Electrical conductivity of the materials according to classical electron theory

According to classical theory (described by drude ) generally free electrons move randomly in all possible directions and no net velocity results. If an electric field E is applied, the electrons are then accelerated with a force eE towards the anode.

So according to Newton’s Law, 

            m (dV/dt)  =  eE      ------(i)  [ as F = ma ]

Where, m = mass of electron        e = charge of electron

            E = applied E-field              V = drift velocity of electron

This electron motion will be counteracted by a frictional force ϒV  due to collisions.

Under this consideration (i) be modified as

           m (dV/dt)  + ϒV =  eE  ------(ii)  

where, ϒ = a constant

For the steady state case(immediate situation just before collisions) we obtain 

V = V_f

 

dV/ dt = 0

Then (ii) reduces to           ϒV_f  =  eE

                                     Or,    ϒ = eE / V_f

Where, V_f = final drift velocity

Putting this value into (ii) we get-

           m (dV/dt)  + (eE / V_f ) V =  eE  

or,     m dV     = [  eE - (eE / V_f ) V  ] dt

or    m V_f    =   [  eE - (eE / V_f ) V  ] t     

[ Here we use the integration range minimum to maximum . That is

for V ;     0 to V_f            for t ;  0 to t   ]

or,    (m V_f² )/ t  =  eEV_f -  eEV

or,   V_f  - V =  (m V_f² )/ eEt

or,       V    =  V_f  -   (m V_f² )/ eEt

                  =   V_f  [ 1 - (m V_f )/ eEt ] ----------(v)

In (v) the factor   (mV_f) / eE has the unit of a time 

which is defined by   τ  = (mV_f) / eE             where, τ = relaxation time

or,  V_f =  (τ e E) / m  ---------------(vi)

Now we know that conduction current density J = N_f V_f e = σE 

Where, σ = conductivity   and N_f = number of free electron

So,      σ = ( N_f V_f e ) / E

               = ( N_f e² τ ) / m     [ from (vi) ]

               = ( N_f e² l ) / Vm     

Where, l = Vτ = mean free path

This is the required relation. So the conductivity is large for a large number of free electrons and for a large relaxation time.
Relaxation time is defined as the average time between two consecutive  collisions. The distance passing by electron during relaxation time is  known as mean free path.