(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.

Explanation

On applying a potential difference V across the ends of a conductor of length l, the electric field
`E = V/l`
Force experienced by an electron F = e E ` = (eE)/(l)` in a direction opposite to that of E.
` therefore ` Acceleration of electron in a direction opposite to that of E will be
` a= F/m = e/m. V/l`
If the average time between two successive collisions suffered by an electron be `tau` , the drift velocity of electron will be `v_d = a tau = e/m. V/l.tau `
But in terms of drift velocity magnitude of electric current is given by
`I = nAev_d = nAe.e/m.V/l.tau = (nAe^2)/(m) tau.V/l`
` therefore ` Conductance `G = I/V = (nAe^2)/(ml) tau`
and conductivity `sigma = (Gl)/(A) = ("ne"^2)/(m) tau`
` therefore ` current density `J = I/A = ("ne"^2 tau V)/(ml) = ("ne"^2 tau)/(m) . E = sigma E`
In term of vectors , we have `vecJ = sigma vecE`