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 67b

Chapter 32, Problem 67b

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?

Verified step by step guidance

Step 1: Understand the given quantities. The motor is connected to a 120 V/60 Hz power line, which means the root mean square (rms) voltage of the power supply is 120 V. The current drawn by the motor is 8.0 A (rms), and the average power dissipation is 800 W.

Step 2: Recall the formula for average power dissipation in an AC circuit:

P = V_{rms} I_{rms} \, \(\cos\)(\(\phi\))

, where

\(\cos\)(\(\phi\))

is the power factor. Rearrange this formula to solve for

\(\cos\)(\(\phi\))

\(\cos\)(\(\phi\)) = \(\frac{P}{V_{rms}\) I_{rms}}

Step 3: Substitute the known values into the formula for

\(\cos\)(\(\phi\))

\(\cos\)(\(\phi\)) = \(\frac{800}{120 \times 8.0}\)

. This will give the power factor of the motor.

Step 4: The rms resistor voltage is the component of the total voltage that is in phase with the current. It can be calculated using the formula:

V_{resistor} = V_{rms} \(\cos\)(\(\phi\))

. Substitute the value of

\(\cos\)(\(\phi\))

obtained in Step 3 into this formula.

Step 5: Simplify the expression to find the rms resistor voltage. This will give the final result for

V_{resistor}

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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 statistical measure of the magnitude of a varying voltage. It represents the equivalent DC voltage that would deliver the same power to a resistor. For AC circuits, the RMS voltage is crucial for calculating power and is typically lower than the peak voltage due to the sinusoidal nature of AC waveforms.

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. It is mathematically expressed as V = I × R. This law is fundamental in analyzing electrical circuits and understanding how voltage, current, and resistance interact.

Power in Electrical Circuits

In electrical circuits, power (P) is the rate at which energy is consumed or converted and is calculated using the formula P = V × I, where V is voltage and I is current. For AC circuits, the average power can also be expressed in terms of RMS values, which allows for accurate calculations of energy dissipation in resistive components. Understanding power is essential for evaluating the efficiency and performance of electrical devices.

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Related Practice

Textbook Question

FIGURE CP32.68 shows voltage and current graphs for a series RLC circuit. What is the resistance R?

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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 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

A motor attached to a 120 V/60 Hz power line draws an 8.0 A current. Its average energy dissipation is 800 W. How much series capacitance needs to be added to increase the power factor to 1.0?

Textbook Question

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?

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