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Earth’s magnetic field is mostly caused by electric currents in the liquid outer core, which is composed of conductive, molten iron. Loops of currents in the constantly moving, liquid iron create magnetic fields.

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However , we can say that the Earth's magnetic field obeys the magnetic dipole equation .

The dipole model of the Earth's magnetic field is a first order approximation of the rather complex true Earth's magnetic field. Due to effects of the interplanetary magnetic field, and the solar wind, the dipole model is particularly inaccurate at high L-shells (e.g., above L=3), but may be a good approximation for lower L-shells. For more precise work, or for any work at higher L-shells, a more accurate model that incorporates solar effects, such as the Tsyganenko magnetic field model, is recommended.

Why all this huge random liquid iron movements seem to generate a magnetic field similar with a magnetic dipole , for low L shells ( below L=3 )? Is just an empirical coincidence ?

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The magnetic field $\vec B=\nabla \times \vec A$ of any localized current $\vec J(\vec r)$ distribution can be approximated by a multi-pole expansion of its vector potential $\vec A$. The lowest order term of this multi-pole expansion gives you a magnetic dipole field $$\vec A \approx \frac {\mu_0}{4\pi}\frac{\vec m \times \vec r}{|\vec r|^3}$$ where $$\vec m=\frac {1}{2} \int_V \vec r' \times \vec J(\vec r')d^3r'$$ is the magnetic dipole moment. Thus the first approximation of earth's magnetic field is a magnetic dipole field.

See, e.g. Jackson, Classical Electrodynamics, 3rd ed., sec. 5.6 Magnetic field of a localized current distribution, Magnetic moment.

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  • $\begingroup$ This is only part of the answer. Being the lowest order term does not mean that the dipole moment is the largest. However, in the expansion of a magnetic field, the dipole term drops off as $1/r^3$, the quadrupole term as $1/r^4$ and so on. The surface of the Earth is at about twice the radius of the Earth's core, so compared to the dipole term, the quadrupole term is half as strong as it is at the surface of the core. See en.wikipedia.org/wiki/…. $\endgroup$
    – A. Newell
    Commented Mar 27, 2018 at 5:09
  • $\begingroup$ @A.Newell - This is a good additional point. The coefficient of the quadrupole term would probably also be important. $\endgroup$
    – freecharly
    Commented Mar 27, 2018 at 13:42
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The generation of the magnetic field in the core is not fully understood, not least because the physics involved (magnetohydrodynamics) is exceedingly complicated.

But we can make some heuristic observations. If you consider two different flowing loops in the core then they both generate magnetic dipoles and they will interact with each other through their magnetic fields. So you cannot consider the core as an assemblage of independent flows. The flows all generate magnetic fields and therefore all interact with each other (now you see why it gets complicated :-).

The overall dipolar nature of the field suggests that the interactions between the flows cause them to line up so their dipoles point in the same direction. I'm not sure to what extent this is obviously the lowest energy configuration for the flow to adopt, but it makes at least qualitative sense that the interactions between individual flows would tend to align them.

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