‘The 12-gauge copper wire in typical residential building has cross-sectional area of 3.31 X 10-6 m? If it carries current of 10.0 A what is the drift speed of the electrons? Assume that each copper atom contributes one free electron to the current The density of copper is 8.95 g/cm?_’
The correct answer and explanation is:
To find the drift speed of electrons in a copper wire, we use the formula: vd=InAev_d = \frac{I}{nAe}
Where:
- vdv_d = drift speed (m/s)
- II = current (A) = 10.0 A
- AA = cross-sectional area (m²) = 3.31×10−6 m23.31 \times 10^{-6} \, \text{m}^2
- ee = elementary charge = 1.6×10−19 C1.6 \times 10^{-19} \, \text{C}
- nn = number density of free electrons (electrons/m³)
Step 1: Find the number density nn
To find nn, we need the number of free electrons per unit volume. First, use the density of copper and its molar mass:
- Density of copper: ρ=8.95 g/cm3=8950 kg/m3\rho = 8.95 \, \text{g/cm}^3 = 8950 \, \text{kg/m}^3
- Molar mass of copper: M=63.5 g/mol=0.0635 kg/molM = 63.5 \, \text{g/mol} = 0.0635 \, \text{kg/mol}
- Avogadro’s number: NA=6.022×1023 atoms/molN_A = 6.022 \times 10^{23} \, \text{atoms/mol}
Now compute number density: n=ρNAM=8950×6.022×10230.0635≈8.49×1028 electrons/m3n = \frac{\rho N_A}{M} = \frac{8950 \times 6.022 \times 10^{23}}{0.0635} \approx 8.49 \times 10^{28} \, \text{electrons/m}^3
Step 2: Plug into the drift velocity formula
vd=10.0(8.49×1028)(3.31×10−6)(1.6×10−19)v_d = \frac{10.0}{(8.49 \times 10^{28})(3.31 \times 10^{-6})(1.6 \times 10^{-19})} vd≈10.04.496×104=2.22×10−4 m/sv_d \approx \frac{10.0}{4.496 \times 10^{4}} = 2.22 \times 10^{-4} \, \text{m/s}
✅ Final Answer:
vd≈2.22×10−4 m/s\boxed{v_d \approx 2.22 \times 10^{-4} \, \text{m/s}}
🧠 Explanation (300 Words):
Drift speed refers to the average velocity that free electrons attain in a conductor when subjected to an electric field. Although electrical energy moves at nearly the speed of light in a wire, individual electrons drift very slowly—on the order of fractions of a millimeter per second.
In a typical copper wire (like 12-gauge used in homes), the current of 10 A might seem large, but copper has a very high electron density, meaning there are trillions of free electrons available in every cubic centimeter to carry the current.
Copper has one free electron per atom. Using its density and molar mass, we calculated that there are about 8.49×10288.49 \times 10^{28} free electrons in each cubic meter of copper. When current flows, each of these electrons contributes just a tiny bit, moving slowly but steadily in the direction opposite to the electric field.
The drift speed calculation uses the idea that the total current is the product of the number of charge carriers, the charge per carrier, the cross-sectional area, and their average velocity. Even though electrons are moving slowly (here, ≈0.22\approx 0.22 mm/s), their vast numbers ensure that a substantial current is maintained.
This low drift speed emphasizes that the signal propagation in a circuit is not due to electrons zooming from point A to B, but rather due to a chain reaction—like pushing a line of marbles—where energy is transferred almost instantaneously, even if each electron moves slowly.