Mobility and Conductivity

According to classical free electron theory the electrons in a metal make random elastic collisions in all directions and the net current is zero.

When a constant electric field is applied to the metal, the electrons are accelerated towards the positive of the field.
During their movement, the random collisions are produced and it has attained a drift velocity V_d in the opposite direction of the applied electric field. The electrons move with an average velocity is called drift velocity.
Drift velocity $V d$ is proportional to the applied electric field $E$ .

i.e V_d∞ E

V_d= μ E

Where $μ$ is called mobility of the electrons.

Mobility is defined as the drift velocity gained by the electron per unit electric field.

μ = V d / E, m 2 V -1 s -1

The steady state drift velocity of the electrons produces a current. (I)

$\begin{aligned} I=\frac{Q}{t} & =\frac{\text { Charge }}{\text { time }} \\ I & =\frac{n e A \ell}{t}=n_{e A V} \end{aligned}$

$\left\lvert\, \because \mathrm{V}_{\mathrm{d}}=\frac{\ell}{\mathrm{t}}\right.$

If ‘ n ‘ is the concentration of free electrons then the current density ‘ J’ can be written as

$\begin{aligned} J=\frac{I}{A} & =\frac{\operatorname{neAV}_{d}}{A} \\ J & =n_{e V} \end{aligned}$

From Ohm’s law $I=\frac{V}{R}$

$\begin{array}{ll} \mathrm{I}=\frac{\mathrm{VA}}{\rho \ell} & \left\lvert\, \mathrm{R}=\frac{\rho \ell}{\mathrm{A}}\right. \\ \mathrm{J}=\frac{\mathrm{I}}{\mathrm{A}}=\frac{\mathrm{VA}}{\rho \ell \mathrm{A}} & \\ \mathrm{J}=\frac{\mathrm{V}}{\rho \ell} & \left\lvert\, \mathrm{E}=\frac{\mathrm{V}}{\ell}\right. \\ \mathrm{J}=\frac{\mathrm{E}}{\rho} & \mathrm{|} \sigma=\frac{1}{\rho} \end{array}$

J = σ E

Where $σ$ is electrical conductivity

$\begin{aligned} \sigma \mathrm{E} & =\mathrm{ne} \mathrm{V}_{\mathrm{d}} \\ \sigma & =\text { ne } \frac{\mathrm{V}_{\mathrm{d}}}{\mathrm{E}} \\ \sigma & =\text { ne } \mu \end{aligned}$

Conclusions

Thus the electrical conductivity is directly proportional to the mobility of electrons.

Mobility of the electron depends on temperature $\mu \infty \frac{1}{T^{\frac{3}{2}}}$
When the temperature of the metal increases, the mobility of the electron decreases and hence the electrical conductivity decreases.
The addition of impurities in the metal decreases the electrical conductivity.