Biomedical Engineering Reference
In-Depth Information
E
∂
∂
+
∂
A
∂
∇×
E
+
t
∇×
A
=∇×
=
0
.
(A.10)
t
Using a scalar potential
ʦ
, we can write
+
∂
A
∂
E
t
=−∇
ʦ
,
(A.11)
because for any scalar field
ʦ
, the identity
∇×
(
∇
ʦ
)
=
0 holds. Thus, we have
=−∇
ʦ
−
∂
A
∂
E
t
.
(A.12)
A.2.1.2
Gauge Transformations
Since electric and magnetic fields are defined as derivatives of the potentials, the
potentials are not uniquely specified by their respective electromagnetic fields. This
flexibility in defining
A
and
ʦ
is referred to as the choice of gauge. Considering again
the relationship
, the vector potential
A
is only
determined up to the gradient of a scalar function, i.e., we can make the replacement
A
∇×∇
ʾ
=
0 for any scalar field
ʾ
ₒ
A
−∇
ʾ
without changing the value of the magnetic field
B
. This implies that,
∂
∂
E
−∇
ʾ)
=−∇
(
ʦ
−
∂
∂
t
)
−
∂
A
ₒ−∇
ʦ
−
t
(
A
(A.13)
∂
t
.
In other words, to keep
E
from being unchanged, we must also make the replacement
of
ʦ
ₒ
ʦ
+
∂ʾ
∂
t
.
(A.14)
This freedom to choose
ʾ
allows us to impose an additional relationship between
A
and
ʦ
. Common choices include the “Coulomb” gauge where
∇·
A
=
0orthe
“Lorentz” gauge, where
0. In the following, we use the
convenient “Gulrajani” gauge [4] for decoupling the differential equations for
A
and
∇·
A
+
μ(∂
ʦ
/∂
t
)
=
, thereby reflecting the fundamental symmetry between electric potentials and
magnetic fields. The Gulrajani gauge is expressed as
ʦ
+
μ
∂
ʦ
∂
∇·
A
t
+
μ˃
ʦ
=
0
.
(A.15)
