/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 Motional emfs in Transportation.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 29: Problem 24

Motional emfs in Transportation. Airplanes and trains move through the earth's magnetic field at rather high speeds, so it is reasonable to wonder whether this field can have a substantial effect on them. We shall use a typical value of 0.50 \(\mathrm{G}\) for the earth's field (a) The French TGV train and the Japanese "bullet train" reach speeds of up to 180 \(\mathrm{mph}\) moving on tracks about 1.5 \(\mathrm{m}\) apart. At top speed moving perpendicular to the earth's magnetic field, what potential difference is induced across the tracks as the wheels roll? Does this seem large enough to produce noticeable effects? (b) The Boeing \(747-400\) aircraft has a wingspan of 64.4 \(\mathrm{m}\) and a cruising speed of 565 \(\mathrm{mph}\) . If there is no wind blowing (so that this is also their speed relative to the ground), what is the maximum potential difference that could be induced between the opposite tips of the wings? Does this seem large enough to cause problems with the plane?

Short Answer

Expert verified
The induced emfs are 6.04 mV for the train and 0.81 V for the airplane. Neither is large enough to cause significant effects.

Step by step solution

01

Convert Units to SI

The Earth's magnetic field is given as 0.50 G (Gauss). We first convert it to Tesla: \(1 \text{ G} = 10^{-4} \text{ T}\), so \(0.50 \text{ G} = 0.50 \times 10^{-4} \text{ T} = 5 \times 10^{-5} \text{ T}\).For velocities, convert speeds from mph to m/s. The train's speed: \(180 \text{ mph} = 180 \times 0.44704 \text{ m/s} \approx 80.47 \text{ m/s}\). For the airplane: \(565 \text{ mph} = 565 \times 0.44704 \text{ m/s} \approx 252.69 \text{ m/s}\). Now proceed to calculate the induced emf for each case.
02

Calculate Induced emf for Train

The potential difference \(V\) induced across the tracks is given by the formula \(V = B \cdot l \cdot v\), where \(B\) is the magnetic field, \(l\) is the distance between the tracks, and \(v\) is the speed of the train.Substitute the values: \(V = (5 \times 10^{-5} \text{ T}) \cdot (1.5 \text{ m}) \cdot (80.47 \text{ m/s}) \).Calculate \(V\):\(V = 6.03525 \times 10^{-3} \text{ V} \approx 6.04 \text{ mV}\).This induced voltage is not large enough to have noticeable effects on the train.
03

Calculate Induced emf for Airplane

Use the same formula for potential difference \(V = B \cdot l \cdot v\), where \(l\) is the wingspan of the airplane, and \(v\) is its speed.Substitute the values:\(V = (5 \times 10^{-5} \text{ T}) \cdot (64.4 \text{ m}) \cdot (252.69 \text{ m/s}) \).Calculate \(V\):\(V = 0.8135264 \text{ V} \approx 0.81 \text{ V}\).This voltage is also small and unlikely to cause noticeable problems with the airplane's operations.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Earth's Magnetic Field
The Earth's magnetic field is a fascinating phenomenon that influences various aspects of life and technology. This magnetic field, also known as the geomagnetic field, originates from the movement of molten iron within the Earth's outer core. As this iron moves, it creates electrical currents. These currents generate the Earth's magnetic field, which extends from the Earth's interior into space. The Earth's magnetic field is crucial because it shields our planet from solar winds and cosmic radiation, protecting the atmosphere from being stripped away. Additionally, it serves as a navigation system for migratory animals such as birds and sea turtles. To understand the impact of this field on transportation, we often use a typical value of 0.50 Gauss (G), which is equivalent to 5 x 10^-5 Tesla (T). Although the Earth's magnetic field is relatively weak, it still can induce an electromotive force (emf) when conductive materials move through it. This principle has practical implications for vehicles moving at high speeds, such as trains and airplanes.
Electromagnetic Induction
Electromagnetic induction is a key concept in physics, describing how an electromotive force (emf) is generated when a conductor moves through a magnetic field. This process, discovered by Michael Faraday in 1831, is the principle behind many electrical devices and technologies we depend on today.When a conductor, such as a metal wire or a train track, moves through a magnetic field, the magnetic flux through the conductor changes. According to Faraday's Law of Electromagnetic Induction, this change in flux induces an emf across the conductor. The magnitude of the induced emf depends on:
  • The strength of the magnetic field (\( B \))
  • The length of the conductor perpendicular to the field (\( l \))
  • The velocity of the conductor with respect to the field (\( v \))
The induced emf can be calculated using the formula: \[V = B \cdot l \cdot v\]In the context of transportation, when trains or airplanes travel through the Earth's magnetic field, they generate a small emf. While this potential difference is typically not large enough to cause any significant effects, understanding this principle is essential for the design and operation of high-speed vehicles, ensuring they remain unaffected by electromagnetic influences.
Physics in Transportation
Physics plays an essential role in modern transportation, influencing the design, efficiency, and safety of vehicles. When we consider high-speed trains and airplanes, physics concepts such as motional electromotive force (emf) become very relevant. When a train moves at high speeds over tracks, the motion through the Earth's magnetic field induces a potential difference across the tracks. Due to the relatively weak magnetic field and moderate speeds, this voltage is often just a few millivolts, such as the 6.04 mV calculated for the French TGV train. This value is negligible and unlikely to impact train operations. Similarly, for an airplane like the Boeing 747-400 flying at 565 mph, the wingspan moving through the Earth's magnetic field produces an induced emf. Calculated to be around 0.81 V between the wingtips, this voltage is still insignificant to affect the plane's functionality or passangers' safety. Understanding these phenomena ensures that engineers and policymakers account for all potential influences on vehicle operation, applying physics to enhance the reliability and safety of vehicles as they move through varying magnetic environments.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A diclectric of permitivity \(3.5 \times 10^{-11} \mathrm{F} / \mathrm{m}\) completely fills the volume between two capacitor plates. For \(t > 0\) the electric flux through the dielectric is \(\left(8.0 \times 10^{3} \mathrm{V} \cdot \mathrm{m} / \mathrm{s}^{3}\right) t^{3}\) . The dielectric is ideal and nonmagnetic; the conduction current in the dielectric is zero. At what time does the displacement current in the dielectric equal 21\(\mu \mathrm{A} ?\)

