Theory of Free Electron Gas Model →
According to the electron gas theory
1. The free electrons are continuous in motion inside the metal. The motion of free electrons are random inside the metal.
2. When the free electrons are collisied to each other then the direction of electrons are changed.
3.
Mean free Path: The length covered by free electrons, between the two successive collisions is called the "Mean Free Path".
4.
Relaxation Time: The time taken between the two successive collisions of free electrons is called the Relaxation time. It is represented by $\tau$.
5.
Drift velocity: When a potential is applied across the metal then these electrons do not move own velocity but it move with an average velocity in the opposite direction of the electric field. This average velocity is called the "Drift Velocity". The drift velocity of electrons depends upon the applied potential.
6.
Mobility of Electrons: When a potential $V$ is applied across the metal then electrostatic force $F$ acts on the electrons i.e
$F=qE $
$F=NeE \qquad(1)$
Where
$N$ →The number of free electrons inside the metal
$E$ → The electric field due to the applied potential
When this electrostatic force is applied to the electrons then these electrons are accelerated with accelerated $a$ i.e
$F=ma \qquad(2)$
From equation $(1)$ and equation $(2)$ we can write
$ma=N\:e\:E$
$a=\frac{N}{m}eE$
$\frac{v_{d}}{\tau}=\frac{N}{m}eE \qquad \left( \because a=\frac{v_{d}}{\tau} \right)$
$v_{d}=\frac{Ne\tau}{m} E$
Where $\mu=\frac{Ne\tau}{m}$. It is known as the mobility of electrons.