## 2. THEORY AND ITS COMPARISON WITH EXPERIMENT |

Let us have a system of coordinates *(x, y, z)* connected with the medium causing
propagation of light. Let the electric field intensity in this medium propagate at speed
*c* in all directions. It is known from Coulomb's law that intensity of the still
standing charge in relation to the system of coordinates *(x, y, z)* decreases with the
square of distance from that charge then represented by hyperboles symmetrical to the charge,
illustrated in section as follows:

Fig. 2.1. The intensity of the stillstanding charge

*r* - distance of the hyperbole point from the beginning

Charge *q* is situated at level *yz* and in the distance of *y _{q}*
from the beginning in the direction of axis

Let us now examine what would happen with the form of curves representing the intensity of the
electric field, if charge *q* will move in a uniform straight line motion in the direction
of the axis *y* at a constant speed *v*. Let's thus join firmly the system of coordinates
*(x', y', z')* with the charge *q*, see fig. 2.2.

*r* - distance of the hyperbole point from the beginning

Fig. 2.2. The system of coordinates *(x', y', z')*

Distance *r'* is measured in direction of axis *y'* from the charge (or from the beginning O'
respectively), while it is valid

r'=r-vt |
(2.1) |

At the moment *t*_{0}=0 both systems become identical.

When , the charge finding itself at the distance of
would emit intensity propagating at speed *c*,
which at the moment of *t* would come to point *r* in time of

(2.2) |

(2.3) |

The index _{id} means that is the case of propagation of the electric field intensity
in direction of the charge motion.

Let the be the distance between the
position of the charge at the moment of
(i.e. when the charge has emitted the intensity to point *r*) and position of the charge at the
moment *t*, when the intensity emitted "has reached" the point *r*.

At the time of the charge will cover the distance of

(2.4) |

This is the distance at which the charge "outrun" the intensity propagated in direction
of the charge motion. Consequently the intensity of the moving charge in relation to the system
of coordinates *(x, y, z)* will change its form in the respective
of various *r*:
it will be deformed (see fig. 2.3)

Fig. 2.3. The intensity of the moving charge in the direction of the motion

It is evident that with increasing distance *r _{i}* (

(2.5) |

(2.6) | |

(2.7) |

*r* are distances of points of hyperbola from the beginning of the non-dashed system,
*r'* are distances of points of hyperbola from the beginning 0' in a dashed system,
*r*, *r'* are variables of the same function
(represented by hyperbolas). In other words, there is distance *r*, that numerically equals
distance. Such distance *r'* numerically
equals distance, both being variables
of the same function . For detail refer to
(2.6) and (2.7). The issue concerns the same Coulomb's law.

By substituing of (2.5) and (2.4) we get

(2.8) |

(2.9) |

(2.10) |

Fig. 2.4. The distance

The charge moving at the speed of *v* parallel to the axis *y* is situated
(at the moment *t*) in the distance of *v.t* from the axis *z*.

At the moment the charge, situated
in the distance of will emit the
intensity to the point *r*.

This intensity will reach at the moment *t* just the point *r* in time of

(2.11) |

(2.12) |

The charge will cover the distance

(2.13) |

at time , while *r'*<0 and
.

This is the distance by which the intensity that propagates in the direction opposite to the movement of the charge, is shifted against the intensity of the stillstanding charge in the direction away from the charge, see fig. 2.5.

Analogically to equations (2.5)-(2.10) we achieve the following:

(2.14) | |

(2.15) | |

(2.16) | |

(2.17) | |

(2.18) | |

(2.19) |

Fig. 2.5. The intensity of the electric field by means of the moving
charge against the direction of motion E^{ad}

Fig. 2.6. The form of intensity for *v* = 0.5*c*

The equations (2.10) and (2.19) are placeable by common equation

(2.20) |

At level *xy*, the section of the intensity hyperboloid is, for the stillstanding charge,
the circle with its centre in the charge, for the moving charge it is the case of all types of Pascal's
screw stocks with the charge at the beginning of the coordinates, see figs. 2.7,
2.8, 2.9 and 2.10.

