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Chapter 22: Problem 72

A planar coil of wire has a single turn. The normal to this coil is parallel to a uniform and constant (in time) magnetic field of \(1.7 \mathrm{T} .\) An emf that has a magnitude of \(2.6 \mathrm{V}\) is induced in this coil because the coil's area \(A\) is shrinking. What is the magnitude of \(\Delta A / \Delta t,\) which is the rate (in \(\left.\mathrm{m}^{2} / \mathrm{s}\right)\) at which the area changes?

Short Answer

Expert verified
The rate of change of area is \( \Delta A/\Delta t = 1.53 \mathrm{m}^2/\mathrm{s} \).

Step by step solution

01

Understand Faraday's Law

Faraday's Law of electromagnetic induction states that the induced electromotive force (EMF) in a coil is directly proportional to the rate of change of the magnetic flux through the coil. The formula can be written as:\[\varepsilon = - \frac{d\Phi}{dt}\]where \(\varepsilon\) is the EMF and \(d\Phi/dt\) is the rate of change of the magnetic flux.
02

Calculate the Change in Magnetic Flux

The magnetic flux \(\Phi\) through the coil is given by:\[\Phi = B \cdot A\cdot \cos(\theta)\]where \(B\) is the magnetic field, \(A\) is the area of the coil, and \(\theta\) is the angle between the magnetic field and the normal to the coil. Since the normal to the coil is parallel to the field, \(\theta = 0\) and \(\cos(\theta) = 1\). Thus, \(\Phi = B \cdot A\).

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Key Concepts

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

Magnetic Flux
Magnetic flux is an essential concept in electromagnetism. It refers to the total magnetic field passing through a given area. Magnetic flux (\(\Phi\)) can be thought of as the number of magnetic field lines that penetrate a certain region. It is a scalar quantity, and its SI unit is the Weber (Wb). The magnetic flux through a surface is calculated using the formula:\[\Phi = B \cdot A \cdot \cos(\theta)\]where:
  • \(B\) is the magnetic field strength in Tesla,
  • \(A\) is the area of the surface in square meters, and
  • \(\theta\) is the angle between the magnetic field and the normal to the surface.
When the normal to the surface is perfectly aligned with the magnetic field, \(\theta = 0\) and \(\cos(\theta) = 1\), meaning the magnetic field is fully effective in penetrating the area. This makes the mathematical expression for flux simply \(\Phi = B \cdot A\).
EMF (Electromotive Force)
Electromotive force, commonly abbreviated as EMF, is not a force, despite the name. It is a measure of the energy provided by a source of electric power, such as a battery or generator, per coulomb of charge. It is measured in volts and can be thought of as the voltage across the terminals of an open circuit. In the context of Faraday's Law of Electromagnetic Induction, EMF is induced when there is a change in magnetic flux over time. The relationship is given by:\[\varepsilon = - \frac{d\Phi}{dt}\]Here:
  • \(\varepsilon\) is the induced EMF,
  • \(d\Phi/dt\) represents the rate of change of magnetic flux.
The negative sign in Faraday's equation is a result of Lenz's Law, which states that the direction of the induced EMF will be such that it opposes the change in magnetic flux that caused it. Lenz's Law is a cornerstone in understanding how electromagnetic systems conserve energy.
Rate of Change of Area
The concept of rate of change of area is crucial in understanding how the size of a coil or conducting loop affects the magnetic flux and induced EMF when the coil is in a changing magnetic field. In our specific problem, the coil's area is decreasing, which affects the overall magnetic flux. According to Faraday's Law, a change in the area of a coil will affect the magnetic flux \(\Phi = B \cdot A\) directly. If the magnetic field \(B\) is uniform and constant, then the only variable causing a change in magnetic flux is the area \(A\).If \(dA/dt\) represents the rate of change of area, the rate of change of flux is given by:\[\frac{d\Phi}{dt} = B \cdot \frac{dA}{dt}\]Thus, if a problem provides the rate of change of area, it can be directly inserted into Faraday's equation to find the induced EMF or vice versa. Understanding this concept helps in the interpretation of how physical changes to a coil's geometry influence electromagnetic induction phenomena.

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Most popular questions from this chapter

During a 72 -ms interval, a change in the current in a primary coil occurs. This change leads to the appearance of a \(6.0-\mathrm{mA}\) current in a nearby secondary coil. The secondary coil is part of a circuit in which the resistance is \(12 \Omega\). The mutual inductance between the two coils is \(3.2 \mathrm{mH}\). What is the change in the primary current?

In each of two coils the rate of change of the magnetic flux in a single loop is the same. The emf induced in coil \(1,\) which has 184 loops, is \(2.82 \mathrm{V}\). The emf induced in coil 2 is 4.23 V. How many loops does coil 2 have?

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 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.

A standard door into a house rotates about a vertical axis through one side, as defined by the door's hinges. A uniform magnetic field is parallel to the ground and perpendicular to this axis. Through what angle must the door rotate so that the magnetic flux that passes through it decreases from its maximum value to one-third of its maximum value?
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