Biomedical Engineering Reference
In-Depth Information
∇×
(
)
=
relationship
B
r
0 holds outside the volume conductor, because there is no
(
)
electric current. Thus,
B
r
can be expressed in terms of the magnetic scalar potential
U
(
r
)
,as
B
(
r
)
=−
μ
0
∇
U
(
r
).
(A.55)
This potential function is derived from
∞
∞
1
μ
0
1
μ
0
U
(
r
)
=
B
(
r
+
τ
e
r
)
·
e
r
d
τ
=
B
0
(
r
+
τ
e
r
)
·
e
r
d
τ,
(A.56)
0
0
where we use the relationship in Eq. (
A.54
). Assuming that a dipole source exists at
r
0
, by substituting Eq. (
A.47
) into Eq. (
A.56
) and performing the integral, we obtain
∞
1
μ
0
(
)
=
B
0
(
+
τ
e
r
)
·
τ
U
r
r
e
r
d
0
e
r
∞
0
1
4
d
τ
=
Q
×
(
r
−
r
0
)
·
3
ˀ
|
r
+
τ
e
r
−
r
0
|
1
4
(
Q
×
r
0
)
·
r
=−
,
(A.57)
ˀ
where
2
=|
r
−
r
0
|
(
|
r
−
r
0
|
r
+|
r
|
−
r
0
·
r
).
(A.58)
is then obtained by substituting
Eq. (
A.57
) into Eq. (
A.55
) and performing the gradient operation. The results are
expressed as
The well-known Sarvas formula [6] for
B
(
r
)
Q
μ
0
4
ˀ
∇
(
×
r
0
)
·
μ
0
4
×
Q
r
r
0
1
(
)
=
=
−
2
(
×
r
0
)
·
∇
B
r
Q
r
ˀ
2
]∇
,
μ
0
=
Q
×
r
0
−[
(
Q
×
r
0
)
·
r
(A.59)
4
ˀ
where
|
r
2
r
−
r
0
|
+
(
r
−
r
0
)
·
r
∇
=
+
2
|
r
−
r
0
|+
2
|
r
|
|
r
|
|
r
−
r
0
|
r
0
.
|+
(
r
−
r
0
)
·
r
−
|
r
−
r
0
|+
2
|
r
(A.60)
|
r
−
r
0
|
becomes zero,
and no magnetic field is generated outside the conductor from a source at the origin.
Also, if the source vector
Q
and the location vector
r
0
are parallel, i.e., if the primary
current source is oriented in the radial direction, no magnetic fields are generated
outside the spherical conductor from such a radial source. This is because the two
We can see that when
r
0
approaches the center of the sphere,
B
(
r
)
