Define Resistivity and conductivity | S.I Unit and Dimensions | Factors on depends

Define Resistivity and conductivity | S.I Unit and Dimensions | Factors on depends

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 : 

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