Derive ohm’s law on the basis of the theory of electron drift…?

To find the relationship between current and drift velocity, consider a conductor of length l and area of cross-section A. If V is the potential difference across the ends of the conductor, then the strength of the electric field is

The acceleration acquired by each electron due to the electric field is

where

is the coulomb’s force experienced by each electron and m is the mass of the electron.

The drift velocity of the electrons is given by

where t, the relaxation time, is the average time that an electron spends between two collisions. It is of the order of 10-14 s.

Now the volume of the conductor is equal to Al and if n is the number of free electrons per unit volume, then, the total number of free electrons in the conductor will be equal to n Al.

Hence, the total charge q = - n Ale

The time taken by free electrons to cross the conductor is

where we have substituted for t and q.

For a given conductor, I a Vd

A small value of drift velocity 10-5m/sec produces a large amount of current, as there are a large number of free electrons in a conductor.

The drift velocity of the electrons Vd is (using E = V / l)

Also I = - neAVd

On substituting for Vd in the above expression we get,

or

Which shows that current is directly proportional to the applied voltage(Ohm’s law).From this, the resistance R can be identified as