Question

A jet plane is travelling west at a speed of 1800 km/h. What is the voltage difference developed between the ends of the wing having a span of 25 m, if the Earth's magnetic field at the location has a magnitude of $5 \times {10^{ - 4}}$T and the dip angle is $30{}^{^0}$?

Answer 1

Hint: In electromagnetic induction when a conductor is moving with certain velocity in the presence of magnetic field then voltage will be generated across the conductor and this is called the induced emf. This happens due to change in the flux which is associated with that conductor. This problem is related to the induced emf.
Formula used:
$\eqalign{
  & \phi = B.A \cr
  & e = BVL\sin \theta \cr} $

Complete answer:
The flux which is associated with the wings of the plane will be changing as it is moving with velocity V in the magnetic field B and the length of the wings be L. It is well known that the earth has a magnetic field associated with it. It has both horizontal and vertical components of magnetic field at every location and the resultant of horizontal component and vertical component at a particular place gives us the resultant magnetic field at that location. Dip is the angle between the resultant magnetic field vector and the horizontal component vector. Horizontal component of the magnetic field always points in the north direction.

$\phi = B.A$

Where $\phi $ is the magnetic flux associated and A is the area that is changing. The above expression is a dot product of magnetic field and area.
We have a term called the motional emf which is induced by the motion of a body in a constant magnetic field. The emf generated in this case is motional emf and its formula generally is given by $e = BVL$
Here e is the induced motional emf and it is to be noted that the B and V and L should be mutually perpendicular to each other.
In the question it is given that plane is travelling in west direction and since wings will be perpendicular to plane travelling direction it means length is along north direction and we know earth’s horizontal magnetic field also will be in north direction and hence we should consider earth’s vertical component of magnetic field which will be $B\sin \theta $ where $\theta $ is the dip angle.
So we have emf generated which will be $e = BVL\sin \theta = 1800 \times \dfrac{5}{{18}} \times 5 \times {10^{ - 4}} \times \sin {30^0} \times 25 = 3.1volts$
Hence 3.1 volts of emf will be generated.

Note:
In these kinds of problems first we should convert all the given quantities into their SI units and then proceed which is why we converted velocity into its SI units. If the emf generated is due to the change in magnetic field then the emf induced will be varying and the emf generated due to motion is not varying and can be considered as a DC source.