Energy is to be transmitted over a pair of copper wires in a...

Energy is to be transmitted over a pair of copper wires in a transmission line at the rate of 20.0 $\mathrm{kW}$ with only a 1.00$\%$ loss over a distance of 18.0 $\mathrm{km}$ at potential difference $\Delta V_{\mathrm{rms}}=1.50 \times 10^{3} \mathrm{V}$ between the wires. Assuming the current density is uniform in the conductors, what is the diameter required for each of the two wires?

Key Concepts

Resistivity and Electrical Resistance

This concept involves understanding the inherent property of a material called resistivity, which directly affects its resistance when used as a conductor. The resistance is calculated using the relationship R = ?L/A, where ? is the resistivity, L is the length, and A is the cross?sectional area. This is fundamental in determining how a conductor’s dimensions influence the current flow and energy losses in power transmission.

Joule (Resistive) Power Loss

Joule heating, or resistive power loss, is the process in which electrical energy is converted into heat as current flows through a resistance. The power loss is quantified by the formula P_loss = I^2R. Minimizing this loss is critical in high-efficiency power transmission, as even small resistances can lead to significant energy dissipation over long distances.

Current Density

Current density is defined as the electric current per unit area (J = I/A) within a conductor. Assuming a uniform current density is important in engineering calculations, as it allows for the use of simple geometric relationships to determine the cross-sectional area required for a wire to safely carry a given current without excessive heating or energy loss.

Power Transmission Efficiency

This concept relates to the need to maximize the efficient delivery of electrical power over distances while minimizing losses. It involves balancing the transmitted voltage, current, and resistive losses to ensure that a desired fraction of the energy reaches the load, which is a fundamental consideration in designing transmission lines.

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