Biomedical Engineering Reference
In-Depth Information
J
S
=
(
r
0
)
=
ʴ(
−
r
0
).
Q
Q
r
(A.45)
Substituting this for the equations for electric potential in Eq. (
A.44
) and magnetic
field in Eq. (
A.40
) in a homogenous infinite volume conductor, we get
1
r
−
r
0
ʦ
(
r
)
=
Q
·
3
,
(A.46)
4
ˀ˃
|
r
−
r
0
|
and
μ
0
4
r
−
r
0
B
(
r
)
=
Q
×
3
.
(A.47)
ˀ
|
r
−
r
0
|
The equations above are the expressions for computing the electric potential and
magnetic field produced by an current dipole at
r
0
embedded in an unbounded homo-
geneous conductive medium.
A.2.3 Magnetic Field from a Bounded Conductor
with Piecewise-Constant Conductivity
We next consider the magnetic field generated by an inhomogeneous conductor. We
assume that the region
V
can be divided into subregions
V
j
,
j
=
1
,...,
, and the
region
V
j
has conductivity
˃
j
. The surface of
V
j
is denoted
S
j
. We also assume that
the conductivity,
˃(
r
)
, is zero outside
V
. We start the derivation from the Bio-Savart
law:
r
μ
0
4
r
−
μ
0
4
r
)
×
3
d
3
r
=
r
)
×
r
)
d
3
r
,
B
(
r
)
=
J
(
J
(
G
(
r
,
(A.48)
r
|
ˀ
|
r
−
ˀ
V
V
where
r
r
−
r
)
=
G
(
r
,
3
.
r
|
|
r
−
r
)
=
r
)
−
˃(
r
)
∇
ʦ
(
r
)
Substituting
J
(
J
S
(
into Eq. (
A.48
), we can obtain
J
S
(
r
)
×
)
=
μ
0
4
r
)
−
˃(
r
)
∇
ʦ
(
r
)
d
3
r
B
(
r
G
(
r
,
ˀ
V
1
˃
j
=
μ
0
4
d
3
r
−
μ
0
4
r
)
×
r
)
r
)
×
r
)
d
3
r
.
J
S
(
(
,
V
j
∇
ʦ
(
(
,
G
r
G
r
ˀ
ˀ
V
j
=
(A.49)
Using
r
)
×
r
)
=∇×[
ʦ
(
r
)
r
)
]
,
∇
ʦ
(
G
(
r
,
G
(
r
,
(A.50)
