Biomedical Engineering Reference
In-Depth Information
B
B
r
J
i
r
σ
3
σ
2
σ
1
G
Fig. 3.28.
A spherical conductor model
n
r
r
/r,
e
r
=
r
/r
,
Using
(
)=
(
r
)
r
)
n
×
(
r
−
·
e
r
= 0
(3.73)
is derived. Thus,
r
)=
μ
0
4
π
r
−
r
)
B
r
(
r
)=
B
0
r
(
r
G
J
i
(
×
3
·
e
r
d
v.
(3.74)
|
r
−
r
|
This means that the radial component of the magnetic field is equal to that of
the magnetic field caused by the applied current only. When we calculate only
the radial component of the magnetic field outside the spherical conductor,
we neglect the volume current. The other components of
B
are affected by
the volume currents. To give all the components of
B
, we use a magnetic
scalar potential:
B
(
r
)=
−
μ
0
∇
U
(
r
)
.
(3.75)
The scalar potential
U
is obtained by a line integral of
∇
U
:
∞
U
(
r
)=
−
∇
U
(
r
+
t
e
r
)
·
e
r
d
t
0
∞
∞
1
μ
0
B
r
(
r
+
te
r
)d
t
=
1
μ
0
=
B
0
(
r
+
te
r
)d
t
0
0
∞
1
4
π
Q
×
d
t
=
(
r
−
r
0
)
·
e
r
3
.
(3.76)
|
r
+
t
e
r
−
r
0
|
0
In the above equations, we assumed a current dipole as the applied current.
The last integral can be computed; that is,
∞
d
t
|r
+
te
r
− r
0
|
r
a
(
ar
+
r
2
=
)
.
(3.77)
3
−
r
0
·
r
0
