/*! 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 5 A 50-loop circular coil has a ra... [FREE SOLUTION] | 91影视

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Chapter 32: Problem 5

A 50-loop circular coil has a radius of \(3.0 \mathrm{~cm}\). It is oriented so that the field lines of a magnetic field are normal to the area of the coil. Suppose that the magnetic field is varied so that \(B\) increases from \(0.10 \mathrm{~T}\) to \(0.35 \mathrm{~T}\) in a time of \(2.0\) milliseconds. Find the average induced emf in the coil. $$ \begin{aligned} \Delta \Phi_{M} &=B_{\text {fimal }} A-B_{\text {initial }} A=(0.25 \mathrm{~T})\left(\pi r^{2}\right)=(0.25 \mathrm{~T}) \pi(0.030 \mathrm{~m})^{2}=7.1 \times 10^{-4} \mathrm{~Wb} \\ |\varepsilon| &=N\left|\frac{\Delta \Phi_{M}}{\Delta t}\right|=(50)\left(\frac{7.1 \times 10^{-4} \mathrm{~Wb}}{2 \times 10^{-3} \mathrm{~s}}\right)=18 \mathrm{~V} \end{aligned} $$

Short Answer

Expert verified
The average induced emf is approximately 18 V.

Step by step solution

01

Understand the Problem

We need to find the average induced electromotive force (emf) in a coil when a magnetic field passing through it changes over time. We have a 50-loop coil with a given radius, and the magnetic field changes from one value to another over a set time period.
02

Calculate the Area of the Coil

The coil is circular, so we begin by calculating its area. The formula for the area of a circle is \(A = \pi r^2\). Here, the radius \(r\) is given as 3.0 cm, which needs to be converted into meters: \(r = 0.030 \, \text{meters}\). Thus, \(A = \pi (0.030)^2\) which approximately equals \(2.83 \times 10^{-3} \, \text{m}^2\).
03

Calculate the Change in Magnetic Flux

The magnetic flux \(\Phi_M\) is given by \(\Phi_M = B \cdot A\), where \(B\) is the magnetic field. We calculate the change in magnetic flux \(\Delta \Phi_M\) as follows: \(\Delta \Phi_M = B_\text{final} \cdot A - B_\text{initial} \cdot A = (0.35 - 0.10) \cdot (2.83 \times 10^{-3})\). The change in flux, therefore, is approximately \(0.25 \cdot 2.83 \times 10^{-3} \, \text{Wb} = 7.075 \times 10^{-4}\) Weber.
04

Compute the Average Induced EMF

Using Faraday's Law, the average induced emf \(|\varepsilon|\) is given by \(N \left| \frac{\Delta \Phi_M}{\Delta t} \right|\), where \(N\) is the number of turns, and \(\Delta t\) is the time duration. Plugging in the values, we get: \(50 \left| \frac{7.075 \times 10^{-4}}{2 \times 10^{-3}} \right|\). This results in an average induced emf of \(17.69 \, \text{V}\), which can be rounded to 18 V as given in the problem.

