Mobility and Conductivity

• 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.