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 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?

Short answer

1. Potential difference is induced across the tracks as the wheels roll is 0.6 mV and this is very small to be noticeable.
2. The maximum potential difference that could be induced between the opposite tips of the wings is 0.813V and this is also very small to be noticeable thus it does not cause problems with the plane.

Step-by-step solution

Concept

A conductor moving in a magnetic field may have a potential difference induced across it, depending on how it is moving. The magnitude of that induced emf is,

$\mathrm{\epsilon }=\mathrm{vBLsin}\left(\mathrm{\varphi }\right)$

Calculation

1. We have a train with a speed of 180 mph moving on tracks about 1.5 m apart, it is moving perpendicularly to the earth's field (B = 0.50 G), so = 90.0°, substitute with the givens we get (note that l m/s = 2237 mph and 1 G =10 T), so we get

$\mathrm{\epsilon }=\mathrm{vBLsin}\left(\mathrm{\varphi }\right)$

Calculation

(a) We have a train with a speed of 180 mph moving on tracks about 1.5 m apart, it is moving perpendicularly to the earth's field (B = 0.50 G), so = 90.0°, substitute with the givens we get (note that l m/s = 2237 mph and 1 G =10 T), so we get

$\begin{array}{l}\mathrm{\epsilon }=\left(180\mathrm{mph}\right)\left(\frac{1\mathrm{m}/\mathrm{s}}{2.237\mathrm{mph}}\right)\left(0.50\times {10}^{-4}\mathrm{T}\right)\left(1.5\mathrm{m}\right)\\ \mathrm{\epsilon }=6.0\times {10}^{-3}\mathrm{V}=6.0\mathrm{mV}\end{array}$

(b) For a Boeing aircraft, which has a wingspan of 644 m and a cruising speed of 565 mph, the induced emf is, $\begin{array}{l}\mathrm{\epsilon }=\left(565\mathrm{mph}\right)\left(\frac{1\mathrm{m}/\mathrm{s}}{2.237\mathrm{mph}}\right)(0.50\times {10}^{-4}\mathrm{T})(64.4\mathrm{m})\\ \mathrm{\epsilon }=0.813V\end{array}$