(a) Estimate the average drift speed of conduction electrons in a copper wire of cross sectional area `1.0xx10^(-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 `9.0xx10^3 kgm^(-3)` and its atomic mass is `63.5 u.` (b) Compare the drift speed obtained with the speed of propagation of electric field along the conductor, which causes the drift motion.

Explanation

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 `v_(d)` is given by Eq. (3.18) `v_(d) = ("I/neA")` Now, `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 metre (assuming one conduction electron 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 `6.0 xx 10^(23)` copper atoms have a mass of 63.5 g,
`n = (6.0 xx 10^(23))/(63.5)xx9.0 xx 10^(8)`
`=8.5 xx 10^(28) m^(-3)`
`v_(d)=(1.5)/(8.5xx10^(28)xx1.6xx10^(-19)xx1.0xx10^(-7))`
`=1.1xx10^(-3)ms^(-1)=1.1"mm s"^(-1)`
(b) (i) At a temperature T, the thermal speed* of a copper atom of
mass M is obtained from `[lt (1//2) Mv^(2) gt = (3//2) k_(B)BT]` 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 vibrational speeds of copper atoms in a conductor. Note that the drift speed of electrons is much smaller, about `10^(-5)` times the typical thermal speed at ordinary temperatures. (ii) An electric field travelling along the conductor has a speed of an electromagnetic wave, namely equal to `3.0 × 10^(8) m s^(-1)` (You will learn about this in Chapter 8). The drift speed is, in comparison, extremely small, smaller by a factor of `10^(-11)`.