/*! 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 43 The maximum strength of the eart... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 22: Problem 43

The maximum strength of the earth's magnetic field is about \(6.9 \times 10^{-5} \mathrm{T}\) near the south magnetic pole. In principle, this field could be used with a rotating coil to generate \(60.0-\mathrm{Hz}\) ac electricity. What is the minimum number of turns (area per turn \(=0.022 \mathrm{m}^{2}\) ) that the coil must have to produce an rms voltage of \(120 \mathrm{V} ?\)

Short Answer

Expert verified
The coil must have at least 111803 turns to generate the required voltage.

Step by step solution

01

Determine the Formula for Induced EMF

The root mean square (rms) voltage generated by a rotating coil in a magnetic field is given by \( V_{rms} = rac{NBA ext{sin}( heta) ext{sin}(2 ext{蟺}ft)}{ ext{鈭�2}} \), but for maximum voltage generation in one complete cycle, sin term will be effectively \( ext{sin}(2 ext{蟺}ft) = ext{sin}(90掳) = 1\). So simplified we have \( V_{rms} = rac{NBA ext{sin}( heta)}{ ext{鈭�2}} \). Since the coil is perpendicular to the magnetic field , \( ext{sin}( heta) = 1 \). Hence the formula is \( V_{rms} = rac{NBA}{ ext{鈭�2}} \).
02

Rearrange the Formula to Solve for N

We rearrange the formula for \( N \) which stands for the number of turns. Thus, \( N = \frac{V_{rms} imes ext{鈭�2}}{BA} \).
03

Insert Known Values

Substitute the known values into the equation:- \( V_{rms} = 120 \text{ V} \)- \( B = 6.9 imes 10^{-5} \text{ T} \)- \( A = 0.022 \text{ m}^2 \)Insert these into: \( N = \frac{120 \times ext{鈭�2}}{6.9 imes 10^{-5} imes 0.022} \).
04

Calculate the Number of Turns

Calculate using the provided values:\[ N = \frac{120 \times ext{鈭�2}}{6.9 imes 10^{-5} imes 0.022} \approx \frac{120 \times 1.414}{0.000001518} \approx \frac{169.68}{0.000001518} \approx 111802106 \].So, the minimum number of turns needed is approximately 111803.

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.

Magnetic Field Strength
Magnetic field strength, denoted as \( B \), is a measure of the magnetic force in a given area. It is measured in teslas (T). The magnetic force can influence various phenomena, including the movement of charged particles and the induction of electricity.
The Earth's magnetic field is relatively weak compared to those artificially generated in laboratories or industrial applications.
Factors Affecting Magnetic Field Strength:
  • Proximity to the source of the magnetic field
  • The medium through which the magnetic field is passing
  • Environmental factors, such as the geomagnetic location
Understanding magnetic field strength is crucial when dealing with electromagnetic induction as it directly influences the voltages generated by a rotating coil. In the given exercise, the magnetic field strength near the South Magnetic Pole is used to help determine the number of turns needed for a coil to produce a specific electrical output.
RMS Voltage
RMS (Root Mean Square) voltage is a statistical measure of the magnitude of a varying voltage. It is particularly useful in alternating current (AC) circuits because it provides a meaningful representation of the circuit鈥檚 power delivery capabilities.
In general terms, the RMS value is a type of "average" voltage of a waveform but more precisely accounts for the waveform's ability to deliver power. Why Use RMS Voltage?:
  • RMS voltage allows for an equivalent DC voltage comparison, simplifying circuit analysis.
  • It is critical for accurately assessing power output and efficiency in AC systems.
In this problem, a rotating coil's induced RMS voltage of 120 V must be achieved using Earth's magnetic field. The RMS voltage formula used involves the number of turns of the coil, the magnetic field strength, and the coil's area.
Rotating Coil
A rotating coil is a fundamental component in the generation of electrical power through electromagnetic induction. The coil, typically made of conductive wire, rotates within a magnetic field and induces an electromotive force (EMF).
Key Characteristics of a Rotating Coil:
  • Number of Turns (\( N \)): More turns increase the induced EMF.
  • Area (\( A \)) of Each Turn: Larger areas boost EMF.
  • Speed of Rotation: Faster rotations generate more voltage.
The basic operation of the coil depends on Faraday's Law, which states that a change in magnetic flux through a closed circuit induces a voltage. In the given problem, a coil rotated within the Earth's magnetic field must have a minimum number of turns to produce the required RMS voltage. Understanding how a rotating coil interacts with a magnetic field is crucial for optimizing its electrical output, as highlighted in the exercise.

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

In a television set the power needed to operate the picture tube comes from the secondary of a transformer. The primary of the transformer is connected to a \(120-\mathrm{V}\) receptacle on a wall. The picture tube of the television set uses \(91 \mathrm{W}\), and there is \(5.5 \mathrm{mA}\) of current in the secondary coil of the transformer to which the tube is connected. Find the turns ratio \(N_{\Omega} / N_{\mathrm{p}}\) of the transformer.

A magnetic field has a magnitude of \(0.078 \mathrm{T}\) and is uniform over a circular surface whose radius is \(0.10 \mathrm{m}\). The field is oriented at an angle of \(\phi=25^{\circ}\) with respect to the normal to the surface. What is the magnetic flux through the surface?

A flat coil of wire has an area \(A, N\) turns, and a resistance \(R .\) It is situated in a magnetic field, such that the normal to the coil is parallel to the magnetic field. The coil is then rotated through an angle of \(90^{\circ},\) so that the normal becomes perpendicular to the magnetic field. The coil has an area of \(1.5 \times 10^{-3} \mathrm{m}^{2}, 50\) turns, and a resistance of \(140 \Omega .\) During the time while it is rotating, a charge of \(8.5 \times 10^{-5} \mathrm{C}\) flows in the coil. What is the magnitude of the magnetic field?

Parts \(a\) and \(b\) of the drawing show the same uniform and constant (in time) magnetic field \(\overrightarrow{\mathbf{B}}\) directed perpendicularly into the paper over a rectangular region. Outside this region, there is no field. Also shown is a rectangular coil (one turn), which lies in the plane of the paper. In part \(a\) the long side of the coil (length \(=L\) ) is just at the edge of the field region, while in part \(b\) the short side (width \(=W\) ) is just at the edge. It is known that \(L / W=\) \(3.0 .\) In both parts of the drawing the coil is pushed into the field with the same velocity \(\overrightarrow{\mathbf{v}}\) until it is completely within the field region. The magnitude of the average emf induced in the coil in part \(a\) is 0.15 V. What is its magnitude in part \(b ?\)

A magnetic field is passing through a loop of wire whose area is \(0.018 \mathrm{m}^{2}\). The direction of the magnetic field is parallel to the normal to the loop, and the magnitude of the field is increasing at the rate of \(0.20 \mathrm{T} / \mathrm{s}\) (a) Determine the magnitude of the emf induced in the loop. (b) Suppose that the area of the loop can be enlarged or shrunk. If the magnetic field is increasing as in part (a), at what rate (in \(\mathrm{m}^{2} / \mathrm{s}\) ) should the area be changed at the instant when \(B=1.8 \mathrm{T}\) if the induced emf is to be zero? Explain whether the area is to be enlarged or shrunk.
See all solutions

Recommended explanations on Physics Textbooks

Engineering Physics

Read Explanation

Translational Dynamics

Read Explanation

Electrostatics

Read Explanation

Fluids

Read Explanation

Torque and Rotational Motion

Read Explanation

Famous Physicists

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.