Define drift velocity of electricity and establish its relation with velocity of the electrons and the intensity of applied electric field.

Explanation

Drift velocity may be defined as that velocity with which a free electron in addition to its random motion gets drifted under the influence of an external field through the body of the conductor.
Relation between drift velocity and electric field : When an electric field is applied across the two ends of a conductor, the electrons possessing negative charge get accelerated in a direction opposite to the electric field. Due to this acceleration, the velocity of an electron increases, but it happens so only for a short time. It is because of the fact that the electron experiences random forces due to vibrating ions in the metal. Therefore, the electron may get deflected or scattered in a wide range of directions due to the action of random forces, Suppose that such a deflection takes place once in a time ton the average, called average relaration time. The velocity acquired by the electron due to external electric field in time equal to t is called drift velocity and it is imposed on the random velocity of electrons. If E is strength of the electric field applied, then
Force on the electron due to electric field = e E
Here, e is charge on electron. If m is mass of the electron, then acceleration produced is given by
`a=(eE)/m " "...(1)`
As the net velocity of an electron due to random motion is zero and the electron is accelerated for an average time `tau` , the additional velocity i.e., drift velocity acquired by the electron,
`v_d = 0 + a tau = atau`
From eqn. (1), substituting the value of a, we have
`upsilon_(d) =(eE)/mtau`
where `tau` = average relaxation time.
Relaxation time. : The short time for which free electron accelerates before it undergoes a collision with the positive ion in conductor, is called a relaxation time.
Relation between drift velocity and current : Consider a conductor of length I and area of cross-section A.

Let V = Potential difference applied across the ends of the conductor
Let n = Number of free electrons per unit volume of the conductor
Al = Volume of the conductor
n Al = Total number of electrons in the conductor
Total charge on the conductor = q = (n Al) e ...(1)
Time taken by the free electrons to cross the length of the conductor,
`t = (l)/(upsilon_d)" "...(2)`
But `I = q/t`
`I=("nAle")/(llupsilon_d)` [Using (1) and (2)]
`I = nAeupsilon_d`