Drive an expression for drift velocity of free electrons in a conductor.

Question

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Answer 1

When conductor is subjected to an electric field $E$ , each electron experience a force
$F = - e E$ and acquires an acceleration of
$a = F / m = - \frac{- e E}{m}$ ..... (1)
Here $m$ = mass of electron, $e$ = charge , $E$ = Electric Field.
The average time difference between two consecutive collisions is known as relaxation time of electron.
$τ = \frac{τ _{1} + τ _{2} + τ _{3} + . . . . .}{n}$ ......(2)
As $v = u + a t$ then drift velocity $v_{d}$ is
$v_{d} = \frac{v _{1} + v _{2} + v _{3} + . . . . + v _{n}}{n}$
$v_{d} = \frac{( u _{1} + u _{2} + . . . . + u _{n} ) + a ( τ _{1} + τ _{2} + . . . . )}{n}$
$v_{d} = 0 + \frac{a ( τ _{1} + τ _{2} + τ _{3} + . . . . + τ _{n} )}{n}$ ( since initially velocity is zero)
$v_{d} = 0 + a τ$
$v_{d} = - \frac{e E}{m} τ$
According to drift velocity expression, the relaxation time is the time interval between successive collisions of an electron on increasing temperature, the electrons move faster and more collisions occur quickly. Hence, relaxation time decreases with increase in temperature which implies that drift velocity also decreases with temperature.

Answer 2

When conductor is subjected to an electric field

E

, each electron experience a force

F = - e E

and acquires an acceleration of

a = F / m = - \frac{- e E}{m}

..... (1)

Here

m

= mass of electron,

e

= charge ,

E

= Electric Field.

The average time difference between two consecutive collisions is known as relaxation time of electron.

τ = \frac{τ _{1} + τ _{2} + τ _{3} + . . . . .}{n}

......(2)

As

v = u + a t

then drift velocity

v_{d}

is

v_{d} = \frac{v _{1} + v _{2} + v _{3} + . . . . + v _{n}}{n}

v_{d} = \frac{( u _{1} + u _{2} + . . . . + u _{n} ) + a ( τ _{1} + τ _{2} + . . . . )}{n}

v_{d} = 0 + \frac{a ( τ _{1} + τ _{2} + τ _{3} + . . . . + τ _{n} )}{n}

( since initially velocity is zero)

v_{d} = 0 + a τ

v_{d} = - \frac{e E}{m} τ

According to drift velocity expression, the relaxation time is the time interval between successive collisions of an electron on increasing temperature, the electrons move faster and more collisions occur quickly. Hence, relaxation time decreases with increase in temperature which implies that drift velocity also decreases with temperature.