Introduction to Electrodynamics · Problem 5.33

The key here is understanding how boundary conditions constrain the fields at a current-carrying surface. Set up coordinates so the current flows in the $x$-direction and $z$ points outward; then use the given equations to relate the tangential and normal components of $\mathbf{B}$ and $\mathbf{H}$ just above and below the surface. The trick is recognizing that the surface current $\mathbf{K}$ acts as a source that discontinuously changes the tangential magnetic field, while the normal component remains continuous—this is exactly what Ampère's law and the no-magnetic-monopole condition enforce at a boundary.

The key here is understanding how boundary conditions constrain the fields at a current-carrying surface. Set up coordinates so the current flows in the $x$-direction and $z$ points outward; then use the given equations to relate the tangential and normal components of $\mathbf{B}$ and $\mathbf{H}$ just above and below the surface. The trick is recognizing that the surface current $\mathbf{K}$ acts as a source that discontinuously changes the tangential magnetic field, while the normal component remains continuous—this is exactly what Ampère's law and the no-magnetic-monopole condition enforce at a boundary.