Vol.3, No.4, 275-284 (2011) Natural Science
http://dx.doi.org/10.4236/ns.2011.34035
A theory about induced electric current and heating in
plasma
Zhiliang Yang*, Rong Chen
Department of Astronomy, Beijing Normal University, Beijing; *Corresponding Author:
Received 29 November 2010; revised 12 January 2011; accepted 12 March 2011.
ABSTRACT
The traditional generalized Ohm’s law in MHD
(Magnetohydrodynamics) does not explicitly
present the relation of electric currents and
electric fields in fully ionized plasma, and leads
to some unexpected concepts, such as “the
magnetic frozen-in plasma”, magnetic recon-
nection etc. In the single fluid model, the action
between electric current and magnetic field is
not considered. In the two-fluid model, the
derivation is based on the two dynamic equa-
tions of ions and electrons. The electric current
in traditional generalized Ohm’s law depends on
the velocities of the plasma, which should be
decided by the two dynamic equations. However,
the plasma velocity, eventually not free, is in-
appropriately considered as free parameter in
the traditional generalized Ohm’s law. In the
present paper, we solve the balance equation
that can give exact solution of the velocities of
electrons and ions, and then derive the electric
current in fully ionized plasma. In the case ig-
noring boundary condition, there is no electric
current in the plane perpendicular to the mag-
netic field when external forces are ignored. The
electric field in the plane perpendicular to
magnetic field do not contribute to the electric
currents, so do the induced electric field from
the motion of the plasma across magnetic field.
The lack of induced electric current will keep
magnetic field in space unaffected. The velocity
of the bulk velocity of the plasma perpendicular
to magnetic field is not free, it is decided by
electromagnetic field and the external forces.
We conclude that the bulk velocity of the fully
ionized plasma is not coupled with the magnetic
field. The motion of the plasma do not change
the magnetic field in space, but the plasma will
be confined by magnetic field. Due to the con-
finement of magnetic field, the plasma kinetic
energy will be transformed into plasma thermal
energy by the Lamor motion and collisions be-
tween the same species of particles inside plas-
ma. Because the electric field perpendicular to
magnetic field do not contribute electric current,
the variation of magnetic field will transfer en-
ergy directly into the plasma thermal energy by
induced electric field. The heating of plasma
could be from the kinetic energy and the varia-
tion of magnetic field.
Keywords: Plasmas; MHD; Electric Current;
Plasma Heating
1. INTRODUCTION
Recently Falthammar [1] commented on the unjustified
use of the motion of magnetic field lines in plasma,
following Alfven, who vigorously warned against the
unjustified use of concept “Frozen-in magnetic field” in
his late years [2]. However, the scientists pay little
attention to the suggestions. The concept of magnetic
field line motion was applied extensively in space
plasma physics and solar physics. Magnetic recon-
nection becomes hot topic in explaining magnetic
activities. The problem is from the generalized Ohm’s
law of MHD (Magnetohydrodynamics) theory which
shows the couple between velocity of plasma and the
magnetic field.
MHD is the physical-mathematical framework that
concerns the dynamics of magnetic fields in electrically
conducting fluids, i.e. in plasmas and liquid metals. One
of the most famous scholars associated with MHD was
the Swedish physicist Hannes Alfven, who receive the
Nobel Prize in Physics in 1970 for fundamental work
and discoveries in MHD with fruitful applications in
different parts of plasma physics. The notion of frozen-in
magnetic field is the result of his work in connection
with the discovery of MHD waves [3].
The central point of the MHD theory is that con-
ductive fluids can support the magnetic field. The pre-

Z. L. Yang et al. / Natural Science 3 (2011) 275-284 276
sence of magnetic fields leads to forces that in turn act
on the fluid, typically a plasma, thereby potentially
altering the geometry and strength of the magnetic fields
themselves. Based on the frame of MHD, a lot of
theories come out, including MHD turbulence, MHD
waves, magneto-convection, MHD reconnection, and
MHD dynamo.
In fact, the MHD theory made little scientific progress
in space science and astrophysics in the past decades.
Problems of space science and astrophysics such as
magnetic generation, are still in puzzle. The key problem
of the MHD theory comes from the generalized Ohm's
law, which is believed to give the electric current in
cosmic plasma.
2. THE TRADITIONAL GENERALIZED
OHM'S LAW AND MHD EQUATIONS
In the standard non-relativistic form, the MHD equa-
tions consist of the basic conservation laws of mass,
momentum and energy together with the induction
equation for the magnetic field. The equations are,
written in SI units:
=0
t
 
u (1)
The equation of motion:
 
