Magnetization and Magnetic Minerals - Quantitative Plate Tectonics - page 81

F D q. v B/ (3.5)

Here, v and q are respectively the velocity of
the particle and its charge. Equation 3.5 shows
that the magnetic force is always orthogonal both
to the direction of motion of the particle and to
the magnetic field vector B .Furthermore, F D 0
when the particle moves in the same direction
of the field. Thus, the equation of motion for
a particle that is moving in a magnetic field
assumes the form:

m d v

dt D q v B

(3.6)

Fig. 3.1 A flow of electric charges walks through a sur-
face S with local velocity v

An interesting class of solutions for Eq. (1.6 )
can be obtained easily for a homogeneous mag-
netic field, for example when B D B k ( k being the
base versor in the direction z ). In this case ( 3.6 )
assumes the following simple form:
8
<

P v x D v y
P v y D v x
P v z D 0

(3.7)

:

where the quantity:

qB
m

(3.8)

is called cyclotron frequency . A solution to this
system of differential equations, with the initial
condition v 0 D v (0), can be obtained in a few
steps. The third equation implies that v z is con-
stant:

Fig. 3.2 Geometry of a current loop

v z .t/ D v 0 z (3.9)

(induction) field B D B ( r ) at any position r as a
function of the cable geometry (Fig. 3.2 ):

Furthermore, taking the derivative of the first
two equations we easily obtain a separation of the
equations:

Z

0 I
4

. r q / d q
k r q k

B .r/ D (3.4)

3

R v x D 2 v x
R v y D 2 v y

C

(3.10)

where the line integral is calculated along the
circuit C and 0 is the magnetic permeability
in the vacuum : 0 D 4 10 7 H/m. In the SI,
the unit of B is the Tesla [1 T D 1Vsm 2 ]. The
magnetic force exerted on a moving charge is
determined by the equation of Lorentz:

Therefore, we obtain:
v x D A sin .t C ®/
v y D A cos .t C ®/

(3.11)