(a) Prove the assumption made above, that if the probability of a collision per unit time is dt/τ, then the average time since the last collision is τ. This follows a similar procedure as Exercise 1.5.3. The probability of a last collision in time interval t is equal to the probability of no collision in all intervals from 0 to t, times the probability of a collision exactly in the range (t, t + dt). Show that this implies that the probability P(t) of no collision in time interval t is equal to e^−t/τ, and therefore the average time since the last collision is τ.

(b) The same argument can equally well be applied to the time until the next collision. Therefore, the mean time between collisions is 2τ, and the average energy gained by an electron between collisions is

Show that this implies that the average rate of energy loss in a resistor with cross-sectional area A and length l is equal to P = IV = V²/R, where R = l/Aσ is the total resistance.

Transcribed Image Text:

m|v² with |v| = (F/m)(2t).