Electromagnetic drift J.L. Fernandez-Chapou1 J. Granados-Samaniego2 J.M. Velázquez-Arcos3 C. A. Vargas4

1Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Av. San Pablo 180. Col. Reynosa Tamaulipas, CP 02200, México D.F., México. e-mail: jlfc@correo.azc.uam.mx,, Tel: +52 5553189026, Fax: +52 5553189540. 2Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Av. San Pablo 180. Col. Reynosa Tamaulipas, CP 02200, México D.F., México. e-mail: jgs3112@yahoo.com.mx, Tel: +52 5553189020, Fax: +52 5553189540. 3Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Av. San Pablo 180. Col. Reynosa Tamaulipas, CP 02200, México D.F., México. e-mail: jmva@correo.azc.uam.mx, Tel: +52 5553189504, Fax: +52 5553189540. 4Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Av. San Pablo 180. Col. Reynosa Tamaulipas, CP 02200, México D.F., México. e-mail: cvargas@correo.azc.uam.mx, Tel: +52 5553182054, Fax: +52 5553189540.

Abstract – We define, by a natural way, a velocity field at a space region occupied by an electromagnetic field. By using that velocity field we can obtain a relativistic expression for drift velocity of an electrically charged particle in presence of an electromagnetic field. Taking in account temporal components and the temporal space of the energy momentum tensor of electromagnetic field, and also his Lorentz invariants, it is possible to define the concept of electromagnetic field “mass density in rest”. Then, we can obtain a very important expression for the velocity field in terms of electromagnetic field energy density and mass density. Also we can show that if we calculate the drift velocity for an electromagnetic

field such that 0E B and B E with the dielectrically permittivity of a transparent media, then drift velocity is the same as light velocity in such media. Key Words – Drift velocity, Electromagnetic field, energy momentum tensor, Lorentz invariants.

1 INTRODUCTION

When we study the Plasma behavior at presence of electromagnetic fields appears the concept of Electromagnetic Drift Velocity. This is the velocity, in the orthogonal direction to the fields, which acquire the electrically charged particles in the presence of a crossed electromagnetic field (EMF). Such velocity is independent of the particle’s properties, that is, there is no dependence of the mass neither electrical charge nor any other property. The value of drift velocity exclusively depends of EMF, and we can say that associated to an EMF exists, in a natural way, a velocity field. For this reason it is possible to see that when we apply an EMF at a spatial region occupied for a Plasma, this will be pushed at full to move it with the drift velocity independent of another effects that may produce the EMF at the inside of Plasma region. In this paper we ___________________________________________

consider only the drift velocity purely electromagnetic. In Plasma Physics scientific Literature it is well known the mathematical expression for drift velocity, which is valid in the case of non-relativistic movement, that is, the particle´s velocities are much smaller than the velocity of light. In this work we write the problem in a full Lorentz´s covariant way, and we obtain a mathematical expression for drift velocity in general that is valid for any value of particle´s velocities. Also, we show some properties of field velocities, which are interesting and directly related with the properties of momentum-energy of EMF. Following an analogy of one particle´s mechanics, we introduce a concept called EMF rest mass density. The second section discusses the invariants of the EMF that are algebraically independent from one another. Such invariants are very useful to study the properties of the EMF in different inertial reference systems. The third section analyses the drift movement deduced of the properties of energy-momentum of the field, looking for this for an inertial system of reference in which the vector of lineal momentum density of the field (Poynting vector) is zero. In analogy with the problem of the mechanics of a particle, this is as a rest inertial system of the EMF which is also not unique. Finally we obtain interesting mathematical expressions about drift velocity in terms of the energy density and rest mass of the EMF.

2 ELECTROMAGNETIC FIELD INVARIANTS

Due that the light velocity is a constant we deduce that the time interval between two events is invariant at a transformation from an inertial system to another. As we will show, in the case of EMF there are two algebraically independents invariants.

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The electromagnetic field defined in Minkowski´s space-time is represented by the anti-symmetric

tensor F F in such a way that his components are written in terms of the three-

dimensional electric E and magnetic B vector

fields by 0iiF E and ij ijk

kF B where ijk is

the alternant three-dimensional symbol. Greek index , , take the values 0, 1, 2 ó 3; while the Latin

index , , ,i j k take the values 1, 2 ó 3. For the EMF

is possible to build the next scalar invariants:

invariant; invariant,F F F F

By electromagnetic tensor definition, in three dimensions, (with c=1), also we have 2 22 =invariantF F B E (1)

4 = invariantF F B E (2)

