla times vec E=0$ and thus $oint vec{E}(vec{r}) dvec{r}=0$ and a closed rectangular integration path at the interface which yields $$E(vec{r})_{1t}=E(vec{r})_{2t} tag 4$$