relation between earth's magnetic field and magnetic moment
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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.:
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
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.:
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
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