SELF

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S.B. Karavashkin and O.N. Karavashkina

3. Visualisation of the field of scalar potential of orbiting proton

The main difficulty in this problem is that in trying to find the direct solution we run into implicit functions.

Suppose the proton revolving with some velocity v_p   in the circular orbit with radius r_p, as shown in Fig. 1.

Fig. 1. Sketch for calculation of the field of orbiting proton

Its orbital frequency is

(11)

Given the space location of proton varies in time, the potential excited by proton at some moment of time t₀  reaches the observation point N with some delay t_N . And t_N depends both on location of N and on proton location at the moment of field measurement. So in general appearance the dependence of scalar potential on time and location of  N  (with negative trial charge equal to that of proton) can be presented as

(12)

where

(13)

is the moment of potential's excitation by proton, and t is the current time.

We can determine t_N (t₃) with the help of sketch shown in Fig. 2.

Fig. 2. Sketch to determine the distance between the proton at the moment of field radiation and the point of field observation

It follows from  OPN  that

(14)

In its turn, proceeding from the fact that proton orbits evenly,

(15)

and given

(16)

we yield

(17)

and

(18)

As we can make sure, the distance  r_N   between the studied point and the field source at the moment of field radiation is described by implicit function; of course, this complicates the task to plot the diagram of dynamic scalar potential, calculating it directly.