Question

Please give detailed relationshio between time and drift velocity with each step explained​

Answer 1

Explanation:

In the Drude model, the relaxation time ττ is the average time between collisions for a charge carrier undergoing uniform acceleration in an electric field. The idea is that when you apply an electric field EE, the charge carriers, each with charge qq, start to accelerate according to Newton's second law,

ma=qEma=qE

After a time tt with no collisions, a charge carrier has the velocity v=qEt/mv=qEt/m. If ττ is the average time between collisions, then 1/τ1/τ is the average rate of collisions (for one charge carrier). Hence, if P(t)P(t) is the probability of accelerating for the time tt without a collision, then the instantaneous rate of change of P(t)P(t) at the time tt is equal to −1/τ−1/τ times P(t)P(t),

dP(t)dt=−1τP(t)dP(t)dt=−1τP(t)

this equation has the normalized (∫∞0dt P(t)=1∫0∞dt P(t)=1) solution,

P(t)=1τe−t/τP(t)=1τe−t/τ

This implies that the average velocity ⟨v⟩⟨v⟩ between collisions is

⟨v⟩=∫∞0dt v(t)P(t)=qEτm∫∞0dt t e−t/τ⟨v⟩=∫0∞dt v(t)P(t)=qEτm∫0∞dt t e−t/τ

Evaluating the integral gives

⟨v⟩=qEmτ⟨v⟩=qEmτ

The drift velocity is just the average velocity of the charge carriers in the conductor. So, as you now see, the drift velocity is proportional to the relaxation time. As the relaxation time increases the drift velocity increases, because the charge carriers have more time to accelerate between collisions.