Suppose we consider the electrons in a conductor to have a random motion, like gas molcules are assumed to have – one of the assumptions of the kinetic theory of gases. If the conductor is of length
and has a voltage
applied across its ends, the electric field is given by
Multiplying both sides by
the elementary unit of electric charge gives
but
will be the force
experienced by the electron so
Applying Newton's second law of motion
gives
where
is the mass of the electron.
The electrons will collide with the atoms of the conductor. If we assume the average time between collisions is
then the electrons will collide with a speed
and the average speed
of the electrons will be half of this –
Substitute this into the electron drift equation
to give
(1)
Ohm's law
gives the resistance
so from (1)
(2)
The resistance R of a material is also given in terms of its resistivity
by
(3)
Equating (2) and (3) gives
but the electron accelerates uniformly from rest so using
to find the distance between collisions
gives
hence