Energy is to be transmitted at the rate of 20.0 kW with only 1.00% loss over a distance of 18.0

step 1 For this solution, the loss power can be expressed through the RMS current and the total resista

Energy is to be transmitted at the rate of 20.0 kW with only...

Energy is to be transmitted at the rate of 20.0 $\mathrm{kW}$ with only 1.00$\%$ loss over a distance of 18.0 $\mathrm{km}$ at potential difference $\Delta V$ . (a) What is the diameter required for each of the two copper wires in the transmission line? Assume the current density is uniform in the conductors. (b) State how the diameter depends on $\Delta V .$ (c) Evaluate the diameter for $\Delta V=1500 \mathrm{V}$ . (d) If you choose to make the diameter $3.00 \mathrm{mm},$ what potential difference is required?

Key Concepts

Relationship between Potential Difference, Current, and Conductor Dimensions

The potential difference (voltage) applied in a transmission system dictates the current drawn for a given power requirement, following P = IV. Since current density is tied to the conductor's area and determines the permissible current without excessive heating, understanding how increasing the voltage can allow a reduction in current—and thereby a smaller conductor cross-section—is pivotal in designing efficient transmission lines.

Power Transmission Efficiency

Efficiency in power transmission requires minimizing energy losses during transmission. By calculating a percentage loss and designing transmission lines accordingly, engineers can determine the necessary parameters (like conductor dimensions) to ensure that the majority of the transmitted power reaches the destination without being wasted as heat.

Electrical Resistivity and Resistance

This concept involves understanding how a material, such as copper, opposes the flow of electric current. The resistance of a conductor is determined by its resistivity, length, and cross?sectional area using the formula R = ?L/A. A clear grasp of resistivity is essential when designing wires for power transmission because the resistance directly affects energy losses due to heat.

Joule Heating and Power Loss in Conductors

Joule heating refers to the process by which electric energy is converted into thermal energy in a resistor when current flows through it. The power loss in a conductor can be calculated using the formula P_loss = I²R. This concept is critical in power transmission, where maintaining low energy loss over long distances is a prime objective.

Current Density and Cross-Sectional Area

Current density is defined as the current per unit area (J = I/A) and represents how the electric current is distributed over the cross section of a conductor. Designing a wire with an appropriate cross-sectional area ensures that the current density remains within safe limits to avoid overheating and excessive power loss. This concept links the geometric properties of the wire to its electrical performance.

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