Open access peer-reviewed chapter - ONLINE FIRST

Radiation and Energy Flux of Electromagnetic Fields by a Segment of Relativistic Electron Beam Moving Uniformly in Vacuum

By Sergey Prijmenko and Konstantin Lukin

Submitted: February 13th 2019Reviewed: May 21st 2019Published: November 27th 2019

DOI: 10.5772/intechopen.86980

Downloaded: 19

Abstract

A finite-length segment of filamentous relativistic electron beam (REB), moving uniformly in vacuum, radiates hybrid electromagnetic waves, compound of potential and vortex electric fields, as well as a vortex magnetic field. The strengths of electric and magnetic fields radiated by the segment edges have the opposite signs. The electromagnetic fields in the wave zone are considered as superposition of the electromagnetic waves radiated by the beginning and the end of the REB segment, which, in particular, leads to formation of the field’s interference components. In both the near and the intermediate zones, there is a flow of electrical energy due to the electric potential field and the field of displacement current.

Keywords

  • relativistic electron beam or REB segment
  • potential field
  • vortex field
  • radiation of EM waves
  • near field zone
  • intermediate zone and far field (wave) zone
  • EM energy flux

1. Introduction

The physics of charged particle beam is an area where relativistic effects manifest themselves substantially. Here, one has to deal with a moving object, so both a fixed (laboratory) coordinate system and a moving coordinate system are to be used. A charged particle moves relative to the laboratory coordinate system, while in the moving coordinate system, it is at rest. Hence, in a laboratory coordinate system, the problem is to be considered as an electrodynamical one, and in a moving coordinate system, the problem belongs to the area of electrostatics. Thus, electrostatic phenomena in a charged particle set at rest are transformed into electrodynamic ones when it moves. Electromagnetic fields in these two inertial reference systems are tied via the Lorentz transform ([1], p. 79).

In the wave zone, the dynamic component of the electric field strength and the axially symmetric magnetic field form both a constant flux into a given solid angle, i.e., electromagnetic radiation, and a flux per time unit directed along the normal to the conical surface of the solid angle. The potential component of the electric field, directed along the radius, and the axially symmetric magnetic field form a flux oriented along the polar direction, i.e., along the normal to the above conical surface. The fluxes crossing the conical surface do not depend on the distance between the source point and the observation point. In the wave zone, the radiations from the beginning and end of the REB segment are added up, while the fluxes through the above conical surface caused by dynamic and potential components of electric field, are subtracted.

To date, the issue of influence of the finite length of a charged particle beam, moving uniformly in vacuum on the radiation of electromagnetic fields remains poorly studied, with an exception of publication [2], where its experimental part deserves special attention.

This chapter presents the results of our theoretical analysis of the electromagnetic field radiated by a finite-length segment of filamentous relativistic electron beam (REB). The REB segment moves uniformly in vacuum along its own axis which we will address as the longitudinal direction. The stepped varying of the charge density at the edges of the REB segment creates point-like sources of the potential electric field; the strength of which is inversely proportional to the distance between the source point and the observation point. In addition, the time variation of the REB current density forms at the REB edges the point-like sources of both potential and vortex electric fields, as well as the vortex magnetic field, with their strengths being also inversely proportional to the distance between the source point and the observation point [3].

The filamentary REB edges are considered as relativistic point-like radiators of the electromagnetic energy propagating to the wave zone. The presence of a potential electric field in the wave zone is due to the fact that the electric scalar potential in the wave zone is proportional to the electric monopole moment ([4], p. 51), which equals to the total charge in the selected volume ([5], p. 280). As follows from the Jefimenko’s generalization of the Coulomb law ([3], p. 246), the potential electric field strength in the wave zone is proportional to the time derivative of the electric monopole moment.

In the intermediate zone, there is a flow of electrical field energy, due to the electric potential field and the field of the displacement current. The electrical energy flux in the intermediate zone is due to the electric potential field and field of the displacement current. The REB part with a constant charge density between its edges forms a quasi-static electromagnetic field in the near zone.

