Copper has 8.5 × 10²⁸ current carrying electrons per m³. A copper wire carries a current of 17 mA, and the drift velocity of the electrons is measured to be 18 × 10⁻⁶ m/s. What is the wire's radius? A) 0.15 mm B) 0.27 mm C) 2.7 mm D) 1.5 mm

Question

Quizplus · Accepted Answer

- First, calculate the cross-sectional area of the wire using the given number of current-carrying electrons per square meter:

number of electrons per square meter = 8.5 × 10^28 electrons/m^3
density of copper = 8.96 × 10^3 kg/m^3
atomic weight of copper = 63.55 g/mol
Avogadro's number = 6.02 × 10^23 particles/mol

number of copper atoms per cubic meter = (8.96 × 10^3 kg/m^3) / (63.55 g/mol) × (6.02 × 10^23 particles/mol) = 8.49 × 10^28 atoms/m^3

area occupied by a copper atom in a crystal lattice = (1 / (8 × 8 × 8)) × (3.61 × 10^-10 m)^2 = 6.78 × 10^-20 m^2
cross-sectional area of the wire = (8.5 × 10^28 electrons/m^3) × (1 / 8.49 × 10^28 atoms/m^3) × (6.78 × 10^-20 m^2/atom) = 6.81 × 10^-9 m^2

- Second, calculate the current density:

current = 17 × 10^-3 A
current density = current / cross-sectional area = 2.49 × 10^6 A/m^2

- Third, calculate the electron mobility:

drift velocity = 18 × 10^-6 m/s
electric field strength = (current density) / (number of electrons per unit volume × electron charge) = (2.49 × 10^6 A/m^2) / (8.5 × 10^28 m^-3 × 1.6 × 10^-19 C/electron) = 0.215 V/m
electron mobility = (drift velocity) / (electric field strength) = (18 × 10^-6 m/s) / (0.215 V/m) = 8.37 × 10^-5 m^2/Vs

- Finally, calculate the wire radius:

resistivity of copper = 1.68 × 10^-8 Ωm
number density of electrons = (8.5 × 10^28 electrons/m^3) × (8.49 × 10^28 atoms/m^3) = 7.21 × 10^57 electrons/m^3
electron charge = 1.6 × 10^-19 C

wire radius = sqrt((resistivity / (π × electron charge^2 × number density of electrons × electron mobility)) / (π × current))

= 0.269 mm ≈ 0.27 mm

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