Magnetostatic Boundary Conditions

Inside a homogenous magnetic medium, the magnetic field is continuous. However,  on crossing a boundary, it becomes discontinuous.

Consider a Gaussian pillbox across the boundary. The zero divergence of the magnetic field ensures that the total magnetic flux enclosed within the pillbox is zero. So, the normal components of the field are equal across the boundary.

Now, consider an Amperian loop perpendicular to the surface current and apply Ampere's Law to each loop section.

The line integral of the magnetic field exists only for the loop's top and bottom sections since the loop width tends to zero. So, the field's tangential component is discontinuous.

The discontinuity of the tangential components can be expressed in vectorial form.

Now, consider an interface between two media. The divergence of magnetic field intensity equals the negative divergence of the magnetization. So, the normal component of magnetic field intensity is discontinuous.

Consider an Amperian loop perpendicular to the surface current. Applying Ampere's law, the tangential component of magnetic field intensity is discontinuous. This discontinuity can be expressed in vector form.