Magnetic field methods

• As for gravity methods, magnetic field is natural, responds to the presence of variable rock types

• Gravity responds to density, magnetic field responds to magnetization, in turn related to magnetic “susceptibility”

• Susceptibility is largely governed by a small number of minerals (magnetite, hematite, pyrrhotite)

• Unlike gravity, the magnetic field can easily be measured from land, sea and air, and there is no need for detailed elevation corrections

• Typically

an easier deisgn criteria than that for gravity instruments (1/108).

Magnetic field methods

Magnetic field methods

• Unlike gravity, the interpretation of magnetic field is more difficult, due to the variable inclination and declination of the earth’s field

Remember this convention:

• Field lines converge on the earth’s magnetic North pole,

• BUT: field lines emerge from the north pole of magnetized minerals (i.e, from a compass, from a magnetized ore body, etc)

Basic magnetic field theory

• In Geol 319/829 we will use the SI conventions for units and equations

• Be aware that other conventions (especially the cgs system are also popular and appear in some textbooks)

Magnetic induction (aka “flux density”, “magnetic field”)

The equation defines magnetic induction in terms of forces acting on moving charges

Basic magnetic field theory

Magnetic induction (aka “flux density”, “magnetic field”)

Units:

Force is measured in N

Units of charge are measured in Coulombs

Therefore, units of B are [N C-1 m-1 s], known as a “Tesla” [T]

• The Tesla is a very large unit – strongest industrial/medical electromagnets ~ 2-3 T• In geophysics we use a nT=10-9 T. The earth’s field near Kingston is approximately

50x10-6 T, or 50,000 nT

Basic magnetic field theory

• Magnetic fields occur naturally (the earth’s field, the field associated with magnetized minerals)

• Fields are also observed in the vicinity of a current carrying wire

Basic magnetic field theory

left

Basic magnetic field theory

Note the similarity to a “dipole field” from a bar magnet.

• in nature all magnetic fields arise in this way, i.e., from the movement of electric charge

• examples are dipole magnetic moment of individual atoms, magnetization of rocks and minerals, even the earth’s magnetic field

Magnetic dipoles

• Dipolar fields are fundamental to understanding magnetic anomalies

• As in gravity, the use of potentials is helpful – it helps us to avoid using vectors, which are more complicated

Magnetic potential

• in gravity we derived the potential from the work done by a point mass

• in magnetics, poles always come in pairs (dipoles)

• we can nevertheless use a magnetic monopole in the following mathematic trick …

Magnetic dipoles

Derivation of the dipole field uses the following mathematical trick:

• assume magnetic monopoles exist

• put two monopoles (of opposite polarity) together to form a dipole

• calculate the potential for the dipole by summing the two monopole

potentials

• find the field by taking the gradient of the potential

Derivation of dipole field

Begin with the magnetic monopole potential:

with

Derivation of dipole field

Create a dipole:

Note: the minus sign appears because of the opposite polarities

Now, obtain the dipole field by

1. Rewriting the equation in terms of (r, θ)

2. Taking the gradient to find:

The result will have two components:

Clarification to Geol 319 Notes(Page 59)

Sherriff and Geldart (1990), and Geol 249 notes: Reverse these signs – the final formulas are then (slightly) different.

Derivation of dipole field

Derivation of dipole field

Make (repeated) use of the “far field” approximation:

Derivation of dipole field

Make (repeated) use of the binomial series:

Derivation of dipole field

We have

and

Thus

Derivation of dipole field

Derivation of dipole field

where m=ql is the “dipole moment”

The next step is to take the gradient of the potential:

Derivation of dipole field

Next lecture: Dipole field

• Emerges from North poles, converges on South poles

• Falls off at a high 1/r3 rate of decay

• Thus we expect that magnetic anomalies are much more localized

• The field anomaly responds mainly to shallow targets