Question

Deduce Ohm’s law using the concept of drift velocity.

OR

Define the term ‘drift velocity’ of charge carriers in a conductor. Obtain the expression for the current density in terms of relaxation time.

OR

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.

OR

(i) On the basis of electron drift, derive an expression for resistivity of a conductor in terms of number density of free electrons and relaxation time. On what factors does resistivity of a conductor depend?

(ii) Why alloys like constantan and manganin are used for making standard resistors?

Answer 1

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 vd the drift velocity of electrons, then the relation between current and drift velocity is

I = – neAv_d …(i)

Where – e is the charge on electron (e = 1.6 × 10^–19 C)

Electric field produced at each point of wire, \(E=\frac{V}{l}\)  (ii)

If τ is relaxation time and E is electric field strength, then drift velocity

Current density \(J(=\frac{I}{A})=\frac{ne^2\,r}{mI}V.\)

This is relation between current density J and applied potential difference V. 

Under given physical conditions (temperature, pressure) for a given conductor

∴ This constant is called the resistance of the conductor (i.e. 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 m3 (n) and the relaxation time (τ) 

Expression for resistivity:

Clearly, resistivity of a conductor is inversely proportional to number density of electrons and relaxation time. 

Resistivity of the material of a conductor depends upon the relaxation time, i.e., temperature and the number density of electrons. 

This is because constantan and manganin show very weak dependence of resistivity on temperature.