Answer to the Question 02/04

FROZEN EARTH

The question was:

What would happen to the magnetic field of Earth if the planet suddenly freezes? (What will be the eventual value of the magnetic field and how long will it take to get there?)

(9/05) Y. Kantor: Although the solution of this problem is rather straightforward, we did not recieve satisfactory solutions up to this point in time. Thus, we decided to publish our "standard" solution of the problem. Some interesting insights into this and relate problems can be found in the lecture notes of Kirk T. McDonald from Princeton University, NJ, USA. The following PDF file contains one of his lectures.


The answer: The magnetic field will decay to negligible values in about a million years.


The solution:
In hot Earth the "dynamo effect" maintains the magnetic field. In a "frozen Earth" the situation becomes much simpler. The current can be caused only by an electric field. In a conductor these two are related via Ohm's law
J=sE,
where s is the conductivity. The current creates the magnetic field: From the Ampere's law (at slowly varying fields) we have (in MKSA (SI) units):
curl B=moJ,
where mo is the permeability of free space. Since the current is changing in time, the magnetic field is also changing. This in turn, generates an electric filed: From Faraday's law we have
curl E=-dB/dt,
where d represents partial derivative. By taking another curl of Ampere's law, and using the fact that curl(curl=-Δ+grad(div, where Δ denotes Laplacian, as well as the fact that div B=0, we find that
ΔB=mos dB/dt.
This diffusion equation demonstrates that the initial magnetic field will diffuse-out/decay in Earth. The exact solution of the equation depends on the detailed initial conditions. However, from simple dimensional analysis we see that the time scale T over which the field decays is related to the length scale L where this field exists. Taking L=10000km (order of magnitude of Earth's diameter), and s=105mho/m (i.e. 0.01 of the conductivity of pure iron), we find that
T=L2mos=0.4 million years
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