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 71b

Chapter 30, Problem 71b

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?

Verified step by step guidance

Step 1: Understand the problem. The LC circuit consists of an inductor and a capacitor. At t = 0 s, the capacitor voltage is at its maximum value, which means the energy in the circuit is entirely stored in the capacitor. The inductor current at this moment is zero because the energy has not yet transferred to the inductor.

Step 2: Recall the energy conservation principle in an LC circuit. The total energy in the circuit remains constant and is shared between the capacitor and the inductor. The energy stored in the capacitor is given by the formula:

E_{C} = \frac{1}{2} C V^{2}

, where C is the capacitance and V is the voltage across the capacitor.

Step 3: Substitute the given values into the capacitor energy formula. The capacitance is

8.0 \times 10^{-12}

F, and the voltage is 25 V. Calculate the energy stored in the capacitor at t = 0 s.

Step 4: Recognize that at t = 0 s, the inductor current is zero because the energy is entirely stored in the capacitor. The current in the inductor will begin to increase as the energy oscillates between the capacitor and the inductor over time.

Step 5: Conclude that the inductor current at t = 0 s is

0

A, based on the initial conditions of the LC circuit and the energy distribution.

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Key Concepts

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

LC Circuit

An LC circuit consists of an inductor (L) and a capacitor (C) connected together, forming a resonant circuit. It oscillates between the energy stored in the electric field of the capacitor and the magnetic field of the inductor. The behavior of the circuit is characterized by its natural frequency, which depends on the values of L and C.

Inductor Current

In an LC circuit, the inductor current represents the flow of electric charge through the inductor. At the moment when the capacitor voltage is at its maximum, the inductor current is at its minimum, which is zero. This is because the energy is fully stored in the capacitor, and the inductor has not yet begun to release its stored energy.

Energy Conservation in LC Circuits

Energy conservation in LC circuits is a fundamental principle where the total energy oscillates between the capacitor and the inductor. At maximum capacitor voltage, all energy is stored in the capacitor, while at maximum inductor current, all energy is stored in the inductor. This interplay allows for continuous oscillation, with energy shifting back and forth between the two components.

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Energy in an LC Circuit

Related Practice

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 P30.76 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

A 50 cm solenoid with 1000 turns has an inductance of 20 mH. Flipping a switch disconnects the inductor from the battery and connects it to a resistor. What is the value of the resistance if the magnetic field decreases by 50% in 150 μs?

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Textbook Question

CALC The current through inductance L is given by $I = I_{0} e^{- t / τ}$ . Find an expression for the potential difference ΔV_L across the inductor.

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Textbook Question

CALC The current through inductance L is given by $I = I_{0} e^{- t / τ}$ . Evaluate ΔV_L at t = 0, 1.0, and 3.0 ms if L = 20 mH, I₀ = 50 mA, and $τ$ = 1.0 ms.

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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