Copper and aluminum are being considered for the cables in a...

Copper and aluminum are being considered for the cables in a high-voltage transmission line where each must carry a current of 50 A. The resistance of each cable is to be $0.15 \Omega$ per kilometer. (a) If this line carries power from Niagara Falls to New York City (approximately $500 \mathrm{km}$ ), how much power is lost along the way in the cable? Compute for each choice of cable material (b) the necessary cable diameter and (c) the mass per meter of the cable. The electrical resistivities for copper and aluminum are given in Table $18.1 ;$ the mass density of copper is $8920 \mathrm{kg} / \mathrm{m}^{3}$ and that of aluminum is $2702 \mathrm{kg} / \mathrm{m}^{3}$.

Copper and aluminum are being considered for the cables in a high-voltage transmission line where each must carry a current of 50 A. The resistance of each cable is to be $0.15 \Omega$ per kilometer.
(a) If this line carries power from Niagara Falls to New York City (approximately $500 \mathrm{km}$ ), how much power is lost along the way in the cable? Compute for each choice of cable material (b) the necessary cable diameter and (c) the mass per meter of the cable. The electrical resistivities for copper and aluminum are given in Table $18.1 ;$ the mass density of copper is $8920 \mathrm{kg} / \mathrm{m}^{3}$ and that of aluminum is $2702 \mathrm{kg} / \mathrm{m}^{3}$.

Key Concepts

Electrical Resistivity

Electrical resistivity is a fundamental material property that quantifies how strongly a material opposes the flow of electric current. It is used in determining a conductor's resistance when combined with its dimensions through the relation R = ?L/A, where ? is the resistivity, L is the length, and A is the cross?sectional area. This concept is crucial for comparing different materials for use in electrical transmission systems.

Resistance Calculation from Material Properties

The calculation of resistance based on a conductor's geometry and the material's resistivity is essential for designing transmission lines. By rearranging the formula R = ?L/A, one can determine the required cross-sectional area (and subsequently, the dimensions such as diameter for a circular cable) that will yield a target resistance per unit length. This is a key step in ensuring the cable meets specific resistance-based performance criteria.

Joule Heating (Power Loss in Conductors)

Joule heating refers to the process by which electrical energy is converted into heat as current flows through a resistor, described by the formula P_loss = I²R. In the context of high-voltage transmission lines, minimizing resistive power losses is crucial for efficient energy transport over long distances. Calculating the power loss helps in understanding and mitigating energy inefficiencies.

Cable Geometry and Cross?Sectional Area

Cable geometry, particularly the cross-sectional area, directly affects a cable's resistance. For circular wires, the area is determined by A = ?(d²/4) where d is the diameter. This concept is critical in designing cables with specified resistance values, as adjusting the diameter alters the resistance and the current carrying capability of the cable.

Material Density and Cable Mass Calculation

Material density, defined as mass per unit volume, is used to calculate the mass of a cable from its geometry. Once the cross-sectional area is determined, the mass per unit length of the cable can be found by multiplying the area by the density. This concept is important for practical mechanical and logistical considerations, such as installation and structural support of transmission lines.

Please give Ace some feedback