=p
t
  

uuuj B

(2)
where

is the mass density and u the fluid bulk
velocity, p is the gas pressure, B the magnetic field,
j the current density, and

is the viscous stress
tensor.The equation for the internal energy, which is
usually written as an equation for the pressure p:
=
ppp Q
t

 
uu (3)
where Q comprises the effects of heating and cooling
as well as thermal conduction and

is the adiabaticity
coefficient. The above equation implies the equation of
state of the ideal ionized gas:

=2 iB pmkT  (4)
It is satisfied for most dilute plasmas.
The induction equation, or Faraday’s law:

2
==
t

 

B
E
uB B (5)
Which is derived by inserting Ohm’s law:
=

 
E
uB j (6)
Here,

is the electrical resistivity. In total, the MHD
equation consist of two vector and two scalar partial
differential equation (or eight scalar equations) that are
to be solved simultaneously.
An early theoretical paper by Lighthill [4] on
magnetized plasma properties in the MHD description
contains in its physics sections criticisms about the
applicability of ideal or resistive the MHD theory for
plasmas. Specifically, ignoring the Hall term in the
generalized Ohm’s law concerned, a simplification still
made almost routinely in magnetic fusion theory. The
objections to classical the MHD theory and their
consequences give rise to the development of Hall MHD
and its application to laboratory and cosmic plasma [5].
According to the MHD equation, we can find that the
key point is the decision of the generalized Ohm’s law.
Witalis [5] stressed that to retain the Hall term, using a
two-fluid plasma description is necessary. Then the
generalized Ohm’s law for fully ionized plasma can be
derived.
For the macroscopic behavior of plasma, Spitzer [6]
gave the detailed discussion. The basic equations are the
equations of ions and electrons together with the
Maxwell’s equations, the equation of continuity, and in
the steady condition 0
t





, the generalized Ohm's
law is expressed as:
=Hn


jE EE  
 (7)
where,
2
==
eei
en
m


 (8)

22
1
=1eei




(9)
22
=1
eei
H
eei





(10)
and =
e
e
e
mc
B
 is the electron larmor frequency, ei

is
the collision frequency between electrons and ions.
According to the above generalized Ohm's law, the
current will depend on the conductivity of magnetized
plasmas. The magnetic field influence on the con-
ductivity of the ‘direct’ current 

and that of Hall’s
current
H

is determined by the parameter eei

 ,
which is nothing other than the turning angle of an
electron on the Larmor circle in the intercollisional time
[7].
In the case 1
eei

 , this corresponds to the weak
magnetic field of dense cool plasmas, so that the current
is scarcely affected by the field:
, 1
H
eei 

 

 (11)

Z. L. Yang et al. / Natural Science 3 (2011) 275-284 277
Thus in a frame of reference associated with the
plasma, the usual Ohm's law with isotropic conductivity
holds.
In the opposite case, when the electrons spiral freely
between collisions, 1
eei

 , we will get the strong
magnetic field and hot rarefied plasma. This plasma is
termed the magnetized one. It is frequently encountered
under astrophysical conditions. In this case,

2
eei Heei


 
 (12)
Hence in the magnetized plasma, for example in the
solar corona, H




. The impact of the mag-
netic field on the direct current is especially strong for
the component resulting from the electric field


E
. The
current in the 

E
direction is considerably weaker than
it would be in the absence of a magnetic field.
In the cosmic conditions, the generalized Ohm’s law
assumes the form of Eq.5 can be used as the appro-
ximation of Eq. 7 [7,8]. This seems to be perfect for the
consistence of single fluid and two-fluid plasma model.
However, Spitzer [6] pointed out the roles of the
generalized Ohm’s law and the dynamic equations have
reversed roles from the usual custom, the equation of
motion determines the current density, while the gene-
ralized Ohm’s law determines the velocity. But in the
research, the generalized Ohm’s law is applied inde-
pendently from the motion equation.
The generalized Ohm’s law with the form of Eq.6 and
Eq.7 was accepted and widely applied in space science
and astrophysics. According to the generalized Ohm’s
law with the form of Eq.6 and the Farady’s law, we can
get the magnetic induction equation:

2
=
t

 

