Specific Resistance (Resistivity)

At a given temperature, the resistance of a conductor is found experimentally to depend on three things: the nature (material) of the conductor, the length of the conductor, and the area of its cross-section. Specifically, resistance $R$ of a conductor of uniform cross-section is found to be:

1. directly proportional to its length $l$, i.e. $R\propto l$, and
2. inversely proportional to its cross-sectional area $A$, i.e. $R\propto \dfrac{1}{A}$. Combining (i) and (ii),

   $R=\rho\frac{l}{A}\qquad\text{--- (11.31)}$

   where $\rho$ (rho), the constant of proportionality, is called the SPECIFIC RESISTANCE or RESISTIVITY of the conductor's material at that temperature. Rearranging Eq. (11.31),

   $\rho=\frac{RA}{l}\qquad\text{--- (11.32)}$

   The SI unit of resistivity is the ohm-metre ($\Omega$m). Numerically, resistivity is the resistance of a sample of the material that has unit length AND unit cross-sectional area at once: when $R=1\,\Omega$, $A=1\,\text{m}^2$ and $l=1\,\text{m}$, then $\rho=1\,\Omega\text{m}$. CONDUCTIVITY, $\sigma$, is defined as the reciprocal of resistivity, $\sigma=\rho^{-1}$, with SI unit $(\Omega\text{m})^{-1}$, i.e. siemens per metre (S m$^{-1}$). Table 11.1 lists representative resistivity values spanning conductors (of order $10^{-8}\,\Omega$m), semiconductors (intermediate values), and insulators (as high as $10^{16}\,\Omega$m) -- a span of roughly 24 orders of magnitude, illustrating just how dramatically materials differ in their ability to conduct. Resistivity is a property of the MATERIAL itself, independent of the sample's particular shape, while resistance $R$ refers to one particular object of specific length and area. In the same spirit, the electric field $\vec{E}$ at a point inside a material (rather than the potential DIFFERENCE across a whole resistor) and the current density $\vec{J}$ at a point (rather than the total current $I$ through a whole resistor) are the local, per-point analogues used for an isotropic material:

   $\frac{E}{J}=\rho\qquad\text{or equivalently}\qquad E=\rho J\qquad\text{--- (11.33)}$

   Dimensionally, the SI unit of $\rho$ can be checked from Eq. (11.33): the unit of $E$ is V/m and the unit of $J$ is A/m$^2$, so the unit of $\rho=E/J$ is $\dfrac{\text{V/m}}{\text{A/m}^2}=\dfrac{\text{V}}{\text{A}}\cdot\text{m}=\Omega\text{m}$, confirming the ohm-metre unit stated above. …