Magnetics

Any discussion of magnetism is complicated because the physical notation for magnetic fields is not standardized.

mks-SI system regards magnetism as an effect of electric currents

- where as -

cgs system begins with forces between poles

Coulomb has shown that the forces between poles obey an inverse distance squared law, i.e.

F p₁p₂/r²

or

F = _op₁p₂/4_rr²

where _o = 4 x 10^-7 Webers/amp-m in the mks-SI system and 1 in the cgs

• _o is the magnetic permeability of free space (i.e. vacuum)
• _r is the relative permeability of the medium

A unit pole strength is defined as the strength that repels an identical pole 1 cm (m) away with a force of 1 dyne (Newton) in free space (i.e. a vacuum). The force is attractive if poles are of opposite sign and repulsive if of like sign.

The magnetic field B due to a pole of strength p₁ at a distance r from the pole is defined as the force per unit positive pole at that point, i.e.

B = _op₁/4_rr²

Magnetic fields can also be defined in terms of a potential like gravity fields. Magnetic potential is defined as the work required to bring a unit pole from infinity to some point in space in the presence of a magnetic field and is given by

V = _op₁/4_rr

The magnetic field in any direction is then given by the partial derivative of the potential in that direction.

However, it is more convenient to think of the magnetic state of a body as resulting from elementary magnets or dipoles.

A dipole (or a bar magnet that acts as a dipole) consists of a positive (or north-seeking pole) and a negative (or a south seeking pole) pole separated by a "small" distance. A pole is where the magnetic effect is strong. A dipole generates its own magnetic field or flux and this field or flux can be mapped out from the directions assumed by a small compass needle (iron filings) suspended in the field.

Note the concentration of filings at the pole and the shape of the field lines. The North end of a compass will align itself with the field such that it points toward the South Pole of the magnet.

The strength of a dipole is termed the magnetic dipole moment, m (note this is a vector)

cgs units - gauss-cm³ (emu)

a bar magnet 1 cm long is said to have a strength of 1 gauss-cm³ if when place in a one gauss field a force of one dyne acting on each end of the dipole is required to keep the magnet perpendicular to the direction of the applied field

a current loop 1 meter in cross-sectional area having a current of 1 Ampere flowing thru it has a strength of 1 Am²

1 Am² = 10³ gauss-cm³

magnetic dipole moment (m) of the Earth is 8 x 10²² Am²

The magnetic potential of a dipole is given by:

V = (_om/4r²)cos

The magnetic field given by:

B = - V(r)

From the above, the radial and tangential components of the magnetic field are:

• B_r = 2(_om/4 r³) sin = 2(_om/4 r³) cos
• B = (_om/4 r³) cos = (_om/4 r³) sin

where is latitude and is co-latitude

The total intensity of the field at a given latitude (colatitude) is given by:

B = B_r² + B ² = (_om/4 r³)(4 cos²+ sin²)^1/2

and the inclination of the field at this location can be derived from the following:

tan I = 2 cot = 2 tan

In actuality, the magnetic field can be either represented by:

• H, the magnetic field intensity (A/m, mks-SI; orested, cgs)

- or -

• B, the magnetic induction (Teslas, mks-SI; gauss, cgs)

1 A/m = 4 x 10^-3 oersteds
1 Tesla = 10⁴ gauss

Magnetic induction (B) originates from all currents both at the microscopic (atomic) and macroscopic level and is considered the number of lines of force per cross-sectional area (i.e. flux density while magnetic field intensity (H) arises from only true currents.

B and H are useful when considering the field in the presence magnetic materials, but the magnetic field of the Earth is measured in a non-magnetic media such as air or water, therefore the equations for free space can be applied to relate B and H

mks-SI:
B = _oH where _o = 4 x 10^-7 Webers/amp-m
cgs:
B = H because _o = 1

when a magnetizable material is placed in a magnetic field it will become magnetized in the direction of the applied field. The intensity of this induced magnetization which we call the induced magnetization, Mi, (or magnetic dipole moment per unit volume ) is proportional to the strength of the field (and in the direction of the field) and will disappear when the field is removed.

Mi = induced magnetization, magnetic polarization, magnetic dipole moment per unit volume
(A/m - mks-SI; orested - cgs)

If Mi is constant and is in the same direction throughout the body, the body is said to be uniformly magnetized by induction.

Magnetic susceptibility

The degree to which a body becomes magnetized is determined by its magnetic susceptibility. This is the fundamental parameter in magnetic prospecting.

• Mi = kH where H has units of A/m (orested in cgs)

- or -

• Mi = kB/_o where B has units of tesla (gauss in cgs)

Susceptibility is both a measure of the ability of a rock to acquire a magnetization but is also a measure of the amount of magnetic material in a rock.
k_SI = 4k_cgs

B = _o(H + Mi) (mks-SI) -or- B = H + 4Mi (cgs)

B = H

where

= _o(1 + k) = _or (mks-SI)

where _r = 1 + k

= 1 + 4k (cgs)

Minerals that make up rocks are either diamagnetic, paramagnetic or ferromagnetic.

(Note: Quantum theory states that two electrons can occupy the same electron shell provided that their spins are in the opposite direction.)

• is associated with the orbital motion of electrons and dominates in an applied field when electrons are completely paired, i.e. filled electron shells
• effect is to oppose the external field disappears when field is removed
• common diamagnetic minerals: quartz, feldspars, salt
• k = -10^-6

• is associated with electron spin and dominates in an applied field when there are unpaired orbital electrons, i.e. unfilled electron shells
• spin moments align with the external field, otherwise randomly distributed
• minerals containing transitional elements are paramagnetic
• transitional elements = Fe, Cu, Zn, Ni, Mn, Cr, Ti
• k = 10^-4 - 10^-6

A paramagnetic material with high susceptibility is said to be a ferromagnet. In these materials the spin magnetic moments of the unpaired electrons between neighboring atoms are magnetically coupled this results in a spontaneous or permanent magnetization. Regions within a magnetic mineral or grain has the same direction of magnetization (i.e. alignment of moments occur) are called magnetic domains.