A long, thin solenoid has 400 turns per meter and radius 1.10 \(\mathrm{cm}\) . The current in the solenoid is increasing at a uniform rate dildt. The induced electric field at a point near the center of the solenoid and 3.50 \(\mathrm{cm}\) from its axis is \(8.00 \times 10^{-6} \mathrm{V} / \mathrm{m}\) . Calculate dildt.

A rectangle measuring 30.0 \(\mathrm{cm}\) by 40.0 \(\mathrm{cm}\) is located inside a region of a spatially uniform magnetic field of 1.25 \(\mathrm{T}\) , with the field perpendicular to the plane of the coil (Fig. 29.29 ). The coil is pulled out at a steady rate of 2.00 \(\mathrm{cm} / \mathrm{s}\) traveling perpendicular to the field lines. The region of the field ends abruptly as shown. Find the emf induced in this coil when it is (a) all inside the field: (b) partly inside the field; (c) all outside the field.

Displacement Current in a Wire. A long, straight, copper wire with a circular cross-scctional area of 2.1 \(\mathrm{mm}^{2}\) carries a current of 16 \(\mathrm{A}\) . The resistivity of the material is \(20 \times 10^{-8} \Omega \cdot \mathrm{m}\) . (a) What is the uniform electric field in the material? (b) If the cur- rent is changing at the rate of 4000 \(\mathrm{A} / \mathrm{s}\) , at what rate is the electric field in the material changing? (c) What is the displacement current density in the material in part (b)? (Hint: Since \(K\) for copper is very close to \(1,\) use \(\epsilon=\epsilon_{0} . )\) (d) If the current is changing as in part (b), what is the magnitude of the magnetic field 6.0 \(\mathrm{cm}\) from the center of the wire? Note that both the conduction current and the displacement current should be included in the calculation of \(B\) . Is the contribution from the displacement current significant?

In a region of space, a magnetic ficld points in the \(+x\) -direction (toward the right). Its magnitude varies with position according to the formula \(B_{x}=B_{0}+b x,\) where \(B_{0}\) and \(b\) are positive constants, for \(x \geq 0\) . A flat coil of area \(A\) moves with uniform speed \(v\) from right to left with the plane of its area always perpendicular to this field. (a) What is the emf induced in this coil while it is to the right of the origin? (b) As viewed from the origin, what is the direction (clockwise or counterclockwise) of the current induced in the coil? (c) If instead the coil moved from left to right, what would be the answers to parts (a) and (b)?
See all solutions

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Read Explanation

Work Energy and Power

Read Explanation

Thermodynamics

Read Explanation

Space Physics

Read Explanation

Electricity and Magnetism

Read Explanation

Geometrical and Physical Optics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.