= Electrical conductivity
{title2=$\sigma$}
<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> is
$$
\mathbf J=\sigma\mathbf E,\qquad [\sigma]=\frac{\mathrm{A/m^2}}{\mathrm{V/m}}=(\Omega\,\mathrm m)^{-1}.
$$
Larger 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. https://openstax.org/books/university-physics-volume-2/pages/9-3-resistivity-and-resistance[Ling, Sanny and Moebs, §9.3: conductivity and resistivity]{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 $E=V/L$. Thus
$$
I=jA=\sigma EA=\frac{\sigma A}{L}V,\qquad R=\frac{L}{\sigma A}.
$$
For the defined <electrical resistance> example $\sigma=5\times10^5\,(\Omega\,\mathrm m)^{-1}$, $L=1\,\mathrm m$, $A=10^{-6}\,\mathrm{m^2}$ and $V=5\,\mathrm V$, the local values are $E=5\,\mathrm{V/m}$ and $j=2.5\times10^6\,\mathrm{A/m^2}$; 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>.
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