Exploring Electrical Conductivity in Metals: Deriving the Classical 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 = Vf    Where, Vf = final drift velocity(max velocity)

dV/ dt = 0

Then (ii) reduces to           ϒ Vf  =  eE
                                     Or,    ϒ = eE / Vf

Putting this value into (ii) we get-

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

or    m Vf    =   [  eE - (eE / Vf ) V  ] t  
   
[ Here we use the integration range minimum to maximum . That is for V ;     0 to Vf     for t ;  0 to t   ]

or,    (m Vf2 )/ t  =  eEVf -  eEV
or,   Vf  - V =  (m Vf2 )/ eEt
or,   V    =  Vf  -   (m Vf2 )/ eEt
             =   Vf  [ 1 - (m Vf )/ eEt ] ------- (v)

In equation (v) the factor   (mVf) / eE has the unit of a time 
which is defined by   τ  = (mVf) / eE            
where, τ = relaxation time

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

Now we know that conduction current density J = Nf Vf e = σE 
Where, σ = conductivity   and Nf = number of free electron

So,      σ = ( Nf Vf e ) / E
               = ( Nf e2 τ ) / m     [ from (vi) ]
               = ( Nf 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.