Problem 24 - University Physics with Modern Physics

Question

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?

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Key Concepts

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.