Define Resistivity and conductivity | S.I Unit and Dimensions | Factors on depends
Resistivity :
$\rho = R \left( \frac{A}{l} \right)$
If $A = 1$, $l = 1$, then $\rho = R$
resistivity of a conductor of a given material is defined as the resistance of the conductor of unit length and unit area of cross-section.
S.I. Unit of Resistivity :
Since, $\rho = R \left( \frac{A}{l} \right)$, therefore, S.I. unit of $\rho$ is $\frac{\text{ohm metre}^2}{\text{metre}}$ or ohm-metre ($\Omega$ m)
Dimensional formula of resistivity :
Resistivity, $[\rho] = \frac{[R] \times [A]}{[L]} = \frac{[ML^2T^{-3}A^{-2}] [L^2]}{[L]} = [ML^3T^{-3}A^{-2}]$
conductivity :
Electrical conductivity or conductivity of a substance is equal to the inverse of its resistivity.
That is,
$ \sigma = \frac{1}{\rho}$
S.I. unit of conductivity :
$\Omega^{-1} m^{-1}$ or $Sm^{-1}$.
Dimensions of Conductivity :
$[\sigma] = \frac{1}{[\rho]} = [M^{-1}L^{-3}T^3A^2]$
$ \rho = \frac{m}{ne^2\tau$ $
and
$ \sigma = \frac{1}{\rho} = \frac{ne^2\tau}{m} \quad \ldots (1)$
Factors on which resistivity ($\rho$) of conductor depends :
we find that resistivity of a conductor depends on (i) $n$, number of electrons/volume and (ii) $\tau$, the relaxation time. That is, (i) $\rho \propto n$ and (ii) $\rho \propto \frac{1}{\tau}$.
(i) Dependence of $\rho$ on $n$ :
Since number of free electrons per unit volume ($n$) is different in different materials, therefore, resistivity of a conductor depends on the nature of the material of the conductor.
(ii) Dependence of $\rho$ on $\tau$ :
Relaxation time ($\tau$) decreases with the increase in temperature of the conductor. Therefore, resistivity of a conductor increases with increase in temperature and vice-versa. That is, $\rho \propto T$, where T is the temperature of the conductor.
Factors affecting conductivity ($\sigma$) of a conductor :
We find that $\sigma \propto$ (i) $n$ and $\sigma \propto$ (ii) $\tau$.
Therefore, conductivity of a conductor depends on
(i) Nature of the material of a conductor: Different materials have different number density of electrons ($n$), so conductivity of different materials is different.
(ii) Temperature of the conductor : When temperature of a conductor increases, the relaxation time ($\tau$) decreases. Hence, conductivity of a conductor decreases. On the other hand, when temperature of a conductor decreases, relaxation time increases.
Hence, the conductivity of a conductor increases. Thus, $\sigma \propto \frac{1}{T}$, where T is the temperature of the conductor.
The variation of conductivity of a conductor with temperature :