Question

relation between vd and mobility expression
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Answer 1

Answer:

In the Drude model, the relaxation time τ 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 E, the charge carriers, each with charge q, start to accelerate according to Newton's second law,

ma=qE

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

dP(t)dt=−1τP(t)

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

P(t)=1τe−t/τ

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

⟨v⟩=∫∞0dt v(t)P(t)=qEτm∫∞0dt t e−t/τ

Evaluating the integral gives

⟨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.

Explanation:

Answer 2

Answer:Conductivity is proportional to the product of mobility and carrier concentration. For example, the same conductivity could come from a small number of electrons with high mobility for each, or a large number of electrons with a small mobility for each.

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