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and
r
D
D
e
;
(1.4)
where
e
is the electric charge density.
It is usually the case that the conduction current in the conducting fluid is much
greater than the displacement current, i.e., j
@
t
D, so that Eq. (
1.1
) is reduced to
r
B
D
0
j
:
(1.5)
The electric and magnetic fields in the above equations are measured in a fixed
or motionless coordinate system, K. Considering a conducting fluid moving at mass
velocity
V
, we first use the reference frame, K
0
, moving with mass velocity
V
of the
fluid. Transformation between two coordinate systems moving at relative velocity
V
causes the transformation of the electromagnetic field. According to Jackson (
2001
),
in a nonrelativistic limit, while as
j
V
j
c, where c stands for the light speed in the
vacuum, the electromagnetic field in the moving frame becomes
E
0
D
E
C
V
B
;
(1.6)
B
0
D
B
and
j
0
D
j
;
(1.7)
where the primed variables are those measured in the moving frame, K
0
, and
the unprimed variables are measured in the motionless frame, K. Transformation
between two reference frames leaves
Maxwell
's equation invariant while the
Ohm
's
law has the form
j
0
D
E
0
;
(1.8)
which is valid only for the K
0
coordinate system. Here denotes the conductivity
coefficient. It should be noted that the Ohm's law given by Eq. (
1.8
) is valid if the
temporal and spatial dispersion in the medium can be neglected.
Substituting Eqs. (
1.6
)-(
1.8
) into Eq. (
1.5
) yields
r
B
D
0
.
E
C
V
B
/:
(1.9)
This implies that there is an electric field only if the conductor moves or the term
@
t
B
in Eq. (
1.2
) is nonzero.
Now we consider the basic equations describing the dynamics of conducting
fluid. The principle of conservation of fluid mass means that the fluid flux into or out
of a volume through its surface must be equal to the rate of mass variations inside
the volume. This mathematical statement in the integral form can be converted to
the differential equation known as the continuity equation, which relates the fluid
mass density and the fluid velocity
V
through (e.g., see Kelley
1989
)
@
t
Cr
.
V
/
D
0:
(1.10)
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