On a web forum I frequent, a person asked if it would be possible to extract energy from the Earth’s magnetic field. He was told no – static magnetic fields can’t transfer energy. For all practical purposes this is true, but in fact we also know that the earth’s magnetic field isn’t static. It changes from day to day and from year to year – and even second to second. The changes are small over small timescales, but in fact the magnetic poles do drift around and the solar wind does perturb the fields and so forth.
Wikipedia gives a reference saying that typical local variations in the magnetic field at the surface are of the order of 1 nanotesla per second. This is pretty small compared to the total field of perhaps 50 microtesla, but it is measurable with sensitive equipment. Can we extract that energy and free ourselves from reliance on coal and oil? We expect the answer is “no” because otherwise someone would have done it, but we can crunch the numbers to make sure. First, Faraday’s law:
Looks bad, but it’s not. In this simple physical situation the calculation above will only involve multiplication.
The right hand side says “Make a closed shape out of a bent wire. A square, a circle, a heart, whatever. Now hold it in place and look at the local magnetic field as it passes through the loop. Take that total magnetic flux and look at the rate at which it’s changing with time.”
The left hand side is just the total potential difference in volts that each electron gains after making one circuit of that loop. It’s that number which will tell us something about how useful this might be as a power source.
Magnetic flux is just the magnetic field multiplied by the area of the loop, assuming the field is perpendicular to and uniform within the loop. We’re interested in the time rate of change of this flux, and let’s say we have a circle with a diameter of 1 meter. The rate of change of the flux is thus (1 nanotesla/second)*(3.14 meters^2).
Which is 3.14×10^-9 volts. Three one-billionths of a volt per square meter of flux-collecting surface. If you tried very hard you might be able to finagle some useful energy out of such a small potential, perhaps with very long superconducting solenoids. But it would be less cost-effective than pretty much any other form of renewable energy by many orders of magnitude.
Still, it was worth a try!