Boundary conditions for ${\bf B}$ and ${\bf H}$

What are the boundary conditions for ${\bf B}$ and ${\bf H}$ at the interface between two magnetic media? The governing equations for a steady-state situation are

$$\nabla\!\cdot\!{\bf B} = 0,$$ (870)

and

$$\nabla\times{\bf H} = {\bf j}_t.$$ (871)

Integrating Eq. (870) over a Gaussian pill-box enclosing part of the interface between the two media gives

$$B_{\perp 2}-B_{\perp 1}= 0,$$ (872)

where $B_\perp$ denotes the component of ${\bf B}$ perpendicular to the interface. Integrating Eq. (871) around a small loop which straddles the interface yields

$$H_{\parallel 2}-H_{\parallel 1} = 0,$$ (873)

assuming that there is no true current sheet flowing in the interface. Here, $H_\parallel$ denotes the component of ${\bf H}$ parallel to the interface. In general, there is a magnetization current sheet flowing at the interface whose density is of amplitude

$$J_m= \frac{B_{\parallel 2}-B_{\parallel 1}}{\mu_0}.$$ (874)

In conclusion, the normal component of the magnetic field and the tangential component of the magnetic intensity are both continuous across any interface between magnetic media.