Question

(a) Define the term 'conductivity' of a metallic wire. Write its SI unit.
(b) Using the concept of free electrons in a conductor, derive the expression for the conductivity of a wire in terms of number density and relaxation time. Hence obtain the relation between current density and the applied electric field $E$.

Answer 1

A. Conductivity - It is the ratio of current density (J) to electric field strength (E). SI Unit is Siemens/meter.

B. Expression for the conductivity of a wire in terms of number density and relaxation time

As we know that

$i=neA{V}_d$

$\frac{V}{R}=neA{V}_d$

$V=neA{V}_d$

$El=neA{V}_d\times \frac{\rho l}{A}$

$E=ne{V}_d\rho$

$E=ne\rho \times \frac{eE}{m}\tau$

$1=\frac{n{e}^2\tau }{m}\rho$

$\frac{1}{\rho }=\frac{n{e}^2\tau }{m}$

$\sigma =\frac{n{e}^2\tau }{m}$ ...(1)

Relation between current density and the applied electric field E

From $1^{st}$ equation of motion

$v=u+at$

$\because V_d=at$

$V_d=\frac{eE}{m}\tau$

$\because i=neA{V}_d$

$\frac{i}{A}=J=$ current density

$J=ne\left(v_d\right)$

$=ne\left(\frac{eE}{m}\tau \right)$

$J=\frac{n{e}^2}{m}\tau \left(E\right)$

$J=\sigma E$

$\sigma =\frac{n{e}^2\tau }{m}$