The Earth's magnetic field does not form a spherical shell around the Earth, but has the shape of the field due to a bar magnet, with a magnetic south pole near the Earth's north pole and a magnetic north pole at the Earth's south pole. The Earth's magnet field is from the magnetic north pole to magnetic south (or geographical south to geographical north) as shown below.
Also shown is that the Earth's magnetic field forms an angle with the Earth's surface at each point. This angle is, for historical reasons, called 'the dip angle', because in the northern hemisphere, where Europeans live, the north seeking pole of a compass dips below the horizontal.
This dip has horizontal and vertical components. If the earth magnetic field at a point has magnitude 4*10^{-5} T and the angle of dip is 70 degrees, then the horizontal component is given by
and the vertical component is
We consider the horizontal component only when considering the voltage induced between the end of a falling conductor, and the vertical component only when considering the voltage induced between the wingtips of an aeroplane travelling horizontally.
Suppose then that a falling rod is oriented horizontally in an east west direction. It cuts the horizontal component of the Earth's magnetic field which points from south to north, at right angles. If the dip angle is
the length of the rod is
it's speed is
and the magnetic field of the Earth has magnitude
then the induced EMF has magnitude
Suppose that a plane is travelling horizontally from north to south (or vice versa). It cuts the vertical component of the Earth's magnetic field at right angles. If the dip angle isthe distance between the wingtips is
it's speed isand the magnetic field of the Earth has magnitude then the induced EMF has magnitude
