Question

drive a relation between electric current and drift velocity and hence deduce Ohm's law and hence obtain the expression for resistivity in terms of number density of free electron and relaxation time​

Answer 1

Explanation:

First one is "relation between electric current and velocity"

Second one is "deduction of ohm's law"

And, the third one is "resistivity in terms of electron density and relaxation"

Answer 2

The Relaxation time of free electrons in a conductor is known as the average time elapsed between two successive collisions.

Ohm's Law:

• Let's take a conductor of length 1m and area A. If a potential difference V is applied across its ends, then the current produced is i. Let, n is the number of electrons per unit volume in the conductor and $V_{d}$ the drift velocity electrons, then the relation between current and drift velocity is written as

                           $I=-neAV_{d}$ ---eqno:1

                        e-- electronic charge, $(q=1.6 X10^{-19} C)$

• Electric field, $E=\frac{V}{I}$---eqno:2
• Let, T be the relaxation time and E is the electric field therefore the drift velocity will be,

                   $V_{d} =\frac{-erE}{m}$----eqno:3

• Now by substituting this in eq 1 we get,

                     $I=-\frac{ne^{2}T }{m} AE$----eqno:4

• from eq2 we can write as,

             $I=\frac{ne^{2}tA }{m} (v)$--eqno:5

• we know that the current density formula as,

                                  $J= \frac{I}{A}$

                                   $=\frac{ne^{2}T }{ml} (V)$

• above is the Relation between current density J and applied potential difference V.

• Under given physical conditions for a given conductor is written

                  $\frac{m}{ne^{2} } \frac{1}{A} = constant$

∴ we can say that the constant is called the resistance of the conductor R

                 $R=\frac{m}{ne^{2} T} \frac{1}{A}$ ----eqno:6

• from eq 5&6 we can write as,

                  $\frac{V}{I} =R$

This is the ohm's law, hence derived.

Expression for resistivity:

as $R=\frac{pl}{A}$---eqno:7

by comparing both eq6&7 we can write as,

resistivity of the conductor

$p=\frac{m}{ne^{2} t} ---eqno:8$

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

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

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