Based on vector analysis we know that it is possible build products with two vectors, polar, axial or one polar and one axial or vice versa, obtaining a true scalar, in the two early cases, and a pseudo scalar at

each one of the last cases. Since E is a polar vector

and B an axial vector, we have that (1) gives a true scalar and (2) a pseudo scalar. The difference is that the first is not modified by a spatial coordinate inversion, while the pseudo scalar becomes a true scalar if we take the product with itself. The invariance of (1) and (2) carry to a series of implications of great utility since it is possible to express the one particle movement into time independent uniform fields and with velocities near to limit velocity, in some of the next cases: uniform electric field, uniform magnetic field, magnetic field parallel to electric field, or both field perpendiculars. So that, if we have a referential system where 0E E

and 0 ,B B 0,E B this will be true at any other inertial reference system, that is, ever it is possible to find an inertial reference system in which E and Bwill be parallel and also 0 0 0E B and

2 2 2 20 0 ,E B E B will be fulfilled. By the other

hand, if 0 0 0E B implies that 0E and 0B are

perpendiculars and if 2 20 0 0,E B is fulfilled, it is

possible to find an inertial reference system in which B is null. If

0 0 0E B and 2 20 0 0E B then E and

B are perpendicular in whatever inertial reference system and it is not possible to find an inertial reference system in which both vectors are zero.

3 DRIFT VELOCITY

In a space-time area where is present an EMF, we define the symmetric tensor T corresponding with

the energy-momentum tensor as 14( ) ,T x F F F F

where F is the electromagnetic field tensor and1, 1, 1, 1 .diag

If this area is occupied only by an EMF free of sources, then T satisfy the continuity equation

, 0.T From Relativity special theory, we know that all the energy forms have an equivalent mass, also we know that all the systems have a rest mass, that in some cases it is null. By means of energy momentum tensor T (x) we define the momentum tetra-vector at a certain reference system like:

3( ) ( ) ; ( )p x P x d P x T v (3)

d3 is the infinitesimal element of a spacialoid three-dimensional hiperplane and v is a unitary tetra- vector perpendicular to the hiperplane. This is the tetra-vector velocity of the hiperplane reference system whit respect to the three-dimensional hiperplane defined as t=cte. The rest mass of a system (anyone not necessarily electromagnetic) is defined as the energy in the reference system where the momentum three vector P is null; the associated reference system is denoted by 0. That is 0

(0) ,M P

(0) 0.P Since P P is invariant, then 2 .M P P (4)

The existence of 0 is guaranteed by the fact that P is temporaloid. So, it is possible to write eq. (3) like ,P M v (5) if 3( ) .M T x d (6)

Now consider the energy-momentum tensor divided by c2=1, which will be called the mass tensor. In the system 0

0 00(0) (0) (0) (0) (0) (0) ,M P V P V M V M (7)

hence for any system ,M V P V M V (8)

For physics conditions, this mass tensor must be symmetric. With the mass tensor M is possible to define the rest system as that system in which

0(0) 0; 1, 2,3,kM k

and conclude that for some system,P MV

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with M given by (8) or (3). In a frame of reference ,we define a field of velocities over a region of space occupied by an electromagnetic field, as the velocity that must have a system of reference 0 with respect to , so that the three-dimensional components with respect to 0 of P are zero, is required in addition that T be diagonalizable (This is true for the EMF always that the invariants of field B2-E2 and E Bthey are not both zero), that is, that

(0) 0P and its

direction and sense coincide with that of P at . This velocity field is both well-defined for a region occupied by an EMF as for a region occupied by matter or any other type of field. What is needed only is to have defined a tensor of energy-momentum of the system. At a region occupied by an EMF without sources, or with various sources distinct to the electro-magnetic, then according to (3.7) and (3.14)

1/ 22 2 31 14 2 ,M F F F F F d (9)

or sustituting

2 2

12

2( ),

4( ),

F F B E

F F E B (10)

so that

222 2 312 4 ,M B E E B d (11)

and the mass density is 222 21

2( ) 4 .m x B E E B (12)

Using the vector transformation rule E and B under a Lorentz transformation

' ,

' ,

E E v B

B B v E

it is possible to find the speed of the system of reference (where 0P ) with respect to the start system, that is, a system of reference with a velocity v that is perpendicular both as E like B that is

parallel to E B and such that ' ' 0E B which indicates that the fields are parallel to this system. Taking the cross product of 'E with 'B and equating to zero yields

2 22 2 2 2

22 2

4

0 2ˆ( ),

E B B E E B

E B E Bv S x (13)

S is a unit vector in the direction of the Poynting vector. The previous expression represents the velocity of the system of reference . But

22 2 2 212 yu E B E B E B E B p so that (13)

takes the following form: 2 2

2ˆ ˆ( ) ( ) ( ),u m u m

p pv x S x S x (14) with

222 212( ) 4 .m B E E B

at once we calculate u2-p2:222 2 2 21

4( ) 4 ,u p B E E B

since B2-E2 and 2

E B are invariants then u2-p2 has

the same value in all inertial reference systems. Particularly in the system 0 where 0,p we have