BuB B
(13)
where 1
=4π 

is the magnetic diffusivity. When the
magnetic diffusivity

is infinite small, the plasma is
called ideal MHD. Alfven noted that the motion of
matter may couple to the deformation of the magnetic
field such that the field lines follow the motion of matter,
and devoted this “frozen-in magnetic field lines”. The
“frozen-in magnetic field lines” theorem and its co-
rollary “ideal MHD” are widely used in space plasma
physics.
One of the most important deviation from ideal MHD
is magnetic reconnection, which is the merging of magn-
etic field lines, as invented by Sweet [9] and Parker [10],
and later applied to the Earth’s magnetosphere by
Dungey [11]. However, the physics involved in the mag-
netic field line merging remains poorly understood.
Though the ideal MHD can be used to simulate the
energy transfer from magnetic field to plasma kinetic
energy [12], the existence of
E
 along auroral mag-
netic field lines, upward directed, as well as downward
directed, is recognized [13]. It requirs a finite parallel
conductivity along the magnetic field B. This may lead
to the breakdown of the single fluid concept of MHD
[14].
The derivation of the generalized Ohm's law can be
found in many classic books and literatures about plasma.
The derivations include the discussions on single fluid
plasma (e.g. [15]), the two-fluid (electrons and ions) [6,7]
plasma and the three-fluid (electrons,ions and neutral
particles).
However, through carefully examination of the deri-
vation, we find the electric current in previous genera-
lized Ohm’s law is not explicitly presented. The velocity
of the plasma should be decided by the electromagnetic
field and external forces, it is not a free parameter.
3. THE ELECTRIC CURRENT IN FULLY
IONIZED PLASMA
To get the generalized Ohm’s law in plasma, we con-
sider the plasma as fluids, which means the distribution
of the particle velocity is Maxwellian distribution. The
generalized Ohm’s law is under the condition that
=0
t


j and the macroscopic velocities of the fluid (or
electron fluid and ion fluid) =0
t


u in the plasma [6].
The generalized Ohm’s law for single fluid is derived
simply from the reference system transformation [15]. It
is first criticized by Lighthill [4] for the lacking of the
Hall term. The mistake eventually is from the physical
consideration. In the single fluid case, the magnetic field
in the reference system is not considered for the reason
that the fluid is still. However, the impact of the
magnetic field is especially strong for the components
resulting from the electric field 

E
perpendicular to
the magnetic field in the fluid system. The primary effect
of the electric field


E
in the presence of the magnetic
field is not the current in the direction 

E
, but rather
the electric drift in the direction perpendicular to both
magnetic field and


E
.
A ful treatment of the plasma can be started from the
kinetic equation for each species of the plasma. For the
fully ionized plasma, they are the kinetic equations for
electrons and ions. One then integrates these equations
over the phase space, defines macroscopic quantities,
and derives various moment equation for each species
[16-18]. These moment equations and macroscopic quan-
tities describe each species as a fluid without invoking
the motion of each individual particle. These equations
can include the interaction among different species.

Z. L. Yang et al. / Natural Science 3 (2011) 275-284 278
They are, for ions,


=
i
iii iii
eeie ie
D
nmP enu
Dt
nm 
 

u
E
BF
uu
(14)
and for electrons,


=
e
eee eee
ieei ei
D
nmP en
Dt
nm 
 

u
E
uB F
uu
(15)
where the parameters ,,,,,, =
ieieieieei
nn PPuu

 are
the density of ions, density of electrons, pressure of the
ion fluid, pressure of the electron fluid, macroscopic ve-
locity of the ion fluid, macroscopic velocity of electron
fluid, collision frequency of ions and electrons. And
vector i
F, and e F denotes other forces, such as
gravity, centrifugal force, and Coriolis force, exerting on
the ion fluid and the electron fluid. We assume that the
ions are singly charged and what charges neutrality
holds in the plasma ==
ie
nnn, and ignoring possible
wave-particle collisions.
To get the Ohm’s law, we neglect the effects from
time derivatives, pressure gradient forces, and additional
forces, which are referred as “other forces”.
In the derivation of generalized Ohm's law, Somov [7]
considered the time derivatives and additional forces as
part of the effective electric field. When there are
magnetic fields, the effect of external forces acting on
charged fluid is completely different to the electric field.
The effects can be seen from the drifts of charged
particles in the magnetic field. In plasma, the drift due to
the electric field causes a macroscopic drift velocity, but
there is no current. However, the drift due to external
forces will cause electric current in plasma—the drift
current [19]. In this paper, we will first neglect the effect
of other forces. The discussion with other forces is in the
later part.
From Eqs.14 and 15, Ignoring the external forces, the
two fluid force balance equations for ions and electrons
in steady state in our discussion will be:

=
ieeiie
en nm 
E
uB uu  (16)

=()
ieeiei
en nm 
E
uBuu  (17)
The two equations are coupled by the electron-ion
collision terms. We define the electric current density as,

