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Chapter 22: Problem 3

In 1996 , NASA performed an experiment called the Tethered Satellite experiment. In this experiment a \(2.0 \times 10^{4}-\mathrm{m}\) length of wire was let out by the space shuttle Atlantis to generate a motional emf. The shuttle had an orbital speed of \(7.6 \times 10^{3} \mathrm{m} / \mathrm{s}\) , and the magnitude of the earth's magnetic field at the location of the wire was \(5.1 \times 10^{-5} \mathrm{T}\) the wire had moved perpendicular to the earth's magnetic field, what would have been the motional emf generated between the ends of the wire?

Short Answer

Expert verified
The motional emf generated is \( 7.752 \times 10^{5} \text{ V} \).

Step by step solution

01

Understanding the Formula for Motional EMF

Motional EMF can be calculated using the formula \( \varepsilon = B \cdot v \cdot L \cdot \sin(\theta) \), where \( B \) is the magnetic field, \( v \) is the speed, \( L \) is the length of the wire, and \( \theta \) is the angle between the velocity vector and the magnetic field. Here, the wire moves perpendicular to the magnetic field, so \( \theta = 90^\circ \), making \( \sin(\theta) = 1 \).
02

Substituting Known Values

Substitute the given values into the formula: \( B = 5.1 \times 10^{-5} \text{ T} \), \( v = 7.6 \times 10^{3} \text{ m/s} \), and \( L = 2.0 \times 10^{4} \text{ m} \). The formula simplifies to \( \varepsilon = 5.1 \times 10^{-5} \cdot 7.6 \times 10^{3} \cdot 2.0 \times 10^{4} \).
03

Calculating the Motional EMF

Perform the multiplication step-by-step: \( 5.1 \times 7.6 = 38.76 \). Now, \( 38.76 \times 2.0 \times 10^{4} = 38.76 \times 2.0 = 77.52 \). Taking the power of 10 into account, \( 77.52 \times 10^{4} = 77.52 \times 10^{4} = 7.752 \times 10^{5} \text{ V} \).

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Key Concepts

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

Tethered Satellite experiment
The Tethered Satellite experiment was a fascinating NASA project that took place in 1996. Its main purpose was to explore the potential for creating electricity in space. This experiment involved letting out a long, 20,000-meter wire from the space shuttle Atlantis. By moving through Earth's magnetic field at a high speed, the wire was expected to generate an electromagnetic force, known as motional EMF.
The idea was to understand more about how objects moving through a magnetic field can create electrical currents. This could have potential applications in generating power in space without relying solely on solar panels. The experiment was a significant step in the exploration of electromagnetic phenomena in space environments. Although it faced some technical challenges, it offered great insights into how electromagnetic fields work in areas beyond our planet.
Magnetic Field
A magnetic field is an invisible field that exerts force on certain objects, specifically those that are magnetic or carry electric charges. In the context of the Tethered Satellite experiment, Earth's magnetic field played a crucial role. It is the primary component needed to generate motional EMF when a conductor, like a wire, moves through it.
To understand how it works, imagine the magnetic field as lines of force surrounding a magnet. The magnetic field's strength is measured in teslas (T). In the experiment, the magnetic field had a strength of 5.1 x 10-5 T.
  • Magnetic fields are created by electric currents.
  • Earth's magnetic field protects the planet from solar wind.
  • The interaction of the wire with the field resulted in the generation of an electric current in the wire due to its movement.
This illustrates how dynamic and influential magnetic fields can be, especially in scientific experiments aimed at exploring space.
Orbital Speed
Orbital speed refers to the speed at which an object travels along its orbit around a larger body, such as a planet or star. In the Tethered Satellite experiment, the space shuttle Atlantis moved at an impressive orbital speed of 7.6 x 103 meters per second. This speed was essential for generating motional EMF.
Orbital speed is influenced by different factors:
  • The gravitational pull of the larger body.
  • The distance from the center of the larger body.
High orbital speeds are necessary to balance the gravitational pull and maintain an orbit. This energy is what allowed the shuttle to traverse the magnetic field effectively, thus inducing an electric current in the 20-kilometer-long wire.
Perpendicular Motion
For motional EMF, having the conductive wire move perpendicular to the magnetic field is a critical factor. In the Tethered Satellite experiment, this perpendicular orientation maximized the EMF generated.
Here's why this matters: when the velocity of the wire is perpendicular to the magnetic field lines, the angle (\(\theta\)) between them is 90 degrees. Thus, \(\sin(\theta)\) becomes 1, which integrates fully into the formula for calculating EMF (\(\varepsilon = B \cdot v \cdot L \cdot \sin(\theta)\)).
  • Perpendicular motion ensures maximum interaction with magnetic fields.
  • It optimizes the amount of EMF induced in the wire.
This arrangement demonstrates the importance of geometric configuration in experiments involving electromagnetic forces. It ensures that the resulting voltage is at its optimum, leading to more efficient energy generation.

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Most popular questions from this chapter

ssm 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 \(\mathrm{m}\) .

A square loop of wire consisting of a single turn is perpendicular to a uniform magnetic field. The square loop is then re-formed into a circular loop, which also consists of a single turn and is also perpendicular to the same magnetic field. The magnetic flux that passes- through the square loop is \(7.0 \times 10^{-3}\) Wb. What is the flux that passes through the circular loop?

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 long, current-carrying solenoid with an air core has 1750 turns per meter of length and a radius of 0.0180 \(\mathrm{m}\) . A coil of 125 turns is wrapped tightly around the outside of the solenoid, so it has virtually the same radius as the solenoid. What is the mutual inductance of this system?

You need to design a \(60.0-\mathrm{Hz}\) ac generator that has a maximum emf of 5500 \(\mathrm{V}\) . The generator is to contain a 150 -turn coil that has an area per turn of 0.85 \(\mathrm{m}^{2}\) . What should be the magnitude of the magnetic field in which the coil rotates?
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