PiMath.de The magnetic field of the earth
Lattice structures of the earth magnetic field
 
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1 - The dipole field of the earth

1.1 - Current loop and dipole

 dipole field of a current loop

  From the school, and from the media, we know the magnetic field of the earth always as a field which corresponds to the field of a rod magnet. It is the so-called dipole field.

The physical attempt for such a magnetic field consists in the consideration of the magnetic field of a current loop.

The mathematical derivation leads to a differential formula in which a so-called elliptical integral appears for which no closed mathematical solution exists - in form of an equation.
Illustration 1.1 - Current loop and dipole    

 

The general proceeding consists in converting the appearing term in the integral into an infinite row.

Simplified:
B = a1·x + a2·x2 + a3·x3 + a4·x4 + ...


Then one goes and simply cuts off this row after the first limb. If one integrates now the left-over, the formula for the dipole field appears.:

 

 formula for the dipole field

 

In the equation stands B for the magnetic flux density, φ (phi) for the latitude, m for the magnetic moment, r for the earth radius and μ for the magnetic permeability.

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

In this case the magnetic moment is the magnetic moment of the earth with m = 6,6845·1022 Am2
One finds here also the value m = 8·1022 Am2

For the earth radius one takes the value from a geodetic system, in this case the WGS84 with: r = 6378155 m

The magnetic permeability μ = 10-7 Vs/Am

(For the derivation of the dipole field see also "Berkeley Physik Kurs 2" from Edward Purcell, page 266-269)

 

 

1.2 - The coordinates for the dipole axis

It is distinguished between the magnetic poles and the geomagnetic poles. The magnetic poles are the places in those the magnetic field vertically stands on the earth surface. The geomagnetic poles are the poles of the dipole field. In the geomagnetic poles the axis of the approximated dipole field cuts the earth surface.
First the dipole axis, so the geomagnetic poles, should be looked here. In older physics books you find often following values for the dipole axis (state in 1980):

 

Name Latitude Longitude
Dipole axis - North +78,8 degrees North -70,9 degrees West
Dipole axis - South -78,8 degrees South +109,1 degrees East

 

For the dipole axis one receives, in the popular-scientific literature, also the coordinates 79 degrees of north in latitude and 70 degrees of west in longitude.
 
In the book the at the moment counting dipole model which is based on the IGRF 1995 is still treated.

 

The dipole axis has tipped over, in comparison to physical axis of rotation, possibly about 11 degrees. This angle must be still taken into consideration in the above equation. If one uses the values of the constants, the following final formula arises for the magnetic flux density which is dependent practically only on the latitude:

 

 Die korrigierte Gleichung für das Dipolfeld μT  (mükroTesla)

 

Remark: 1 μT = 1000 γ    (1 mükroTesla = 1000 Gamma)

 

The mathematical approach shown here for reaching the dipole equation can be looked on account of the cut-off rest limbs, merely as a first approximation. Particularly as still to be seen in the book, that the dipole model is not constant in the course of time, in its spatial position, and must be adapted.
If one starts integrating the remaining limbs of the infinite row (from the current loop consideration), one receives the quadrupol field, the octupol field etc.

 

 Multipole development for a magnetic field
 
Illustration 1.2 - The multi- pole development for the magnetic field

 

By changing the terms in a row the whole process is an approximative, the solution is only one approximation. In addition, is always to be considered that the described approach is a purely mathematical one. Besides, the question by physical relevance stays open !!!
If one takes actual values of the field, there can be shown, with the magnetic poles, that the dipole model is not enough to explain the earth magnetic field.

 

 

1.3 - The declension of the earth-magnetic field

The field vector B of the earth magnetic field is described by three dimensions independent of each other which are called elements of the magnetic earth field. There exist two different systems of representation. Here only the system from Illustration.1.3 is shown and explained because of simplicity.
 
 Field elements of the earth magnetic field
Illustration 1.3 - Field elements of the magnetic field
 
The horizontal angle between the direction from magnetic pole to the geographic north pole is called declension D or also horizontal magnetic variation. The horizontal direction to the magnetic poles can be determined directly with a compass.

For the vertical deviation of the magnetic field vector of the parallel orientation to the earth surface the concept inclination I is used. If one so twists the compass so that its axis lies horizontal, we have an inclinatorium before ourselves - it misses the angle between the earth surface and the magnetic field.
The fact that there is this angle and that it is depend of the measuring place is known for over 400 years. The fact that in the pole the magnetic field stands vertically on the earth surface and in the equator, however, in parallel with the earth surface has led with to the dipole model.

And the vertical component of the magnetic field vector is called vertical intensity Z.

(see also " Grundlagen der Geophysik " from Hans Berckhemer, page 138)
 
If one takes the "International Geomagnetic Reference Field" briefly IGRF of 1980 for the declension so the following illustration 1.4 arises (to the IGRF see also chapter 2.2)

 

 Declension of the earth magnetic field   By analysis of the declension, so of the horizontal false variation of the earth field, two areas can be won. Since the south and the north-magnetic pole, the places in those the magnetic field vertically on the earth surface stands and, hence, exists here no horizontal variation. These both areas define the magnetic poles.

The inclination can be pulled up alternatively for the declension of course also to the pole determination.
Illustration 1.4 - The declension of the earth magnetic field - IGRF 1980    

 

 

1.4 - The determination of the coordinates for the magnetic poles (1980)

The coordinates of the real poles can be found, while one determines the edges of the pole areas from illustration 1.4 and forms from it the averages:

 

Name Latitude Longitude
South-magnetic pole +75 degrees North -103,5 degrees West
North-magnetic pole -65,5 degrees South +141 degrees East

 

However, in physics books or in the popular-scientific literature one also finds the following coordinates:

 

Name Latitude Longitude
South-magnetic pole +75 degrees North -101 degrees West
North-magnetic pole -67 degrees South +143 degrees East

 

If one summarises both results, the areas in which the magnetic poles lie nowadays arise:

The coordinates for the magnetic pole areas

Name Latitude Longitude
South-magnetic pole 75 degrees North 101-103,5 degrees West
North-magnetic pole 65,5-67 degrees South 141-143 degrees East

 

In the book the at the moment counting pole areas (2005) are still treated.

 

 

1.5 - The consequence

If one compares the found areas, from chapter 1.4 with the coordinates of the dipole axis from chapter 1.2, one recognises:

1) The pole areas do not subtend themselves as one would expect it after the dipole theory.

2) The available pole areas do not lie yet near the dipole axis.

 

From it two possible conclusions can be drawn:

1) other influences must be here which disturb or distort the dipole field. The dipole model shows therefore only one partial view of the complete earth-magnetic field. The spatial change of the dipole axis in the course of time likewise speaks for it.

2) the dipole model does not fit here and one would have to keep a lookout after a new approach.

The dipole model also does not last around to describe, the complete magnetic field of the earth with its four extreme areas in the next chapter. The result is:

 

The dipole model only is not sufficient,
to be able to explain the actual earth magnetic field.

 

And with the multi-pole development of the earth field the question by physical relevance stays open !!!

Remark:
Already Gauß and Weber recognised in 1838, by their experiments with the earth magnetic field that the magnetic field cannot be simply explained by the model of a stick magnet or a current loop. (see in addition also
chapter 2.7)

 

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