Derivation of ohm's law by free electron model

“Derivation of ohm's law by free electron model”:

 In above figure shows a current carrying conductor of length l , area of cross section  A and free electron density be “n” electrons/m³.  When we applied potential V across the conductor then electric field E is produced, in the influence of electric field  electrons are moving in opposite direction to the field by the Drift velocity due to which the current I flows through it.

Electric force experienced by electron
F = q.E

Since, charge on electron, q = -e
Hence, F = -e.E
           |F|= e.E


If the mass of an electron be “m” and acceleration of it between two consecutive collision be “a” then by second law of motion
F = m.a

Comparing above two forces

m.a = e.E

Since, E = ∆V/∆X = v/l

m.a = e.(v/l)

a = e.v/m.l

By first equation of motion
V = U + at

Here , U = 0 and t = τ (Relaxation time)

Hence, V = 0 + (e.v/m.l)τ

By the definition :
Drift velocity, Vd = (U+V)/2

Vd = {0 + (e.v.τ/m.l)}/2

Vd = e.v.τ/2m.l

We know that,
the current flowing through any conductor
I = n.e.A.Vd

Putting the value of Vd

I = n.e.A (e.v.τ/2m.l)

I = (n.e².A.v.τ)/2m.l

2m.l×I = n.e².A.v.τ

v = (2m.l/n.e².A.τ)×I

 If the physical condition of the conductor remain the same then the quantity (2m.l/n.e².A.τ) is constant term this is known as resistance provided by the conductor. It is denoted by “R”. It's unit is “ohm”.

Hence, v = RI
        v ∝ I
This is ohm's law.