Biomedical Engineering Reference
In-Depth Information
z
B 0
I
y
r
v
q
x
F
FIGURE 16.30 Lorentz force on a charge revolving about a static magnetic field. Vectors for velocity, v, force, F ,
and magnetic field,
B 0 , are shown at the position of the charge.
The classical angular frequency of the charge can be expressed in terms of the velocity of
the charge and the radius, which can be rewritten from the previous equation to give
o c ¼ v = r ¼ v mv = qB
Þ¼ð q = m Þ B
ð
16
:
53a
Þ
0
0
From the definition of the classic gyromagnetic ratio, Eq. (16.50), comes an important equa-
tion in MRI for the classical frequency,
v c ¼ 2
p ¼ ð
2
g c Þ B
¼ g c B
0
0
ð
16
:
53b
Þ
2
p
p
which shows that the orbital frequency of the charge is proportional to the applied mag-
netic field. Unfortunately, this is not exactly what is needed for MRI because classical elec-
tromagnetic theory is for a charge that does not revolve on its own axis. The charge of
interest in MRI is for an electron, which has its own individual spin. This situation is anal-
ogous to the revolution of the earth around the sun in combination with the revolution of
the earth about its own axis. To obtain this important equation, an explanation of spin states
is necessary from quantum mechanics.
16.3.3 Spin States
Based on the previous discussion, one could expect that the electron spinning on its own
axis would create a miniature magnetic field; consequently, it would behave like a magnetic
dipole with its own north and south poles.
Permanent bar magnets are dipoles that have a strong polarization in the form of north
and south poles. If two equal permanent bar magnets are placed in the north to south setup
shown in Figure 16.31a, they are strongly attracted and are said to have a strong attractive
force between them. If they are placed close to each other in a north-north (or a south-
south) configuration as in Figure 16.31b, the magnets are forced apart by a strong repulsive
force. These two arrangements of magnets are two positions in which the strongest forces
are stabilized in equilibrium.
If a number of the tiny magnetic dipoles are placed in a strong static magnetic field,
B 0 ,
then they will align either with the direction of the field (parallel) or lock into a position
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