Surface current density ($\mathbf{K}$)

A surface current density is electric current per unit transverse length confined to a surface. Integrating a thin layer's current density through its thickness gives $\mathbf{K}$.

A current sheet idealizes electric current as supported on a surface. It produces a jump in the tangential magnetic field.

Two broad perfect conductors of width $w$, separated by gap $d\ll w$, carry opposite steady electric currents $I$ and connect through a terminating resistor $R$. Away from edges, Ampère's circuital law and the superposition principle give gap magnetic field magnitude ${\mu }_{0}I/w$. Each conductor is an equipotential, and Ohm's law sets their electric potential difference to $IR$. Planar symmetry and charge-free Gauss's law then give uniform gap electric field magnitude $IR/d$. Injection parallel to the upper current makes the electric and magnetic Lorentz forces opposite. Their cancellation selects speed $Rw/({\mu }_{0}d)$ independent of nonzero electric charge and of operating current; nonrelativistic kinetic energy is $m[Rw/({\mu }_{0}d){]}^2/2$.

The Lorentz force per unit area on an ideal current sheet is computed using the mean of its limiting magnetic fields. The sheet's own local magnetic field is equal and opposite on its two sides and cancels in this mean. Equivalently, integrate the Lorentz force density through a thin regularized layer: its self-field contribution averages to zero, leaving the external field acting on the surface current density.

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.

For a thin layer with essentially uniform tangential electric field, integrating electrical conductivity through its thickness gives the sheet conductivity tensor ${\mathbf{\sigma }}_s$. The two-dimensional form of Ohm's law is $\mathbf{K}={\mathbf{\sigma }}_s{\mathbf{E}}_{\parallel }$, with surface current density $\mathbf{K}$. Its SI unit is $\mathrm{S}=A/V$: amperes per metre divided by volts per metre. When response is the same in every in-plane direction, the sheet conductivity reduces to a scalar $\Sigma$; for uniform bulk electrical conductivity $\sigma$ and thickness $t$, $\Sigma =\sigma t$.