Explanation:
let's go through the calculations step by step:
Given:
Cross-sectional area of the wire, A = 3.10 * 10^-6 m^2
Current, I = 5.50 A
Density of aluminum, ρ = 2.70 g/cm^3
Charge of an electron, q = 1.6 * 10^-19 C
Step 1: Calculate the mass of the wire
Mass = density * volume
Volume = A * L, where L is the length of the wire
Let's assume the length of the wire, L = 1 m
Volume = (3.10 * 10^-6 m^2) * (1 m) = 3.10 * 10^-6 m^3
Mass = (2.70 g/cm^3) * (3.10 * 10^-6 m^3) = 8.37 * 10^-6 g
Step 2: Convert the mass to moles of aluminum
Molar mass of aluminum (Al) = 26.98 g/mol
Number of moles = mass / molar mass
Number of moles = (8.37 * 10^-6 g) / (26.98 g/mol) = 3.10 * 10^-7 mol
Step 3: Calculate the number of aluminum atoms
Avogadro's number = 6.022 * 10^23 atoms/mol
Number of aluminum atoms = number of moles * Avogadro's number
Number of aluminum atoms = (3.10 * 10^-7 mol) * (6.022 * 10^23 atoms/mol) = 1.87 * 10^17 atoms
Step 4: Calculate the number of conduction electrons
Since each aluminum atom contributes one conduction electron,
Number of conduction electrons = number of aluminum atoms * (1 electron/atom)
Number of conduction electrons = (1.87 * 10^17 atoms) * (1 electron/atom) = 1.87 * 10^17 electrons
Step 5: Calculate the drift velocity of the electrons
Drift velocity, v = I / (nAq), where n is the number of conduction electrons per unit volume
We are given the cross-sectional area, A, and the current, I, from which we have all the necessary values
v = (5.50 A) / ((1.87 * 10^17 electrons) * (3.10 * 10^-6 m^2) * (1.6 * 10^-19 C))
Calculating this expression will give us the drift velocity of the electrons in meters per second.