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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 that describes how an electromotive force (emf) is induced in a coil due to changes in magnetic flux through it. This law states that whenever the magnetic environment of a coil changes, it induces an emf, which can lead to a current if the circuit is closed. The formula for Faraday鈥檚 Law is given as:\[|\varepsilon| = N \left| \frac{\Delta \Phi_{M}}{\Delta t} \right|\]where:
  • \( |\varepsilon| \) is the induced emf,
  • \( N \) is the number of loops or turns in the coil,
  • \( \Delta \Phi_{M} \) is the change in magnetic flux,
  • \( \Delta t \) is the time over which the change takes place.
In practical terms, this means that if you have a coil with multiple loops, changes in the magnetic field will induce a larger emf due to the sum of contributions from each loop.
magnetic flux
Magnetic flux quantifies the amount of magnetic field passing through a given area. It is denoted by \( \Phi_{M} \) and is calculated using the formula:\[\Phi_{M} = B \cdot A \cdot \cos(\theta)\]Here:
  • \( B \) is the magnetic field strength,
  • \( A \) is the area through which the field lines pass,
  • \( \theta \) is the angle between the magnetic field lines and the normal to the surface.
In our exercise, the coil is normal to the magnetic field, making \( \theta = 0 \) and \( \cos(0) = 1 \). Thus, the flux simply becomes \( \Phi_{M} = B \cdot A \). Understanding magnetic flux helps in comprehending how the magnetic field interacts with the coil to induce an emf. It鈥檚 essential in determining the strength of the induced current.
circular coil
A circular coil, like the one in the exercise, is a loop or series of loops shaped in a circle. The shape and number of coils affect how efficiently magnetic fields interact with the coil. The area \( A \) of each loop within the coil influences the magnetic flux. It is calculated as:\[A = \pi r^2\]where \( r \) is the radius of the coil. Larger coils capture more flux, while more loops amplify the induced emf. In our problem, each loop has a radius of \( 3.0 \text{ cm} \) (or \( 0.030 \text{ m} \)), and with 50 loops, the coil has significant capacity to induce a higher emf as a response to flux changes.Circular coils are common in many electromagnetic devices due to their efficiency in flux capture and ease of construction.
magnetic field variation
Magnetic field variation refers to changes in the strength of the magnetic field around a coil. In our context, this variation is what drives the induction of emf according to Faraday's Law. In the given exercise, the magnetic field changes from \( 0.10 \text{ T} \) to \( 0.35 \text{ T} \) in a short span of \( 2 \text{ ms} \). This increase in magnetic field results in a corresponding increase in magnetic flux through the coil. The rapid rate of change is crucial as it influences the magnitude of the induced emf - a faster change results in a higher emf. The relationship between magnetic field variation and induced emf is direct: the greater and quicker the change, the larger the induced current potential. This principle is employed in generating electricity in power plants and other electromagnetic applications.

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Most popular questions from this chapter

A copper bar \(30 \mathrm{~cm}\) long is perpendicular to a uniform magnetic field of \(0.80 \mathrm{~Wb} / \mathrm{m}^{2}\) and moves at right angles to the field with a speed of \(0.50 \mathrm{~m} / \mathrm{s}\). Determine the emf induced in the bar. $$ |\varepsilon|=B L v=\left(0.80 \mathrm{~Wb} / \mathrm{m}^{2}\right)(0.30 \mathrm{~m})(0.50 \mathrm{~m} / \mathrm{s})=0.12 \mathrm{~V} $$

Imagine a 100 -turn flat coil much like that in Fig. \(32-9(a)\). It is in a uniform downward B-field that is decreasing uniformly at a rate of \(0.020 \mathrm{~T}\) every second. The area of the coil is \(0.25 \mathrm{~m}^{2}\). (a) Determine the emf across the coil. (b) Which terminal is at the higher voltage? [Hint: Draw a diagram. Use Eq. (32.4); don't worry about the minus sign, and only concern yourself with what is happening inside the coil. Study the previous four problems.]

A flux of \(9.0 \times 10^{-4} \mathrm{~Wb}\) is produced in the iron core of a solenoid. When the core is removed, a flux (in air) of \(5.0 \times 10^{-7} \mathrm{~Wb}\) is produced in the same solenoid by the same current. What is the relative permeability of the iron?

Depicts a two-turn horizontal coil in a uniform downward B-field. Assume the field is increasing. (a) What is the direction of the induced magnetic field in the coil and why? (b) What is the direction of the induced current in the coil and why? ( \(c\) ) Which terminal is at a higher voltage? [Hint: Draw a diagram. Only concern yourself with what is happening inside the area of the coil.]

A room has its walls aligned accurately with respect to north, south, east, and west. The north wall has an area of \(15 \mathrm{~m}^{2}\), the east wall has an area of \(12 \mathrm{~m}^{2}\), and the floor's area is \(35 \mathrm{~m}^{2}\). At the site the Earth's magnetic field has a value of \(0.60 \mathrm{G}\) and is directed \(50^{\circ}\) below the horizontal and \(7.0^{\circ}\) east of north. Find the magnetic flux through the north wall, the east wall, and the floor.
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