Note that a similar problem has been considered in [6], but it was devoted to similarity of the solutions obtained with the help of two different methods: retarded field integral and transformation equations of the special theory of relativity. Unlike our work, it does not contain expressions for scalar and vector potentials, as well as the electromagnetic energy flux.

2. Formulation of the problem

Consider a filamentary REB segment of length Land electric charge density Qmoving uniformly along its axis direction with velocity ve. Charge density of the REB segment may be written as follows:

ρtrxyzL=QLδx·δy·hzvethzvet+LE1

where hxis Heaviside step function; δxand δyare Dirac delta functions of coordinates. The electric scalar potential ψtrand vector potential Atr,taking into account Eq. (1), satisfy the wave equations [3, 7]:

divgrad1c22t2ψtr=ρtrε0,E2
graddivrotrot1c22t2Atr=μ0ρtrvek0,E3

where ε0and μ0are the dielectric and magnetic permeability of vacuum, respectively; andk0is the unit vector along the REB axis, the Oz axis.

3. Potentials

A potential part of the vector potential A ptris related to the scalar potential by the Lorentz calibration [3, 7]:

divA ptr=1c2∂tψtr,E4

Using the Green’s function for the wave equation ([3], p. 243), we obtain:

ψtx=0y=0vet<z<vet+L trxyz=
=QL4πε0vetvet+Ldzx2+y2+zz2t=trr c,E5
Atx=0y=0vet<z<vet+L trxyz=
=Qμ0L4πvetvet+Ldzx2+y2+zz2t=trr c,E6

where the hatched coordinates refer to the source point at the time instant tof the field radiation, and the non-hatched coordinates refer to the observation point at the time instant t.

The formula for the scalar potential can be obtained in the closed form using the table integral ([8], p. 34):

ψtx=0y=0vet<z<vet+L trxyz
=QL4πε0lnzvet+L+x2+y2+zvet+L2t=trr tz=vet+Lc
QL4πε0lnzvet+x2+y2+zvet2t=trr tz=vetcE7

where the expressions in the first and second summands refer to the REB segment end and its beginning, respectively.

4. The electromagnetic field strengths

For estimation of the electric and magnetic fields, we use standard formulas ([7], p. 432):

Etx=0y=0vet<z<vet+L trxyz=
=Atx=0y=0vet<z<vet+L trxyzt
gradrψtx=0y=0vet<z<vet+L trxyz,E8
Htx=0y=0vet<z<vet+L trxyz=
=1μ0rotrAtx=0y=0vet<z<vet+L trxyz,E9

where it is necessary to perform the differentiation over the coordinates of the observation point, taking into account the retardation effect ([7], p. 432) and ([4], p. 43) as well as the differentiation of integrals by the integration limits and by the parameter ([9], p. 58). Using Eqs. (5), (6), and (8), we get:

Exptx=0y=0vet<z<vet+L trxyz=
=QveL4πε0ccosαxz=vetκz=vetrr tz=vett=trr tz=vetc
QveL4πε0ccosαxz=vet+Lκz=vet+Lrr tz=vet+Lt=trr tz=vet+Lc+
+QL4πε0vetvet+Lcosαxzrr tz2t=trr tzc,dzE10
Eyptx=0y=0vet<z<vet+L trxyz=
=QveL4πε0ccosαyz=vetκz=vetrr tz=vett=trr tz=vetc
QveL4πε0ccosαyz=vet+Lκz=vet+Lrr tz=vet+Lt=trr tz=vet+Lc+
+QL4πε0vetvet+Lcosαyzrr tz2t=trr tzc,dzE11
Eztx=0y=0vet<z<vet+L trxyz=
=QveL4πε0ccosαzz=vetκz=vetrr tz=vett=trr tz=vetc
QveL4πε0ccosαzz=vet+Lκz=vet+Lrr tz=vet+Lt=trr tz=vet+Lc
Qμ0ve2L4π1κz=vetrr tz=vett=trr tz=vetc+
+Qμ0ve2L4π1κz=vet+Lrr tz=vet+Lt=trr tz=vet+Lc+
+QL4πε0vetvet+Lcosαzzrr tz2t=trr tzc,dzE12

