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Chapter 30: Problem 9

At the instant when the current in an inductor is increasing at a rate of 0.0640 A/s, the magnitude of the self-induced emf is 0.0160 V. (a) What is the inductance of the inductor? (b) If the inductor is a solenoid with 400 turns, what is the average magnetic flux through each turn when the current is 0.720 A?

Short Answer

Expert verified
(a) Inductance is 0.25 H. (b) Magnetic flux is 0.00045 Wb per turn.

Step by step solution

01

Understand the Problem and Identify Formulas

To solve for the inductance \( L \), we have the formula for the self-induced emf, \( \epsilon = -L \frac{dI}{dt} \), where \( \frac{dI}{dt} \) is the rate of change of current. We need to calculate \( L \) since \( \epsilon = 0.0160 \) V and \( \frac{dI}{dt} = 0.0640 \) A/s.
02

Solve for the Inductance

Rearrange the formula for self-induced emf to solve for \( L \): \( L = \frac{-\epsilon}{\frac{dI}{dt}} \). Substitute the given values: \( L = \frac{-0.0160}{0.0640} = -0.25 \). The inductance \( L \) is 0.25 H (since the value of the inductance is generally considered as positive).
03

Set Up for Flux Calculation in a Solenoid

To find the average magnetic flux \( \Phi \) through each turn when the current is 0.720 A, use the relation \( \Phi = \frac{L \cdot I}{N} \), where \( L \) is the inductance, \( I \) is the current, and \( N \) is the number of turns in the solenoid.
04

Calculate the Magnetic Flux

Substitute the known values into the formula: \( \Phi = \frac{0.25 \times 0.720}{400} \). Simplify to find \( \Phi = 0.00045 \text{ Wb (Webers)} \) as the average magnetic flux through each turn when the current is 0.720 A.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Self-Induced EMF
Self-induced EMF is a phenomenon in which an electromotive force (EMF) is generated within a circuit due to the change in current flowing through it. This occurs because the changing current produces a varying magnetic field, which induces a voltage back into the circuit itself. This concept is described by Faraday's Law of Electromagnetic Induction.
  • This induced EMF opposes the change in current because of Lenz's law.
  • The formula used to calculate self-induced EMF is \( \epsilon = -L \frac{dI}{dt} \), where \( \epsilon \) is the EMF, \( L \) is the inductance, and \( \frac{dI}{dt} \) is the rate of change of current.
A key point to remember is that the negative sign in the formula indicates opposition to changes in the current flow. Understanding how self-induced EMF operates helps in designing circuits, especially those containing inductors.
Inductance
Inductance is a measure of how effective an inductor is at inducing an EMF due to a change in current. It represents the ability of an inductor to store energy in its magnetic field.
  • Inductance \( L \) is measured in henries (H).
  • The larger the inductance, the greater the ability to oppose changes in current.
The inductance is affected by factors such as:
  • The number of turns in the coil or solenoid.
  • The cross-sectional area of the coil.
  • The material of the core around which the coil is wound.
In practical applications, inductors with high inductance are used to control the flow of AC or store energy in some forms of power supplies.
Magnetic Flux
Magnetic flux \( \Phi \) quantifies the total magnetic field passing through a given area, often measured in webers (Wb). It's a fundamental concept in electromagnetics to describe the strength and orientation of the magnetic field through a surface.
  • Magnetic flux is calculated using the formula \( \Phi = \frac{L \cdot I}{N} \), where \( L \) is the inductance, \( I \) is the current, and \( N \) is the number of turns.
For a solenoid, the magnetic flux through each loop is relevant in understanding how effectively it can store energy. Changes in magnetic flux are crucial for inducing EMF in nearby circuits, emphasizing its role in transformers and induction processes.
Solenoid
A solenoid is a type of inductor arranged in a helical coil. It generates a magnetic field when an electric current flows through it, and is commonly used in various applications such as electromagnets, inductors, and motors.
  • The strength of the magnetic field created by a solenoid is largely dependent on the number of turns and the current passing through it.
  • A solenoid with a larger number of turns or a greater current will have a stronger magnetic field.
Solenoids are essential in electromagnetic systems where it is crucial to control the magnetic field, such as in switches and relays. The magnetic field inside a solenoid is generally uniform and concentrated along the axis, making them ideal for controlled induction applications.

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Most popular questions from this chapter

An \(L\)-\(C\) circuit containing an 80.0-mH inductor and a 1.25-nF capacitor oscillates with a maximum current of 0.750 A. Calculate: (a) the maximum charge on the capacitor and (b) the oscillation frequency of the circuit. (c) Assuming the capacitor had its maximum charge at time \(t = 0\), calculate the energy stored in the inductor after 2.50 ms of oscillation.

An air-filled toroidal solenoid has 300 turns of wire, a mean radius of 12.0 cm, and a cross-sectional area of 4.00 cm\(^2\). If the current is 5.00 A, calculate: (a) the magnetic field in the solenoid; (b) the self inductance of the solenoid; (c) the energy stored in the magnetic field; (d) the energy density in the magnetic field. (e) Check your answer for part (d) by dividing your answer to part (c) by the volume of the solenoid.

A 35.0-V battery with negligible internal resistance, a 50.0-\(\Omega\) resistor, and a 1.25-mH inductor with negligible resistance are all connected in series with an open switch. The switch is suddenly closed. (a) How long after closing the switch will the current through the inductor reach one-half of its maximum value? (b) How long after closing the switch will the energy stored in the inductor reach one-half of its maximum value?

A 6.40-nF capacitor is charged to 24.0 V and then disconnected from the battery in the circuit and connected in series with a coil that has \(L =\) 0.0660 H and negligible resistance. After the circuit has been completed, there are current oscillations. (a) At an instant when the charge of the capacitor is 0.0800 \(\mu\)C, how much energy is stored in the capacitor and in the inductor, and what is the current in the inductor? (b) At the instant when the charge on the capacitor is 0.0800 \(\mu\)C, what are the voltages across the capacitor and across the inductor, and what is the rate at which current in the inductor is changing?

A coil has 400 turns and self-inductance 7.50 mH. The current in the coil varies with time according to \(i = (680 \, \mathrm{mA}) \mathrm{cos} (\pi{t}/0.0250 \, \mathrm{s})\). (a) What is the maximum emf induced in the coil? (b) What is the maximum average flux through each turn of the coil? (c) At \(t = 0.0180\) s, what is the magnitude of the induced emf?
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