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
dV/ dt = 0
Then (ii) reduces to ϒVf = eE
Or, ϒ = eE / Vf
Where, Vf = final drift velocity
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 (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.
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.