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Interstellar Magnetic Field The Voyager I spacecraft moves through interstellar space with a speed of \(8.0 \times 10^{3} \mathrm{m} / \mathrm{s} .\) The magnetic field in this region of space has a magnitude of \(2.0 \times 10^{-10} \mathrm{T} .\) Assuming that the \(5.0-\mathrm{m}\) -long antenna on the spacecraft is at right angles to the magnetic field, find the induced emf between its ends.

Short Answer

Expert verified
The induced emf is \(8.0 \times 10^{-6} \, \text{V}\).

Step by step solution

01

Understand the formula for induced emf

The formula to find the induced electromotive force (emf) in a conductor moving through a magnetic field is given by \( \text{emf} = B \cdot L \cdot v \cdot \sin(\theta) \), where \( B \) is the magnetic field strength, \( L \) is the length of the conductor (antenna), \( v \) is the speed of the conductor, and \( \theta \) is the angle between the velocity and the magnetic field direction.
02

Determine the angle \( \theta \)

Since the antenna is at right angles to the magnetic field, the angle \( \theta \) is 90 degrees. Therefore, \( \sin(\theta) = \sin(90^\circ) = 1 \).
03

Substitute the values into the formula

Substitute \( B = 2.0 \times 10^{-10} \, \text{T} \), \( L = 5.0 \, \text{m} \), \( v = 8.0 \times 10^{3} \, \text{m/s} \), and \( \sin(90^\circ) = 1 \) into the formula: \[ \text{emf} = (2.0 \times 10^{-10} \, \text{T}) \cdot (5.0 \, \text{m}) \cdot (8.0 \times 10^{3} \, \text{m/s}) \cdot 1 \]
04

Calculate the induced emf

Compute the product: \[ \text{emf} = 2.0 \times 5.0 \times 8.0 \times 10^{-10} \times 10^{3} = 80 \times 10^{-10+3} = 80 \times 10^{-7} \, \text{V} \] This simplifies to \[ 8.0 \times 10^{-6} \, \text{V} \].

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Key Concepts

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

Magnetic Field
A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. This invisible field is represented by magnetic field lines that flow from the north pole to the south pole and are denoted by the symbol \( B \) in physics.

Magnetic fields arise in environments where electric currents and magnetic strengths are present. Earth's magnetic field is a classic example, protecting our planet from cosmic radiation and influencing navigational compasses.
  • The strength of a magnetic field is measured in Tesla (T).
  • An important property of magnetic fields is that they can induce electromotive force (emf) in a conductor moving through them.
  • This induction of emf is a manifestation of the fundamental electromagnetic force in physics.
In the context of the Voyager I spacecraft, the magnetic field in interstellar space is relatively weak, measuring just \(2.0 \times 10^{-10} \text{ T} \). Despite its low magnitude, it is significant enough to influence the spacecraft's systems and create an induced emf when interacting with the spacecraft's antenna.
Antenna
An antenna is a conductor that transmits and receives electromagnetic waves. It plays a critical role in communication systems, including those on spacecraft like Voyager I. The efficiency of an antenna to induce or receive signals depends largely on its design and orientation.

For the Voyager I spacecraft, the antenna is a straight conductor measuring \(5.0 \text{ m} \) in length.
  • Length and shape of the antenna are crucial to its operation and affect the efficiency of signal transmission and reception.
  • The position of the antenna relative to the magnetic field is essential for maximizing the induced emf.
  • In this case, as the antenna is perpendicular to the magnetic field, it achieves optimal conditions for maximum emf induction.
This configuration ensures the spacecraft can capture the weak electromagnetic influences in interstellar space effectively, translating them into useful electrical signals.
Voyager I Spacecraft
The Voyager I spacecraft is one of humanity's farthest-traveling probes, launched by NASA in 1977 to explore the outer planets and beyond. Embarking on a mission that extends into interstellar space, it has provided unprecedented insights into distant cosmic regions.

Key attributes of the Voyager I spacecraft include:
  • It travels at a speed of \(8.0 \times 10^{3} \text{ m/s} \), allowing it to cross vast distances spanning light-years.
  • Voyager I sends back data to Earth via its antenna; this information is critical for understanding space environments beyond our solar system.
  • As it moves through the magnetic field of interstellar space, the spacecraft's systems, including its antenna, exploit these magnetic influences to maintain functionality and data transmission.
Voyager I continues to communicate with Earth, transcending its original mission timeline due to its robust engineering and effective use of electromagnetic principles.

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Most popular questions from this chapter

A conducting rod slides on two wires in a region with a magnetic field. The two wires are not connected. Is a force required to keep the rod moving with constant speed? Explain.

Predict/Explain A metal ring is dropped into a localized region of constant magnetic field, as indicated in Figure \(23-30\). The magnetic field is zero above and below the region where it is finite. (a) For each of the three indicated locations \((1,2,\) and 3 ), is the induced current clockwise, counterclockwise, or zero? (b) Choose the best explanation from among the following: Clockwise at 1 to oppose the field; zero at 2 because the field is uniform; counterclockwise at 3 to try to maintain the field. II. Counterclockwise at 1 to oppose the field; zero at 2 because the field is uniform; clockwise at 3 to try to maintain the field. III. Clockwise at 1 to oppose the field; clockwise at 2 to maintain the field; clockwise at 3 to oppose the field.

A cubical box \(22 \mathrm{cm}\) on a side is placed in a uniform \(0.35-\mathrm{T}\) magnetic field. Find the net magnetic flux through the box.

Airplane emf A Boeing KC-135A airplane has a wingspan of \(39.9 \mathrm{m}\) and flies at constant altitude in a northerly direction with a speed of \(850 \mathrm{km} / \mathrm{h}\). If the vertical component of the Earth's magnetic field is \(5.0 \times 10^{-6} \mathrm{T}\), and its horizontal component is \(1.4 \times 10^{-6} \mathrm{T},\) what is the induced emf between the wing tips?

A car drives onto a loop detector and increases the downward component of the magnetic field within the loop from \(1.2 \times 10^{-5} \mathrm{T}\) to \(2.6 \times 10^{-5} \mathrm{T}\) in \(0.38 \mathrm{s} .\) What is the induced emf in the detector if it is circular, has a radius of \(0.67 \mathrm{m},\) and consists of four loops of wire? A. \(0.66 \times 10^{-4} \mathrm{V}\) B. \(1.5 \times 10^{-4} \mathrm{V}\) C. \(2.1 \times 10^{-4} \mathrm{V}\) D. \(6.2 \times 10^{-4} \mathrm{V}\)

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