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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 'I. 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 \(\mathrm{m}^{2} / \mathrm{s}\) ) at which the area changes?

Short Answer

Expert verified
The magnitude of \( \frac{\Delta A}{\Delta t} \) is approximately \( 1.53 \) \( \mathrm{m}^{2}/\mathrm{s} \).

Step by step solution

01

Understanding the Formula

The rate at which the area changes is linked to electromagnetic induction. According to Faraday's law of induction, the induced emf is equal to the rate of change of magnetic flux through the coil. The formula is given by \( \text{emf} = -\frac{d \Phi}{dt} \), where \( \Phi = B \cdot A \) is the magnetic flux and \( B \) is the magnetic field strength.
02

Relating Variables

Since the normal to the coil is parallel to the magnetic field, the entire coil's area contributes to the magnetic flux. So, \( \Phi = B \cdot A \). The induced emf is related to the change in area: \( 2.6 = -B \cdot \frac{\Delta A}{\Delta t} \).
03

Substituting Known Values

Substitute the given values into the formula: \( 2.6 = -1.7 \cdot \frac{\Delta A}{\Delta t} \). We need to solve for \( \frac{\Delta A}{\Delta t} \).
04

Solving for Rate of Change of Area

Rearrange the equation to find \( \frac{\Delta A}{\Delta t} \): \( \frac{\Delta A}{\Delta t} = \frac{-2.6}{1.7} \). Then, calculate \( \frac{\Delta A}{\Delta t} \).
05

Calculating the Final Answer

Perform the division: \( \frac{\Delta A}{\Delta t} = \frac{-2.6}{1.7} \approx -1.53 \). Therefore, the magnitude of the rate of change of area is approximately \( 1.53 \) \( \mathrm{m}^{2} / \mathrm{s} \).

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Key Concepts

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

Electromagnetic Induction
The magic behind many electrical phenomena lies in electromagnetic induction. At its core, it's the process of generating an electromotive force (emf) by changing the magnetic environment around a conductor. The heart of this concept is Faraday's Law of Induction, which describes how moving a conductor within a magnetic field, or changing the field near it, can produce electricity.
One everyday use of electromagnetic induction is in generators, where mechanical energy is converted into electrical energy.
This conversion happens as the magnetic field changes over time, producing an emf that drives the current through the circuit. Without electromagnetic induction, many of our gadgets and appliances wouldn't be functional.
Magnetic Flux
Magnetic flux can be imagined as the number of magnetic field lines passing through a certain area.It's a measure that links the area with the magnetic field's strength at a right angle to that area. The formula for magnetic flux is given by \( \Phi = B \cdot A \cdot \cos(\theta) \), where \( B \) is the magnetic field strength, \( A \) is the area through which the lines pass, and \( \theta \) is the angle between the field and the normal to the surface.
When the field is perpendicular, \( \theta = 0 \) and thus \( \cos(\theta) = 1 \). This simplifies the magnetic flux to \( B \cdot A \).
  • Magnetic flux helps in understanding the amount of magnetic field interacting with a particular area.
  • A changing magnetic flux over time is crucial for inducing emf according to Faraday's law.
Induced EMF
Induced emf, or electromotive force, is essentially the electricity generated through electromagnetic induction.Hubbed around Faraday鈥檚 law, it states that this emf is directly related to the rate of change of magnetic flux through a coil.The formula for this is \( \text{emf} = -\frac{d\Phi}{dt} \), reflecting the negative sign from Lenz's Law, which emphasizes the directional nature to oppose the change causing it.
The practical applications of induced emf are expansive 鈥� it's why transformers work, how we have electricity from wind turbines, and the basis of inductive charging technology.When you consider the experience of emf flowing, you are diving into a fundamental principle of how change is translated into electrical force.
Rate of Change of Area
The rate of change of area (\( \Delta A / \Delta t \) ) suggests how rapidly a surface is expanding or shrinking over time.In contexts like our coil example, shrinking the area of a loop within a magnetic field changes the magnetic flux and can induce an emf.Faraday's law can then be utilized to connect this rate of area change with the induced emf, where changes to the area equate proportional changes to the magnetic flux.To compute this in our exercise, use \( \text{emf} = -B \cdot \frac{\Delta A}{\Delta t} \).
This concept emphasizes that not only is the field important but how the shape or size of your surface interacts with it.A faster change in area in this context means a stronger induced emf, making it essential to understand, especially in designing efficient electrical systems.

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Most popular questions from this chapter

A piece of copper wire is formed into a single circular loop of radius \(12 \mathrm{cm} .\) A magnetic field is oriented parallel to the normal to the loop, and it increases from 0 to \(0.60 \mathrm{T}\) in a time of \(0.45 \mathrm{s}\). The wire has a resistance per unit length of \(3.3 \times 10^{-2} \Omega / \mathrm{m} .\) What is the average electrical energy dissipated in the resistance of the wire?

The earth's magnetic field, like any magnetic field, stores energy. The maximum strength of the earth's field is about \(7.0 \times 10^{-5} \mathrm{T} .\) Find the maximum magnetic energy stored in the space above a city if the space occupies an area of \(5.0 \times 10^{8} \mathrm{m}^{2}\) and has a height of \(1500 \mathrm{m}\).

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 \(b ?\)

Two \(0.68-\mathrm{m}\) -long conducting rods are rotating at the same speed in opposite directions, and both are perpendicular to a 4.7-T magnetic field. As the drawing shows, the ends of these rods come to within \(1.0 \mathrm{mm}\) of each other as they rotate. Moreover, the fixed ends about which the rods are rotating are connected by a wire, so these ends are at the same electric potential. If a potential difference of \(4.5 \times 10^{3} \mathrm{V}\) is required to cause a 1.0 -mm spark in air, what is the angular speed (in \(\mathrm{rad} / \mathrm{s}\) ) of the rods when a spark jumps across the gap?

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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