Biomedical Engineering Reference
In-Depth Information
ʩ
And
is a d-dimensional domain (volume in 3d). The unknown function is the
electrostatic potential
u
and the given data is the charge distribution
f
.
Next, the Gauss' law, also known as Gauss flux theorem, which is a law relating
the distribution of electric charge to the resulting electric field is used. In its integral
form, the law states that, for any volume
V
in space, with boundary surface
∂
V
,the
following equation holds:
ˁ
Q
v
E
·
ndS
=
·
dV
=
,
(3)
∂
V
∂
V
where the left hand side is called the “electric flux” through
∂
V
,
E
is the electric
field
V
ndS
is a surface integral with an outward facing surface with a normal
orientation. The surface
∂
V
is the surface bounding the volume
V
,
Q
V
is the total
electric charge in the volume V,
∂
is the electric constant - a fundamental physical
constant. Again using divergence theorem, the differential form of the Gauss'law is:
=
ˁ
∇·
E
,
(4)
where
is the charge density. Since the line integral of the electric field around any
close curve in space equals 0
ˁ
E
(
·
dl
=
0
)
,
E
=−∇
u
.
(5)
Eliminating by substitution, we have a form of the Poisson equation:
=−
ˁ
Δ
u
.
(6)
In practice
E
can be calculated as the difference between the elecric potential at two
positions divided by the distance between those positions as follows:
∂
u
u
1
−
u
2
x
=
E
=
.
(7)
∂
Δ
Therefore, if we know the electric field, we can calculate the voltage at different
points. The current density in the medium
J
can be obtained from the Ohm law as
follows:
J
=
˃
E
,
(8)
where
˃
is the electrical conductivity of the material in the model (S/m).
