Question

What is meant by drift velocity of free electrons? Derive Ohm’s law on the basis of theory of electron drift.

Or

Explain how does the resistivity of a conductor depend upon (i) number density ‘n’ of free electrons, and (ii) relaxation time τ.

Or

Define the term resistivity and write its S.I. unit. Derive the expression for the resistivity in terms of number density of free electrons and relaxation time.

Or

Show that the resistance of a conductor is given by:

R = \(\frac{ml}{ne^2\tau A}\)

Answer 1

Drift Velocity is defined as the velocity of free electrons with which they get drifted towards the positive terminal under the influence of the external field.

Resistivity (specific resistance) (ρ): If the material of a conductor is defined as the resistance of unit length and unit area of cross-section of the conductor.

And ρ = \(\frac{m}{ne^2\tau }\)

Thus resistivity of the material of a conductor depends upon

(i) ρ ∝ \(\frac{1}{n}\)  i.e. resisitivity is inversely proportional to the number density of electrons (number of electons per unit volume) inside the conductor.

(ii) ρ ∝ \(\frac{1}{\tau}\)  i.e. resistivity is inversely proportional to the average relaxation time of the free electrons in the conductor.

The value of n depends upon the nature of conductor and τ depends upon the temperature of the conductor.

Ohm’s law from drift velocity of electrons.

Since drift velocity vd and current I flowing in a conductor are related by the relation:

v_d = \(\frac{I}{ne A}\) .........................(1)

Also drift velocity in terms of average relaxation time r is given by

This is Ohm’s law.

But \(\frac{V}{I} \)  = R, the resistance of the conductor

So Eq. (3) becomes