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?
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 55a
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?
Verified step by step guidance1
Step 1: Understand the problem. The goal is to find the instantaneous current i in a series RLC circuit when the voltage ε equals its peak value ε₀. This involves calculating the impedance Z of the circuit and using Ohm's law to determine the current.
Step 2: Calculate the angular frequency ω of the oscillator. The angular frequency is given by ω = 2πf, where f is the frequency of the oscillator. Substitute f = 5.0 kHz into the formula to find ω.
Step 3: Determine the reactance of the inductor (X_L) and the capacitor (X_C). The inductive reactance is given by X_L = ωL, where L is the inductance of the inductor. The capacitive reactance is given by X_C = 1 / (ωC), where C is the capacitance of the capacitor. Use the values of L = 3.3 mH and C = 480 nF to calculate X_L and X_C.
Step 4: Calculate the total impedance Z of the circuit. The impedance in a series RLC circuit is given by Z = √(R² + (X_L - X_C)²), where R is the resistance. Substitute the values of R, X_L, and X_C into the formula to find Z.
Step 5: Use Ohm's law to find the peak current I₀. Ohm's law states that I₀ = ε₀ / Z, where ε₀ is the peak voltage. Substitute ε₀ = 5.0 V and the calculated Z into the formula to find I₀. Since ε = ε₀, the instantaneous current i is equal to the peak current I₀.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
RLC Circuit
An RLC circuit is an electrical circuit consisting of a resistor (R), inductor (L), and capacitor (C) connected in series or parallel. The behavior of the circuit is influenced by the values of these components, which determine the circuit's impedance and resonance frequency. In this case, the series RLC circuit will respond to an alternating current (AC) source, such as the 5.0 kHz oscillator mentioned.
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09:40
LRC Circuits
Impedance
Impedance is the total opposition that a circuit offers to the flow of alternating current, combining resistance (R) and reactance (X). In an RLC circuit, the impedance can be calculated using the formula Z = √(R² + (X_L - X_C)²), where X_L is the inductive reactance and X_C is the capacitive reactance. Understanding impedance is crucial for determining the current in the circuit when a voltage is applied.
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08:40Impedance in AC Circuits
Instantaneous Current
Instantaneous current refers to the current flowing through a circuit at a specific moment in time. In an AC circuit, this current varies sinusoidally with time, and its maximum value occurs when the voltage is at its peak. To find the instantaneous current when ε = ε₀, one must consider the phase relationship between voltage and current, which is influenced by the circuit's impedance.
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05:38Intro to Current
Related Practice
Textbook Question
The tuning circuit in an FM radio receiver is a series RLC circuit with a 0.200 μH inductor. The receiver is tuned to a station at 104.3 MHz. What is the value of the capacitor in the tuning circuit?
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Textbook Question
The tuning circuit in an FM radio receiver is a series RLC circuit with a 0.200 μH inductor. FM radio stations are assigned frequencies every 0.2 MHz, but two nearby stations cannot use adjacent frequencies. What is the maximum resistance the tuning circuit can have if the peak current at a frequency of 103.9 MHz, the closest frequency that can be used by a nearby station, is to be no more than 0.10% of the peak current at 104.3 MHz? The radio is still tuned to 104.3 MHz, and you can assume the two stations have equal strength.
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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 average power dissipated?
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
Show that the power factor of a series RLC circuit is cos ϕ=R/Z.
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