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Ch. 29 - Electromagnetic Induction and Faraday's Law
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
Chapter 28, Problem 49

If 75 MW of power at 45 kV (rms) arrives at a town from a generator via transmission lines of total resistance 3.0 Ω, calculate (a) the emf at the generator end of the lines, and (b) the fraction of the power generated that is wasted in the lines.

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Step 1: Start by calculating the current (I) in the transmission lines using the formula for power: \( P = V_{rms} \cdot I \), where \( P \) is the power delivered (75 MW) and \( V_{rms} \) is the root mean square voltage (45 kV). Rearrange the formula to solve for \( I \): \( I = \frac{P}{V_{rms}} \).
Step 2: Calculate the power loss in the transmission lines due to resistance using the formula \( P_{loss} = I^2 \cdot R \), where \( R \) is the total resistance of the transmission lines (3.0 Ω). Substitute the value of \( I \) from Step 1 into this formula.
Step 3: Determine the emf at the generator end of the lines. The total voltage drop across the transmission lines is given by \( V_{drop} = I \cdot R \). Add this voltage drop to the delivered voltage \( V_{rms} \) to find the emf: \( V_{generator} = V_{rms} + V_{drop} \).
Step 4: Calculate the total power generated by the generator. The total power is the sum of the power delivered to the town and the power lost in the transmission lines: \( P_{total} = P + P_{loss} \).
Step 5: Find the fraction of power wasted in the transmission lines. This is given by \( \text{Fraction wasted} = \frac{P_{loss}}{P_{total}} \). Substitute the values of \( P_{loss} \) and \( P_{total} \) from the previous steps to compute this fraction.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This relationship is expressed as V = IR. Understanding this law is crucial for calculating the voltage drop across the transmission lines due to their resistance.
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Power Loss in Resistive Circuits

In electrical circuits, power loss due to resistance can be calculated using the formula P_loss = I^2R, where P_loss is the power lost, I is the current, and R is the resistance. This concept is essential for determining how much power is wasted in the transmission lines, which directly affects the efficiency of power delivery from the generator to the town.
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RMS Voltage

RMS (Root Mean Square) voltage is a way of expressing the effective value of an alternating current (AC) voltage. It represents the equivalent direct current (DC) voltage that would deliver the same power to a load. In this question, the given 45 kV (rms) voltage is critical for calculating the current and subsequently the emf at the generator end.
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