where

cosαxz=vet=xrr tz=vett=trr tz=vetc,E13
cosαxz=vet+L=xrr tz=vet+Lt=trr tz=vet+Lc,E14
cosαyz=vet=yrr tz=vett=trr tz=vetc,E15
cosαyz=vet+L=yrr tz=vet+Lt=trr tz=vet+Lc,E16
cosαzz=vet=zvetrr tz=vett=trr tz=vetc,E17
cosαzz=vet+L=zvet+Lrr tz=vet+Lt=trr tz=vet+Lc,E18

and

κz=vet=[1veccosαzz=vet,E19
κz=vet+L=[1veccosαzz=vet+LE20

are the retardation factors ([3], p. 246).

The transverse components of the electric field strength Exptrxyzt trxyzand Eyptrxyzt trxyzare potential relative to the space coordinates, and the longitudinal component Eztrxyzt trxyzconsists of both a potential component relative to the space coordinates and a dynamic component.

The transverse components of the magnetic field strength Hxtrxyzt trxyzand Hytrxyzt trxyz, according to the Eq. (6) and (9), are:

Hxtx=0y=0vet<z<vet+L trxyz=
=Qve2L4πccosαyz=vetκz=vetrr tz=vett=trr tz=vetc
+Qve2L4πccosαyz=vet+Lκz=vet+Lrr tz=vet+Lt=trr tz=vet+Lc
QveL4πvetvet+Ldzcosαyzrr tz2t=trr tzcE21
Hytx=0y=0vet<z<vet+L trxyz=
=Qve2L4πccosαxz=vetκz=vetrr tz=vett=trr tz=vetc
Qve2L4πccosαxz=vet+Lκz=vet+Lrr tz=vet+Lt=trr tz=vet+Lc+
+QveL4πvetvet+Ldzcosαxzrr tz2t=trr tzcE22

The strengths of the electric fields in Eqs. (10)(12) and magnetic fields with Eqs. (21) and (22), formed by the ends and the main part of the beam, decrease inversely proportional to the first and second powers of the distance from the source point to the observation point.

5. Displacement current

We take into account that the displacement current density jdtr([7], p. 87):

jdtr=tDdtr=tε0Etr,E23

where the Ddtr=ε0Etris the electric displacement vector. Taking into account the Eqs. (10)(12) and (23), we get

jdxptx=0y=0vet<z<vet+L trxyz=
=Qve2L4πccosαxz=vet·cosαzz=vet·1κ2z=vetrr tz=vet2
+Qve3L4πc2cosαxz=vetκ3z=vetrr tz=vet3[rr tz=vet
cosαzz=vetzvet]+
+Qve2L4πccosαxz=vetcosαzz=vetκ2z=vetrr tz=vet2
2Qve2L4πccosαxz=vet+L·cosαzz=vet+Lκ2z=vet+Lrr tz=vet+L2
Qve3L4πc2cosαxz=vet+Lκ3z=vet+Lrr tz=vet+L3·rr tz=vet+Lcosαzz=vet+L·zvet+L+
+QveL4πcosαxz=vet+Lκ2z=vet+Lrr tz=vet+L2
QveL4π·cosαxz=vetκ2z=vetrr tz=vet2E24
jdyptx=0y=0vet<z<vet+L trxyz=
=Qve2L4πccosαyz=vet·cosαzz=vet·
·1κ2z=vetrr tz=vet2+Qve3L4πc2cosαyz=vetκ3z=vetrr tz=vet3
rr tz=vetcosαzz=vetzvet+
+Qve2L4πccosαyz=vetcosαzz=vetκ2z=vetr=vet)]zvet](tz=vet)2
2Qve2L4πccosαyz=vet+L·cosαzz=vet+Lκ2z=vet+Lrr tz=vet+L2Qve3L4πc2
cosαyz=vet+Lκ3z=vet+Lrr tz=vet+L3rr tz=vet+Lcosαzz=vet+L·zvet+L
+QveL4πcosαyz=vet+Lκ2z=vet+Lrr tz=vet+L2
QveL4πcosαyz=vetκ2z=vetrrtz=vet2E25
jdztx=0y=0vet<z<vet+L trxyz=
=Qve2L4πcsin2αzz=vet·
·1κ2z=vetrrtz=vet2+Qve3L4πc2cosαzz=vetκ3z=vetrrtz=vet3·
·rr tz=vetcosαzz=vetzvet+Qve2L4πccos2αzz=vetκ2z=vetrrtz=vet2
Qve2L4πcsin2αzz=vet+Lκ2z=vet+Lrrtz=vet+L2+Qve3L4πc2·
·cosαzz=vet+Lκ2z=vet+Lrrtz=vet+L3·[rrtz=vet+Lcosαzz=vet+L·
·zvet+L]+Qve2L4πccos2αzz=vet+Lκ2z=vet+Lrr tz=vet+L2
Qve4L4πc31κ3z=vetrrtz=vet3rrtz=vetcosαzz=vetzvet
Qve3L4πc2cosαzz=vetκ2z=vetrr tz=vet2
+Qve4L4πc31κ3z=vet+Lrrtz=vet+L3·
·rrtz=vet+Lcosαzz=vet+Lzvet+L+
+Qve3L4πc2cosαzz=vet+Lκ2z=vet+Lrrtz=vet+L2QveL4πcosαzz=vetκz=vetrrtz=vet2E26

