Final answer:
The drift speed of an electron inside the aluminum wire, given the provided parameters and assuming one conduction electron per atom, is calculated to be 396,000 n/s.
Step-by-step explanation:
Calculating the Drift Speed of Electrons in an Aluminum Wire
To calculate the drift speed of electrons in an aluminum wire carrying a current of 12 A, with a wire radius of 4.0 mm (0.004 m), and aluminum density of 2.7 g/cm³ (2.7 x 10³ kg/m³), and a molar mass of aluminum of 27 g/mol, we use the equation I = nqAvd, where I is the current, n is the number of free electrons per volume, q is the charge of an electron (-1.60 x 10⁻¹⁹ C), A is the cross-sectional area of the wire, and vd is the drift speed of the electrons.
First, convert density of aluminum to kg/m³:
2.7 g/cm³ = 2700 kg/m³.
Next, find the number of moles in 1 kg of aluminum:
1 kg / 27 g/mol = 1 kg / 0.027 kg/mol = 37.04 mol.
Now, find the number of free electrons in 1 kg of aluminum by using Avogadro's number:
37.04 mol x 6.0×10²³ mol⁻¹ = 2.22×10²¸ free electrons.
Then, calculate the number of electrons per cubic meter. Since density is 2700 kg/m³, there are:
2700 kg/m³ x 2.22×10²¸ electrons/kg = 5.99×10²¹ electrons/m³.
The wire's cross-sectional area, A, is πr², where r is radius. So, A = π(0.004 m)² = 5.03×10⁻µ m².
Now, use these values in the drift speed formula:
12 A = (5.99×10²¹ electrons/m³)(-1.60 x 10⁻¹⁹ C/electron)(5.03×10⁻µ m²)vd.
Solving for vd gives us:
vd = 12 A / [(5.99×10²¹ electrons/m³)(-1.60 x 10⁻¹⁹ C/electron)(5.03×10⁻µ m²)]
vd = 0.396 m/s or 396000 n/s.
The drift speed of an electron inside the aluminum wire is 396,000 n/s.