Ohm's law states that the potential difference across an ohmic component equals its current times its resistance: .
Electrical electrical conductivity relates current density to the local electric field in the rest frame of an isotropic ohmic medium: . It is a bulk material coefficient, distinct from a sheet surface electrical conductivity. A moving conductor instead uses the moving-conductor Ohm law.
A material whose current density is proportional to its electric field, with proportionality given by its electrical conductivity.
In a homogeneous Ohmic conductor, the charge continuity equation and Gauss's law imply exponential decay of the bulk charge density, . 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 and the corresponding free-charge convention, or for total charge in the vacuum form of Gauss's law.
The Drude model describes classical charge carriers accelerated by an applied electric field and randomized by collisions of constant mean waiting time . Its mean drift obeys , and harmonic conductivity is in the negative-frequency exponential convention.
The Drude model relative permittivity is . At frequencies large compared with the collision rate but below , its leading real part is negative and the physical spatial wave-number branch has positive imaginary part. The field therefore decays into the conductor; close to the threshold the full complex response must be retained.
Joule heating converts electrical work into internal energy. A resistor dissipates ; in an ohmic continuum the local rate per volume is , with resistivity . Wind-driven currents are a proposed source of heating of an inflated giant planet.
