Define relaxation time of the free electrons drifting in a conductor.How is it related to the drift velocity of free electrons? Use this relation to deduce the expression for the electrical resistivity of the material.

Explanation

The resistance time or mean free time `(t_0)` of a free electron inside a conductor is defined as the average time spent by the electron between two successive collisions.
It is assumed that just after a collision, the electron velocity is reduced to zero. Then the force on it due to the applied electric field E is eE.So the acceleration of the electron is `(eE)/m` and the velocity attained by it just before the next collision `= (eE)/mt_0`.Therefore, the average or drift velocity of free electrons inside a conductor is
`v=(0+(eE)/3t_0)/2or,v=(eE)/(2m)t_0`
or,`t_0=(2m)/(eE)tv`.....(1)
we also know, `v=1/( n e A)`
So, `1/(n e A)=(eE)/(2m)t_0or,E=(2m)/(n e^2 At_0)l`
Now, if the potential difference applied between the ends of a conductor of length l, then `E=V/l`.
then its resistance, `R=V/l=(El)/l=(ml)/(me^2At_0)=rho1/A`
So the electrical resistivity of the material is
`rho=(2nm)/(n e^2t_0)`
Also, substituting `t_0` from (1), we have
`rho=(eE)/(n e^2v)=E/(n ev)`....(3)