 Magnetic-field jump across a surface current

A surface current density $\mathbf{K}$ concentrated on an interface creates a discontinuity of tangential magnetic field. Let the unit normal $\mathbf{n}$ point from side $-$ to side $+$. In SI units, the microscopic Ampere-Maxwell law givesHere the surface current includes both free current and bound current. An infinitesimal rectangular integration loop straddling the sheet derives the tangential jump. The no-monopole equation separately gives $\mathbf{n}\cdot ({\mathbf{B}}_{+}-{\mathbf{B}}_{-})=0$.

In a macroscopic material description, the magnetic field intensity instead satisfiesWriting a jump in $\mathbf{B}$ using only free current without accounting for magnetization would mix these two descriptions. In Gaussian units, the total-current magnetic-field jump is $4\pi {\mathbf{K}}_{\mathrm{total}}/c$.

For an isolated uniform sheet in vacuum, reflection symmetry givesFor example, current along $+x$ on $z=0$ gives a field along $-y$ above the sheet and $+y$ below it. The directions agree with the right-hand rule.