 Solution

A thin pillbox applied to $\nabla \cdot \mathbf{B}=0$ and a thin rectangular contour applied to the magnetostatic Ampère law giveThe normal points from medium1 to medium2, and ${\mathbf{K}}_{\mathrm{free}}$ is a free sheet current. Without such a current, tangential $H$ is continuous; normal $B$ is continuous independently. The remaining components need not be continuous because $\mathbf{B}={\mu }_0(\mathbf{H}+\mathbf{M})$ contains the magnetization.

For linear isotropic media with no free sheet current, $B_{1n}=B_{2n}$ and $B_{1t}/{\mu }_1=B_{2t}/{\mu }_2$. If ${\theta }_i$ measures a field-line angle to the normal, $\tan {\theta }_2/\tan {\theta }_1={\mu }_2/{\mu }_1$. This magnetic field-line refraction follows from the two different component conditions, not from continuity of the complete vector field. Bound surface currents are already represented in the constitutive relation and must not be added again as free currents.