Magnetization and Magnetic Minerals - Quantitative Plate Tectonics - page 84

with the magnetic colatitude. These expressions
show that the magnitude of B depends from the
inverse cube of the distance from the origin:

4 r 3 3cos 2 ™ C 1 1=2

0 m

B.r/ D (3.24)

We note that at any fixed distance from the ori-
gin, the magnitude of B is maximum at the poles
(™ D 0or™ D ) and attains its minimum along
the magnetic equator (™ D /2). Equations 3.22
and 3.23 can be used to determine the compo-
nents of the Earth's magnetic field in a local ref-
erence frame (see Sect. 2.3 ) . In the next chapter,
we shall see discuss the application of the mag-
netic dipole model to the representation of the
geomagnetic field. For the moment, it is sufficient
to say that in a local coordinate system the term
( 3.22 ) corresponds to the vertical component of
the field, Z , while the horizontal component, H ,
will be obtained by ( 3.23 ).

Fig. 3.6 Force lines of the magnetic dipole field B gener-
ated by a small current loop

3.3 Maxwell's Equations for

the Magnetic Field

The four Maxwell's equations of classical Elec-
trodynamics express, in a concise form, all the ba-
sic features of the electromagnetic fields and their
relation with the electric and magnetic sources.
Together with Lorentz's equation, they furnish a
complete description about the origin of the elec-
tromagnetic fields, their interaction with charged
particles, and their evolution in time. Two of these
equations describe the sources of magnetic fields.
The first of them is a differential form of the law
of Gauss :

Fig. 3.7 Components of the dipole field generated by a
magnetic dipole directed as the z -axis

3 m r
r 5

r r 3

0
4

B .r/ Š (3.21)

This expression shows that at any location r
the field vector B ( r ) can be decomposed in two
orthogonal components, one directed radially as
r , and a component that is tangent to the sphere
of radius r (Fig. 3.7 ). They are, respectively,

r B D 0 (3.25)

This equation simply states that there are no
magnetic charges in the physical world. It also
implies that a vector field exists, A D A ( r ), such
that:

0 m cos™
2 r 3

B r .r/ D (3.22)

0 m sin ™
4 r 3

B ™ .r/ D (3.23)

B Dr A (3.26)

The field A D A ( r ) is called vector potential .
The second of Maxwell's equations devoted to

where ™ is the angle between r and m .Inthe
case of the geomagnetic field, this angle coincides