Ch 32: AC Circuits

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 32: AC Circuits

Problem 65a

Chapter 32, Problem 65a

You're the operator of a 15,000 V rms, 60 Hz electrical substation. When you get to work one day, you see that the station is delivering 6.0 MW of power with a power factor of 0.90. What is the rms current leaving the station?

Verified step by step guidance

Step 1: Recall the formula for real power (P) in an AC circuit: \( P = V_{rms} \cdot I_{rms} \cdot \text{power factor} \), where \( V_{rms} \) is the root mean square voltage, \( I_{rms} \) is the root mean square current, and the power factor accounts for the phase difference between voltage and current.

Step 2: Rearrange the formula to solve for \( I_{rms} \): \( I_{rms} = \frac{P}{V_{rms} \cdot \text{power factor}} \).

Step 3: Substitute the given values into the formula: \( P = 6.0 \times 10^6 \ \text{W} \), \( V_{rms} = 15,000 \ \text{V} \), and \( \text{power factor} = 0.90 \).

Step 4: Perform the division: First, calculate the denominator \( V_{rms} \cdot \text{power factor} \), then divide the real power \( P \) by this value to find \( I_{rms} \).

Step 5: The result of the calculation will give you the rms current \( I_{rms} \) in amperes, which represents the current leaving the station.

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Key Concepts

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

RMS Voltage

RMS (Root Mean Square) voltage is a measure of the effective voltage of an alternating current (AC) system. It represents the equivalent direct current (DC) voltage that would deliver the same power to a load. In this context, the 15,000 V rms indicates the effective voltage level at which the electrical substation operates.

Power Factor

Power factor is a dimensionless number between 0 and 1 that represents the ratio of real power flowing to the load, to the apparent power in the circuit. A power factor of 0.90 indicates that 90% of the power is being effectively used for work, while the remaining 10% is reactive power, which does not perform any useful work but is necessary for maintaining the voltage levels in the system.

Electrical Power Calculation

Electrical power in an AC circuit can be calculated using the formula P = V * I * PF, where P is the real power in watts, V is the rms voltage, I is the rms current, and PF is the power factor. This relationship allows us to determine the current flowing through the system when the power, voltage, and power factor are known, which is essential for understanding the operation of the electrical substation.

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Textbook Question

A motor attached to a 120 V/60 Hz power line draws an 8.0 A current. Its average energy dissipation is 800 W. What is the motor's resistance?

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Textbook Question

A motor attached to a 120 V/60 Hz power line draws an 8.0 A current. Its average energy dissipation is 800 W. What is the rms resistor voltage?

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Textbook Question

A generator consists of a 12-cm by 16-cm rectangular loop with 500 turns of wire spinning at 60 Hz in a 25 mT uniform magnetic field. The generator output is connected to a series RC circuit consisting of a 120 Ω resistor and a 35 μF capacitor. What is the average power delivered to the circuit?

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Textbook Question

Commercial electricity is generated and transmitted as three-phase electricity. Instead of a single emf ε = ε₀ cos ωt, three separate wires carry currents for the emfs ε₁= ε₀ cos ωt, ε₂= ε₀ cos(ωt+120°), and ε₃= ε₀ cos(ωt−120°). This is why the long-distance transmission lines you see in the countryside have three parallel wires, as do many distribution lines within a city. Show that the potential difference between any two of the phases has the rms value 3–√ ε_rms, where ε_rms is the familiar single-phase rms voltage. Evaluate this potential difference for ε_rms= 120 V. Some high-power home appliances, especially electric clothes dryers and hot-water heaters, are designed to operate between two of the phases rather than between one phase and neutral. Heavy-duty industrial motors are designed to operate from all three phases, but full three-phase power is rare in residential or office use.

Textbook Question

Commercial electricity is generated and transmitted as three-phase electricity. Instead of a single emf, three separate wires carry currents for the emfs ε₁ = ε₀ cos ωt, ε₂ = ε₀ cos(ωt +120°), and ε₃ = ε₀ cos(ωt−120°) over three parallel wires, each of which supplies one-third of the power. This is why the long-distance transmission lines you see in the countryside have three wires. Suppose the transmission lines into a city supply a total of 450 MW of electric power, a realistic value. What would be the rms current in each wire if the transmission voltage were ε₀ = 120 V rms?

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Textbook Question

Commercial electricity is generated and transmitted as three-phase electricity. Instead of a single emf, three separate wires carry currents for the emfs ε₁ = ε₀ cos ωt, ε₂ = ε₀ cos(ωt +120°), and ε₃ = ε₀ cos(ωt−120°) over three parallel wires, each of which supplies one-third of the power. This is why the long-distance transmission lines you see in the countryside have three wires. Suppose the transmission lines into a city supply a total of 450 MW of electric power, a realistic value. In fact, transformers are used to step the transmission-line voltage up to 500 kV rms. What is the current in each wire?

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