(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.

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=neAVd
VR=neAVd
V=neAVdE/l=neAVd×ρl/A=neVdρE=neρ×eEmτ
1=ne²τ/mρ
ρ=ne²τ/m
σ=ne²τ/m.. (1)
Relation between current density and the applied electric field E
From 1st equation of motion
v=u+at
⇒Vd=at
Vd=eE/mτ
⇒i=neAVd
i/A=J=current density
J=ne(vd)=ne(eE/mτ)
J=ne²/mτ(E)
J=σE
σ=ne²τ/m