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A Study of Geomagnetism using Scilab and IBM Data Explorer

A friend I walk with is keen on micronavigation his skill with map and compass is a good match for anyone wielding a GPS. This dependency on the earths magnetic field and our need to "correct for magnetic variation" makes me inquisitive about the nature of the earths magnetic field. As illustrated by the links below it comes as no surprise that a vast amount of intellectual effort has been expended in understanding geomagnetism since the early days of William Gilbert (1600) [2] upto modern day MHD computations modelling the geodynamo [1].

I have investigated a simple model of the earths magnetic field based on the idea that the earths magnetic field may be represented by a single current loop. We use numerical integration with the Biot-Savart law.

Radius of the current loop is 4000km
Loop carries a current of 1500MA

The following scilab script computes the magnetic field due to a current loop over a 22x22x22 region of total size 11R_e where R_e is the radius of the earth (R_e=6371km).Such a crrent loop is illustrated below.

For each computed field point the Biot Savart law is used to compute the field due to the loop, the numerical integration over the current elements is performed using Simpsons rule. The scilab script to perform this computation is linked below. The script requires a function routine to compute the simpson rule integration, these are contained in the zip file geomagresources.zip. The resources file also contains the net file used to visualise the results with IBM data explorer.

Results calculated using the scilab model and using the bfield2.net data explorer network are shown below.

Data explorer output

The scilab script file generates general data scriptions that may be read by IBM data explorer. Using IBM data explorer to run the visual program bfield2.net we may view streamlines representing the magnetic field lines and indivdual magnetic field vectors at each spatial location. The visualisation has a control panel labelled controls that may be used to explore these different aspects of the data set. The results have been compared to the results generated using geomagnetic models provided by the geophysical data centre [7] and are representative within at least an order of magnitude. MHD models of the geodynamo enable an understanding of palaeomagnetism and the evolution of the earths mgnetic field. Results calculated using the MHD geomagnetic dynamo are shown below [1].

In 1838 the German mathematician and magnetician Frederick Gauss developed a method of representing the magnetic field in terms of a converging series of spherical harmonics, whose terms were functions of latitude, longitude and radial distance from the centre of the earth [5]. There exist a wide range of techniques for the computation of magnetic fields, this is important in a wide range of medical, scientific and technological disciplines. An interesting method is one that uses an expansion of spherical harmonics in reciprocal space [9].

In the final blog entry in this series of three we will investigate charged particle motion in the earths magnetic field.

Links

Keywords: data explorer, geomagnetism, math, plasma, scilab

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Mike Griffiths

Using Scilab/Matlab to Model Particle Motion Under the Lorentz Force

This is the first of three articles about charged particle motion in electromagentics fields and the earths magnetic field. The articles will include simple demonstrations built using the matlab clone scilab and the visualisation tool IBM Open data explorer, both of these are open source applications. The first article will describe a simple scilab based application for modelling charged particle motion. The Lorentz force can be used to model a wide range of systems and phenomena including

Motion of particles in colliders and their detectors e.g. the CERN Large Hadron Collider
Understanding the solar interior and atmosphere
Understanding the charge particles in the ionosphere e.g. the borealis
Confinement of plasmas for experimental fusion reactors
Focusing of beams for electron microscopy

The Lorentz force is the force on point charges due to electric and magnetic fields the elctric field gives rise to a linearly increasing force relationship between the charge and the elctric field intensity. The force generated through the magnetic field is such that it is perpendicular to the plane formed by the particle velocity vector and the magnetic field. This explains the action of the vecotor cross product term.

The cross product term can be undersood from the relativistic nature of the elctromagnetic interaction. It is important to remember that the relativistically covariant Maxwell equations and the special theory of relativity enable us to understand the unified nature of the single electromagnetic interaction. When the lorentz transformations are applied to the electric field we have a cross product relationship between the velocity of the particle and the magnetic field it therfore appears that the magnetic interaction is generated by a relativistic effect, an article in the links provides a good description.

The scilab script uses the lorentz force to update the position of a particle in a constant and uniform electromagnetic field. The equations of motion are solved using a simple Euler integration step. When executed, the script starts a number of dialogs in turn requesting the user to

Define the initial velocity
Define the b field
Define the e field
Provide a tile for the plot
The plost is drawn
The user is asked if they want to save the plot, if yes and OK are clicked a file save dialog opens.

The particle mass and charge are hard coded at the start of the script but can be altered if the user requires. Not surprisingly there is a lot of information about the lorentz force and charged particle motion, including some interesting video content on utube, one such link in the useful links section below

Useful Links

Scilab script requires the lorentz force function file

Matlab script requires the lorentz force function file

Scilab home page

IBM Open data explorer

Wikipedia on Lorentz force

Wikibook on the lorentz force

Science world info about the lorentz force

Article about relativistic transformation of electromagnetic fields

Utube demonstration of Lorentz force

Keywords: lorentz, matlab, scilab

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