Biomedical Engineering Reference
In-Depth Information
This equation shows that the magnetic field in an unbounded homogeneous medium
is expressed by the famous Biot-Savart law with replacing the total current
J
with
the source (impressed) current
J
S
. In other words, the ohmic current
J
E
does not
contribute to the magnetic field in an unbounded homogeneous medium.
The expression for the scalar potential in Eq. (
A.37
) can be simplified in a fol-
lowing manner. Let us use the identity
r
)
∇
·
r
)
J
S
(
J
S
(
1
V
∇
·
d
3
r
=
d
3
r
+
r
)
·∇
d
3
r
.
J
S
(
(A.41)
r
|
r
|
r
|
|
r
−
|
r
−
|
r
−
V
V
The Gauss theorem is expressed as
r
)
r
)
J
S
(
J
S
(
V
∇
·
d
3
r
=
n
dS
,
r
|
·
|
−
r
|
|
−
r
r
S
r
)
becomes zero on
S
, the surface integral on the right-hand side of the equation above
becomes zero, and we can then get the relationship:
where
n
is again the outward unit normal to the surface
S
. If we assume that
J
S
(
∇
·
r
)
J
S
(
1
d
3
r
=−
r
)
·∇
d
3
r
.
J
S
(
(A.42)
r
|
r
|
|
r
−
|
r
−
V
V
Therefore, using
r
1
r
−
∇
r
|
=
3
,
(A.43)
|
r
−
|
r
−
r
|
Eq. (
A.37
) is rewritten as
r
1
r
−
r
)
·
3
d
3
r
.
ʦ
(
)
=
J
S
(
r
(A.44)
r
|
4
ˀ˃
|
r
−
V
The above equation is the expression to compute the electric potential in an
unbounded homogeneous medium.
A.2.2.4
Dipoles in an Unbounded Homogeneous Medium
The transmembrane current density
J
S
arises due to concentration gradients, Source
models for
J
S
fall into two categories: rigorous and phenomenological. Although a
rigorous source model accounts reasonably accurately for each of the microscopic
currents, it is difficult to derive such source models. A phenomenological source
model is one which produces the same external fields, but is artificial in the sense
that it does not actually reflect the microscopic details of the problem.
A representative phenomenological source model is the dipole model. The dipole
is the simplest source for both
ʦ
and
B
and can be written as:
