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Ch. 30 - Inductance, Electromagnetic Oscillations, and AC Circuits
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 30 - Inductance, Electromagnetic Oscillations, and AC CircuitsProblem 67
Chapter 29, Problem 67

At t = 0, the current through a 60.0-mH inductor is 50.0 mA and is increasing at the rate of 78.0 mA/s. What is the initial energy stored in the inductor, and how long does it take for the energy to increase by a factor of 8.0 from the initial value?

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Step 1: Recall the formula for the energy stored in an inductor, which is given by \( U = \frac{1}{2} L I^2 \), where \( U \) is the energy, \( L \) is the inductance, and \( I \) is the current. Substitute the given values \( L = 60.0 \ \text{mH} = 60.0 \times 10^{-3} \ \text{H} \) and \( I = 50.0 \ \text{mA} = 50.0 \times 10^{-3} \ \text{A} \) to calculate the initial energy stored in the inductor.
Step 2: To determine the time it takes for the energy to increase by a factor of 8, note that the energy stored in the inductor is proportional to the square of the current, \( U \propto I^2 \). If the energy increases by a factor of 8, then \( I^2 \) must increase by a factor of 8. Solve for the new current \( I_f \) using \( I_f^2 = 8 I^2 \), and then find \( I_f = \sqrt{8} I \).
Step 3: Use the given rate of change of current, \( \frac{dI}{dt} = 78.0 \ \text{mA/s} = 78.0 \times 10^{-3} \ \text{A/s} \), to calculate the time it takes for the current to increase from \( I \) to \( I_f \). The relationship is \( \Delta I = \frac{dI}{dt} \cdot \Delta t \), where \( \Delta I = I_f - I \). Rearrange to solve for \( \Delta t \): \( \Delta t = \frac{\Delta I}{\frac{dI}{dt}} \).
Step 4: Substitute \( I_f = \sqrt{8} I \) and \( I = 50.0 \times 10^{-3} \ \text{A} \) into the equation for \( \Delta I \), and then calculate \( \Delta t \) using the given rate of change of current.
Step 5: Summarize the results: The initial energy stored in the inductor is calculated using the formula in Step 1, and the time it takes for the energy to increase by a factor of 8 is determined using the steps outlined above. Ensure all units are consistent and properly converted during calculations.

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Key Concepts

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

Inductance and Energy Storage

Inductance is a property of an inductor that quantifies its ability to store energy in a magnetic field when an electric current flows through it. The energy (U) stored in an inductor can be calculated using the formula U = (1/2) L I^2, where L is the inductance in henries and I is the current in amperes. In this case, the inductor has a value of 60.0 mH, and the initial current is 50.0 mA.
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Rate of Change of Current

The rate of change of current through an inductor affects the induced electromotive force (emf) and the energy stored in the inductor. According to Faraday's law of electromagnetic induction, the induced emf (ε) is proportional to the rate of change of current (di/dt). In this scenario, the current is increasing at a rate of 78.0 mA/s, which will influence how quickly the energy in the inductor increases.
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Energy Increase Factor

The energy increase factor refers to how many times the initial energy stored in the inductor increases over time. If the energy increases by a factor of 8.0, it means the new energy stored is 8 times the initial energy. To find the time it takes for this increase, one must consider the relationship between current, inductance, and the energy formula, and how the current changes over time due to the given rate of change.
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