Biomedical Engineering Reference
In-Depth Information
A.2.2
Magnetic Field and Electric Potential in an Unbounded
Conducting Medium
A.2.2.1
Potential Equations
We have shown that using the vector and scalar potentials, the Maxwell's equations
in Eqs. (
A.3
) and (
A.4
) are rewritten, respectively, as
B
=∇×
A
,
(A.16)
=−∇
ʦ
−
∂
A
∂
t
.
E
(A.17)
Let us express the other two Maxwell's equations using these potentials and the
source current
J
S
. Let us substitute the two equations above into Eq. (
A.5
), resulting
in
2
A
∂
−
μ
∇
∂
ʦ
∂
t
−
μ
∂
∇×
(
∇×
A
)
=
μ
J
t
2
.
(A.18)
The relationship in Eq. (
A.6
) is expressed using the potentials such that
J
S
−
˃
∇
ʦ
−
˃
∂
A
∂
J
=
t
.
(A.19)
Substituting Eq. (
A.19
)into(
A.18
), and using the identity
∇×
(
∇×
A
)
=
2
A
, we obtain
∇
(
∇·
A
)
−∇
2
A
∂
+
μ
∂
ʦ
∂
−
μ
∂
−
μ˃
∂
A
∂
2
A
∇
−∇·
∇·
A
t
+
μ˃
ʦ
t
=−
μ
J
S
,
(A.20)
t
2
and finally
2
A
∂
−
μ
∂
−
μ˃
∂
A
∂
2
A
∇
t
=−
μ
J
S
,
(A.21)
t
2
where the gauge relationship in Eq. (
A.15
)isusedinEq.(
A.20
). From Maxwell's
equation in Eq. (
A.2
), a similar derivation leads to
2
ʦ
−
μ
∂
ʦ
−
μ˃
∂
ʦ
∂
t
=−
ˁ
2
∇
.
(A.22)
t
2
∂
Finally, we use Eqs. (
A.6
), (
A.7
), and (
A.11
), to obtain the time derivative of the
charge
ˁ
, expressed as
