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9. NEW RELATIVISTIC THEORY OF PARTICLE DYNAMICS

          For derivation of main expressions of particle dynamics following from new space-time theory we shall use Einstein's method  [3]. Einstein A. Zur Electrodynamik bewegten Korper // Annalen der Physik. - 1905. - B., 17. - s. 891 – 921.

           Let some particle having a charge e_o and rest mass m_o be at some time moment at rest in the inertial referenve frame B, which is moving at a speed u in the positive direction of  X axis of the inertial reference frame A. Let this particle be in the electromagnetic field, which source is at rest in the inertial reference frame A. Then we can suppose, that motion of this particle in the inertial reference frame B obeys hereafter to the equations

.(9.1)

where

.(9.2)

E_x', E_y', E_z' are components of intensity of the electric field acting upon the particle, which is at rest in the inertial reference frame B;  E_x,  E_y, E_z, B_y, B_z are the components of electric field intensity and magnetic field induction created by a source (being at rest in the inertial reference frame A) in that point of reference frame A, in which the particle under consideration is at each specific moment of time. At that expressions (9.2) are derived applying transformation (6.11) to Maxwell's equations in the reference frame A for electromagnetic field, the source of which is at rest in the inertial reference frame A (similarly to a method, which was used for derivation of expressions (8.6) by applying transformation (6.9) in section 8).

            Substituting expressions (9.2) into equations (9.1) we have

.(9.3)

The right-hand parts of equations (9.1) and (9.3) are the forces (in the inertial reference frame B) acting upon a particle with a charge e_o, which is at rest in the inertial reference frame B. That is why in this equations the formula (8.22) for dependence of charge upon speed  is not used. But in the right-hand parts of equations (9.3) the forces acting upon a particle are expressed using components of electromagnetic field vectors measured in the reference frame A.

            Let us now represent the left-hand parts of equations (9.3) using coordinates and time measured in the inertial reference frame A. For this purpose let us use transformation (6.9) (because we now consider the events happening to a particle, which is at rest in the inertial reference frame B).

            Having taken twice the time t'  derivatives from each of three last equations of transformation  (6.9) and substituting in the resulting expressions (after differentiation) the values

we have

..(9.4)

Now let us substitute the right-hand parts of expressions (9.4) instead of left-hand parts of the equations (9.3). As a result we have

.(9.5)

where, as before,

If E_z and B_y are the only components of electromagnetic field , then from three expressions (9.5) we shall have only the last

(9.6)

The curvature of a particle trajectory under action of this deflection field takes place in the plane xz and the radius R of the trajectory curvature we can determine from the formula

(9.7)

If only a magnetic field with induction B_y is present, from equations (9.6) and (9.7) we shall obtain the expression for the radius of   particle trajectory curvature in the transversal magnetic field

(9.8)

If only electric ffield with intensity E_z is present, from equations (9.6) and (9.7) we shall obtain the expression for the radius of particle trajectory curvature in the transversal electric field

 (9.9)

In the special relativity theory the analogous formulas for formulas (9.8) and (9.9) are the expressions

(9.10)

(9.11)

where V is the speed of particle motion according to the special relativity theory, which can not exceed the constant c_o.

From expressions (9.8) and (9.9) we  have

.  (9.12)

And from expressions (9.10) and (9.11) we have

 (9.13)

The formula (9.12) from the new space-time theory coincides with the formula (9.13) from the special relativity theory, if between "V-speed" from the sspecial relativity theory and "u-speed" from the new space-time theory there dependences V = u (1 + u²/c_o²) ^-1/2 (7.12) and u = V (1 - V²/c_o²) ^-1/2 (7.13).

If only electric field with intensity Е_x is present, from three expressions (9.5) we shall have only the first one, which can be written in the form

 (9.14)

             Let a particle with charge e_o and rest mass m_o be initially at rest in the origin of the reference frame A. At a certain moment on this particle an eccelerating electric field begins acting, the source of this electric field is at rest in the inertial reference frame A and the vector of electric intensity of this field is parallel to X axis of the inertial reference frame A. Then within an infinitesimal line segment dx, within which the particle acceleration may be considered as constant, this particle will take from the electrostatic field the energy amounting to

(9.15)

Having substituted instead of the expression in the right-hand part of the formula (9.15) the left-hand part of the equation (9.14), we shall obtain

 (9.16)

But in the expression (9.16) it is possible, considering the value c_u as a constant, to perform the following transformations

 (9.17)

where b_u = u/c_u. That is why the expression (9.16) may be written in the form

 (9.18)

The full energy extracted by the particle from the electrostatic field and converted into the particle kinetic energy we can determine if we shall perform integration of the expression (9.18) within the limits from zero to b_u

   (9.19)

Having performed the integration we obtain

.  (9.20)

The appearance of dependence (9.20) of a particle kinetic energy upon the particle speed in the new space-time theory coincides with the appearance of the analogous dependence from the special relativity theory. But in the formula (9.20)   the value b_u is determined by the formula

(9.21)

and in the special relativity theory instead of b_u we have the value b, which is determined by the formula

(9.22)

By the way, if in the formula (9.22) we shall substitude the expression V = u (1 + u²/c_o²) ^-1/2 (7.12), we shall obtain the formula (9.21). Consequently, taking into account the formula (7.12), the dependence (9.20) of particle kinetic energy upon its speed in the new space-time theory coincides with analogous dependence from the special relativity theory. But, having substituted the formula (9.21) into the formula (9.20), we shall obtain

 (9.23)

Then, if we, as before, shall consider, that

(9.24)

is the rest energy of  a particle, the formula (9.23) can be considered as the difference between the particle full energy

 (9.25)

Having squared the both parts of the equation (9.25) we have the expression

   (9.26)

which can be considered as relation between the full energy of the particle and its momentum in the new space-time theory

 (9.27)

is the momentum of a particle in the new space-time theory.

Having substituted the expression u = V (1 - V²/c_o²) ^-1/2 (7.13) in the formula (9.28), we obtain the expression

(9.29)

Having solved the expression (9.23) with respect to a particle speed, we obtain the dependence of a particle speed upon its kinetic energy in the new space-time theory

 (9.30)

Substituting the formula (9.30) into the formula (9.8) we obtain the dependence of a radius of charged particle trajectory curvature in the transversal magnetic field upon the kinetic energy of the particle, which is valid in the new space-time theory

 (9.31)

This dependence coincides with the appropriate dependence from the special relativity theory, which determines the operation of cyclic accelerators of elementary particles. The formula (9.31) can be also written in the form

 (9.32)

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