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

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

$$\nabla\!\cdot\!{\bfm B} = 0,$$ (549)

and

$$\nabla\wedge{\bfm H} = {\bfm j}_t.$$ (550)

Integrating Eq. (3.128) over a Gaussian pill-box enclosing part of the boundary surface between the two media gives

$$({\bfm B}_2 - {\bfm B}_1)\!\cdot\!{\bfm n}_{21} = 0,$$ (551)

where ${\bfm n}_{21}$ is the unit normal to this surface directed from medium 1 to medium 2. Integrating Eq. (3.129) around a small loop which straddles the boundary surface yields

$$({\bfm H}_2 - {\bfm H}_1)\wedge {\bfm n}_{21} = 0,$$ (552)

assuming that there is no true current sheet flowing in this surface. In general, there is a magnetization current sheet flowing in the boundary surface whose density is given by

$${\bfm J}_m = {\bfm n}_{21}\wedge({\bfm M}_2 - {\bfm M}_1),$$ (553)

where ${\bfm M}_1$ is the magnetization in medium 1 at the boundary, etc. It is clear that the normal component of the magnetic field and the tangential component of the magnetic intensity are both continuous across any boundary between magnetic materials.