Question

A particular aluminum transmission line has a resistance per length of 0.068 Ω/km. Assume aluminum has a resistivity of 2.65 × 10⁻⁸ Ω⋅m and a density of 2.70 × 103 kg/m³. A.) What is the mass of one kilometer of this line, in kilograms? b.) What is the mass, in kilograms per kilometer, of a copper line having the same resistance as the aluminium one? Copper has a resistivity of 1.72 × 10-8 Ω⋅m and a density of 8.90 × 10³ kg/m³.

Answer 1

To find the mass of one kilometer of the aluminum transmission line, use the formula mass = volume x density. Calculate the cross-sectional area using the resistivity and resistance per length, then substitute it into the mass formula. Follow the same steps to find the mass per kilometer of a copper line with the same resistance.

Step-by-step explanation:

To find the mass of one kilometer of the aluminum transmission line, we can use the formula:

mass = volume x density

The volume of a wire can be calculated using its cross-sectional area, which is given by:

cross-sectional area = π x (diameter/2)^2

Using the resistivity of aluminum and the resistance per length, we can calculate the cross-sectional area:

diameter = 2 x (resistivity/resistance per length)

Finally, we can substitute the cross-sectional area into the mass formula to find the mass per kilometer of the aluminum line.

To find the mass per kilometer of a copper line with the same resistance, we can follow the same steps, using the resistivity and density of copper.

Answer 2

To find the mass of one kilometer of an aluminum transmission line, determine the cross-sectional area using the given resistivity and resistance per length, then calculate the volume, and multiply by the density of aluminum. Repeat the process for copper using its respective resistivity and density to obtain mass per kilometer for a line with the same resistance.

Step-by-step explanation:

Calculating the Mass of Aluminum and Copper Transmission Lines

To calculate the mass of one kilometer of the aluminum transmission line with a resistance per length of 0.068 Ω/km, we use the formula for the resistance of a cylindrical wire: R = ρL/A, where ρ is the resistivity, L is the length, and A is the cross-sectional area. Given that the resistivity of aluminum (ρ) is 2.65 × 10⁻₈ Ω⋅m, we can solve for the cross-sectional area (A). Once we have the cross-sectional area, we can calculate the volume of one kilometer of wire by multiplying A by the length (1000 m). Using the density of aluminum (2.70 × 10³ kg/m³), we multiply the volume by the density to get the mass. For copper, we perform the same calculation using its resistivity (1.72 × 10⁻₈ Ω⋅m) and density (8.90 × 10³ kg/m³) to find the mass per kilometer of a copper line with the same resistance as the aluminum one.

Therefore, the formula to find the mass (m) in kg per km for both materials would be m = ρ × (L/A) × density. Remember that for the copper line, we need to ensure the resistance is the same, so A will differ between aluminum and copper.