Explain drift of electron and drift velocity. Derive equation of current in term of cross-section of conductor.

Explanation

`rArr` In conductor free electron collide with ions and direction of their motion is continuously changing. Let there be N electron in conductor (i = 1, 2, 3, ... , N). At given time velocity of ith electron be `v_(l)` then
its averege speed `((1)/(N) sum_(i=1)^(N) v_(i) = 0 )` will be zero.
`rArr ` When electric field is present then electron will be accelerated.
`rArr` Acceleration of electron , `vec(a) = ("force " )/("mass") = (- e vec(E))/(m)` which is for very short duration.
`rArr `Charge of electron is ( - e) and mass be m.
`rArr` Let last collision of `i^(th)` electron at time t. after this collision let `t_(i)` time is elapsed and after this velocity at time t be `vec(v_(t))` then,
`vec(v_(t)) = vec(v_(i))` + at
`therefore vec(v_(t)) = 0 - (e vec(E))/(m) t " " [ because v_(i) = 0 ] `

`rArr` At time t average velocity of electron wi1l be average velocity of `v_(i)` velocity of all electron and it will be zero because `v_(i)` is zero for all electron.
`rArr` Collision among electron not at definite interval hence direction of velocity of electron will be changing.
`rArr` Let average time interval between two successive collision be `tau` which is called relaxation time
`rArr` Average velocity of N electron gives drift velocity `vec(v_(d))`
`therefore vec(v_(d)) = 0 - (e vec(E))/(m) . (t)_("average")`
`therefore v_(d) = - (e vec(E))/(m) . tau`
Which is called drift velocity of electron.