=ie en juu (18)
In all the previous derivations, the authors define the
macroscopic velocity of plasma flow as [6,7,17].
=ii ee
ie
mm
mm


uu
u (19)
With the relationship of ie
mm, the plasma flow
velocity becomes
=e
ie
i
m
m
uuu (20)
By substituting (18) and (20) into Eqs.16 and 17, then
subtracting (16) from (17), one then gets the form of
Ohm's law the same as Eq.7.
In fact, the Ohm’s law is not completed. Besides the
approximation of the Eq.20, we can have the additional
relation for the current. Adding Eqs.16 and 17 together
directly will give the result for =0jB . Some authors
have discussed the results. Early in 1956, Spitzer dis-
cussed the dynamics of the full ionized plasma, and a
later version in 1962 [6]. In the discussion of the
macroscopic behavior of a plasma, Spitzer obtains the
simpler equations in the conditions that ignoring the
terms in ei mm, t


j
and t


u, considering the changes
so slow that inertial effects are negligible. With the
gravity and the pressure gradient kept, Spitzer obtained
the relationship of,
=p



jB (21)
where p is the pressure,

is the density and

is the
gravity potential. Spitzer suggested that the Eq.21 be the
equation of motion and determine the current density
[6].
However, in the case that we derive the generalized
Ohm’s law, the pressure gradient and gravity potential is
always ignored. This will lead to the result of,
=0

jB (22)
In this case, there will be no any current perpendicular to
the magnetic field, a contradicted result to the
generalized Ohm’s law of Eq. 7. If the pressure gradient
is considered, the generalized Ohm’s law will be
different, which will be discussed later in Part 3.
In principle, we don’t need to get the solution by
above method. The balance Eqs.16 and 17 can comple-
tely determine the velocities of ion fluid and electron
fluid in the slowly varied electromagnetic field system.
Setting the magnetic field in the direction of z, so
=
z
B
B, we can have the electric field perpendicular to
magnetic field in the direction
x
, that is =
x

E
E, and
=
z
E
E
. The direction


E
B is the direction of y.
Then we can have Eq.16 and 17 written in the form of
components in the direction of
x
, y and z.

=0
x
eiyei ixex
e
e
m 
Euuu
 (23)


=0
eixei iyey
 uuu 
(24)

Z. L. Yang et al. / Natural Science 3 (2011) 275-284 279

=0
x
eeyei exix
e
e
m
 
Euuu
 (25)

=0
eexei eyiy
uuu 
(26)
 
=0
zeiizez
e
e
mEuu
 (27)
 
=0
zeieziz
e
e
m
Euu  (28)
where =, =
ie
ie
ee
mc mc
B
B
 . The solutions of Eqs.23-26
are,
==0
ex ix
uu (29)
And
2
==
ey iy

E
B
uu
B
(30)
As we have defined, the current in the plane per-
pendicular to the magnetic field is zero, since the ions
and electrons have exactly the same drift velocity . It is
consistent with the Eq. 22 as the other forces are ignored
(the force-free case). Then we have only the current in
the direction parallel to the magnetic field. Along the
magnetic field line, it is the normal Ohm's law since
there is no effect by the magnetic field. So we can
conclude that in the fully ionized plasma, when the other
forces besides the electromagnetic field are ignored, the
electric current in the plasma will be:

2
== 
B
EB
jE B 

To be consistent with the common Ohm’s law, we notice
the relation:



22
=


BEBE BB
E
BB
Then we can have the generalized Ohm’s law as:

2
=



E
BB
jE B
 (31)
and the plasma velocities is determined by the following
relation,
2
==
ee ii
ei
mm
mm


E
B
B


(32)
In the above method, we can decide the electric
current and the plasma velocity by the electric field and
the magnetic field in any reference system.
4. THE ELECTRIC CURRENT FROM
EXTERNAL FORCES IN FULLY
IONIZED PLASMA
The previous Ohm’s law is always confused when the
external forces are included. In this part, we will discuss
the case when external forces are included in the fully
ionized plasma. We consider the pressure gradient, grav-
ity or friction as the external forces i f and e f for the
fluid of ion and electron respectively.
=ii
i
e
P
nm

F
f, =ee
e
e
P
nm

F
f. In the nearly steady case,
the balance equations for the ion fluid and electron fluid
in x, y plane will be:

=
iieiie
e
e
m 
E
uB fuu  (33)

=
eeeiei
e
e
m

E
uB fuu  (34)
In the direction of magnetic field will be:
 
=
z
izei izez
e
e
m
E
fuu  (35)
 
=
z
ezei eziz
e
e
m
 
E
fuu  (36)
Here, we assume the magnetic field in the direction of
z, the electric field perpendicular to magnetic field in
the direction
x
, and


E
B is in the direction of y.
And Eqs.32-35 can be written as:

=0
x
eiyei ixexix
e
e
m 
Euuuf
 (37)


=0
eixei iyeyiy
 uuuf 
(38)