Fig. 2.7, 2.8. At level *(x, y)* section of the "hyperoloid" of the intensity
for various speeds of the moving charge have a shape of all types of Pascal's screw stocks with charge
at the beginning of the coordinates

Fig. 2.9, 2.10. At level *(x, y)* section of the "hyperoloid" of the intensity
for various speeds of the moving charge have a shape of all types of Pascal's screw stocks with charge
at the beginning of the coordinates

In the period from 1901 to 1906, Kaufmann wrote a number of works, the most coherent of them seems to
be^{(1)}: concerning experimental evidence of "the changeability of mass with speed".
We shall revalue his experiment and will prove - on the basis of the theory given in the preceeding section
2.1.1 - the subject is the influence of intensity of the moving charge on the magnitude
of the deviation of influence of intensity of the moving charge on the magnitude of the deviation of beta-rays
in the crossed electromagnetic field, and not the changeability of mass with speed.

The attempt is done through a short correct description for sake of qualitative examination of the experiment, utilising some of the measured and calculated values given by Kaufmann in (1).

Beta-rays from Ra source, moving at speed are simultaneously deflected in the crossed electric and magnetic field, see diagram in fig. 2.11.

Fig. 2.11. Kaufmann's Experiment - diagram

The device is situated in the evacuated glass vessel. The rays go out from the Ra source, pass the electric screen and create a small spot on a photographic plate.

When the electric field will be created on the condenser plates PP', the additional stripe in the *y*-
direction will arise apart from the non-diverting middle spot close to 0 (consisting of gamma and little diverting
).

When the entire device is situated between the poles of the U-shaped magnet (with the electric field switched
off), the stripe will arise in the direction of the axis *z*.

While at the magnetic field we have the movement of electrons along circles expressed in the following equations

(2.21*) |

in the electric field we first have the movement along straight line

(2.22*) |

(2.23*) |

(2.24*) |

The values *E* used in the calculations

(2.25*) |

Considering out theory on and the values (2.25*) be multiplied by we achieve deviations identical with the results of Kaufmann's experiment, see table 1. Thus the theory under 2.1.1 concerning the intensity of the moving charge of the electric field may be regarded experimentally confirmed.

1631 V | 2603 V | 3250 V | |

y_{b}[cm] |
0.1236 0.1119 |
0.1493 0.1302 |
0.1664 0.1616 |

y[cm] | 0.23626 | 0.3873 | 0.4985 |

y_{T}[cm] |
0.0629 | 0.09947 | 0.12557 |

y_{T}-theoretical value (our new theory): |

^{(1)}Annalen der Physik, Vierte Folge, Band 19, Leipzig 1906, Verlag von
Johann Ambrosius Barth, page 487-552

Let us take the equation (2.20) in the vector form:

(2.21) |

The force acting on the moving electric charge is

(2.22) |

It is known, in line with the classical theory, that a magnetic field is created by the moving charges
and electric currents. The result is that the moving charge creates its own magnetic field of induction
*B** _{q}*. It continues in this field in motion. According to Lorentz, the force acting on the moving
charge in the electromagnetic field at speed

(2.23) |

Intensity ** E** of the electric field according to Lorentz equals to our intensity

As the forces acting on the acting on the moving charge are equal, it must be valid

(2.24) |

i.e.

(2.25) |

while the vectorial equation is in force

(2.26) |

(2.27) |

Based on (2.27), Maxwell's equations which are always valid (not only in static) acquire the form:

1.

(...Gauss law) | ||

as | ||

2. | ||

there are no magnetic charges |

As in the statics

further

We use (2.29) and except of constant it is valid

(2.31) |

(...Faraday's law) | (2.32) |

(2.33) |

Total magnetic field

(2.34) |

(2.35) |

On the own magnetic field B_{Q} of the charge moving at speed *v*
it is possible to write:

(2.36) |

because (2.29),(2.31) and (2.27) are valid and because

(2.37) | ||

i.e. | (2.38) |

Let's accept the following designation

The equation (2.27) can be written in the following form:

(2.39) |

where *H*_{stat} is the intensity of the static magnetic field and
corresponding angles
relate to angles .

while

i.e.

or | respectively | (2.40) |

we get | (2.41) | |

as | (2.42) | |

Then | (2.43) |

Where from | (2.44) | |

It is possible to write the equation (2.38) also in the following form: | ||

(2.45) |