Biomedical Engineering Reference
In-Depth Information
(
)
μ
Il
Δ×
r
(8.21)
m
Br
()
=
x
x
3
4
π
r
10
−7
N/A
2
is the magnetic
permeability of the vacuum. Equation (8.21) is referred as the Biot-Savart Law,
named after two French physicists. The factor
where
μ
m
is the “magnetic permeability” and
μ
m
0
=
4
π
×
Δ
l
×
r
is the cross product of the
vector
l
with the vector
r
(radius direction is pointing to the magnetic field point).
The effect of the cross product is to make
B
perpendicular to the direction of the
current and radius direction. The direction of
B
at any point is tangent to a circle
centered on and perpendicular to the current element.
B
is said to be an axial vec-
tor field because of its axial (cylindrical) symmetry with respect to its source. The
orientation of the
B
field around the source current (clockwise or counterclockwise)
is determined by the direction of the current (the flow of positive ions). The magni-
tude of the magnetic field is
Δ
μ
Il
Δ
sin( )
θ
(8.22)
Br
()
=
m
x
2
4
π
r
where
is the angle between the current element and the position vector of the field
point. The magnetic force exerted by
B
on a length
L
of test current
I
is
θ
(8.23)
FILB
=
sin
θ
where
is the angle between
B
and the test current. If
B
is parallel to
I
, then
F
is
zero. The direction of
F
is perpendicular to both
B
and the test current. This situa-
tion is different from that of the electric and gravitational forces, which act along
a vector parallel to their respective fields (pointing from one source to another).
In further contrast with the case of the electric force, parallel currents attract and
antiparallel currents repel. Hence, magnetic fields and forces are inherently 3D. The
point in the center of the magnet is called the isocenter of the magnet. The magnetic
field at the isocenter is
B
0
.
θ
8.5.3.1 Frequency Response
Each magnetic moment also experiences a torque due to the field, resulting in pre-
cession about the
z
-axis, and the moment describes a cone in the
xy
plane [Figure
8.9(c)]. The angular frequency of precession, known as the Larmor frequency (
f
R
),
is proportional to the external magnetic field,
B
, that is,
(8.24)
f
=
γ
B
R
where
is the gyromagnetic ratio of the nucleus, a quantity unique to each atom.
For hydrogen,
γ
10
4
rad/s-gauss. The nuclei of different
elements and even of different isotopes of the same element have very different fre-
quencies (Table 8.2). In biological materials, the magnetic nuclei of
1
H,
13
C,
23
Na,
γ
is 42.57 MHz/T or 2.675
×
