$(a)$ Estimate the average drift speed of conduction electrons in a copper wire of cross-sectional area $1.0 \times 10^{-7} \; m^{2}$ carrying a current of $1.5 \; A$. Assume that each copper atom contributes roughly one conduction electron. The density of copper is $9.0 \times 10^{3} \; kg/m^{3}$ and its atomic mass is $63.5 \; u$.
$(b)$ Compare the drift speed obtained above with,$(i)$ thermal speeds of copper atoms at ordinary temperatures,$(ii)$ speed of propagation of electric field along the conductor which causes the drift motion.

$(A)$ The drift speed $v_{d}$ is given by the formula $v_{d} = \frac{I}{neA}$.
Given: $I = 1.5 \; A$, $A = 1.0 \times 10^{-7} \; m^{2}$, $e = 1.6 \times 10^{-19} \; C$.
The number density $n$ is calculated as:
$n = \frac{\text{Density} \times N_{A}}{\text{Atomic Mass}} = \frac{9.0 \times 10^{3} \; kg/m^{3} \times 6.022 \times 10^{23} \; atoms/mol}{63.5 \times 10^{-3} \; kg/mol} \approx 8.5 \times 10^{28} \; m^{-3}$.
Substituting these values:
$v_{d} = \frac{1.5}{8.5 \times 10^{28} \times 1.6 \times 10^{-19} \times 1.0 \times 10^{-7}} \approx 1.1 \times 10^{-3} \; m/s = 1.1 \; mm/s$.
$(b) (i)$ Thermal speed of copper atoms at $300 \; K$ is approximately $2 \times 10^{2} \; m/s$. The drift speed is about $10^{-5}$ times smaller than the thermal speed.
$(ii)$ The electric field propagates at the speed of light, $c = 3.0 \times 10^{8} \; m/s$. The drift speed is smaller by a factor of $10^{-11}$ compared to this speed.