Textbook Question
What is the reactance of a 3.00-H inductor at a frequency of 80.0 Hz?
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Ch 31: Alternating Current
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610
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Table of contents
- Ch 01: Units, Physical Quantities & Vectors41
- Ch 02: Motion Along a Straight Line67
- Ch 03: Motion in Two or Three Dimensions47
- Ch 04: Newton's Laws of Motion23
- Ch 05: Applying Newton's Laws59
- Ch 06: Work & Kinetic Energy14
- Ch 07: Potential Energy & Conservation8
- Ch 08: Momentum, Impulse, and Collisions18
- Ch 09: Rotation of Rigid Bodies42
- Ch 10: Dynamics of Rotational Motion48
- Ch 11: Equilibrium & Elasticity27
- Ch 12: Fluid Mechanics31
- Ch 13: Gravitation35
- Ch 14: Periodic Motion32
- Ch 15: Mechanical Waves25
- Ch 16: Sound & Hearing32
- Ch 17: Temperature and Heat36
- Ch 18: Thermal Properties of Matter38
- Ch 19: The First Law of Thermodynamics33
- Ch 20: The Second Law of Thermodynamics22
- Ch 21: Electric Charge and Electric Field36
- Ch 22: Gauss' Law24
- Ch 23: Electric Potential31
- Ch 24: Capacitance and Dielectrics18
- Ch 25: Current, Resistance, and EMF26
- Ch 26: Direct-Current Circuits30
- Ch 27: Magnetic Field and Magnetic Forces18
- Ch 28: Sources of Magnetic Field10
- Ch 29: Electromagnetic Induction28
- Ch 30: Inductance17
- Ch 31: Alternating Current21
- Ch 32: Electromagnetic Waves1
- Ch 33: The Nature and Propagation of Light20
- Ch 34: Geometric Optics34
- Ch 35: Interference19
- Ch 36: Diffraction25
- Ch 37: Special Relativity20
- Ch 38: Photons: Light Waves Behaving as Particles15
- Ch 39: Particles Behaving as Waves26
- Ch 40: Quantum Mechanics I: Wave Functions21
- Ch 41: Quantum Mechanics II: Atomic Structure25
- Ch 42: Molecules and Condensed Matter18
- Ch 43: Nuclear Physics16
- Ch 44: Particle Physics and Cosmology23
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
Chapter 31, Problem 15a
You have a 200-Ω resistor, a 0.400-H inductor, and a 6.00-μF capacitor. They are connected to form an L-R-C series circuit. What is the impedance of the circuit?
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Step 1: Recall the formula for impedance in an L-R-C series circuit: \( Z = \sqrt{R^2 + (X_L - X_C)^2} \), where \( R \) is the resistance, \( X_L \) is the inductive reactance, and \( X_C \) is the capacitive reactance.
Step 2: Calculate the inductive reactance \( X_L \) using the formula \( X_L = \omega L \), where \( \omega \) is the angular frequency and \( L \) is the inductance. Note that \( \omega = 2\pi f \), where \( f \) is the frequency of the circuit.
Step 3: Calculate the capacitive reactance \( X_C \) using the formula \( X_C = \frac{1}{\omega C} \), where \( C \) is the capacitance and \( \omega \) is the angular frequency.
Step 4: Substitute the values of \( R \), \( X_L \), and \( X_C \) into the impedance formula \( Z = \sqrt{R^2 + (X_L - X_C)^2} \).
Step 5: Simplify the expression to find the impedance \( Z \). Ensure all units are consistent (e.g., convert \( \mu F \) to \( F \) and use SI units for resistance, inductance, and capacitance).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Impedance in AC Circuits
Impedance is the total opposition that a circuit offers to the flow of alternating current (AC) and is represented as a complex number. It combines resistance (R) and reactance (X), where reactance arises from inductors and capacitors. The formula for impedance in an L-R-C series circuit is Z = √(R² + (X_L - X_C)²), where X_L is the inductive reactance and X_C is the capacitive reactance.
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Impedance in AC Circuits
Inductive Reactance
Inductive reactance (X_L) is the opposition to the change of current in an inductor and is given by the formula X_L = 2πfL, where f is the frequency of the AC source and L is the inductance in henries. This reactance increases with frequency, meaning that at higher frequencies, inductors resist changes in current more strongly, affecting the overall impedance of the circuit.
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Capacitive Reactance
Capacitive reactance (X_C) is the opposition to the change of voltage across a capacitor and is calculated using the formula X_C = 1/(2πfC), where C is the capacitance in farads. Unlike inductive reactance, capacitive reactance decreases with increasing frequency, which means that capacitors allow more current to pass at higher frequencies, influencing the total impedance in the circuit.
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Related Practice
Textbook Question
You have a 200-Ω resistor, a 0.400-H inductor, and a 6.00-μF capacitor. Suppose you take the resistor and inductor and make a series circuit with a voltage source that has voltage amplitude 30.0 V and an angular frequency of 250 rad/s. What is the impedance of the circuit?
Textbook Question
What is the inductance of an inductor whose reactance is 120 Ω at 80.0 Hz?
Textbook Question
A resistor with R = 300 Ω and an inductor are connected in series across an ac source that has voltage amplitude 500 V. The rate at which electrical energy is dissipated in the resistor is 286 W. What is (a) the impedance Z of the circuit; (b) the amplitude of the voltage across the inductor; (c) the power factor?
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Textbook Question
The power of a certain CD player operating at 120 V rms is 20.0 W. Assuming that the CD player behaves like a pure resistor, find the maximum instantaneous power.
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Textbook Question
You have a 200-Ω resistor, a 0.400-H inductor, and a 6.00-μF capacitor. They are connected to form an L-R-C series circuit. What is the current amplitude?