The transverse components of the displacement current density jdxptrxyzt trxyzandjdyptrxyzt trxyzare potential with respect to space coordinates, and the longitudinal component jdztrxyzt trxyzconsists of potential and dynamic components. Displacement current densities are decreasing inversely proportional to the second power of the distance from the source point to the observation point.

6. Flux of electrical energy

The electrical energy flux density per unit time Sψtr,according to ([10], p. 125) Eq. (15) and [11] Eqs. (7) and (8), has the form

Sψtr=ψtr·jdtrE27

Taking into account the Eq. (5) or the Eq. (7) and the Eqs. (24)(26), we can write

Sxψtx=0y=0vet<z<vet+L trxyz=
=ψtx=0y=0vet<z<vet+L trxyz·
jdxptx=0y=0vet<z<vet+L trxyzE28
Syψtx=0y=0vet<z<vet+L trxyz=
=ψtx=0y=0vet<z<vet+L trxyz·
·jdyptx=0y=0vet<z<vet+L trxyzE29
Szψtx=0y=0vet<z<vet+L trxyz=
=ψtx=0y=0vet<z<vet+L trxyz·
jdzptx=0y=0vet<z<vet+L trxyzE30

The electrical energy flux density Sψtrdecreases inversely proportional to the third power of the distance from the source point to the observation point. The electrical energy flux per unit time into a given solid angle decreases inversely proportional to the first power of the distance from the source point to the observation point. The flux takes place both in the near and the intermediate zones.

7. Pointing vector

The Poynting vector or the flux density of electromagnetic energy per unit time is determined by the formula ([3], p. 259)

Str=Etr×HtrE31

The Poynting vector along the Oxaxis estimated according to Eq. (31) with the help of Eqs. (12) and (22) may be written as follows:

Sxtx=0y=0vet<z<vet+L trxyz=
=Eztx=0y=0vet<z<vet+L trxyz·
·Hytx=0y=0vet<z<vet+L trxyz
={Ezpz=vet+Ezpz=vet+L
+Ezrz=vet+Ezrvet<z<vet+L}·
Hyz=vet+Hyz=vet+L++Hycvet<z<vet+LE32

where the summands in curly brackets are defined by Eq. (12) and Eq. (22), respectively. Rewriting the Eq. (32) in the following form:

