A line having a total resistance of 0.20 Ωdelivers 10.00 kW...

A line having a total resistance of $0.20 \Omega$ delivers $10.00 \mathrm{~kW}$ at $250 \mathrm{~V}$ to a small factory. What is the efficiency of the transmission?The line dissipates power due to its resistance. Consequently we'll need to find the current in the line. Use $\mathrm{P}=V I$ to find $I=\mathrm{P} / V$. Then Power lost in line $=I^{2} R=\left(\frac{\mathrm{P}}{V}\right)^{2} R=\left(\frac{10000 \mathrm{~W}}{250 \mathrm{~V}}\right)^{2}(0.20 \Omega)=0.32 \mathrm{~kW}$ $$ \text { Efficiency }=\frac{\text { Power delivered by line }}{\text { Power supplied to line }}=\frac{10.00 \mathrm{~kW}}{(10.00+0.32) \mathrm{kW}}=0.970=97.0 \% $$

A line having a total resistance of $0.20 \Omega$ delivers $10.00 \mathrm{~kW}$ at $250 \mathrm{~V}$ to a small factory. What is the efficiency of the transmission?The line dissipates power due to its resistance. Consequently we'll need to find the current in the line. Use $\mathrm{P}=V I$ to find $I=\mathrm{P} / V$. Then
Power lost in line $=I^{2} R=\left(\frac{\mathrm{P}}{V}\right)^{2} R=\left(\frac{10000 \mathrm{~W}}{250 \mathrm{~V}}\right)^{2}(0.20 \Omega)=0.32 \mathrm{~kW}$
$$
\text { Efficiency }=\frac{\text { Power delivered by line }}{\text { Power supplied to line }}=\frac{10.00 \mathrm{~kW}}{(10.00+0.32) \mathrm{kW}}=0.970=97.0 \%
$$

Key Concepts

Ohm's Law

Ohm's Law states that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to its resistance. This fundamental principle of electrical engineering is used to relate voltage, current, and resistance, making it essential for analyzing circuits and determining current flow in power transmission scenarios.

Joule Heating

Joule heating refers to the process where electrical energy is converted into heat energy due to the resistance in a conductor. The power dissipated as heat in a resistor, calculated by the formula I²R, is a critical concept when assessing losses in transmission lines and understanding how resistance impacts overall energy efficiency.

Electrical Power

Electrical power is the rate at which electrical energy is transferred by an electric circuit, and it is calculated by the product of voltage and current. This concept is central to understanding how much energy is available to do useful work in a system and forms the basis for determining operational requirements in power delivery systems.

Transmission Efficiency

Transmission efficiency measures the performance of a power delivery system by comparing the useful power output to the total power supplied. It accounts for losses incurred, primarily due to resistive heating in the transmission lines, and is calculated as the ratio of power delivered to the load over the sum of power delivered and power lost. This concept is key in evaluating and optimizing power distribution systems.

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