Question

The cylindrical aluminum wire in the figure below has a radius of 0.1 mm and a length of 30.0 m. A 1.0 A current is going through it from left to right. You are given rhoAluminum = 2.8×10⁻⁸ Ω−m.

a) Compute the resistance of the wire.

Answer 1

To calculate the resistance of the aluminum wire, we use the resistivity formula with the given resistivity, length, and radius to find the cross-sectional area. After calculating the area, we find the resistance to be approximately 26.7×10⁻⁸ Ω.

Step-by-step explanation:

To calculate the resistance of the cylindrical aluminum wire, we can use the resistivity formula R = ρL/A, where ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area of the wire. Given the resistivity of aluminum ρAluminum = 2.8×10⁻⁸ Ω-m, the length L = 30.0 m, and radius of the wire r = 0.1 mm, we can find the cross-sectional area with the formula A = πr².

First, we convert the radius from millimeters to meters: r = 0.1 mm = 0.0001 m. Then we calculate the cross-sectional area,

A = π(0.1² · 10⁻⁸ m²) = π(0.01· 10⁻⁸ m²) = π · 10⁻⁸ m²

Using the area, we can now compute the resistance of the wire:

R = ρL/A = (2.8×10⁻⁸ Ω-m) · (30.0 m) / (π · 10⁻⁸ m²) = (84×10⁻⁸ Ω-m) / (π · 10⁻⁸ m²) = 26.7×10⁻⁸ Ω

Therefore, the resistance of the wire is approximately 26.7×10⁻⁸ Ω.