Magnetostatic Boundary Conditions

The zero divergence of the magnetic field ensures that the total magnetic flux entering and leaving a Gaussian pillbox at the boundary must be equal. This means the normal components of the magnetic field on both sides of the boundary are identical, making the normal component continuous regardless of the media properties.

Surface currents at the boundary create a discontinuity in the tangential component of the magnetic field. Using an Amperian loop perpendicular to the surface current, Ampere's Law shows that the tangential field component jumps by an amount equal to the product of vacuum permeability and the surface current density.

For magnetic field intensity, the normal component is discontinuous when magnetization differs between media, while the tangential component remains continuous if no free surface currents exist. In contrast, for magnetic field itself, the normal component is continuous due to zero divergence, but the tangential component is discontinuous at surface currents.

Magnetic field lines emerging from a ferromagnetic medium into air approach a direction nearly normal to the interface. This occurs because the permeability ratio causes the refraction angle to approach zero, forcing magnetic flux lines to exit perpendicular to the boundary rather than at oblique angles.

The magnetic vector potential is continuous across any boundary between two media, similar to the scalar potential in electrostatics. However, the derivative of the vector potential is discontinuous at the boundary, reflecting the discontinuity in the tangential component of the magnetic field.

The ratio of tangential to normal components of magnetic field intensity is proportional to the permeability of each medium. This relationship mirrors Snell's Law for dielectrics, where higher permeability in one medium causes the magnetic field to refract more parallel to the interface, similar to how refractive index affects light bending.

The divergence of magnetic field intensity equals the negative divergence of magnetization at the interface. When magnetization differs between the two media, this creates a discontinuity in the normal component of magnetic field intensity, even though the normal component of the magnetic field itself remains continuous.