State Ohm' law and deduce it from the knowledge of drift velocity of free electrons in a conductor carrying current.

Explanation

Relaxation time of free electrons drifting in a conductor is the average time elapsed between two successive collisions.
Deduction of Ohm’s Law: Consider a conductor of length l and cross-sectional area A. When a potential difference V is applied across its ends, the current produced is I. If n is the number of electrons per unit volume in the conductor and `v_d` the drift velocity of electrons, then the relation between current and drift velocity is
`I=-neV_D`.............(i)
where –e is the charge on electron `(e=1.6 times 10^-16C)`
Electric field produced at each point of wire `E=V/I`.........(ii)
If is relaxation time and E is electric field strength, then drift velocity `v_d=(eTe)/m`..........(iii)
Substituting this value in (i), we get )`I=-neA(- (et)/m E) or I=(ne^2t)/m AE`………(iv)
As `E=V/I` [From (i)]
)]`therefore I=(ne^2 t AV)/m V/I or V/I=m/(ne^2t).l/A`
Current density `J(=I/A)=(ne^2C)/(ml)`
This is relation between current density J and applied potential difference V.
`m/(ne^2t).l/A= constant` This constant is called the resistance of the conductor (i.e. R)
i.e, `R=m/(ne^2t).l/A…..(vi) From (v) and (vi) V/I=R`
This is Ohm’s law. From equation (vi) it is clear that the resistance of a wire depends on its length (l), cross-sectional area (A), number of electrons per `m^2(n)` and the relaxation time (t).
Expression for resistivity
As `r=(pl)/A`.....(viii)
Comparing (vi) and (viii), we get , Resistivity of a conductor `p=m/(ne^2t)`.....(ix)
Clearly, resistivity of a conductor is inversely proportional to number density of electrons and relaxation time