Estimate the average drift speed of conduction electrons in a copper wire of cross-sectional area `1.0 xx 10^(-7) m^(2)` carrying a current of 1.5 A. Assume that each copper atom contributes roughly one conduction electron. The density of copper is `9.0 xx 10^(3) kg//m^(3)` and its atomic mass is 63.5 u.

Explanation

(a) The direction of drift velocity of conduction electrons is opposite to the electric field direction, i.e., electrons drift in the direction of increasing potential. The drift speed `u _(d)` is given by `u _(d) =(l//n eA)`
No, `e = 1.6 xx 10 ^(-19) C, A = 1.0 xx 10^(-7) m ^(2).I = 1. 5 A.` THe density of conduction electrons, n is equal to the number of atoms per cubic meter (assuming one conduction enectron per Cu atom as is reasonable from its valence electron count of one). A cubic metre of copper has a mass of `9.0 xx 10 ^(3)Kg.` Since `6.0xx 10^(23)` copper atoms have a mass of `63.5g.`
`n = (6.0 xx 10 ^(23))/(63.5) xx 9.0 xx 10 ^(6)`
` = 8.5 xx 10 ^(28) m^(-3)`
which gives,
`u _(d) = ( 1.5)/(8.5 xx 10 ^(28) xx.16 xx 10 ^(-19) xx 1.0 xx 10 ^(-7))`
`=1.1 xx 10 ^(-3) ms ^(-2) =1.1 mm s ^(-1)`
(b) (1) At a temperature T.the thermal speed of a copper atom of mass M is obyained from` [ lt (1//2) Mv ^(2)gt = (3//2) K _(B) T]` and is thus typically of the order of `sqrt (k_(B) T//M).` where `k _(B)` is the Boltzmann constant. For copper at 300 K. this is about `2 xx 10 ^(2)` m/s. This figure indicates the random vibrattonal speeds of copper atoms in a conductor. Note that the drift speed of electrons is much smaller, about `10 ^(-5)` times the tpyical thermal speed at ordinary temperatures.
(i) An electric field travelling along the conductor has a speed of an electronagetic wave, namely equal to `3.0 xx 10 ^(8) m s ^(-1)` The drift speed is, in comparison, extremely small, smaller by a factor of `10 ^(-11).`