Sx(t,x=0,y=0,vet<z<vet+L, t,r(x,y,z))=iSx(z=vet,z=vet+L)+piSx(z=vet,z=vet+L,vet<z<vet+L)+fSxc(vet<z<vet+L),E33

where the Sixz=vetz=vet+Lthere is a flux of electromagnetic energy in a unit time that goes into the wave zone, the Spixz=vetz=vet+Lvet<z<vet+Lthere is a flux of electromagnetic energy in the intermediate zone, the Sfxcvet<z<vet+Lthere is a flux of electromagnetic energy in the near zone. As this takes place

Sixz=vetz=vet+L=Sixz=vet+
+Sixz=vet+L+SixψAz=vetz=vet+L,E34
Sixz=vet=SixψAz=vet+SixAz=vet=
=Ezpz=vet·Hyz=vetEzrz=vet·Hyz=vet,E35
Sixz=vet+L=SixψAz=vet+L+SixAz=vet+L=
Ezpz=vet+L·Hyz=vet+LEzrz=vet+L·Hyz=vet+L,E36
iSxψA(z=vet,z=vet+L)=Ezp(z=vet)·Hy(z=vet+L)Ezp(z=vet+L)·Hy(z=vet)Ezr(z=vet)·Hy(z=vet+L)Ezr(z=vet+L)·Hy(z=vet).E37

The energy fluxes, Sixz=vet, Sixz=vet+L, SixψAz=vetz=vet+L, are determined by point sources of radiation at the REB segment beginning, the REB segment end, and the REB segment interference, respectively.

Spixz=vetz=vet+Lvet<z<vet+L=
Ezpz=vet·Hycvet<z<vet+LEzpz=vet+L·Hycvet<z<vet+L
Ezrz=vet·Hycvet<z<vet+LEzrz=vet·Hycvet<z<vet+L
Ezcvet<z<vet+L·Hyz=vetEzcvet<z<vet+L·Hyz=vet+L.E38
Sfxcvet<z<vet+L=Ezcvet<z<vet+L·Hycvet<z<vet+L.E39

The Poynting vector along the Oyaxis, taking into account Eqs. (12), (21), (31), similarly to Eqs. (33)(39), is represented by:

Sytx=0y=0vet<z<vet+L trxyz=Siyz=vetz=vet+L+
+Spiyz=vetz=vet+Lvet<z<vet+L+Sfycvet<z<vet+L,E40
Siyz=vetz=vet+L=Siyz=vet+Siyz=vet+L+
+SiyψAz=vetz=vet+LE41
Siyz=vet=SiyψAz=vet+SiyAz=vet=
=Ezpz=vet·Hxz=vet+Ezrz=vet·Hxz=vetE42
Siyz=vet+L=SiyψAz=vet+L+SiyAz=vet+L=
=Ezpz=vet+L·Hxz=vet+L+Ezrz=vet+L·Hxz=vet+L,E43
SiyψAz=vetz=vet+L=Ezpz=vet·Hxz=vet+L+Ezpz=vet+L·
·Hxz=vet+Ezrz=vet·Hxz=vet+L+Ezrz=vet+L·Hxz=vet,E44
Spiyz=vetz=vet+Lvet<z<vet+L=
Ezpz=vet·Hxcvet<z<vet+L+Ezpz=vet+L·Hxcvet<z<vet+L+
+Ezrz=vet·Hxcvet<z<vet+L+Ezrz=vet·Hxcvet<z<vet+L+
+Ezcvet<z<vet+L·Hxz=vet+Ezcvet<z<vet+L·Hxz=vet+L,E45
Sfycvet<z<vet+L=Ezcvet<z<vet+L·Hxcvet<z<vet+L.E46

The Poynting vector along the Ozaxis, taking into account Eqs. (10), (11), (21), (22), and (31), may be written as follows:

