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[Si] has been ignored in textbooks|1, 2] and in the specialized literature. This neglect leads to its replacement by charge and current densities, which are phenomenological quantities. We name this tensor the source part because it is responsible for the source terms which, as seen below, will appear in Maxwell’s equations.

Figure 5 [Si] has been ignored in textbooks|1, 2] and in the specialized literature. This neglect leads to its replacement by charge and current densities, which are phenomenological quantities. We name this tensor the source part because it is responsible for the source terms which, as seen below, will appear in Maxwell’s equations.

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FIGURE 1 — Schematics of the three frames of coordinates in the geometrical space. M is the point of observation. Tensors leading to Maxwell’s equations are expressed in the laboratory frame. Electromagnetic particles tensors are first described at M in spherical coordinates r,6,y with M fixed with respect to the origin Op. A particle spinning around the Z axis is described by the tensors obtained from a local Lorentz transformation at M. Note that M close to Op» allows the description of the structure of the particle. The long range behavior of the particle is obtained when M is far from Op.
FIGURE 1 — Schematics of the three frames of coordinates in the geometrical space. M is the point of observation. Tensors leading to Maxwell’s equations are expressed in the laboratory frame. Electromagnetic particles tensors are first described at M in spherical coordinates r,6,y with M fixed with respect to the origin Op. A particle spinning around the Z axis is described by the tensors obtained from a local Lorentz transformation at M. Note that M close to Op» allows the description of the structure of the particle. The long range behavior of the particle is obtained when M is far from Op.
‘TABLE 1 — Synopsis of the different electromagnetic tensors. Tensors in the upper part are obtained from the gradient a, of the 4-potential A’ at point M. a; is split into its symmetric and antisymmetric parts si and i in the coordinate frame where M is at rest wit respect to the origin (a.s. splitting). Then these tensors are transformed into their covariant form 84; and f;; by lowering the index 7 with the metric tensor (r.l. operations). A Lorentz transform L is performed on these tenors to find those of the spinning particle. A; and ay; in the lower part are obtained by the lowering index operation acting on A* or a,’. Then ax; is split into its symmetric part S,; and its antisymmetric part F,; which is the usual electromagnetic tensor. The non-commutativity of the r.l. and a.s. operations is fundamental.
‘TABLE 1 — Synopsis of the different electromagnetic tensors. Tensors in the upper part are obtained from the gradient a, of the 4-potential A’ at point M. a; is split into its symmetric and antisymmetric parts si and i in the coordinate frame where M is at rest wit respect to the origin (a.s. splitting). Then these tensors are transformed into their covariant form 84; and f;; by lowering the index 7 with the metric tensor (r.l. operations). A Lorentz transform L is performed on these tenors to find those of the spinning particle. A; and ay; in the lower part are obtained by the lowering index operation acting on A* or a,’. Then ax; is split into its symmetric part S,; and its antisymmetric part F,; which is the usual electromagnetic tensor. The non-commutativity of the r.l. and a.s. operations is fundamental.
