Question

Transforming fields Part A had us show E*B and E^2-C^2B^2 are invariant under lorentz transformations. I am not interested in that, I am looking for the following:

(b) The magnetic field of the earth is predominantly vertical in the polar regions. It was suggested to use this field to measure the speed of an aircraft by measuring the voltage induced in a wire suspended in the aircraft. What orientation should the wire have relative to the aircraft? Estimate the magnitude of the voltage induced in a 1 m wire.

Answer 1

According to Lorentz transformation, the electric field in the moving plane due to the magnetic field of the earth is proportional to the cross product of velocity and the magnetic field which is perpendicular to the velocity vector

Hence the wire should be perpendicular to the magnetic field. Since the magnetic field of the earth is vertical in the polar regions, the wire's orientation should be horizontal.

Assuming the speed of the plane is much smaller than the speed of the light

The induced voltage is

given L = 1 m

Assuming the magnetic field at the poles is about B =  65 microteslas

and the speed of the plane is about 900 km/s = 900,000/3600 m/s = 250 m/s

the induced voltage is