PiMath.de Planetary Systems of the Earth 1
Classisc Systems
  Copyright © Klaus Piontzik  
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4.0 - Earth magnetic field

dipole field

  From the school and from the media, we always know the earth's magnetic field as a field that corresponds to the field of a bar magnet. It is due to the so called dipole field.
Historical this view of the earth's magnetic field explains the behavior of a needle of in-clination. For historical reasons, this view of the magnetic field explains the behavior of an inclination needle. The needle is perpendicular to the earth's surface at the pole and horizontal to the earth's surface at the equator.
Illustration 4.0.1 – dipole field    


Taking measured values of the field, ít can be shown graphically that four magnetic poles (Illustration 3.4.2) exist and that the dipole model is not sufficient to explain exactly the real earth's magnetic field.


 current loop

  The physical approach for such a dipole field is the con-templation of the magnetic field of a so called current loop.
The mathematical derivation leads to a differential equation in which a so-called elliptic integral occurs, for which no closed mathematical solution - in the form of an equation - exists.
Illustration 4.0.2 – current loop    


The mathematically commonly used approach is converting the appearing term in the integral in an infinite sequence:

Simplified can be written:

infinite series

Then just cut off this sequences after the first link. Now integrate the rest, so the general equation for the dipole field, which only depends on the geographical latitude φ is created.


4.0.1 - Equation: Equation for the dipole field µ T


In the equation stands:
B for the magnetic flux density
φ for the latitude
m for the magnetic moment
r for the Earth's radius
µ for die magnetic permeability

m, r, µ are constants that are defined as follows:

The magnetic moment in this case is the earth's magnetic moment with m = 6.6845*1022 Am2
One also finds the value m = 8*1022 Am2

For the earth's radius you take the value from a geodetic system, in this case the WGS84
which is given as: r = 6378155 m

The magnetic permeability µ = 10-7 Vs/Am

(For the derivation of the dipole field see also "Berkeley Physics Course 2" by Edward Purcell, pages 266-269)


The mathematical approach presented here to the achievement of the dipole equation can be considered due to the cut off limbs of the rest, merely as a first approximation.

If you start to integrate the remaining limbs of the infinite sequence (from the current loop consideration), so you get the quadrupole field, the Octupole field, etc. Overall, this is called multi pole forming.

 multi pole forming

Illustration 4.0.3 – multi pole forming
See "Grid Structures of the Earth's Magnetic Field," Chapter 1, The Earth's Dipole Field, pages 17-23.

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Der Autor - Klaus Piontzik