Outline Magnetic dipole moment

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

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:

Magnetic Potential 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:

V Br

or:

V V V

r r r

B r

Biot-Savart law:

2

ˆm

dC I

r

r r

B r ÑCm depends on the unit system:emu: Dimensionless Cm = 1SI: Cm = m0/4p = 10-7 H/m

is a unit vector along rr̂

“Henry”, unit of inductance

Magnetic Potential Therefore, the change in the potential:

3 3m m

dV C I C I d

r r

r r r r

r r rÑ Ñ

The first equation here uses: The second equation assumes an infinitesimal loop and uses

ˆr

r

r

r r=

The contour integral gives twice the area of the loop: Hence the change in V over distance Dr:

2d s r r nÑ

3 3 2

ˆ2 2m m m

I sV C C C

r r r

n r m r m r

0d rÑ

Magnetic Potential Therefore, the magnetic potential equals:

2

ˆmV C constr

m rr

The const is usually taken zero, so that 0V

Note two properties different from the monopole potential for gravity: 1. The potential decays with distance as 1/r2

2. It also depends on the direction (zero within the plane orthogonal to m and opposite signs in the directions parallel and opposite to m)

Note that the potential decreases along the direction of m,and hence B is generallyoriented along m

Poisson’s relation The dipole field (V1/r2) equals a derivative of a

“monopole field” (V 1/r) with respect to the positions of the source (r0) or observation point (r) :

00 0 0

, m m

m mV C C

r r n n

r r r r r 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):

mC m

G

H r n g rr

Magnetic Monopoles The dipole field can be presented as produced by two

monopoles separated by an arbitrary (small) distance a:

1 1, ,2 2

a aV V q V q

r r n r n

1 , m

qV q C

rrwhere

and is the pole strength m

qa

Magnetic filed of a uniform magnetization

The magnetic potential of a single magnetic moment m:

2

ˆmV Cr

m rr

Consider m uniformly distributed in space with density M:

of a volume dVm 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:

4 mV C r M r

and therefore magnetic field:

4 mCB r M

Magnetic Induction Magnetic moments within the medium (remanent or induced)

cause magnetic field (take SI units now, Cm = m0/4p):

0H M

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):

0 0 B H M

If the magnetization is induced: then:

M H

0 1 B H H

m = m0(1+k) is called magnetic permeability

k is called magnetic susceptibility

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:

2 0n nu un n

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)

2 0

Total field In magnetic observations, the “total field” is typically measured Total field is a scalar quantity – variation of the magnitude of magnetic

field:

ambient ambientT B B B

DT equals the projection of the anomaly field vector DB onto the direction of the ambient (geomagnetic) field :

ambientB̂

2

ambient ambient

ambient ambient ambientambient ambient2 2

ambientambient ambient

21 1 1 1

T

B B B

B B B B BB B B

BB B

This ratio is the unit

vectorambientB̂ As a projection onto a constant direction,

DT equals satisfies the Laplace equation, i.e., it is a potential field