3.3.3 Electric charge. This section briefly describes the method to find the electric charge Q associated with an e.m. particle. The main steps are as follows : 1- The second couple of Maxwell’s equation (63) relates the electric displacement D to the charge density p. We use this equation here to define p. For the moment p is an intermediate quantity that will be integrated to obtain the total charge of the particle and used later in the case of a spinning particle. 2- There are few fundamental principles that fix the dynamics of a physical system. The first of them is the principle of least action which is expressed by the Euler-Lagrange equations :
3.3.3 Electric charge. This section briefly describes the method to find the electric charge Q associated with an e.m. particle. The main steps are as follows : 1- The second couple of Maxwell’s equation (63) relates the electric displacement D to the charge density p. We use this equation here to define p. For the moment p is an intermediate quantity that will be integrated to obtain the total charge of the particle and used later in the case of a spinning particle. 2- There are few fundamental principles that fix the dynamics of a physical system. The first of them is the principle of least action which is expressed by the Euler-Lagrange equations :
The expression of the potential, as seen by a non-rotating observer at M, is : Let us apply the transformation to solution e. We split the tensor(28) into its symmetric and antisym- metric parts which we transform into their covariant forms to use both direct and reciprocal spaces. The Lorentz transform gives the rotating tensors [f;,;] and [3,;]. The field tensor is :
The expression of the potential, as seen by a non-rotating observer at M, is : Let us apply the transformation to solution e. We split the tensor(28) into its symmetric and antisym- metric parts which we transform into their covariant forms to use both direct and reciprocal spaces. The Lorentz transform gives the rotating tensors [f;,;] and [3,;]. The field tensor is :
Electric and magnetic inductions are given by the derivatives of the Lagrangian with respect to the com- ponents of the fields E /c and B ; One applies the chain rule and equations (60) to obtain : Other components Dy , Dz, Hy and Hz are obtained from circular permutations of 2, y, z. We use the special notation (Dx , Dy , Dz ) to make a distinction from the usual displacement vector (Dx , Dy , Dz) which appears in Maxwell’s equations. These components are defined from the derivatives of £ with respect to the electric field. These quantities are proportional (Dx = cDx,...). The dimension of Dx is a density of dipoles in 4-space : Q L/(L*) while that of Dx is Q L~? T7?. We have also used a lower index notation Dx , Dy ,... to stress the fact that the components of the induction pseudovectors are those of a type [D*,] tensor :
Electric and magnetic inductions are given by the derivatives of the Lagrangian with respect to the com- ponents of the fields E /c and B ; One applies the chain rule and equations (60) to obtain : Other components Dy , Dz, Hy and Hz are obtained from circular permutations of 2, y, z. We use the special notation (Dx , Dy , Dz ) to make a distinction from the usual displacement vector (Dx , Dy , Dz) which appears in Maxwell’s equations. These components are defined from the derivatives of £ with respect to the electric field. These quantities are proportional (Dx = cDx,...). The dimension of Dx is a density of dipoles in 4-space : Q L/(L*) while that of Dx is Q L~? T7?. We have also used a lower index notation Dx , Dy ,... to stress the fact that the components of the induction pseudovectors are those of a type [D*,] tensor :
The second expression is obtained after simplification by eq.(73). Equating each component of the 4- vector of eq.(74) gives : Dr = (Dx , Dy , Dz) is the symbol for the induction in the reciprocal space.
The second expression is obtained after simplification by eq.(73). Equating each component of the 4- vector of eq.(74) gives : Dr = (Dx , Dy , Dz) is the symbol for the induction in the reciprocal space.
One sees that [sii] 4 [Sei] and [xi] A [f,;]- It is this non-commutativity which leads to the two branches of electromagnetism : The first branch is based on the tensors(1) (expressed in the cartesian frame of coordinates) :
One sees that [sii] 4 [Sei] and [xi] A [f,;]- It is this non-commutativity which leads to the two branches of electromagnetism : The first branch is based on the tensors(1) (expressed in the cartesian frame of coordinates) :
[Fy;] is the standard electromagnetic tensor. Fields are defined from the usual equations (60). 7 Appendix B. This appendix gives explicit formulas for the potentials and the corresponding tensors for the first 5 even electromagnetic particles. Odd solutions are obtained by changing wt into wt — 7/2.
[Fy;] is the standard electromagnetic tensor. Fields are defined from the usual equations (60). 7 Appendix B. This appendix gives explicit formulas for the potentials and the corresponding tensors for the first 5 even electromagnetic particles. Odd solutions are obtained by changing wt into wt — 7/2.
Formulas for the g_; solution are obtained by changing ¢ into y — 7/2 in the preceding expressions.
Formulas for the g_; solution are obtained by changing ¢ into y — 7/2 in the preceding expressions.