The question is related to the conversion of magnetic energy into electrical one and whether the process can be achieved in places such as the Van Allen radiation belt.
So, lets us take this by parts: First the magnetic to electric energy conversion. Well, according to the first law of thermodynamics energy cannot be “created or destroyed”, but we can indeed convert it from one from to another one. It turns out that we can use some kinetic energy to move, say, a magnet. In turn this kinetic energy can be converted to electrical energy thanks to the properties of electromagnetism, in particular to the so-called Faraday’s law. Faraday discovered that, when moving a permanent magnet into and out of a coil of wire, an electrical current was induced in the wire while the magnet was in motion.
Now, to the Van Allen radiation belt: the belt is part of the Earth’s magnetosphere. Ok, ok… The magnetosphere is the part of space near a celestial object in which charged particles are controlled by the magnetic field generated by the object itself. So the Van Allen belts extend from an altitude of about 1,000 to 60,000 kilometers above the surface in which region radiation levels vary. In order to convert magnetic energy to electrical, as mentioned above, we requiere the magnetic field to be in movement or vary. It is generally accepted that in that context, the Earth is effectively a permanent magnet and thus to generate electric power from that, you have to move electric conductors (wires) thought the field in the right direction and with the right orientation of the conductor. Not an easy task…
However, one can perhaps take advantage of the variations of the magnetic field. In Nature 439, 799-801 (16 February 2006) it has been reported that
“… Earth’s magnetic field is weak: it varies from about 25 microtesla (T) at the Equator to 75 T at the poles, with geomagnetic field lines inclined, in Europe and North America, at an angle of about 60° to the (horizontal) surface. The field is not constant: currents in the ionosphere and disturbances from Earth’s interior produce slow daily variations in the field with amplitudes of some 25 nanotesla (nT), and superimposed on these are further oscillations with periods of a few seconds and amplitudes of about 1 nT.”
Using the very crude approximation that there are variations of 1nT per second, and take a circular area of with radius of 1 metre we would end up with a voltage of Volts or approximately 3.1415 nano volts. Or in other terms we would get about 3 one-billionth’s of a volt per square meter of flux… probably not a lot of usable energy and thus maybe not that cost effective.