=0
x
eeyei exixex
e
e
m
 
Euuuf
 (39)


=0
eexei eyiyey
uuuf  (40)
 
=0
zeiizez iz
e
e
mEuuf
 (41)
 
=0
zeieziz ez
e
e
m
Euuf  (42)
Then, from Eqs.37-40 we can get the velocities of
ions and electrons in the plane perpendicular to magnetic
field as,


2
=eyiyeiex e
ex
e
ff f
u



(43)

Z. L. Yang et al. / Natural Science 3 (2011) 275-284 280

2
=exixeiex e
ey s
e


ff f
u



(44)

2
=exixeiiy e
ix
e
ff f
u



(45)

2
=eyiyeiix e
iy s
e


ff f
u



(46)
where the velocity 2 =
s


E
B
B. So we can get,
=ey iy
ix ex
e


ff
uu

(47)
=ex ix
iy ey
e


ff
uu

(48)
And if we use Eq.41 subtract 42 directly we can get
the velocity difference of electrons and ions in the
direction z:

1
=2
iz ezzizez
ei iei
e
m
uuEf f

(49)
By multiplying ne, we can get the electric current in
the plane perpendicular to the magnetic field.
== ey iy
x
e
ne

ff
jj

(50)
== ex ix
Hy
e
ne  ff jj

(51)

2
== 2
z
zizez
ei eei
ne ne
mjjEf f

 (52)
The velocity of the plasma can be decided by the
definition:
=.
ee ii
ei
mm
mm




 (53)
The electric current  j is in the direction of electric
field perpendicular to magnetic field, or Pedersen current,
and
H
j is the Hall current. The solution suggests that
the current has no relationship with the collisions and
electric field in the plane perpendicular to the magnetic
field. The electric current depends on the external forces
and magnetic field only. When the external forces are
ignored, the electric current in the plane perpendicular to
magnetic field disappear.
If we keep the pressure gradient as the only external
force, then =i
i
e
P
nm

f, =e
e
e
P
nm


f. Supposing
=ie PPP, we can have the electric currents as:

==
y
x
ee
eP
m


jj

(54)

==
x
Hy
ee
eP
m

jj

(55)
The above equations show the current density in the
whole space, where the plasmas are fully ionized and the
gas gradients are considered.
5. UNDERSTANDING THE INDUCED
ELECTRIC CURRENT IN
CONDUCTOR
If we take into consideration conductor border, there
is only one charged particle in conductor, whereas the
other one keeps still. In the case of ordinary conductor,
this charged particle is electron, and could also be hole
carrying positive charge. We could get the equation of
electric current by discussing the movement of charged
particles in electromagnetic field. Here we assume
charged particle is electron and static particle is ion
(normally fixed lattice).
=0
x
eey eiex
e
e
m
 
EVV
 (56)
=0
y
eex eiey
e
e
m
EVV
 (57)
where, we assume the magnetic filed is in z direction,
and resultant field of electric field and induced electric
field is in the direction of x,
y
E
is the electric field
caused by accumulation of electric charge in conductor
border, also is named Hall electromotive force. In equi-
librium, the macroscopic velocity of electron in y
direction is zero: =0
ey
V. We get:
=
x
ex
eei
e
m

E
V

(58)
=ex
yc
V
E
B (59)
If density of electron number is n, then the electric
current and Hall electromotive force in conductor is:
=
x
x
j
E

(60)
11
==
yx
nec nec


Ej
B
j
B (61)
So, when induced electric current is produced by the
movement of conductor in magnetic field, electric current
and electric field is in accord with Ohm’s law because of
the effect of Hall electromotive force in conductor
border.

Z. L. Yang et al. / Natural Science 3 (2011) 275-284 281
If we ignore such an effect,
y
E
in Eq.57 is zero, and
macroscopic velocity of electron in y direction is not
zero, then we get electron’s velocities in both x and y
direction:
2
2
=
1
ex x
e
eei
ei
e
m




VE


(62)

22
=e
ey x
eei e
e
m
 VE


(63)
and the electric current in conductor is:
2
2
=
1
x
x
e
ei

E
j



(64)
and
== =
yey xx
eeieei
ee
ne mc mc 
j
V
j
B
j
B

(65)
The former electric current is usually called Padersen
current and the latter one Hall current. According to the
two equations, the electric current in the direction of
electric field is affected by magnetic field, and accord
with Ohm’s law in form. However, if the magnetic field
is large enough, it could affect the current to be very
tiny.
Back to the Farady law of electromagnetic induction,
when conductor moves in magnetic field, we could get
induced electric field =

