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 8
(a) Compute the reactance of a 0.450-H inductor at frequencies of 60.0 Hz and 600 Hz. (b) Compute the reactance of a 2.50-μF capacitor at the same frequencies. (c) At what frequency is the reactance of a 0.450-H inductor equal to that of a 2.50-μF capacitor?
Verified step by step guidance1
To find the reactance of an inductor, use the formula for inductive reactance: \( X_L = 2\pi f L \), where \( f \) is the frequency and \( L \) is the inductance. For part (a), calculate \( X_L \) at 60.0 Hz and 600 Hz using \( L = 0.450 \) H.
For the reactance of a capacitor, use the formula for capacitive reactance: \( X_C = \frac{1}{2\pi f C} \), where \( C \) is the capacitance. For part (b), calculate \( X_C \) at 60.0 Hz and 600 Hz using \( C = 2.50 \times 10^{-6} \) F.
To find the frequency at which the reactance of the inductor equals that of the capacitor, set \( X_L = X_C \). This gives the equation \( 2\pi f L = \frac{1}{2\pi f C} \).
Solve the equation \( 2\pi f L = \frac{1}{2\pi f C} \) for \( f \). This involves rearranging the equation to find \( f = \frac{1}{2\pi \sqrt{LC}} \).
Substitute the given values of \( L = 0.450 \) H and \( C = 2.50 \times 10^{-6} \) F into the equation \( f = \frac{1}{2\pi \sqrt{LC}} \) to find the frequency at which the reactances are equal.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inductive Reactance
Inductive reactance is the opposition that an inductor presents to alternating current, due to the magnetic field created by the current. It is calculated using the formula X_L = 2πfL, where X_L is the inductive reactance, f is the frequency, and L is the inductance. This concept is crucial for determining how an inductor behaves at different frequencies.
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Capacitive Reactance
Capacitive reactance is the opposition that a capacitor presents to alternating current, due to the electric field created between its plates. It is calculated using the formula X_C = 1/(2πfC), where X_C is the capacitive reactance, f is the frequency, and C is the capacitance. Understanding capacitive reactance is essential for analyzing how a capacitor affects AC circuits at various frequencies.
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Frequency and Reactance Equality
The frequency at which the reactance of an inductor equals that of a capacitor is determined by setting their reactance formulas equal: 2πfL = 1/(2πfC). Solving this equation gives the frequency at which the inductive and capacitive reactances are equal, highlighting the balance point in AC circuits where inductive and capacitive effects cancel each other out.
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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
A capacitor is connected across an ac source that has voltage amplitude 60.0 V and frequency 80.0 Hz. What is the phase angle Φ for the source voltage relative to the current? Does the source voltage lag or lead the current?
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Textbook Question
A capacitor is connected across an ac source that has voltage amplitude 60.0 V and frequency 80.0 Hz. What is the capacitance C of the capacitor if the current amplitude is 5.30 A?
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Textbook Question
A capacitance C and an inductance L are operated at the same angular frequency. (a) At what angular frequency will they have the same reactance? (b) If L = 5 00 mH and C = 3.50 μF, what is the numerical value of the angular frequency in part (a), and what is the reactance of each element?
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Textbook Question
What is the inductance of an inductor whose reactance is 120 Ω at 80.0 Hz?