Ohm's Law, Resistance, Resistivity and Conductivity

Ohm's law states that, provided the physical conditions of a conductor (in particular, its temperature) remain unchanged, the current $I$ flowing through it is directly proportional to the potential difference $V$ applied across its ends:

The constant of proportionality, $R$, is called the electrical resistance of the conductor. Its SI unit is the ohm ($\Omega$), defined so that $1\ \Omega = 1\ \text{V/A}$ -- a conductor has a resistance of one ohm if a potential difference of one volt across it drives a current of one ampere through it.

Resistance depends both on the conductor's material and on its geometry (length and cross-sectional area). For a conductor of uniform cross-section, length $l$ and area $A$, it is found experimentally (and can be derived from the microscopic relation $I = nAe\mu E$ of Section 3.4) that

where the constant of proportionality $\rho$ is called the resistivity (or specific resistance) of the material. Resistivity depends only on the nature of the material (and its temperature) -- NOT on the conductor's particular length or thickness -- which is what makes it useful as a genuine material property, tabulated once for a substance and then usable for a wire of any size cut from it. Its SI unit is the ohm-metre ($\Omega\,\text{m}$), obtained by rearranging $\rho = RA/l$.

Combining $R = \rho l/A$ with $I = nAe\mu E$ and $V = El$ (for a uniform field along a wire of length $l$) shows that resistivity can be written microscopically as

connecting the macroscopic, measurable resistivity directly to the microscopic free-electron density $n$ and relaxation time $\tau$. …