E
VB. We usually compute
the electric current through Ohm’s law. It is correct when
we have considered the effect of Hall electromotive
force but incorrect when we ignore such an effect. Beca-
use the relationship between induced electric current and
magnetic field is more complicated, the induced current
is very tiny when electric field is strong.
Then the electric current in conductor can not be
computed by the normal induced electric field and the
induced electric current equation:

=
j
EVB  is
conditional on considering the border Hall electric field
and ions keeping still.
6. THE NATURE OF JOULE'S LAW
When electricity flows through a substance, the rate of
evolution of heat in watts equals the resistance of the
substance in ohms times the square of the current in
amperes. This is Joule’s law, a basic relation in physics.
As experimentally determined and announced by J. P.
Joule, the law states that when a current of voltaic
electricity is propagated along a metallic conductor, the
heat evolved in a given time is proportional to the
resistance of the conductor multiplied by the square of
the electric intensity. Today the law would be stated as
2
=
H
RI , where
H
is rate of evolution of heat in
watts, the unit of heat being the joule; R is resistance
in ohm’s; and
I
is current in amperes. This statement
is more general than the one sometimes given that
specifies that R be independent of
I
.
In the basic electromagnetism, Joule’s law is accu-
rately derived and discussed. The normal discussion is as
follow. Recalling the work done on charge q by an
electric field in moving, the charge along some contour
C is:

=d
C
Wq 

E
rl (66)
When a steady stream of charges (electric current)
flowing along contour C, we need to determine the rate
of work per unit time, the power applied by the field to
the current. The time derivative of work is power, so we
can have,


dd
== d
ddC
W
Pq
tt 

E
rl (67)
Since the electric field is static, we can have,

2
d
=d==
dC
q
PIVIR
t
Erl (68)
where V is the electric potential difference between
either end of the contour. This power is transformed into
the heat in the conductor when electric current flowing
inside.
Considering now the power delivered in some volume,
the volume of a resistor, we can get the power density
as:


33
=VoltmAm=WmpEJ (69)
In history, the Joule’s law is discovered by the
experiment of electric current flowing along conductor,
and is expressed by the square of electric current. We
may have the sense that the heat is produced by electric
current. If we look inside the physics processes in the
conductor, we can find the heat is the work done by the
electric field. In the case the macroscopic electric current
is zero, the electric field will contribute energy to the
conductor.
We can have a simple derivation of the Ohm’s law and
Joule’s law in the view of charged particles moving in
the electric field. The electric current flowing in cond-
uctor is that the charge particles (e.g. electrons) move in
the conductor. Supposing the electrons moves with the
average velocity e

in the action of electric field, the
electric current density can be expressed as:
=ee ne

j

(70)
where e n and e represent the density and charge of

Z. L. Yang et al. / Natural Science 3 (2011) 275-284 282
electron. The collision frequency of electrons with ions
or neutral atoms is supposed to be =1

 ,

is the
free motion time. In every collision, the electron will
lose its moment ee
mu and its energy as 2 12 ee mu. For
steady electric current, we have,
=
ee
me

uE (71)
Therefore, we can get the electric current as:
=jE

(72)
where
2
=e
e
ne
m

 (73)
is the electric conductivity, and
=1


(74)
is the resistivity of the conductor. Eq.71 is the ohm’s law.
with the above discussion, we can also get the heating
from the electron to ions or atoms. In every collision, the
energy from electron to ions or atoms is 2 12 ee mu. In
unit volume, the energy from electrons to conductor is
2
12 eee nmu in the time interval

. Therefore in unit
time and unit volume, the heating energy which is from
the electric field is,
2
=h
E
 (75)
The equation can be written as
2
=hj

(76)
This is the Joule’s law.
Though the heating can be expressed as the two forms
by electric field or electric current, the heating comes
from the electric field accelerating charged particles and
the collision transporting kinetic energy to thermal
energy. When there is macroscopic electric current, the
heating can be represented by electric current. However,
when the macroscopic electric current is not presented,
the heating can be happened by the electric field.
7. THE ENERGY TRANSPORT IN
PLASMA
From the discussions in Section 3 and Section 4, we
can find that the electric field perpendicular to the mag-
netic field cannot cause macroscopic electrical current in
plasma. This leads to the failure of the concept of
magnetic frozen-in and so do the dynamo and magnetic
reconnection, which are widely applied in space science
and astrophysics.
One may worry about the fact that the energy released
and eruption concerning the magnetic field, such as the
plasma heating and the magnetic activities in solar
physics and space science. However, we can easily und-
erstand the physics processes in plasma with out the
concept of “magnetic frozen-in”.
Recalling the charged particle drift in electromagnetic
field, we notice that the drift velocities of the particles
represent the motion of the guiding center. The drift
velocity depends only on the electric field and magnetic
field, and is not related to the initial velocities of the
particles. Eventually, the particle is experiencing the
motion of drift and Lamor motion. The Lamor velocity
depends on the initial velocity of the particle. For the
collective effect of the particles, the Lamor motion
inside the particles will increase the thermal velocities of
the particles through collision, it does not affect the
macroscopic motion.
Now we consider the plasma with initial velocity enter
the region with magnetic field. The external electric field
is always screened. The plasma will be captured by the
magnetic field, the drift velocity of the plasma is zero
perpendicular to the magnetic field. The kinetic energy
of the plasma perpendicular to magnetic field is
transformed to the thermal energy of the plasma. With


