Magnetic units, and the UBC-GIF magnetic dipole applet

• m = moment
• r = distance from dipole, and "r^hat" is a unit vector in the radial direction
• θ = angular distance (in polar coordinates), and "θ^hat" is a unit vector in the angular direction.
• u = magnetic permeability of free space

The magnetic flux density (B in Tesla) for a dipole when working in polar coordinates is

.

The corresponding magnetic field intensity (H in Amps/metre) for a dipole is

.

The equation for B gives the magnetic flux density at position r, due to a dipolar magnetic source that has magnetic moment, m. The 4 is an inherent part of the relation when it is derived within the SI system of units. B is in Teslas and m is in Am². The magnetization of a body is m per unit volume, giving units of A/m.

Note that the 4 that appears when converting between emu and SI units is not part of the above equation. This equation is derived by considering the net effect of two magnetic poles of equal but opposite strength, or equivalently, by considering the net effect of an elemental circular current loop.

The important point is that the "amplitude" of the flux density in Tesla is like , multiplied by whatever the size of the second term happens to be for the particular location in the coordinate system.

Try this with the UBC-GIF dipole applet:

As a simple example, consider the field strength 2m away from a dipole of magnetic moment, 10Am², when inclination and declination are both 0 degrees (so the cos + sin term is 1): H = 10/(4×2³) = 0.0995Am².

Since B = u₀H, and u₀ = 4 × 10^-7 = 1257 × 10^-9, H = 0.0995Am² gives B = 125 nT (in air or free space). (The value will, in fact, be negative at the magnetic equator.)

Note that the UBC-GIF dipole applet is only calculating the field at discrete points (hence, the blocky appearance). So the results of calculating the field above a dipole will not be quite correct unless the distance above the measurement location is more than 5 or so units above the dipole. Try, for example, setting depth to 19 so the total distance between dipole and measurement is 20. The maximum anomaly of 0.125 nT is more accurately obtained. Unfortunately, you can't set the data spacing to less than one with this applet. In general, it is good practice to avoid asking numerical codes for answers when you use values that are similar to the discretization intervals.

References

For references regarding magentics and magnetic units, your best bet is to refer to a standard text on geophysical magnetics. The first two books listed below are available via amazon.com. There are also many websites put up by schools and colleges with courses in physics and electromagnetics. Search using, for example, "magnetic AND flux AND field AND units".

• One highly recommended text is "Potential Theory in Gravity and Magnetic Applications" (R. Blakely, Cambridge University Press).
• Another that might help is "Applied Geophysics 2nd edition" (Telford, Geldart and Sherrif , Cambridge University Press 1990).
• The "Encyclopedic Dictionary of Exploration Geophysics" (RE Sheriff) is available from the SEG (http://seg.org).