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Chapter 22: Problem 2

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 \(5.1 \times 10^{-5} \mathrm{~T}\). If 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 is \( 7.75 \times 10^{3} \text{ V} \).

Step by step solution

01

Understand the Formula for Motional EMF

The formula for calculating the motional electromotive force (emf) is given by \( \text{emf} = B \cdot v \cdot L \cdot \sin(\theta) \), where \(B\) is the magnetic field strength, \(v\) is the velocity of the wire, \(L\) is the length of the wire, and \(\theta\) is the angle between the velocity and the magnetic field. Since the wire is moving perpendicular to the Earth's magnetic field, \( \theta = 90^\circ \), and \( \sin(90^\circ) = 1 \). Thus, the formula simplifies to \( \text{emf} = B \cdot v \cdot L \).
02

Substitute Given Values

Now substitute the given values into the simplified formula. Here, \( B = 5.1 \times 10^{-5} \text{ T} \), \( v = 7.6 \times 10^{3} \text{ m/s} \), and \( L = 2.0 \times 10^{4} \text{ m} \). Thus, the expression for the motional emf becomes: \( \text{emf} = 5.1 \times 10^{-5} \times 7.6 \times 10^{3} \times 2.0 \times 10^{4} \).
03

Calculate the Motional EMF

Perform the calculation for the motional emf. Multiply the values: \( \text{emf} = 5.1 \times 10^{-5} \times 7.6 \times 10^{3} \times 2.0 \times 10^{4} \). Simplifying, first multiply the numbers without exponents: \( 5.1 \times 7.6 \times 2.0 = 77.52 \). Now, handle the exponents: \( 10^{-5} \times 10^{3} \times 10^{4} = 10^{2} \). Combining these results gives \( 77.52 \times 10^{2} \), which equals \( 7752 \).
04

Express the Final Answer

After performing the calculations, the motional emf across the wire is \( 7.75 \times 10^{3} \text{ volts} \).

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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 project by NASA in collaboration with the Italian Space Agency. Conducted in 1996 aboard the space shuttle Atlantis, it involved deploying a long, conductive wire into orbit to explore motional electromotive force (emf). This experiment sought to better understand how emf is generated when a conductor moves through a magnetic field. Since Atlantis carried a 20,000-meter wire, this provided a significant opportunity to analyze how motion and magnetic fields interact on a large scale.
The experiment was mainly about harnessing natural phenomena like Earth's magnetic field to generate electricity. It helps demonstrate how motion through a magnetic field can convert kinetic energy into electrical energy, a concept fundamental in both astrophysics and electrical engineering.
Overall, the Tethered Satellite Experiment not only aimed to gather scientific data but also inspired potential future technologies for sustainable energy generation in space.
Orbital Speed
Orbital speed is a critical aspect of celestial mechanics. For any object in orbit, such as the space shuttle Atlantis, the speed determines the stability and altitude of the orbit. In the Tethered Satellite Experiment, the orbital speed of the shuttle was given as 7,600 meters per second. This speed is necessary to balance the gravitational pull of the Earth, allowing the shuttle to remain in its orbital path.
When an object orbits Earth, it is in a constant state of free fall towards the planet. However, because of its high speed, it moves forward instead of crashing into the surface, effectively "falling around" the Earth. This constant balance between gravity and forward motion is what defines orbital speed.
  • High speeds are essential for maintaining an orbit.
  • Orbital speed influences the generation of motional emf since it affects how the wire cuts through Earth's magnetic field.
Understanding orbital speed is essential for satellite deployment, maintaining space station orbits, and conducting experiments like those involving tethered satellites.
Earth's Magnetic Field
Earth's magnetic field plays a significant role in many natural and technological processes. It is generated by the movement of molten iron in Earth's outer core and acts like a giant magnet surrounding the planet. In space, this magnetic field protects us from solar winds and cosmic radiation.
For the Tethered Satellite Experiment, the Earth's magnetic field was crucial. Measured at 5.1 x 10^-5 Tesla at the location of the tethered wire, it facilitated the generation of motional emf as the wire moved through it. The interaction of the wire with the magnetic field induces an electric current, a fundamental principle in electromagnetism.
  • The strength of the magnetic field influences the amount of emf generated.
  • Magnetic fields are a vital part of designing devices such as dynamos and transformers.
Overall, Earth's magnetic field is not just a natural wonder but a powerful tool for technological innovations in space and beyond.
Perpendicular Motion
Perpendicular motion refers to the movement of an object at an angle of 90 degrees to a given reference. In the case of the Tethered Satellite Experiment, the reference was Earth's magnetic field. Calculating motional emf involves perpendicular motion because the maximum emf is generated when the angle between the velocity of the moving wire and the magnetic field is 90 degrees. At this angle, the sine factor of the emf formula becomes one, maximizing the induced voltage.
This concept is a cornerstone of Faraday's Law of Electromagnetic Induction, stating that the induced emf in a closed loop is proportional to the rate of change of magnetic flux through the loop. In simpler terms, if you want to maximize the electrical energy produced, you need to move the conductor perpendicularly through the magnetic field.
  • Perpendicular motion ensures the most efficient energy conversion.
  • It is an essential principle in designing electrical generators and alternators.
Understanding perpendicular motion is crucial for both students and engineers dealing with magnetic fields and electrical generation.

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Most popular questions from this chapter

Electric doorbells found in many homes require \(10.0 \mathrm{~V}\) to operate. To obtain this voltage from the standard 120-V supply, a transformer is used. Is a step-up or a stepdown transformer needed, and what is its turns ratio \(N_{\mathrm{s}} / N_{\mathrm{p}}\) ?

Mutual induction can be used as the basis for a metal detector. A typical setup uses two large coils that are parallel to each other and have a common axis. Because of mutual induction, the ac generator connected to the primary coil causes an emf of \(0.46 \mathrm{~V}\) to be induced in the secondary coil. When someone without metal objects walks through the coils, the mutual inductance and, thus, the induced emf do not change much. But when a person carrying a hand gun walks through, the mutual inductance increases. The change in emf can be used to trigger an alarm. If the mutual inductance increases by a factor of three, find the new value of the induced emf.

The drawing shows a copper wire (negligible resistance) bent into a circular shape with a radius of \(0.50 \mathrm{~m} .\) The radial section \(B C\) is fixed in place, while the copper bar \(A C\) sweeps around at an angular speed of \(15 \mathrm{rad} / \mathrm{s}\). The bar makes electrical contact with the wire at all times. The wire and the bar have negligible resistance. A uniform magnetic field exists everywhere, is perpendicular to the plane of the circle, and has a magnitude of \(3.8 \times 10^{-3} \mathrm{~T}\). Find the magnitude of the current induced in the \(\operatorname{loop} A B C .\)

The resistances of the primary and secondary coils of a transformer are 56 and \(14 \Omega\), respectively. Both coils are made from lengths of the same copper wire. The circular turns of each coil have the same diameter. Find the turns ratio \(N_{\mathrm{s}} / N_{\mathrm{p}}\).

A house has a floor area of \(112 \mathrm{~m}^{2}\) and an outside wall that has an area of \(28 \mathrm{~m}^{2}\). The earth's magnetic field here has a horizontal component of \(2.6 \times 10^{-5} \mathrm{~T}\) that points due north and a vertical component of \(4.2 \times 10^{-5} \mathrm{~T}\) that points straight down, toward the earth. Determine the magnetic flux through the wall if the wall faces (a) north and (b) east. (c) Calculate the magnetic flux that passes through the floor.
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