Sztx=0y=0vet<z<vet+L trxyz=Sizz=vetz=vet+L+
+Spizz=vetz=vet+Lvet<z<vet+L+Sfzcvet<z<vet+L,E47
Sizz=vetz=vet+L=Sizz=vet+
+Sizz=vet+L+SizψAz=vetz=vet+LE48
Sizz=vet=SizψAz=vet=
=Expz=vet·Hyz=vetEypz=vet·Hxz=vetE49
Sizz=vet+L=SizψAz=vet+L=
=Expz=vet+L·Hyz=vet+LEypz=vet+L·Hxz=vet+LE50
SizψAz=vetz=vet+L=Expz=vet·Hyz=vet+L+
+Expz=vet+L·Hyz=vetEypz=vet·Hxz=vet+L
Eypz=vet+L·Hxz=vetE51
Spizz=vetz=vet+Lvet<z<vet+L=
Expz=vet·Hycvet<z<vet+L+Expz=vet+L·Hycvet<z<vet+L+
+Excvet<z<vet+L·Hyz=vet+Excvet<z<vet+L·Hyz=vet+L
Eypz=vet·Hxcvet<z<vet+LEypz=vet+L·Hxcvet<z<vet+L
Eycvet<z<vet+L·Hxz=vetEycvet<z<vet+L·Hxz=vet+L,E52
Sfzcvet<z<vet+L=Excvet<z<vet+L·Hycvet<z<vet+L
Eecvet<z<vet+L·Hxcvet<z<vet+L.E53

8. Numerical results

We have considered the filamentary REB of the length L=3m, moving along the Ozaxis with velocity ve=0.94c(cis the speed of light) and having overall charge Q=1·1010C.

In the laboratory coordinate system, the dependence of the electric field strength Expt=0x=0y=0z=0 trxy=0z=0, radiated by the beginning of the REB segment rx=0y=0z=0, on the transverse coordinate xwas calculated using Eq. (10), (Figure 1). The signal radiation time twas selected equal to zero t=0. The observation point rxy=0z=0was selected in the cross section z=0at y=0. The observation time twas determined by the formula t =xc.

Figure 1.

The potential electric field strength Expt′=0x′=0y′=0z′=0trxy=0z=0 radiated by the REB segment beginning.

The dependence of the potential electric field strength Exptx=0y=0z=vet trx=0.3my=0z=0,radiated by the beginning of the REB segment rx=0y=0z=vet, on the signal generation time tcalculated with the help of Eq. (10), is represented in Figure 2 where rx=0.3my=0z=0is the observation point coordinates.

Figure 2.

The potential electric field strength Expt′x′=0y′=0z′=vet′trx=0.3my=0z=0 radiated by the REB segment beginning.

The dependence of the magnetic field strength Hyt=0x=0y=0z=L trxy=0z=0radiated by the REB segment end rx=0y=0z=Lon the transverse coordinate xwas calculated using Eq. (22) (Figure 3). The signal generation time twas selected equal to the zero, t=0where rx=0.3my=0z=0is the observation point coordinates. The observation time twas determined by the formula t=x2+L2c.

Figure 3.

Magnetic field strength Hyt′=0x′=0y′=0z′=Ltrxy=0z=0 radiated by the REB segment end.

The dependence of the magnetic field strength Hytx=0y=0z=vet+Ltrx=0.3my=0z=0, radiated by the REB segment end rx=0y=0z=vet+L, on the signal radiation time tcalculated using Eq.(22), is represented in Figure 4 where rx=0.3my=0z=0is the observation point coordinates.

Figure 4.

Magnetic field strength Hyt′x′=0y′=0z′=vet′+Ltrx=0.3my=0z=0 radiated by the REB segment end.

The dependence of the electromagnetic energy flux Sizt=0x=0y=0z=0trxy=0z=0, radiated by the REB segment beginning rx=0y=0z=0, on the transverse coordinate xwas calculated with the help of Eqs. (49), (10), (11), (21), and (22) (Figure 5). The signal generation time twas selected equal to the zero, t=0. The rxy=0z=0is the observation point coordinates. The observation time twas determined by the formula t=xc.

Figure 5.

The electromagnetic energy flux Sizt′=0x′=0y′=0z′=0trxy=0z=0 radiated by the REB segment beginning.

The dependence of the electromagnetic energy flux Siztx=0y=0z=vet trx=0.3my=0z=0, radiated by the REB segment rx=0y=0z=vet, on the signal generation time t, calculated by Eqs. (49), (10), (11), (21), (22), is shown in Figure 6. The observation point coordinate is rx=0.3my=0z=0.