• ferromagnetic - spin moments parallel
• antiferromagnetic - spin moments equal and anti-parallel
• ferrimagnetic - spin moments anti-parallel but unequal

Domains are separated by domain walls therefore a single grain can be composed of a single domain (~ 1 m) to many domains

in larger grains the number of domains is a function of minimizing energy and is controlled by the shape of the grain, crystalline anisotropy of the grain, and by magnetostrictive forces (strain)

Minerals loose their spontaneous magnetization as they pass through their Curie temperature at which point thermal agitation exceeds magnetic ordering.

Ferromagnetic, ferrimagnetic, and antiferrimagnetic minerals show hysteresis, i.e. irreversibility of magnetic behavior with applied field.

The minerals that are largely responsible for the magnetic properties of rocks are within the ternary system FeO-TiO₂-Fe₂O₃, i.e. iron-titanium oxides.
Pyrrhotite (FeS_1+x, 0 < x <0.15) when present also contributes.

Some rocks have a permanent magnetization in addition to an induced magnetization. Remanent magnetization is a primary magnetization acquired at or shortly after the rock formed. The direction of this magnetization, Mr, may not necessary be in the direction of the induced vector.

The ratio between Mr and Mi is called the Konigsberger ratio, Q. For rocks where Q < .2 we ignore Jr.

Plutonic --> course grained metamorphic rocks: .1 < Q < 10

Volcanic --> fine grained metamorphic rocks:1 < Q < 100

Types of remanent magnetization

1. Thermoremanent (TRM)
2. Depositional (DRM)
3. Chemical (CRM)
4. Viscous remanent magnetization (VRM)

The magnetic field at any point on the Earth's surface can be described by 3 vectors and 2 angles.

The 3 vectors are:

1. Horizontal field (H)
2. Vertical field (Z)
3. Total field (T)

The 2 angles are:

1. Declination (D)
2. Inclination (I)

Equations relating these 3 vectors and 2 angles include:

H = T cosI
Z = T sinI = H tanI
X = H cosD
Y = H sinD

X² + Y² = H²
X² + Y² + Z² = T² = H² + Z²

Isomagnetic maps: contour maps of equal

• declination -> isogonic
• inclination -> isoclinic
• magnetic field value -> isodynamic

Magnetic dip poles -> locations on the Earth's surface where I = ±90^o

The magnetic dip poles presently located at:

• 75^oN, 101^oW (North magnetic pole)
• 67^oS, 143^oE (South magnetic pole)

Geomagnetic poles -> location on the Earth's surface where the poles of a theoretical dipole (a mathematical best fit solution - IGRF) would be located

The geomagnetic poles are presently located at:

• 78.5^oN, 70^oW (North geomagnetic pole)
• 78.5^oS, 110^oE (South geomagnetic pole)

Time perturbations of Earth's magnetic field:

1. Long term changes -> reversals
2. Secular changes -> greater that one year
   Shown below is the historic record of geomagnetic field direction at Greenwich, England.


3. Diurnal variations -> daily
   associated with ionization of the upper atmosphere by solar radiation in sympathy with tidal effects of the Earth and moon;
   variation is generally 20 - 80 nT.
4. Magnetic storms ->
   rapid and violent changes associated with sunspot activity;
   can exceed a 1000 nT of change over 24 hrs.

1. Diurnal variation correction

• assume linear rate of change
• use recording base station magnetometer
• algebraic difference between "t" and "t" = 0 is the correction

2. Temperature correction

• older instruments using permanent magnets were temperature dependent
• most modern instruments are temperature compensated and therefore correction is negligible
• changes due to temperature are generally incorporated into the diurnal variation correction

3. Terrain correction

• topography may give rise to magnetic anomalies
• no general rules for terrain correction
• usually anomalies showing strong correlation with topography are regarded as less significant

4. Normal correction

• removal of International Geomagnetic Reference Field (IGRF)

- or -

• setting base equal to zero and removing this value from the data; this is equivalent to removing IGRF and brings all base readings to a common value

Rules for drawing (sketching) D Z and D H anomalies

D Z = Z - Z_o
D H = H - H_o

1. If the vertical component of the magnetic field of the anomalous body reinforces the vertical component of the Earth’s magnetic field along the profile line, then D Z at that location has a positive value.
2. If the vertical component of the magnetic field of the anomalous body opposes the vertical component of the Earth’s magnetic field along the profile line, then D Z at that location has a negative value.
3. If the magnetic field of the anomalous body at a given location along the profile line is horizontal, then D Z is zero at that location.
4. If the magnetic field of the anomalous body at a given location along the profile line is vertical then D Z is a maximum (positive or negative depending on whether the magnetic field reinforces or opposes the Earth’s field) at that location.

1. If the horizontal component of the magnetic field of the anomalous body reinforces the horizontal component of the Earth’s magnetic field along the profile line, then D H at that location has a positive value.
2. If the horizontal component of the magnetic field of the anomalous body opposes the horizontal component of the Earth’s magnetic field along the profile line, then D H at that location has a negative value.
3. If the magnetic field of the anomalous body at a given location along the profile line is vertical, then D H is zero at that location.
4. If the magnetic field of the anomalous body at a given location along the profile line is horizontal then D H is a maximum (positive or negative depending on whether the magnetic field reinforces or opposes the Earth’s field) at that location.