Ch 30: Electromagnetic Induction

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 30: Electromagnetic Induction

Problem 76a

Chapter 30, Problem 76a

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?

Verified step by step guidance

Step 1: Analyze the circuit configuration. The circuit consists of a 30 V battery, a 10 Ω resistor, a 20 Ω resistor, and a 10 mH inductor. The switch has been open for a long time, meaning the inductor has no initial current flowing through it. When the switch is closed at t = 0 s, the inductor will oppose any sudden change in current due to its inductive property.

Step 2: Understand the behavior of the inductor immediately after the switch is closed. At t = 0 s, the inductor behaves like an open circuit because it resists any instantaneous change in current. Therefore, no current flows through the branch containing the inductor at this instant.

Step 3: Simplify the circuit for t = 0 s. Since the inductor acts as an open circuit, the 20 Ω resistor is effectively disconnected from the circuit. The current flows only through the 10 Ω resistor connected directly to the battery.

Step 4: Apply Ohm's Law to calculate the current through the 10 Ω resistor. The voltage across the resistor is equal to the battery voltage (30 V). Using Ohm's Law, \( I = \frac{V}{R} \), where \( V \) is the voltage and \( R \) is the resistance.

Step 5: Conclude that the current through the 20 Ω resistor is zero immediately after the switch is closed because the inductor prevents current from flowing through its branch at t = 0 s.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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. This relationship is expressed mathematically as I = V/R. Understanding this law is crucial for calculating the current through resistors in a circuit.

Inductance and Inductive Reactance

Inductance is a property of an electrical component, typically a coil or inductor, that opposes changes in current. When the switch is closed, the inductor initially resists the change in current, creating a transient response. The inductive reactance (XL) can be calculated using the formula XL = 2πfL, where f is the frequency and L is the inductance. However, immediately after closing the switch, the inductor behaves like a short circuit.

Transient Response in RL Circuits

In an RL circuit, when a switch is closed, the current does not instantly reach its maximum value due to the inductor's opposition to changes in current. Instead, the current increases gradually, following an exponential curve characterized by the time constant τ = L/R, where L is the inductance and R is the total resistance in the circuit. Analyzing the transient response is essential for understanding the behavior of the circuit immediately after the switch is closed.

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Related Practice

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?

Textbook Question

An electric oscillator is made with a 0.10 μF capacitor and a 1.0 mH inductor. The capacitor is initially charged to 5.0 V. What is the maximum current through the inductor as the circuit oscillates?

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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. After the switch has been closed for a long time, what is the current in the circuit? Call this current I₀.

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Textbook Question

An LC circuit is built with a 20 mH inductor and an 8.0 pF capacitor. The capacitor voltage has its maximum value of 25 V at t = 0 s. What is the inductor current at that time?

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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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