representing the increase of the thermal energy of
the plasma in unit volume, we can get:
2
1
=2




(77)
where


is the velocity of plasma perpendicular to
magnetic field,

is the density of the plasma.
Because there is no electric current generated in the
processes, the external magnetic field does not change
by the motion of plasma.
When plasma is in varied magnetic field, an induced
electrical field will be in the plasma. From Farady’s law,
=t



B
E, the induced electric field will be always
perpendicular to the magnetic field. In basic electro-
magnetism, the variation of the magnetic field will cause
eddy electric field, “the eddy current dissipation” will
happen in the plasma. In metal conductor, the eddy
current dissipation is owing to the small resistivity and
strong eddy current.
In fact, the eddy current is not necessary for the
dissipation. The induced electric field directly contribute
energy to the thermal energy of the conductor or plasma
in the varied magnetic field. Generally, the charge
particles in electric field will gain kinetic energy and
form electric current. However, the existence of the
magnetic field will act on the current and turn the current
into small circles. The average effect of the current will
be zero. But the kinetic energy of the particles will be
transformed into the thermal energy due to the collision
of the particles. The final effect of the induced electric
field for the particles is heating the plasma or conductor.

Z. L. Yang et al. / Natural Science 3 (2011) 275-284 283
In plasma, the induced electric field is always per-
pendicular to magnetic field, so the current by the in-
duced electric field will be turned around by the
magnetic field and forms small circles. The radius of the
current is much small. There will be no macroscopic
current due to the overlap of the circled electric current.
However, the induced electric field will contribute
energy to the thermal energy of the plasma. Since the
induced electric field depends on the variation of mag-
netic field, the increase of the thermal energy of the
plasma will come from the variation of the magnetic
field. We can suppose the increase of the thermal energy
as:
2
2
=ind t 





B
E (78)
Though the transfer of magnetic field to thermal
energy may depends on the collisions between charged
particles, there is no macroscopic electric current inside
the plasma.
From the above discussions, we can conclude that the
heating of the plasma in magnetic field may comes from
the motion of plasma across magnetic field, and the
variation of magnetic field.
Eventually, the processes of plasma heating by
variation of magnetic field is observed in the solar
atmosphere. Zhang, Zhang & Zhang [20] studied the
relation of CME with magnetic field variation. Then
found that all of the CME have the variation of magnetic
field and maybe the source of CME eruption.
8. CONCLUSIONS
In the present paper, we get the electric current and
the plasma velocity in the fully ionized plasma by
solving the dynamic equations of charged particles. The
macroscopic electric current is decided by the difference
of the averaged velocities of ions and electrons. The
electric current and plasma velocity are completely de-
cided by electromagnetic field and external forces.
When the external forces i f and e f acting on ions
and electrons respectively, including the pressure gra-
dient, the friction and gravitation, are considered, the
electric current in the plane perpendicular to magnetic
field will be the form of Eqs.50-52, the electric field
perpendicular to magnetic field does not contribute to
the macroscopic electric current. The macroscopic electric
current depends only on the external forces.
The electric current perpendicular to the magnetic
field do not depend on the electric field. The induced
electric field in magnetic field does not contribute to the
electric current, since the induced electric field is always
perpendicular to the magnetic field. In any reference
system with velocity

, the induced electric field
'

B

is perpendicular to the magnetic field, and will
not contribute to the macroscopic electric current, so
cannot distort the magnetic field in space. The velocity
of the plasma is not coupled with magnetic field.
When the external forces are ignored, there is no
electric current in the plane perpendicular to magnetic
field. The Pedersen current and Hall current are zero.
The plasma has a velocity,
2
=

