Textbook Question
A series RLC circuit consists of a 75 Ω resistor, a 0.12 H inductor, and a 30 μF capacitor. It is attached to a 120 V/60 Hz power line. What is the peak current I?
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Ch 32: AC Circuits
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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Table of contents
- Ch 01: Concepts of Motion35
- Ch 02: Kinematics in One Dimension75
- Ch 03: Vectors and Coordinate Systems58
- Ch 04: Kinematics in Two Dimensions60
- Ch 05: Force and Motion23
- Ch 06: Dynamics I: Motion Along a Line53
- Ch 07: Newton's Third Law27
- Ch 08: Dynamics II: Motion in a Plane52
- Ch 09: Work and Kinetic Energy56
- Ch 10: Interactions and Potential Energy50
- Ch 11: Impulse and Momentum56
- Ch 12: Rotation of a Rigid Body71
- Ch 13: Newton's Theory of Gravity52
- Ch 14: Fluids and Elasticity44
- Ch 15: Oscillations55
- Ch 16: Traveling Waves37
- Ch 17: Superposition39
- Ch 18: A Macroscopic Description of Matter53
- Ch 19: Work, Heat, and the First Law of Thermodynamics60
- Ch 20: The Micro/Macro Connection53
- Ch 21: Heat Engines and Refrigerators34
- Ch 22: Electric Charges and Forces40
- Ch 23: The Electric Field39
- Ch 24: Gauss' Law32
- Ch 25: The Electric Potential63
- Ch 26: Potential and Field57
- Ch 27: Current and Resistance42
- Ch 28: Fundamentals of Circuits39
- Ch 29: The Magnetic Field47
- Ch 30: Electromagnetic Induction60
- Ch 31: Electromagnetic Fields and Waves32
- Ch 32: AC Circuits63
- Ch 33: Wave Optics32
- Ch 34: Ray Optics57
- Ch 35: Optical Instruments30
- Ch 36: Special Relativity38
- Ch 37: The Foundations of Modern Physics25
- Ch 38: Quantization49
- Ch 39: Wave Functions and Uncertainty31
- Ch 40: One-Dimensional Quantum Mechanics35
- Ch 41: Atomic Physics18
- Ch 42: Nuclear Physics27
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 32, Problem 51c
A series RLC circuit consists of a 75 Ω resistor, a 0.12 H inductor, and a 30 μF capacitor. It is attached to a 120 V/60 Hz power line. What is the average power dissipated?
Verified step by step guidance1
Step 1: Calculate the angular frequency (ω) of the AC power source using the formula ω = 2πf, where f is the frequency of the power line. For this problem, f = 60 Hz.
Step 2: Determine the inductive reactance (X_L) using the formula X_L = ωL, where L is the inductance of the inductor (0.12 H).
Step 3: Calculate the capacitive reactance (X_C) using the formula X_C = 1 / (ωC), where C is the capacitance of the capacitor (30 μF or 30 × 10⁻⁶ F).
Step 4: Compute the total impedance (Z) of the circuit using the formula Z = √(R² + (X_L - X_C)²), where R is the resistance (75 Ω).
Step 5: Calculate the average power dissipated using the formula P_avg = (V_rms² / Z) × (R / Z), where V_rms is the root mean square voltage of the power source (120 V).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Impedance in RLC Circuits
Impedance is the total opposition that a circuit offers to the flow of alternating current (AC) and is represented as a complex number. In a series RLC circuit, the impedance combines the resistance (R), inductive reactance (XL), and capacitive reactance (XC). The formula for impedance is Z = √(R² + (XL - XC)²), where XL = 2πfL and XC = 1/(2πfC). Understanding impedance is crucial for calculating current and power in AC circuits.
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08:40
Impedance in AC Circuits
Average Power in AC Circuits
The average power dissipated in an AC circuit is calculated using the formula P = VIcos(φ), where V is the root mean square (RMS) voltage, I is the RMS current, and φ is the phase angle between the voltage and current. The phase angle is determined by the impedance of the circuit and affects how much of the power is actually used for work versus being stored in the reactive components. This concept is essential for determining how much power is effectively consumed by the circuit.
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05:37Power in AC Circuits
Resonance in RLC Circuits
Resonance occurs in RLC circuits when the inductive reactance equals the capacitive reactance (XL = XC), resulting in maximum current flow and minimal impedance. At resonance, the circuit can draw maximum power from the source, and the phase angle φ becomes zero, meaning the voltage and current are in phase. While this concept is not directly needed for calculating average power, it provides insight into the behavior of the circuit under different frequencies, which can affect power dissipation.
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05:23Resonance in Series LRC Circuits
Related Practice
Textbook Question
A series RLC circuit consists of a 75 Ω resistor, a 0.12 H inductor, and a 30 μF capacitor. It is attached to a 120 V/60 Hz power line. What is the phase angle ϕ?
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Textbook Question
In FIGURE P32.54, what is the current supplied by the emf when (a) the frequency is very small and (b) the frequency is very large?
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Textbook Question
A series RLC circuit consists of a 50 Ω resistor, a 3.3 mH inductor, and a 480 nF capacitor. It is connected to a 5.0 kHz oscillator with a peak voltage of 5.0 V. What is the instantaneous current i when ε = ε0?
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Textbook Question
Show that the power factor of a series RLC circuit is cos ϕ=R/Z.
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