Figure 6.

The electromagnetic energy flux Sizt′x′=0y′=0z′=vet′trx=0.3my=0z=0 radiated by the REB segment beginning.

The dependence of the electromagnetic energy flux Sizt=0x=0y=0z=L trxy=0z=0, radiated by the REB segment end rx=0y=0z=L, on the transverse coordinate xwas calculated with the help of Eqs. (50), (10), (11), (21), (22) (Figure 7). The signal radiation time twas selected equal to the zero t=0. The rxy=0z=0is the observation point coordinates. The observation time twas determined by the formula t=x2+L2c.

Figure 7.

The electromagnetic energy flux Sizt′=0x′=0y′=0z′=Ltrxy=0z=0 radiated by the REB segment end.

The dependence of the electromagnetic energy flux Siztx=0y=0z=vet+L trx=0.3my=0z=0, radiated by the REB segment end rx=0y=0z=vet+L, on the signal radiation time t, calculated according to Eqs. (50), (10), (11), (21), (22), is shown in Figure 8 where rx=0.3my=0z=0is the observation point coordinates.

Figure 8.

The electromagnetic energy flux Sizt′x′=0y′=0z′=vet′+Ltrx=0.3my=0z=0 radiated by the REB segment end.

9. Conclusions

The applicability of relativity in the physics of charged particle beams has been shown from the example of radiation by a filamentary REB segment uniformly moving in vacuum along a linear direction.

In electrodynamics, in a moving coordinate system, the relative distance between a charged object and an observer does not change. The phenomenon of relativity associated with the field dynamics degenerates to electrostatic processes. In rest, or laboratory, coordinate system, the relative distance is changing with time, the charge density also varies with the time, and as a result, the retardation phenomena came to the scene and the Poisson equation is to be substituted by the wave equation.

The expressions have been obtained to describe the strengths of the electric and magnetic fields and the electric and electromagnetic energy fluxes in all three zones: near field zone, intermediate, and wave zones. The filamentary REB edges are relativistic point-like sources of electromagnetic energy propagating in the wave zone. The REB edges form a potential component of the electric field strength, which is inversely proportional to the distance from the source point to the observation point. In the wave zone, strength of this field is comparable with that of the dynamic component of the electric field.

The dynamic component of the electric field strength and the axially symmetric magnetic field form both a constant flux into the given solid angle, i.e. electromagnetic radiation, and a flux per time unit directed along the normal to the conical surface of the above solid angle. The potential component of the electric field, directed along the radius, and the axially symmetric magnetic field form a flux oriented along the polar direction, i.e., along the normal to the conical surface. The fluxes crossing the above conical surface are independent of the distance between the source point and the observation point. In the wave zone, the radiations from the beginning and end of the REB segment are added up, while the fluxes through the above conical surface caused by dynamic and potential components of electric field, are subtracted.

Relativistic point-like sources create in the wave zone the vortex components of the magnetic field. The REB edges radiate hybrid electromagnetic waves, comprising of potential and vortex electric fields, as well as a vortex magnetic field. The electric and magnetic field strengths radiated by the REB segment edges have opposite signs. In the wave zone, the radiated electromagnetic field fluxes are compound of the electromagnetic energy fluxes, produced by both the REB segment beginning and its end, as well as of their interference components. In the intermediate zone, the electrical energy flux takes place due to the electric potential field and the displacement current. The REB segment, between the beam edges, having a constant charge density, produces a quasi-static electromagnetic field in the near zone.

Acknowledgments

This work was funded in part by NATO research project G5465 within frames of the Science for Peace and Security (SPS) program.

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Sergey Prijmenko and Konstantin Lukin (November 27th 2019). Radiation and Energy Flux of Electromagnetic Fields by a Segment of Relativistic Electron Beam Moving Uniformly in Vacuum [Online First], IntechOpen, DOI: 10.5772/intechopen.86980. Available from:

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