a "4-pole" decreasing like 1/r4, plus an 8-pole decreasing like 1/r5, and so forth. The field of an isolated "monopole" would presumable decrease like 1/r2, the way gravity does--but no such single pole was ever observed, they always come (at the very least) in pairs.
The new tools for better observation and description of the Earth's magnetic field led to better, world-wide observations. Gauss and Weber organized a "Magnetic Union" for setting up observatories, and Humboldt enlisted Russia's Czar to create a chain of them across Siberia. The greatest help however came from the British empire, whose "Magnetic Crusade" led by Sir Edward Sabine set up stations from Canada to Tasmania (then known as "Van Diemen's Land"). The vast network not only made possible the first global models of the field, but also demonstrated the world-wide character of magnetic storms.
One can compare today's magnetic models, some of them based on satellite observations, to the ones started by Gauss more than 150 years ago. One trend then stands out: the dominant "dipole" field is getting weaker, at about 5% per century (the rate might have increased since 1970). In the unlikely event that the trend continues unchanged, about 1500-2000 years from now the magnetic polarity of the Earth would reverse.
However, the magnetic field of Earth is not likely to disappear, because the high electrical conductivity of the core demands that the magnetic flux entering the core can only change very, very slowly (more about "flux" below). As the late Ned Benton has shown (together with Coerte Voorhies), even though such conservation does not allow flux entering the core to disappear, it may be redistributed--perhaps replacing the big bundle of the dipole with a number of more complex ones (4-pole, 8-pole, etc.). Such replacement yields weaker fields on the surface, and meanwhile the dominant "dipole" pattern may (sometimes) reverse, later growing again to dominate the pattern but (possibly) with the main magnetic poles reversed.
In the mathematical description of magnetic fields, the magnetic flux through some given area is an important concept. The prescription for deriving the flux entering the area--here, core surface--is
(1) divide the area into patches--those where field lines enter and those where where they leave.
(2) Ignore the latter areas. Divide the area where the lines enter into small sub-areas.
(3) Multiply each sub-area by the component of the magnetic vector perpendicular to the surface, and
(4) Add all products together. What you get is the magnetic flux entering the core.
If you had done the same with areas where field lines exit from the core, it can be shown that you would have got the same result, but with minus sign, denoting flux leaving instead of entering. If you ignore the sign and mark every contribution as positive, you get exactly the same flux. Calculating the "total unsigned flux" at the core surface gives the sum of the two--the "total unsigned flux," equal to twice the entering magnetic flux, and therefore, also (very very nearly) unchanged with time. The "total unsigned flux" is however easier to derive (which is why geomagneticians prefer working with it), because you no longer need to divide the surface into irregular patches of incoming and outgoing field lines--every bit of area is used in the calculation.
"Carl Friedrich Gauss, Titan of Science"
This definitive biography of Gauss by Waldo G Dunnington, 479 pp., was first published in 1955.
Republished in 2004 by the Mathematical Association of America, with an addition by Jeremy Gray as well as a contribution by Fritz-Egbert Dohse, bringing the page count to 586.
Tidbit. Gauss had 6 children, and he quarreled with one son over the cost of a party which Gauss refused to pay. In protest, the son left Germany in 1830 and joined the US army. After ending his service, he settled in Missouri and was joined there by a brother; as a result, many of Gauss'es descendants today live in the US.
Next Stop: 11. The Sun's magnetic cycle
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