Textbook Question
The switch in FIGURE P30.76 has been open for a long time. It is closed at t = 0 s. What is the current through the 20 Ω resistor after the switch has been closed a long time?
Ch 30: Electromagnetic Induction
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 30, Problem 77a
The switch in FIGURE P30.77 has been open for a long time. It is closed at t = 0 s. After the switch has been closed for a long time, what is the current in the circuit? Call this current I0.
Verified step by step guidance1
Step 1: Analyze the circuit diagram. The circuit consists of a battery with voltage ΔV_bat, a resistor R, and an inductor L connected in series. The switch has been open for a long time, meaning no current was flowing initially, and the inductor had no stored energy.
Step 2: Understand the behavior of the inductor. When the switch is closed at t = 0 s, the inductor initially opposes the change in current due to its inductance. Over time, the current increases and the inductor's opposition diminishes as it reaches a steady state.
Step 3: At steady state (after the switch has been closed for a long time), the inductor behaves like a short circuit because the rate of change of current (dI/dt) becomes zero. This means the inductor no longer opposes the current flow.
Step 4: Apply Ohm's Law to the circuit at steady state. Since the inductor acts as a short circuit, the total resistance in the circuit is just R. The current I_0 can be calculated using the formula: .
Step 5: Conclude that the steady-state current I_0 depends only on the battery voltage ΔV_bat and the resistance R. To find the numerical value of I_0, substitute the given values of ΔV_bat and R into the formula.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inductance
Inductance is a property of an electrical circuit that opposes changes in current. It is primarily associated with coils or loops of wire, where a changing current generates a magnetic field that induces an electromotive force (EMF) in the opposite direction. This phenomenon is crucial in understanding how circuits behave when switches are closed or opened, particularly in circuits containing inductors.
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Steady State in RL Circuits
In an RL circuit, the steady state refers to the condition after the circuit has been closed for a long time, where the current reaches a constant value. At this point, the inductor behaves like a short circuit, and the entire voltage from the battery is dropped across the resistor. Understanding this concept is essential for determining the final current in the circuit after the switch is closed.
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Ohm's Law
Ohm's Law states that the current (I) 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. Mathematically, it is expressed as V = IR. This law is fundamental in analyzing circuits, as it allows for the calculation of current when the voltage and resistance are known, particularly in the context of the steady state in RL circuits.
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Related Practice
Textbook Question
CALC The rectangular loop in FIGURE CP30.81 has 0.020 Ω resistance. What is the induced current in the loop at this instant?
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Textbook Question
The switch in FIGURE P30.76 has been open for a long time. It is closed at t = 0 s. What is the current through the 20 Ω resistor immediately after the switch is closed?
Textbook Question
In recent years it has been possible to buy a 1.0 F capacitor. This is an enormously large amount of capacitance. Suppose you want to build a 1.0 Hz oscillator with a 1.0 F capacitor. You have a spool of 0.25-mm-diameter wire and a 4.0-cm-diameter plastic cylinder. How long must your inductor be if you wrap it with 2 layers of closely spaced turns?
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Textbook Question
The switch in FIGURE P30.77 has been open for a long time. It is closed at t = 0 s. Find an expression for the current I as a function of time. Write your expression in terms of I0, R, and L.
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Textbook Question
The 300 μF capacitor in FIGURE P30.75 is initially charged to 100 V, the 1200 μF capacitor is uncharged, and the switches are both open. What is the maximum voltage to which you can charge the 1200 μF capacitor by the proper closing and opening of the two switches?
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