E
B
uB (79)
It is the global drift of the plasma in electromagnetic
field. In plasma, the electric field is screened, the drift
velocity is zero.
When plasma with average velocity

enter mag-
netic field, the plasma will be confined by the magnetic
field. The kinetic energy of the plasma perpendicular to
magnetic field will be transported into plasma thermal
energy. The macroscopic velocity perpendicular to mag-
netic field is zero. It is one process for plasma heating.
When plasma is in varied magnetic field, the induced
electric field will heat plasma. The macroscopic electric
current is not presented for the existence of magnetic
field. The heating process is not from Ohm's dissipation,
but directly from the action of induced electric field and
magnetic field. The increase of the plasma thermal
energy depends on the variation of magnetic field with
time.
In this paper, we have not considered the boundary
condition of the plasma and the inhomogeneous mag-
netic field. The effect of the boundary and inhomo-
geneous field will have effect on the electric current in
the plasma. The work will be done in future research.
REFERENCES
[1] Falthamma, C.G. (2006) Comments on the motion of
magnetic field lines. Americal Journal of Physics, 74,
454-455. doi:10.1119/1.2180285
[2] Alfven, H. (1976) On frozen-in field lines and field-line
reconnection. Journal Geophysical of Research, 81,
4019-4021. doi:10.1029/JA081i022p04019
[3] Alfven, H. (1942) Existence of electromanetic-hydron-
amic waves. Nature, 150, 405.
doi:10.1038/150405d0
[4] Lighthill M.J. (1960) Studies on magneto-hydrodynamic
waves and other anisotropic wave motions. Philosophical
Transactions of the Royal Society, 252A, 397-430.
[5] Witalis, E.A. (1986) Hall magnetohydrodynamics and its
applications to laboratory and cosmic plasma. IEEE
Transactions on Plasma science, 6, 842-848.
doi:10.1109/TPS.1986.4316632
[6] Spitzer L.Jr. (1962) Physics of fully ionized gases, sec-
ond revised edition. Dover Publications, Inc., New York.
[7] Somov, B.V. (2000) Cosmic plasma physics. Kluwer

Z. L. Yang et al. / Natural Science 3 (2011) 275-284 284
Academic Publishers, Dordrecht, pp. 181-190.
[8] Volkov, T.F. (1966) Hydrodynamic description of a colli-
sionless plasma, In: Leontovich M.A., Ed., Reviews of
Plasma Physics, Consultant Bureau, New York, 4, 1-21.
[9] Sweet, P.A. (1958) The neutral point theory of solar
flares. In: Lehnert B., Ed., Electromagnetic Phenomena
in Cosmical Physics, Cambridge University Press, Lon-
don.
[10] Parker, E.N. (1957) Sweet's mechanism for merging
magnetic fields in conducting fluids. Journal Geophysi-
cal of Research, 62, 509-520.
doi:10.1029/JZ062i004p00509
[11] Dungey, J.W. (1961) Interplanetary fields and the auroral
zone. Physical Review Letters, 6, 47-48.
doi:10.1103/PhysRevLett.6.47
[12] Paschmann, G..O., Sonnerup, B.U., Papamastroakis, I.
Sckopke, N., Haerendel, G., Bame, S.J., Asbridge, J.R.,
Gosling, J.T., Russell, C.T. and Elphic, R.C. (1979)
Plasma acceleration at the earth’s magnetopause: Evi-
dence for reconnection. Nature Lond, 282, 243-246.
doi:10.1038/282243a0
[13] Paschmann, G. and Treumann. R. (2003) Auroral plasma
physics. In: Paschmann G., Haaland, S., Treumann, R.,
Ed., ISSI Space Science Series, 15, ISBN 1-4020-0936
-1.
[14] Yamauchi, M. and Blomberg, L. (1997) Problems on
mappings of the convection and on the fluid concept.
Physics and Chemistry of the Earth, 22, 709-714.
doi:10.1016/S0079-1946(97)00200-0
[15] Parker, E.N. (1979) Cosmical magnetic fields. Clarendon
Press, Oxford, pp. 31-32.
[16] Gomobsi, T.I. (1994) Gaskinetic theory. Cabridge Uni-
versity, New York.
[17] Song, P., Gombosi, T.I. and Ridley, A.J. (2001) Three-
fluid Ohm’s law. Journal of Geophysical Research, 106,
8149-8156. doi:10.1029/2000JA000423
[18] Pandey, B.P. and Wardle, M. (2008) Hall magnetohydro-
dynamics of partially ionized plasmas. Mononly Notices
of the Royal Astronomical Society, 385, 2269-2278.
doi:10.1111/j.1365-2966.2008.12998.x
[19] Chen, F.F. (1974) Introduction to plasma physics. Plenum
Press, New York.
[20] Zhang, Y., Zhang, M. and Zhang, H.Q. (2007) A statisti-
cal study on the relationship between surface field varia-
tion and CME initiation. Advances in Space Research, 39,
1762-1766. doi:10.1016/j.asr.2007.02.003