You're Chief Financial Officer for a power company, and you consult your engineering department in an effort to minimize power-line losses. Your power plant produces \(60-\mathrm{Hz}\) power at \(365 \mathrm{kV} \mathrm{rms}\) and \(200 \mathrm{A} \mathrm{rms},\) and delivers it via transmission lines with total resistance \(100 \Omega .\) You ask the engineers for the percentage of power that's lost. They reply that it depends on the power factor. What's the percentage loss for power factors of (a) 1.0 and (b) \(0.60 ?\)

Understanding Ohm's law is fundamental when calculating power-line losses. It explains the relationship between voltage, current, and resistance, which is crucial in electrical circuits. Ohm's law states that the current (\(I\right)\) through a conductor between two points is directly proportional to the voltage (\(V\right)\) across the two points and inversely proportional to the resistance (\(R\right)\) of the conductor. The formula is defined as
\[ I = \frac{V}{R} \]
In contexts involving power lines, this formula is manipulated to express the power loss (\(P_{loss}\right)\) as \[ P_{loss} = I^2 * R \]
which accounts for losses solely due to the resistance of the wire. A higher current or resistance directly leads to increased losses, highlighting why power companies often transmit electricity at high voltages and low currents.

When discussing electricity transmission, apparent power is a term that comes into play. Apparent power (\(S\right)\) is the product of the root mean square (rms) voltage and current in an AC circuit. It is measured in volt-amps (\(VA\right)\) and is given by the formula:
\[ S = V * I \]
However, this doesn't reflect the actual energy consumed or used in a circuit, as it combines both real and reactive power, which are the components of power that do actual work and store energy, respectively. The resulting unit is mega volt-amps (\(MVA\right)\) for large-scale power systems like the one in the given exercise. Understanding the difference between apparent, real, and reactive power is key to optimizing and understanding energy systems.

The power factor (\(PF\right)\) is a dimensionless number between -1 and 1 that describes how effectively the electrical power is being converted into useful work output. It is a vital concept in the electricity industry since it affects the efficiency of power delivery.

The power factor is the ratio of real power (\(P\right)\) that is used to do work and the apparent power (\(S\right)\) that is supplied to the circuit. The formula for power factor is defined as:
\[ PF = \frac{P}{S} \]
A power factor of 1, or 100%, means that all the power is being effectively converted into work with no reactive power in the system. A lower power factor indicates the presence of reactive power, which leads to inefficiencies in the system. This is why power companies sometimes charge higher rates for customers with a low power factor, as it requires more apparent power to supply the same amount of real power.

Real power (\(P\right)\) is the capacity of the circuit for performing work in a given time. Unlike apparent power, real power indicates the actual amount of power consumed by all the resistive components of the circuit, which includes lights, heaters, and other appliances that convert electricity into heat, light, or motion. Real power is measured in watts (\(W\right)\) and when dealing with large quantities such as in the energy sector, in megawatts (\(MW\right)\). The presence of a power factor indicates that a system might have both resistive and inductive or capacitive elements, where the latter two do not contribute to real power but do factor into apparent power.

Finally, power loss percentage is a critical metric for power companies as it indicates the efficiency of power transmission. It is the ratio of power lost due to resistance in the power lines relative to the total power generated or transmitted. The formula to calculate power loss percentage is:
\[ \text{Power Loss Percentage} = \left(\frac{P_{loss}}{P}\right) * 100\% \]
This percentage is significantly influenced by the power factor because a low power factor can increase the current requirement for the same amount of real power, thus leading to greater losses. In financial terms, higher power loss percentages mean more wasted energy and increased costs for production and infrastructure maintenance. Enhancing the power factor can reduce these losses and improve the overall fiscal and environmental efficiency of power transmission.