Question

Using the concept of free electrons in a conductor , derive the expression for the conductivity of a wire in terns of number density and relaxtion time . Hence obtain tge relation between current density and appiled eletric field e

Answer 1

Consider a conductor of length L and cross sectional area A , when potental difference v is applied across its end the current produced is i. If n is the number of electrons per unit volume in conductor and Vd is the drift velocity then,

I = -nAeVd           Equaltion 1.

Where e is charge on electron , n  is no of density

Electric field produced at each point, E = V/L           Equation 2

if t is relaxation time and e is the electric field strength then drift velocity

Vd  =  -etE / m                          Equation 3

Substituting value in equation 1:

I = - neA(-etE / m )

I = (n e2t / m ) AE                    Equation 4  

As e = V/L then,

I =   n e2t A / m  x V/L

Or

V/I = m/n e2t  x L/A

From other law R = V/I = m/n e2t  x L/A              Equation 5