Ohmic conductor ($\mathbf{J}=\sigma \mathbf{E}$)

A material whose current density is proportional to its electric field, with proportionality given by its electrical conductivity.

In a homogeneous Ohmic conductor, a steady electric current density obeys $\nabla \cdot \mathbf{j}=0$ and $\nabla \times \mathbf{j}=0$. Uniform axial current in a straight cylinder follows if its side is insulated and its flat end faces are equipotentials. The electric potential then solves Laplace's equation with those boundary conditions, and uniqueness makes it linear in the axial coordinate. Steadiness alone does not impose these contact conditions: nonuniform end potentials can produce nonuniform current even with an insulating side. Bending also generally redistributes current; in an annular conductor the azimuthal current density is proportional to the inverse radius.

In a homogeneous Ohmic conductor, the charge continuity equation and Gauss's law imply exponential decay of the bulk charge density, $\rho (t)=\rho (0)e^{-\sigma t/ϵ}$. For an isolated finite body, bulk charge moves to its boundary; the accumulating surface charge density preserves total electric charge. The formula uses a constant permittivity $ϵ$ and the corresponding free-charge convention, or ${ϵ}_0$ for total charge in the vacuum form of Gauss's law.

A uniform cylindrical volume charge relaxes exponentially under Ohm's law and the continuity equation for electric charge. In a cylinder of radius $R$ initially carrying no surface charge, the radial conduction current deposits ${\rho }_s(t)=R{\rho }_0(1-e^{-\sigma t/{ϵ}_0})/2$ on its surface. Total charge per unit length remains $\pi R^2{\rho }_0$, so the exterior radial electric field is time independent even though the interior field decays. Neglecting the deposited surface charge violates charge conservation.