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Electrical conductivity $\sigma$ measures the local response of current density to electric field. In the rest frame of an isotropic Ohmic conductor, the local form of Ohm's law isLarger positive electrical conductivity gives larger current density at the same electric field. It is a bulk material coefficient, distinct from the geometry-dependent electrical conductance of a whole component and from a sheet's surface conductivity. Electrical resistivity is ${\rho }_{\mathrm{el}}=1/\sigma$ in this scalar model. ^[ref].

The local and whole-wire forms fit together particularly neatly. For uniform current in a straight homogeneous wire, with length $L$, wire cross-sectional area $A$, constant electrical conductivity and terminal electric potential difference $V$, the axial electric field has magnitude $E=V/L$. ThusFor the defined electrical resistance example $\sigma =5\times 10^5(\Omega \mathrm{m}{)}^{-1}$, $L=1\mathrm{m}$, $A=10^{-6}{\mathrm{m}}^2$ and $V=5\mathrm{V}$, the local values are $E=5V/m$ and $j=2.5\times 10^{6}A/{\mathrm{m}}^2$; integrating gives $I=2.5\mathrm{A}$. One material law therefore supplies both a spatial description and a terminal relation.

The scalar form requires isotropy. In an anisotropic material the response can instead be a tensor, $J_i={\sum }_j{\sigma }_{ij}{E}_j$, and current density need not be parallel to electric field; uniaxial conductivity tensor is one existing example. These statements concern the material's rest frame. A moving electrical conductor instead uses the moving-conductor Ohm law.

In the wing frame the unperturbed plasma moves with velocity $-u_0{\mathbf{e}}_x$. The ideal magnetohydrodynamics condition $\mathbf{E}+\mathbf{u}\times \mathbf{B}/c=0$ therefore produces the motional electric fieldThe wing's surface conductivity then gives the surface current densityIts direction also follows directly from the Lorentz force on the wing's mobile charges.