Problem 28 - University Physics with Modern Physics

Question

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 G for the earth's field. (a) The French TGV train and the Japanese "bullet train" reach speeds of up to 180 mph moving on tracks about 1.5 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 m and a cruising speed of 565 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 potential difference is 6.03 mV for the train and 0.814 V for the plane, both too small to cause noticeable effects.

Step 1: Convert the Speeds to Meters per Second

To convert mph (miles per hour) to m/s (meters per second), use the conversion factor 1 mile = 1609.34 meters and 1 hour = 3600 seconds.For the train: 180 mph = \(180 \times \frac{1609.34}{3600} \approx 80.47\) m/s.For the plane: 565 mph = \(565 \times \frac{1609.34}{3600} \approx 252.67\) m/s.

Step 2: Calculate the Induced EMF for the Train

Use the formula for the electromotive force (EMF) induced in a conductor moving perpendicular to a magnetic field: \( E = B \cdot v \cdot L \), where:- \( B = 0.50 \) G = \(0.50 \times 10^{-4}\) T (since 1 G = 10^{-4} T is the conversion factor)- \( v = 80.47 \) m/s (train speed)- \( L = 1.5 \) m (distance between the tracks)Thus, \( E = 0.50 \times 10^{-4} \times 80.47 \times 1.5 \approx 6.03 \times 10^{-3} \) V or 6.03 mV.

Step 3: Calculate the Induced EMF for the Plane

For the plane, using the same formula for EMF: \( E = B \cdot v \cdot L \), we have:- \( B = 0.50 \times 10^{-4} \) T- \( v = 252.67 \) m/s (plane speed)- \( L = 64.4 \) m (wingspan)Thus, \( E = 0.50 \times 10^{-4} \times 252.67 \times 64.4 \approx 0.814 \) V.

Step 4: Assess the Potential Differences

For the train, the induced potential difference is 6.03 mV, which is very small and unlikely to have any noticeable effects on the train or its systems. For the plane, the induced potential difference is approximately 0.814 V, which is still relatively small compared to operating voltages in aircraft systems and is unlikely to cause any problems.

Key Concepts

Earth's Magnetic Field

The Earth's magnetic field is an omnipresent force that we might not regularly notice, but it significantly affects our environment. It is a geomagnetic field that extends from the Earth's interior out into space, where it interacts with solar winds. For vehicles like planes and trains, the magnetic field can induce small electrical currents as they move.

This field is measured in units called gauss (G), and on average, it presents a value of approximately *0.50 G*. To put this into perspective, one gauss is equivalent to *10,000 teslas (T)*, the standard unit for magnetic fields used in physics. Therefore, the Earth's magnetic field at the surface is about *5 × 10^{-5} T*.

While vehicles move at high speeds through this field, it can create interesting effects like induced electomotive forces. Understanding how this field interacts with conductors (like the metal bodies of trains or planes) helps in analyzing potential effects, although in practice, these differences are often quite minimal.

Electromotive Force (EMF)

Electromotive force (EMF) is a fundamental concept in electromagnetism. It refers to the potential difference generated in a conductor when it moves through a magnetic field. When a conductor like a train or plane moves perpendicular to the Earth's magnetic field, a voltage is induced across it, which is calculated using the simple formula:

\[ E = B \cdot v \cdot L \] where *E* is the induced EMF, *B* is the magnetic field strength (in teslas), *v* is the speed of the conductor (in meters per second), and *L* is the length of the conductor (in meters) moving through the field.

For example, the French TGV train, moving at about 80.47 m/s with a track gauge of 1.5 m, experiences a small EMF of roughly 6.03 millivolts. This happens due to the relatively weak Earth's magnetic field and the short distance between the tracks.

Similarly, high-speed planes like the Boeing 747, with a much longer wingspan of 64.4 m, experience a larger EMF—approximately 0.814 volts—due to their higher speed and longer conductive pathway through the magnetic field. Nonetheless, this induced EMF is minor enough not to interfere with electronic systems onboard.

Potential Difference

Potential difference, often known as voltage, is the measure of electric potential energy per unit charge between two points. It provides the "push" needed for electrical current to flow through a circuit. When dealing with high-speed vehicles and the Earth's magnetic field, the term "induced potential difference" becomes relevant.

As in the context of a train moving across train tracks, or the wings of an aircraft in motion, the potential difference can be induced by the magnetic field as the conductor (or parts of the vehicle) moves through it. However, whether this induced potential results in significant effects depends on the magnitude of the potential difference generated.

For example, a potential difference of 6.03 mV induced across a train's tracks is negligible compared to operating voltages used in railway systems, which usually range from 20V to several hundreds of volts.

Similarly, for aircraft like the Boeing 747, an induced potential difference of 0.814 V is quite small relative to voltages encountered during standard operating conditions of aircraft systems. This means that the induced potential difference is unlikely to affect normal technological operations or safety of such vehicles.