1. Outline Magnetic dipole moment Magnetization Magnetic induction
Dipoles and monopoles
Magnetization
Magnetic susceptibility and permeability
Magnetic induction
Poisson’s equation and potential for magnetic field
Potential fields in practice:
Vertical component of gravity
Each of the vertical components of magnetic field
Total field

2. Magnetic dipole moment
There exist no magnetic “charges” or ”monopoles”
…but for static fields, they can be constructed mathematically
The elementary magnetic source is the dipole formed by a loop of current:

3. Magnetic Potential or: Consider a shift Dr of the observation point
If curl (B) = 0 (no current at the observation point), field B can be presented as a gradient of a potential V:
or:
Biot-Savart law:
is a unit vector
along r
Cm depends on the unit system:
emu: Dimensionless Cm = 1
SI: Cm = m0/4p = 10-7 H/m
“Henry”, unit of inductance

4. Magnetic Potential Therefore, the change in the potential:
The first equation here uses:
The second equation assumes an infinitesimal loop and uses
The contour integral gives twice the area of the loop:
Hence the change in V over distance Dr:

5. Magnetic Potential Therefore, the magnetic potential equals:
Note that the potential
decreases along
the direction of m,
and hence B is generally
oriented along m
Therefore, the magnetic potential equals:
The const is usually taken zero, so that
Note two properties different from the monopole potential for gravity:
The potential decays with distance as 1/r2
It also depends on the direction (zero within the plane orthogonal to m and opposite signs in the directions parallel and opposite to m)

6. Poisson’s relation
The dipole field (V1/r2) equals a derivative of a “monopole field” (V 1/r) with respect to the positions of the source (r0) or observation point (r) :
This is simply the spatial derivative in the direction of the magnetic moment, m
Therefore, the magnetic field of a structure with given direction of magnetization n is related to its gravity field (Poisson’s relation):

7. Magnetic Monopoles
The dipole field can be presented as produced by two monopoles separated by an arbitrary (small) distance a:
where
and is the pole strength

8. Magnetic filed of a uniform magnetization
The magnetic potential of a single magnetic moment m:
Consider m uniformly distributed in space with density M:
Then, similarly to Gauss’s theorem in gravity, integration of 1/r2 over all sources in the volume gives the potential uniformly decreasing in the direction of M:
and therefore magnetic field:

9. Magnetic Induction
Magnetic moments within the medium (remanent or induced) cause magnetic field (take SI units now, Cm = m0/4p):
This field adds to the filed B = m0H produced by external currents. The sum is called the magnetic induction field (this field acts on our magnetometer):
k is called
magnetic susceptibility
If the magnetization is induced:
then:
m = m0(1+k) is called magnetic permeability

10. Components as potential fields
If f(x) is a potential field (satisfies the Laplace equation, ), then its gradient evaluated in a fixed direction is also a potential field:
This means that:
Vertical component of gravity satisfies Laplace equation
Each of the three components of magnetic field satisfies Laplace equation
However, when measured on the near-spherical Earth, the “vertical” components are actually radial derivatives
These do not rigorously satisfy the Laplace equation
The error is small (as the ratio of the survey size to the radius of curvature)

11. Total field
In magnetic observations, the “total field” is typically measured
Total field is a scalar quantity – variation of the magnitude of magnetic field:
DT equals the projection of the anomaly field vector DB onto the direction of the ambient (geomagnetic) field :
This ratio
is the unit vector
As a projection onto a constant direction, DT equals satisfies the Laplace equation, i.e., it is a potential field