Biomedical Engineering Reference
In-Depth Information
∂ˁ
∂
t
=−∇·
J
=−∇·
(˃
E
+
J
S
)
−∇
ʦ
−
∂
A
∂
=−
˃
∇·
−∇·
J
S
t
ʦ
+
˃
∂
∇·
A
2
=
˃
∇
−∇·
J
S
∂
t
∂
∂
μ
∂
ʦ
∂
2
=
˃
∇
ʦ
−
˃
t
+
μ˃
ʦ
−∇·
J
S
t
2
ʦ
−
μ˃
∂
ʦ
2
∂
ʦ
∂
2
=
˃
∇
−
μ˃
t
−∇·
J
S
,
(A.23)
t
2
∂
where we again use the gauge relationship in Eq. (
A.15
).
A.2.2.2
Derivation of Potentials
To facilitate the solution of the equations for potentials, we can exploit linearity and
also assume harmonic time dependence with angular frequency
. This assump-
tion does not require that the signals have perfect sinusoidal dependence, and is
merely a mathematical convenience that helps subsequent discussion for determin-
ing the quasi-static regime. Namely, we express the solutions of the partial differential
equations in Eqs. (
A.21
), (
A.22
), and (
A.23
), such that
ˉ
A
e
i
ˉ
t
(
t
,
r
)
=
A
(
r
)
,
(A.24)
(
e
i
ˉ
t
t
,
r
)
=
ʦ
(
r
)
,
(A.25)
e
i
ˉ
t
ˁ(
t
,
r
)
=
ˁ(
r
)
,
(A.26)
J
S
(
e
i
ˉ
t
t
,
r
)
=
J
S
(
r
)
.
(A.27)
Substituting
A
and
J
S
(
into Eq. (
A.21
), we obtain the following partial
differential equation of space alone,
(
t
,
r
)
t
,
r
)
2
A
k
2
A
∇
(
)
+
(
)
=−
μ
J
S
(
),
r
r
r
(A.28)
where,
k
2
=−
i
ˉμ(˃
+
i
ˉ)
.
Substitution of
(
t
,
r
)
and
ˁ(
t
,
r
)
into Eq. (
A.22
) also results in
)
=−
ˁ(
r
)
2
k
2
∇
ʦ
(
r
)
+
ʦ
(
r
.
(A.29)
Substitution of
(
, and
J
S
(
t
,
r
)
,
ˁ(
t
,
r
)
t
,
r
)
into Eq. (
A.23
) gives the relationship:
2
2
2
i
ˉˁ (
r
)
=
˃
∇
ʦ
(
r
)
−
μ˃(
i
ˉ)
ʦ
(
r
)
−
μ˃
(
i
ˉ)
ʦ
(
r
)
−∇·
J
S
(
r
),
