/*! 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 4 A single loop of wire with an ar... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 21: Problem 4

A single loop of wire with an area of 0.0900 \(\mathrm{m}^{2}\) is in a uniform magnetic field that has an initial value of 3.80 \(\mathrm{T}\) , is perpendicular to the plane of the loop, and is decreasing at a constant rate of 0.190 \(\mathrm{T} / \mathrm{s}\) . (a) What emf is induced in this loop? (b) If the loop has a resistance of \(0.600 \Omega,\) find the current induced in the loop.

Short Answer

Expert verified
(a) Emf induced is 0.0171 V. (b) Induced current is 0.0285 A.

Step by step solution

01

Understanding the Problem

We need to calculate the induced electromotive force (emf) in a circular loop when the magnetic field through it is changing. According to Faraday's Law of Electromagnetic Induction, the induced emf is equal to the negative change in magnetic flux divided by the change in time.
02

Faraday's Law Equation

The induced emf can be calculated using Faraday's Law: \( \text{emf} = - \frac{d\Phi_B}{dt} \), where \( \Phi_B = B \times A \) is the magnetic flux, \( B \) is the magnetic field, and \( A \) is the area of the loop. Since \( B \) is decreasing at 0.190 T/s, we use \( \frac{dB}{dt} = -0.190 \ \mathrm{T/s} \).
03

Calculate Change in Magnetic Flux

The change in magnetic flux \( \frac{d\Phi_B}{dt} \) is the product of the area \( A = 0.0900 \ \mathrm{m}^2 \) and the rate of change of the magnetic field \( \frac{dB}{dt} \). Thus, \( \frac{d\Phi_B}{dt} = A \times \frac{dB}{dt} = 0.0900 \ \mathrm{m}^2 \times -0.190 \ \mathrm{T/s} = -0.0171 \ \mathrm{T \cdot m^2/s} \).
04

Calculate Induced emf

Using Faraday's Law, \( \text{emf} = - \frac{d\Phi_B}{dt} = - (-0.0171) = 0.0171 \ \mathrm{V} \). The negative sign indicates direction according to Lenz's Law, but here we are interested in magnitude.
05

Calculate Induced Current Using Ohm's Law

Ohm's Law (\( V = I \cdot R \)) relates voltage, current, and resistance. Here, \( V \) is the induced emf (0.0171 V), \( R = 0.600 \ \Omega \). Solving for current \( I \), \( I = \frac{V}{R} = \frac{0.0171 \ \mathrm{V}}{0.600 \ \Omega} = 0.0285 \ \mathrm{A} \).

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
The foundation of electromagnetic induction is explained by Faraday's Law. It states that the electromotive force (emf) induced in a circuit is directly proportional to the rate of change of magnetic flux through the circuit. An emf can be thought of as the voltage generated to oppose the change in the magnetic environment of a loop.

Faraday's Law is mathematically expressed as: \[ \text{emf} = - \frac{d\Phi_B}{dt} \], where \( \Phi_B \) is the magnetic flux through a loop and \( t \) represents time. The negative sign in the equation is a nod to Lenz's Law, indicating the direction of the induced emf opposes the change in flux.

To simplify, if the magnetic field through a loop changes consistent with time, this will induce an emf according to the above relationship. This principle is widely applied in generators and transformers, playing a critical role in electrical engineering.
magnetic flux
Magnetic flux helps in understanding how much magnetic field passes through a certain area. It's like counting how many magnetic lines penetrate a given surface. The magnetic flux \( \Phi_B \) through a loop of area \( A \) located in a magnetic field \( B \) is calculated as:
\[\Phi_B = B \times A \times \cos(\theta)\]
where \( \theta \) is the angle between the magnetic field and the normal to the surface.

For simplicity, when the magnetic field is perpendicular to the plane of the loop, \( \theta = 0 \) degrees and \( \cos(\theta) = 1 \), making our equation \( \Phi_B = B \times A \). Changing magnetic flux through a loop is the essence of electromagnetic induction, leading to the creation of an emf.

In the context of a changing magnetic environment, understanding how flux varies helps predict the behavior of induced emfs.
Ohm's Law
Once the voltage or emf across a circuit element is known, Ohm's Law comes into play to find the induced current. This basic law relates voltage (V), current (I), and resistance (R) in a simple linear relationship: \[V = I \times R\]
In our case, the voltage \( V \) is the induced emf we calculated using Faraday's Law. With the loop's resistance \( R \), Ohm's Law can be rearranged to find the current: \[I = \frac{V}{R}\]

Ohm's Law is an essential building block in understanding how electrical circuits respond to various forms of voltage inputs, whether from a battery or an induced emf.

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 \(12.0 \mathrm{~V}\) dc battery having no appreciable internal resistance, a \(150.0 \Omega\) resistor, an \(11.0 \mathrm{mH}\) inductor, and an open switch are all connected in series. After the switch is closed, what are (a) the time constant for this circuit, (b) the maximum current that flows through it, (c) the current \(73.3 \mu\) s after the switch is closed, and (d) the maximum energy stored in the inductor?

When a certain inductor carries a current \(I,\) it stores 3.0 \(\mathrm{mJ}\) of magnetic energy. How much current (in terms of \(I )\) would it have to carry to store 9.0 \(\mathrm{mJ}\) of energy?

Self-inductance of a solenoid. A long, straight solenoid has \(N\) turns, a uniform cross-sectional area \(A,\) and length \(l .\) Use the definition of self- inductance expressed by Equation 21.13 to show that the inductance of this solenoid is given approximately by the equation \(L=\mu_{0} A N^{2} / l .\) Assume that the magnetic field is uniform inside the solenoid and zero outside. (Your answer is approximate because \(B\) is actually smaller at the ends than at the center of the solenoid. For this reason, your answer is actually an upper limit on the inductance.)

A circular area with a radius of 6.50 \(\mathrm{cm}\) lies in the \(x\) -y plane. What is the magnitude of the magnetic flux through this circle due to a uniform magnetic field \(B=0.230 \mathrm{T}\) that points (a) in the \(+z\) direction? (b) at an angle of \(53.1^{\circ}\) from the \(+z\) direction? (c) in the \(+y\) direction?

An air-filled toroidal solenoid has a mean radius of 15.0 \(\mathrm{cm}\) and a cross-sectional area of 5.00 \(\mathrm{cm}^{2} .\) When the current is \(12.0 \mathrm{A},\) the energy stored is 0.390 \(\mathrm{J} .\) How many turns does the winding have?
See all solutions

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Read Explanation

Medical Physics

Read Explanation

Modern Physics

Read Explanation

Dynamics

Read Explanation

Electrostatics

Read Explanation

Electricity

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.