Earth's magnetic field and planes.

A Boeing 777 is flying due north at v = 900 k m / h v=900km/h (typical cruising speed) and at an altitude where the vertical component of the Earth's magnetic field is B = 50 μ T B=50\mu T . The wingspan of the 777 is about 60m. Determine the induced voltage in Volts across the tips of the wings.

Details and assumptions

  • Although wings have a complicated shape as in the figure, for the purposes of this problem you may model the wings as a straight conducting rod of length L = 60 m L=60m if you wish.


The answer is 0.75.

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2 solutions

Kee Wei Lee
May 20, 2014

First we convert the quantities into their standard units. So, v = 900 k m / h = ( 900 1000 ) / ( 60 60 ) m / s = 250 m / s v=900km/h=(900*1000)/(60*60)m/s=250m/s Also it is good to note that B = 50 1 0 6 T B=50*10^{-6} T By Faraday's Law the induced voltage will be equal to the product of the magnetic field, length of the conducting rod and the speed at which the rod is traveling. So induced V = B L v = ( 50 1 0 6 ) ( 60 ) ( 250 ) = 0.75 V V=BLv=(50*10^{-6})(60)(250)=0.75 V

David Mattingly Staff
May 13, 2014

Let us focus on a electron in the wing, which is free to move since the wing is a conductor. The magnetic force acting on this electron has a component directed along the wing given by F m = e v B . F_{m}= e v B . since the velocity and magnetic field are perpendicular. If everything has come to equilibrium and the electrons are stationary there must exist an electric field along the wing canceling this magnetic force. Thus e E = e v B E = v B V = v B L = 0.75 V e E= e v B \rightarrow E= v B \rightarrow V=vB L=0.75 V

The origin of this electric field is easy to understand. Suppose that initially a conducting rod is at rest in a magnetic field. If the rod starts moving, then the magnetic force will cause some electrons to accumulate at the tip of the rod. This situation is similar to that of a capacitor. This redistribution of charges is the reason why an electric field is created. The electrons will drift towards the end of the rod until the electric force cancels the magnetic force.

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