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Chapter 20: Problem 51

Suppose you wanted to build an electric generator using the Earth's magnetic field. Assume it has a strength of \(0.040 \mathrm{mT}\) at your location. Your generator design calls for a coil of 1000 windings rotated at exactly \(60 \mathrm{~Hz}\). The coil is oriented so that the normal to the area lines up with the Earth's field at the end of each cycle. (a) What must the coil diameter be to generate a maximum voltage of \(170 \mathrm{~V}\) (required in order to average \(120 \mathrm{~V}\) )? Does this seem like a practical design? (b) Some generators operate at \(50 \mathrm{~Hz}\). How would this change the coil diameter? (c) What number of windings would make this coil arrangement a "manageable" size?

Short Answer

Expert verified
The coil diameter is approximately 1.198 m for 60 Hz. At 50 Hz, it is about 1.312 m. Reducing windings decreases the size.

Step by step solution

01

Understanding the Problem

We need to calculate the coil diameter required to generate a maximum voltage of 170 V using Earth's magnetic field. Given a magnetic field strength of 0.040 mT, 1000 windings, and a rotation frequency of 60 Hz, we will use the formula for the maximum induced EMF: \( E = NAB\omega \), where \( N \) is the number of turns, \( A \) is the area, and \( \omega \) is the angular frequency of the coil.
02

Deriving Angular Frequency

The angular frequency \( \omega \) is calculated using \( \omega = 2\pi f \), where \( f \) is the frequency. For a frequency of 60 Hz: \( \omega = 2\pi \times 60 = 120\pi \) rad/s.
03

Calculating Area

We rearrange the formula \( E = NAB\omega \) to solve for the area (A): \( A = \frac{E}{NB\omega} \). Substituting the values: \( A = \frac{170}{1000 \times 0.040 \times 10^{-3} \times 120\pi} \approx 1.129 \text{ m}^2 \).
04

Finding Diameter from Area

The area of a circle is \( A = \pi r^2 \), where \( r \) is the radius. Rearranging for \( r \), we have \( r = \sqrt{\frac{A}{\pi}} \). Substituting the values: \( r \approx \sqrt{\frac{1.129}{\pi}} \approx 0.599 \text{ m} \). The diameter \( d = 2r \) is approximately 1.198 m.
05

Evaluating 50 Hz Effect

If the frequency changes to 50 Hz, \( \omega = 2\pi \times 50 = 100\pi \) rad/s. Re-calculate the area: \( A = \frac{170}{1000 \times 0.040 \times 10^{-3} \times 100\pi} \approx 1.355 \text{ m}^2 \). Calculate the new diameter: \( d = 2 \times \sqrt{\frac{1.355}{\pi}} \approx 1.312 \text{ m} \).
06

Managing Size with Windings

To make the coil manageable, we can reduce the number of windings to decrease the required diameter. For example, if we have 500 windings instead of 1000, calculate the new area \( A = \frac{170}{500 \times 0.040 \times 10^{-3} \times 120\pi} \) and find the corresponding diameter using the same process. This would significantly reduce the coil size.

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Key Concepts

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

Electromagnetic Induction
Electromagnetic induction is the fundamental concept behind the operation of electric generators. It occurs when a conductor, such as a coil of wire, moves through a magnetic field or when the magnetic field around the conductor changes, inducing an electromotive force (EMF) within the conductor. This principle was discovered by Michael Faraday and is described by Faraday's law of induction. The law states that the induced voltage in a coil is proportional to the rate of change of the magnetic flux through the coil.

In the exercise, this principle is applied as the coil is rotated in the Earth's magnetic field. The continuous change in angle between the coil and the magnetic field lines causes a change in the magnetic flux, which in turn induces a voltage across the coil. The formula used, \( E = NAB\omega \), showcases the relationship where \( E \) is the induced EMF, \( N \) is the number of windings, \( A \) is the coil area, \( B \) is the magnetic field strength, and \( \omega \) is the angular frequency.