222 2 2 210 4( ) 4 ,u m B E E B

that represents the density of energy in the system 0,which is called mass density. From equation u2-p2=m2, solving for p and substituting into (14) gives the following expression:

2 20ˆ ˆ( ) ,u m u m

u mu mv x S S

(15)and multiplying by u m

2 2

0ˆ .u m v u m S p (16)

Consider a uniform EMF and assume that there is a particle in the EMF, such particles gain a speed average equal to (0) .v Call at this velocity average drift velocity VD of the particle in question (i.e. v(0) VD). In accord to (12) and (16)

ˆ,U MD U MV S (17)

If the field is not uniform, then the drift velocity is different at every point in the region of space. If there are supraluminals charged particles and these are able to interact with the EMF, then perhaps they will move with a speed of drift equal to

ˆ.U M

D U MV S (18) Usually the root in (15) eliminates, which results with VD>1, whereas any physical meaning might be. In the case of supraluminal particles the forces tends to decelerate them. (The speed of the supraluminals particles may decrease approaching c as much as you like under the action of a force for a long time). For this reason, it is possible to assume that to interact with the cross EMF, it could be them slowing down until reaching the speed expressed in (18). In the case that

2 2

0,0,

E B

B E

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the motion of a particle in a uniform EMF is reduced to consider the motion in an inertial reference system conveniently chosen in which only is present a uniform electric field. Substituting into (13) and choosing the sign that gives us v<1,

,BEv

It is possible to note that the value of the speed with which you must move the inertial reference system only depends on the rate between the magnetic field and electric field, is independent of the electrical charge and the energy of the particles involved. When is fullfilled

2 20 0,E B y B Ethe analysis of the movement is reduced to consider an inertial reference system in which the intensity of the electric field is reduced to zero, and only the magnetic field is present. To make a similar calculation considering the expression for drift velocity we found that

2 ; with orthogonal to B.E B ED D BB

V V E

Is immediately analyzed the case when 2 2cos 0 0,E B EB y B E

Using the expression found before for drift velocity ˆ,U M

D U MV S

and calculating U and M,2 21 1

2 2; cos ,U E M E

Then the drift velocity is 1212

1/ 21 cos1 cos

ˆ.DV S

In the case 2 20 0,E B y B E

Tensor energy momentum is not diagonalizable and as a result there is no drift speed since E and B are perpendiculars in any inertial reference system and it is not possible to find a system in which any or both vector field E and B be zero. However particles are constantly accelerated in the direction perpendicular to the plane determined by vectors E and B so that the speed of those tends to c = 1, speed of light, in an infinite time. In a transparent material with >1 and

0,B E y E BIt is possible to know the drift velocity by calculating M and U,

1/ 2222 21 12 24 1M B E E B E

2 2 21 12 2 1 .U E B E

substituting into the expression of the drift velocity we have

1/ 21 ,cv c

from Optics is known that n

and that for almost all transparent dielectric materials is possible considering that 1, therefore

,nthen

.cD nV

Note that this velocity coincides precisely with the velocity of light in the material. This result seems curious and is consistent with the result that would be obtained by making tends to 1. In this case

,E B then VD c and M 0. The invariants of the

EMF of a plane electromagnetic wave are zero, so that M=0 and formally, VD=c as occurs with the photon. Then we found that for a EMF in a transparent medium drift velocity also coincides with the velocity of the photon in that medium. Of course this does not mean that the photon acquires mass moving in such médium because that is actually observed is the velocity of diffusion in the medium, because these are absorbed and reissued by the atoms moving always at speed c in interatomic spaces.

4 CONCLUSIONS

We have presented a study where, based on the properties of energy-momentum Tensor and invariants of electromagnetic field, we have deduced a mathematical equation for Drift velocity of an electrical charged particle at presence of arbitrary electric and magnetic fields. From analyzed cases, the most interesting one is that when the fields are orthogonal, since the resultant trajectories are helicoidal or cicloidal. Also we have obtained an expression for the drift velocity in terms of mass and energy densities, whose implications must be analyzed deeper. We found an attractive result when we introduce the definition of refraction index in terms of electrical permittivity and magnetic permeability. This result may be understood like that the photon movement is a result of drift electromagnetic.

References

[1] A.O Barut, Electrodynamics and Classical Theory of Fields (Dover, New York, 1980)

[2] J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1980)

[3] L. D. Landau and E. M. Lifshitz, Teoría Clásica de Campos, 2da. Ed. (Reverté, Barcelona, 1973)

[4] F. Rohrlich, Am. J. Phys. 28, 639 (1960)

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