/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 33 At the instant when the current ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 21: Problem 33

At the instant when the current in an inductor is increasing at a rate of 0.0640 \(\mathrm{A} / \mathrm{s}\) , the magnitude of the self-induced emf is 0.0160 \(\mathrm{V} .\) What is the inductance of the inductor?

Short Answer

Expert verified
The inductance is 0.25 henrys.

Step by step solution

01

Identify the Given Values

We are given that the rate of change of current, \( \frac{di}{dt} \), is 0.0640 A/s, and the induced emf, \( \varepsilon \), is 0.0160 V.
02

Use the Formula for Induced EMF

We know from Faraday's law of electromagnetic induction that the magnitude of the self-induced emf (\( \varepsilon \)) in an inductor is given by the formula \( \varepsilon = L \cdot \frac{di}{dt} \), where \( L \) is the inductance.
03

Rearrange the Equation to Solve for Inductance

Rearrange the formula \( \varepsilon = L \cdot \frac{di}{dt} \) to solve for \( L \):\[L = \frac{\varepsilon}{\frac{di}{dt}}\]
04

Substitute the Given Values

Substitute the known values into the equation: \( L = \frac{0.0160 \, \text{V}}{0.0640 \, \text{A/s}} \).
05

Calculate the Inductance

Perform the division: \( L = \frac{0.0160}{0.0640} = 0.25 \).
06

Conclusion

The inductance of the inductor is 0.25 henrys.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Faraday's Law
Faraday's Law is a fundamental principle in electromagnetism. It describes how electric currents are induced in a conductor due to changing magnetic fields. Imagine a magnetic field passing through a loop of wire. When the strength of this magnetic field changes, it creates an electromotive force (EMF) in the wire. This is a brilliant concept because it lays the groundwork for how we understand generators and transformers.

In this context, the law can be described mathematically as:
  • The induced EMF is proportional to the rate of change of the magnetic flux.
  • This relationship is captured by the equation \( \varepsilon = - \frac{d\Phi}{dt} \), where \( \varepsilon \) is the induced EMF and \( \Phi \) is the magnetic flux.
  • The negative sign shows that the induced current will oppose the change in magnetic flux, as per Lenz's Law.
In our scenario of an increasing current, Faraday's Law specifically helps relate this changing magnetic field to the observed self-induced EMF.
Self-induced EMF
Self-induced EMF occurs when the changing current in a coil affects its own magnetic field, inducing an EMF that opposes the change of the current. It is one of the key aspects of an inductor's function. When you have a coil and current flowing through it, the current generates a magnetic field.
  • As the current changes, so does the magnetic field, inducing a voltage across the coil.
  • This self-induced EMF follows the same fundamental principle as Faraday's Law but happens within the conductor rather than across multiple wires or conductors.
  • The formula \( \varepsilon = L \cdot \frac{di}{dt} \) shows this relationship, where \( L \) is the inductance and \( \frac{di}{dt} \) is the rate of change of current.
In practical applications, this self-induction is crucial in devices such as electrical transformers and chokes, as it helps manage and stabilize current flows.
Rate of change of current
The rate of change of current, expressed as \( \frac{di}{dt} \), is essential in understanding how quickly the current flowing through an inductor is increasing or decreasing. This rate directly influences the self-induced EMF in a conductor. Here鈥檚 how it works:
  • A rapid increase in current can lead to a stronger self-induced EMF, which may oppose this change more significantly.
  • The magnitude of self-induced EMF depends on both the rate of change of the current and the inductance of the coil.
  • This means devices like inductors need to be carefully designed to handle specific current changes effectively, preventing unwanted spikes that could harm electrical circuits.
Understanding \( \frac{di}{dt} \) is crucial for anyone working with electric circuits, helping to predict how changes in current will affect other components.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A coil of wire with 200 circular turns of radius 3.00 \(\mathrm{cm}\) is in a uniform magnetic field along the axis of the coil. The coil has \(R=40.0 \Omega\) . At what rate, in teslas per second, must the magnetic field be changing to induce a current of 0.150 \(\mathrm{A}\) in the coil?

You need a transformer that will draw 15 \(\mathrm{W}\) of power from a 220 \(\mathrm{V}\) power line, stepping the voltage down to \(6.0 \mathrm{V}(\mathrm{rms}),\) (a) What will be the current in the secondary coil? (b) What should be the resistance of the secondary circuit? (c) What will be the equivalent resistance of the input circuit?

\(\bullet\) A circular loop of wire with a radius of 12.0 \(\mathrm{cm}\) is lying flat on a tabletop. A magnetic field of 1.5 \(\mathrm{T}\) is directed vertically upward through the loop (Figure 21.49 ). (a) If the loop is removed from the field region in a time interval of 2.0 \(\mathrm{ms}\) , find the average emf that will be induced in the wire loop during the extraction process. (b) If the loop is viewed looking down on it from above, is the induced current in the loop clockwise or counterclockwise?

In a physics laboratory experiment, a coil with 200 turns enclosing an area of 12 \(\mathrm{cm}^{2}\) is rotated from a position where its plane is perpendicular to the earth's magnetic field to one where its plane is parallel to the field. The rotation takes 0.040 s. The earth's magnetic field at the location of the laboratory is \(6.0 \times 10^{-5} \mathrm{T.}\) (a) What is the total magnetic flux through the coil before it is rotated? After it is rotated? (b) What is the average emf induced in the coil?

A 15.0\(\mu \mathrm{F}\) capacitor is charged to 175\(\mu \mathrm{C}\) and then connected across the ends of a 5.00 \(\mathrm{mH}\) inductor. (a) Find the maximum current in the inductor. At the instant the current in the inductor is maximal, how much charge is on the capacitor At this instant, what is the current in the inductor? (c) Find the maximum energy stored in the inductor. At this instant, what is the current in the circuit?
See all solutions

Recommended explanations on Physics Textbooks

Particle Model of Matter

Read Explanation

Magnetism

Read Explanation

Linear Momentum

Read Explanation

Astrophysics

Read Explanation

Physical Quantities And Units

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.