Key aspects of the exercise include understanding how altering frequency or the number of windings can affect the generated voltage. This exploration deepens your grasp of how electromagnetic induction can be optimized to achieve a desired voltage output.
Magnetic Field Strength
Magnetic field strength is a crucial factor in determining the effectiveness of an electric generator. It is measured in teslas (T) or, more commonly for everyday contexts, in milli-teslas (mT). It represents the intensity of the magnetic field at a given point and influences how much voltage is induced in a coil when it rotates within this field.

In the context of the exercise, the Earth's magnetic field of \(0.040 \mathrm{mT}\) is used. While this is relatively weak compared to artificial magnets used in commercial generators, it still allows for the design of a functioning generator, albeit with certain limitations. Since the field strength is directly proportional to the induced voltage \( E = NAB\omega \), a stronger magnetic field would increase the induced voltage for the same coil design.

Understanding this concept helps clarify the relationship between magnetic field strength and generator efficiency. Designers of generators must balance between achieving a high enough magnetic field strength and the practical considerations of building and operating the generator in varied environments.
Coil Windings
Coil windings are pivotal in the design and functionality of electric generators. The coil acts as the nucleating point for electromagnetic induction, where the number of windings directly impacts the amount of induced EMF. More windings mean more turns through which magnetic field lines can pass, increasing the total induced voltage.

Analyzing the exercise, we see a proposed design with 1000 windings. This number is selected based on balancing the induced voltage and the physical constraints of managing larger coils. However, reducing the number of windings, such as to 500, requires changes in other design parameters to maintain the same voltage output. For instance, with fewer windings, either the coil diameter must increase, or the rotation speed (frequency) must adjust to generate the same voltage.

When designing a generator, engineers must consider the manageability of the coil size along with the desired electrical output. Extensive windings can make a coil bulky and impractical, which is why adjusting the number of windings against other factors like frequency and magnetic field strength is essential for a practical and efficient generator design.

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Most popular questions from this chapter

An ideal transformer has 840 turns in its primary coil and 120 turns in its secondary coil. If the primary coil draws 2.50 A at \(110 \mathrm{~V}\), what are (a) the current and (b) the output voltage of the secondary coil?

The armature of an ac generator has 100 turns. Each turn is a rectangular loop measuring \(8.0 \mathrm{~cm}\) by \(12 \mathrm{~cm} .\) The generator has a sinusoidal voltage output with an amplitude of \(24 \mathrm{~V}\). (a) If the magnetic field of the generator is \(250 \mathrm{mT}\), with what frequency does the armature turn? (b) If the magnetic field was doubled and the frequency cut in half, what would be the amplitude of the output?

(a) In May 2008 , the United States successfully landed a spacecraft named Phoenix near the northern polar regions of Mars. Immediately upon landing, the craft sent a message indicating that all had gone well. Using the astronomical data in the appendix of this book, determine the shortest amount of time it took this signal to reach the Earth. (b) If the Phoenix transmitter sent out spherical electromagnetic waves with a power of \(100 \mathrm{~W}\), how many watts per square meter would arrive at the Earth, assuming that Mars was in its closest location to Earth? (c) A radio signal was sent to a deep space probe traveling in the plane of the solar system. Earth received a response 3.5 days later. Assuming the probe computers took 4.5 hours to process the signal instructions and to send out the return message, was the probe within the solar system? (Assume a solar system radius of about 40 times the Earth-Sun distance.)

The secondary coil of an ideal transformer has 450 turns, and the primary coil has 75 turns. (a) Is this transformer a (1) step-up or (2) step-down transformer? Explain your choice. (b) What is the ratio of the current in the primary coil to the current in the secondary coil? (c) What is the ratio of the voltage across the primary coil to the voltage in the secondary coil?

The starter motor in an automobile has a resistance of \(0.40 \Omega\) in its armature windings. The motor operates on \(12 \mathrm{~V}\) and has a back emf of \(10 \mathrm{~V}\) when running at normal operating speed. How much current does the motor draw (a) when running at its operating speed, (b) when running at half its final rotational speed, and (c) when starting up?
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